Subgroups — C₂×#22574542 #31763424

Order 17152.

8814 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22574542	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16864133	C₂	1	4	yes	maximal, complemented
3	C₂×#16864132	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16864066	C₂	1	2	yes	maximal, complemented
5	C₂×#16864040	C₂	1	4	yes	maximal, complemented
6	C₂×#16864031	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₄×C₂²⋊D₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×#16864000	C₂	1	4	yes	maximal, complemented
9	C₂×#16863959	C₂	1	4	yes	maximal, complemented
10	C₂×#16863955	C₂	1	4	yes	maximal, complemented
11	C₂×#16863930	C₂	1	2	yes	maximal, complemented
12	C₂²×#127×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#11761627	C₂²	1	4	yes	complemented
14	C₂³×C₂²⋊D₆₇	C₂²	1	4	yes	complemented
15	C₂²×D₄×D₆₇	C₂²	1	4	yes	complemented
16	#16864133	C₂²	1	64	yes	complemented
17	#16864132	C₂²	1	16	yes	complemented
18	#16864066	C₂²	1	32	yes	complemented
19	#16864040	C₂²	1	64	yes	complemented
20	#16864031	C₂²	1	16	yes	complemented
21	D₄×C₂²⋊D₆₇	C₂²	1	64	yes	complemented
22	#16864000	C₂²	1	64	yes	complemented
23	#16863959	C₂²	1	64	yes	complemented
24	#16863955	C₂²	1	64	yes	complemented
25	C₂×#14524039	C₂²	1	4	yes	complemented
26	#16863930	C₂²	1	32	yes	complemented
27	C₂×#14524037	C₂²	1	4	yes	complemented
28	C₂×#14524035	C₂²	1	2	yes	complemented
29	C₂×#14524031	C₂²	1	2	yes	complemented
30	C₂×#14524030	C₂²	1	4	yes	complemented
31	C₂×#14524026	C₂²	1	4	yes	complemented
32	C₂×#14524024	C₂²	1	4	yes	complemented
33	C₂×#14524024	C₂²	1	8	yes	complemented
34	C₂×#14524024	C₂²	1	4	yes	complemented
35	C₂×#14524022	C₂²	1	2	yes	complemented
36	C₂×#14524017	C₂²	1	2	yes	—
37	C₂×#14524015	C₂²	1	8	yes	complemented
38	C₂×#14524015	C₂²	1	8	yes	complemented
39	C₂×#14524015	C₂²	1	8	yes	complemented
40	C₂×#14524015	C₂²	1	8	yes	complemented
41	C₂×#14524014	C₂²	1	4	yes	complemented
42	C₂×#14524014	C₂²	1	4	yes	complemented
43	C₂×#14524012	C₂²	1	8	yes	complemented
44	C₂×#14524011	C₂²	1	8	yes	complemented
45	C₂×#14524011	C₂²	1	4	yes	complemented
46	C₂×#14524011	C₂²	1	4	yes	complemented
47	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	complemented
48	C₂×C₄×D₂₆₈	C₂²	1	2	yes	complemented
49	C₂×D₄×Dic₆₇	C₂²	1	4	yes	complemented
50	C₂×D₄×Dic₆₇	C₂²	1	4	yes	complemented
51	C₂×C₄×C₂²⋊D₆₇	C₂²	1	4	yes	complemented
52	C₂×#14524000	C₂²	1	4	yes	complemented
53	C₂²×C₂²⋊Dic₆₇	C₂²	1	4	yes	complemented
54	C₂³×D₄×C₆₇	C₂²	1	4	yes	complemented
55	C₂³×D₄×C₆₇	C₂²	1	2	yes	complemented
56	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
57	C₂×#127×C₆₇	C₂²	1	4	yes	complemented
58	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic, complemented
59	C₂×#127×C₆₇	C₂²	1	2	yes	complemented
60	C₂×#127×C₆₇	C₂²	1	16	yes	complemented
61	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic, complemented
62	C₂⁴×D₆₇	C₂³	1	4	yes	—
63	C₂×#11761627	—	4	16	no	—
64	C₂×#11761627	C₂³	1	32	yes	complemented
65	C₂²×Dic₁₃₄	C₂³	1	2	yes	—
66	C₂²×D₂₆₈	C₂³	1	2	yes	—
67	C₂²×C₂²⋊D₆₇	—	4	16	no	—
68	C₂²×C₂²⋊D₆₇	C₂³	1	4	yes	—
69	C₂²×C₂²⋊D₆₇	C₂³	1	8	yes	—
70	C₂²×C₂²⋊D₆₇	C₂³	1	8	yes	—
71	C₂²×C₂²⋊D₆₇	C₂³	1	8	yes	—
72	C₂²×C₂²⋊D₆₇	C₂³	1	32	yes	complemented
73	C₂×D₄×D₆₇	—	4	16	no	—
74	C₂×D₄×D₆₇	C₂³	1	32	yes	complemented
75	C₂³×Dic₆₇	C₂³	1	8	yes	—
76	C₂³×Dic₆₇	C₂³	1	4	yes	—
77	C₂²×C₄×D₆₇	C₂³	1	4	yes	—
78	#14524039	C₂³	1	32	yes	complemented
79	#14524037	C₂³	1	32	yes	complemented
80	#14524035	C₂³	1	16	yes	complemented
81	#14524031	C₂³	1	16	yes	complemented
82	#14524030	C₂³	1	32	yes	complemented
83	#14524026	C₂³	1	32	yes	complemented
84	#14524024	C₂³	1	32	yes	complemented
85	#14524024	C₂³	1	64	yes	complemented
86	#14524024	C₂³	1	32	yes	complemented
87	#14524022	C₂³	1	16	yes	complemented
88	#14524017	C₂³	1	16	yes	—
89	#14524015	C₂³	1	64	yes	complemented
90	#14524015	C₂³	1	64	yes	complemented
91	#14524015	C₂³	1	64	yes	complemented
92	#14524015	C₂³	1	64	yes	complemented
93	#14524014	C₂³	1	32	yes	complemented
94	#14524014	C₂³	1	32	yes	complemented
95	#14524012	C₂³	1	64	yes	complemented
96	#14524011	C₂³	1	64	yes	complemented
97	#14524011	C₂³	1	32	yes	complemented
98	#14524011	C₂³	1	32	yes	complemented
99	C₄×Dic₁₃₄	C₂³	1	16	yes	complemented
100	C₄×D₂₆₈	C₂³	1	16	yes	complemented
101	D₄×Dic₆₇	C₂³	1	32	yes	complemented
102	D₄×Dic₆₇	C₂³	1	32	yes	complemented
103	C₄×C₂²⋊D₆₇	C₂³	1	32	yes	complemented
104	#14524000	C₂³	1	32	yes	complemented
105	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
106	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
107	C₂×Dic₆₇⋊C₄	C₂³	1	4	yes	—
108	C₂×Dic₆₇⋊C₄	C₂³	1	4	yes	—
109	C₂×Dic₆₇⋊C₄	C₂³	1	4	yes	—
110	C₂×Dic₆₇⋊C₄	C₂³	1	4	yes	—
111	C₂×C₂²⋊Dic₆₇	C₂³	1	4	yes	—
112	C₂×C₂²⋊Dic₆₇	C₂³	1	4	yes	—
113	C₂×C₂²⋊Dic₆₇	C₂³	1	4	yes	—
114	C₂×C₂²⋊Dic₆₇	C₂³	1	8	yes	—
115	C₂×C₂²⋊Dic₆₇	C₂³	1	8	yes	—
116	C₂×C₂²⋊Dic₆₇	C₂³	1	32	yes	complemented
117	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
118	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
119	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
120	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
121	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
122	C₂⁵×C₆₇	C₂³	1	4	yes	—
123	C₂²×D₄×C₆₇	—	4	16	no	—
124	C₂²×D₄×C₆₇	—	4	8	no	—
125	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
126	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
127	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
128	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
129	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
130	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
131	C₂²×D₄×C₆₇	C₂³	1	32	yes	complemented
132	C₂²×D₄×C₆₇	C₂³	1	16	yes	complemented
133	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
134	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
135	#127×C₆₇	C₂³	1	8	yes	—
136	#127×C₆₇	C₂³	1	32	yes	complemented
137	#127×C₆₇	C₂³	1	8	yes	complemented
138	#127×C₆₇	C₂³	1	16	yes	complemented
139	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
141	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
142	C₂×C₄²×C₆₇	C₂³	1	8	yes	complemented
143	C₂³×D₆₇	—	4	16	no	—
144	C₂³×D₆₇	—	4	8	no	—
145	C₂³×D₆₇	C₂⁴	1	4	yes	—
146	C₂³×D₆₇	C₂⁴	1	16	yes	—
147	#11761627	—	4	64	no	—
148	C₂×Dic₁₃₄	—	2	8	no	—
149	C₂×Dic₁₃₄	C₂⁴	1	8	yes	—
150	C₂×D₂₆₈	—	2	8	no	—
151	C₂×D₂₆₈	C₂⁴	1	8	yes	—
152	C₂×C₂²⋊D₆₇	—	2	16	no	—
153	C₂×C₂²⋊D₆₇	—	4	16	no	—
154	C₂×C₂²⋊D₆₇	—	4	16	no	—
155	C₂×C₂²⋊D₆₇	—	4	16	no	—
156	C₂×C₂²⋊D₆₇	—	4	64	no	—
157	C₂×C₂²⋊D₆₇	C₂⁴	1	16	yes	—
158	C₂×C₂²⋊D₆₇	C₂⁴	1	32	yes	—
159	C₂×C₂²⋊D₆₇	C₂⁴	1	32	yes	—
160	C₂×C₂²⋊D₆₇	C₂⁴	1	32	yes	—
161	D₄×D₆₇	—	4	64	no	—
162	C₂²×Dic₆₇	—	4	8	no	—
163	C₂²×Dic₆₇	—	4	16	no	—
164	C₂²×Dic₆₇	C₂⁴	1	4	yes	—
165	C₂²×Dic₆₇	C₂⁴	1	4	yes	—
166	C₂²×Dic₆₇	C₂⁴	1	4	yes	—
167	C₂²×Dic₆₇	C₂⁴	1	32	yes	—
168	C₂²×Dic₆₇	C₂⁴	1	16	yes	—
169	C₂×C₄×D₆₇	—	2	16	no	—
170	C₂×C₄×D₆₇	C₂⁴	1	16	yes	—
171	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
172	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
173	Dic₆₇⋊C₄	C₂⁴	1	16	yes	—
174	Dic₆₇⋊C₄	C₂⁴	1	16	yes	—
175	Dic₆₇⋊C₄	C₂⁴	1	16	yes	—
176	Dic₆₇⋊C₄	C₂⁴	1	16	yes	—
177	C₂²⋊Dic₆₇	C₂⁴	1	16	yes	—
178	C₂²⋊Dic₆₇	C₂⁴	1	16	yes	—
179	C₂²⋊Dic₆₇	C₂⁴	1	16	yes	—
180	C₂²⋊Dic₆₇	C₂⁴	1	32	yes	—
181	C₂²⋊Dic₆₇	C₂⁴	1	32	yes	—
182	(C₂×D₆₇)⋊C₄	C₂⁴	1	16	yes	—
183	(C₂×D₆₇)⋊C₄	C₂⁴	1	16	yes	—
184	(C₂×D₆₇)⋊C₄	C₂⁴	1	16	yes	—
185	(C₂×D₆₇)⋊C₄	C₂⁴	1	16	yes	—
186	C₄×Dic₆₇	C₂⁴	1	16	yes	—
187	C₂⁴×C₆₇	—	2	16	no	—
188	C₂⁴×C₆₇	—	4	16	no	—
189	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
190	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
191	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
192	C₂⁴×C₆₇	C₂⁴	1	16	yes	—
193	C₂×D₄×C₆₇	—	2	16	no	—
194	C₂×D₄×C₆₇	—	2	16	no	—
195	C₂×D₄×C₆₇	—	4	64	no	—
196	C₂×D₄×C₆₇	—	4	16	no	—
197	C₂×D₄×C₆₇	—	4	4	no	—
198	C₂×D₄×C₆₇	—	4	8	no	—
199	C₂×D₄×C₆₇	—	4	4	no	—
200	C₂×D₄×C₆₇	—	4	32	no	—
201	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
202	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
203	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
204	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
205	C₂×D₄×C₆₇	C₂⁴	1	32	yes	—
206	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
207	C₂²×C₄×C₆₇	—	2	4	no	—
208	C₂²×C₄×C₆₇	—	2	8	no	—
209	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
210	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
211	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
212	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
213	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
214	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
215	C₄²×C₆₇	C₂⁴	1	4	yes	—
216	C₄²×C₆₇	C₂⁴	1	8	yes	—
217	C₄²×C₆₇	C₂⁴	1	4	yes	—
218	C₂³×C₆₇	—	2	4	no	—
219	C₂³×C₆₇	—	2	16	no	—
220	C₂³×C₆₇	—	2	16	no	—
221	C₂³×C₆₇	—	2	8	no	—
222	C₂³×C₆₇	—	2	8	no	—
223	C₂³×C₆₇	—	2	32	no	—
224	C₂³×C₆₇	—	4	32	no	—
225	C₂³×C₆₇	—	8	32	no	—
226	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
227	C₂³×C₆₇	C₂⁵	1	8	yes	—
228	C₂³×C₆₇	C₂⁵	1	8	yes	—
229	C₂³×C₆₇	C₂⁵	1	2	yes	—
230	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
231	C₂²×D₆₇	—	2	8	no	—
232	C₂²×D₆₇	—	2	8	no	—
233	C₂²×D₆₇	—	2	8	no	—
234	C₂²×D₆₇	—	4	32	no	—
235	C₂²×D₆₇	—	4	16	no	—
236	C₂²×D₆₇	—	8	32	no	—
237	C₂²×D₆₇	C₂⁵	1	8	yes	—
238	D₄×C₆₇	—	2	32	no	—
239	D₄×C₆₇	—	2	32	no	—
240	D₄×C₆₇	—	4	32	no	—
241	D₄×C₆₇	—	4	8	no	—
242	D₄×C₆₇	—	4	16	no	—
243	D₄×C₆₇	—	4	8	no	—
244	C₂×C₄×C₆₇	—	2	8	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	16	no	—
249	C₂×C₄×C₆₇	H(5,𝔽₂)	1	8	yes	—
250	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
251	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
252	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
253	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
254	C₂²⋊D₆₇	—	2	32	no	—
255	C₂²⋊D₆₇	—	4	32	no	—
256	C₂²⋊D₆₇	—	4	32	no	—
257	C₂²⋊D₆₇	—	4	32	no	—
258	C₂×Dic₆₇	—	2	8	no	—
259	C₂×Dic₆₇	—	2	8	no	—
260	C₂×Dic₆₇	—	2	8	no	—
261	C₂×Dic₆₇	—	4	16	no	—
262	C₂×Dic₆₇	—	4	32	no	—
263	C₂×Dic₆₇	C₂⁵	1	8	yes	—
264	C₂×Dic₆₇	C₂⁵	1	8	yes	—
265	C₂×Dic₆₇	C₂⁵	1	8	yes	—
266	D₂₆₈	—	2	16	no	—
267	C₄×D₆₇	—	2	32	no	—
268	Dic₁₃₄	—	2	16	no	—
269	C₂²×C₆₇	—	2	4	no	—
270	C₂²×C₆₇	—	2	8	no	—
271	C₂²×C₆₇	—	2	16	no	—
272	C₂²×C₆₇	—	2	8	no	—
273	C₂²×C₆₇	—	2	8	no	—
274	C₂²×C₆₇	—	2	16	no	—
275	C₂²×C₆₇	—	4	16	no	—
276	C₂²×C₆₇	—	4	16	no	—
277	C₂²×C₆₇	—	8	32	no	—
278	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
279	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
280	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
281	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
282	C₂×D₆₇	—	2	8	no	—
283	C₂×D₆₇	—	2	8	no	—
284	C₂×D₆₇	—	2	8	no	—
285	C₂×D₆₇	—	4	16	no	—
286	C₂×D₆₇	—	8	32	no	—
287	C₄×C₆₇	—	2	2	no	—
288	C₄×C₆₇	—	2	2	no	—
289	C₄×C₆₇	—	2	4	no	—
290	C₄×C₆₇	C₂×H(5,𝔽₂)	1	8	yes	—
291	Dic₆₇	—	2	8	no	—
292	Dic₆₇	—	2	8	no	—
293	Dic₆₇	—	2	8	no	—
294	C₂×#3543	—	67	1	no	maximal, Sylow
295	C₂×C₆₇	—	2	4	no	—
296	C₂×C₆₇	—	4	8	no	—
297	C₂×C₆₇	—	4	8	no	—
298	C₂×C₆₇	#3543	1	4	yes	complemented
299	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
300	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	2	yes	—
301	D₆₇	—	4	8	no	—
302	#3543	—	67	32	no	—
303	C₂×#560	—	67	4	no	—
304	C₂×#559	—	67	1	no	—
305	C₂×#550	—	67	2	no	—
306	C₂×#546	—	67	4	no	—
307	C₂×D₄²	—	67	4	no	—
308	C₂×D₄²	—	67	1	no	—
309	C₂×#540	—	67	4	no	—
310	C₂×#535	—	67	4	no	—
311	C₂×#534	—	67	4	no	—
312	C₂²×#127	D₆₇	1	1	yes	characteristic, complemented
313	C₂×#529	—	67	2	no	—
314	C₆₇	C₂×#3543	1	1	yes	characteristic, complemented, Sylow
315	C₂²×#41	—	67	4	no	—
316	C₂³×D₄	—	67	4	no	—
317	C₂³×D₄	—	67	4	no	—
318	C₂³×D₄	C₂×D₆₇	1	2	yes	complemented
319	C₂³×D₄	C₂×D₆₇	1	4	yes	complemented
320	#560	—	67	64	no	—
321	#559	—	67	16	no	—
322	#550	—	67	32	no	—
323	#546	—	67	64	no	—
324	D₄²	—	67	64	no	—
325	D₄²	—	67	16	no	—
326	#540	—	67	64	no	—
327	#535	—	67	64	no	—
328	#534	—	67	64	no	—
329	C₂×#127	—	67	4	no	—
330	C₂×#127	C₂×D₆₇	1	1	yes	characteristic
331	C₂×#127	C₂×D₆₇	1	4	yes	complemented
332	C₂×#127	C₂×D₆₇	1	1	yes	characteristic, complemented
333	C₂×#127	C₂×D₆₇	1	2	yes	complemented
334	C₂×#127	C₂×D₆₇	1	16	yes	complemented
335	#529	—	67	32	no	—
336	C₂×#126	—	67	4	no	—
337	C₂×#125	—	67	2	no	—
338	C₂×#124	—	67	4	no	—
339	C₂×#124	—	67	2	no	—
340	C₂×#123	—	67	4	no	—
341	C₂×#123	—	67	4	no	—
342	C₂×#123	—	67	8	no	—
343	C₂×#123	—	67	4	no	—
344	C₂×C₂²⋊Q₈	—	67	2	no	—
345	C₂×#121	—	67	8	no	—
346	C₂×#121	—	67	8	no	—
347	C₂×#121	—	67	8	no	—
348	C₂×#121	—	67	4	no	—
349	C₂×#121	—	67	4	no	—
350	C₂×#121	—	67	8	no	—
351	C₂×#121	—	67	2	no	—
352	C₂×#120	—	67	8	no	—
353	C₂×#120	—	67	4	no	—
354	C₂×#120	—	67	4	no	—
355	C₂×#120	—	67	8	no	—
356	C₂×C₄×Q₈	—	67	2	no	—
357	C₂×C₄×D₄	—	67	4	no	—
358	C₂×C₄×D₄	—	67	4	no	—
359	C₂×C₄×D₄	—	67	4	no	—
360	C₂×C₄×D₄	—	67	2	no	—
361	C₂×#117	—	67	4	no	—
362	C₂²×C₂²⋊C₄	—	67	4	no	—
363	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic, complemented
364	C₂⁵	—	67	4	no	—
365	C₂⁵	C₂²×D₆₇	1	4	yes	—
366	C₂×#41	—	268	16	no	—
367	C₂×#41	—	67	32	no	—
368	C₂²×Q₈	—	67	2	no	—
369	C₂²×D₄	—	4	16	no	—
370	C₂²×D₄	—	4	8	no	—
371	C₂²×D₄	—	268	16	no	—
372	C₂²×D₄	—	268	16	no	—
373	C₂²×D₄	—	67	8	no	—
374	C₂²×D₄	—	67	8	no	—
375	C₂²×D₄	—	67	8	no	—
376	C₂²×D₄	—	67	32	no	—
377	C₂²×D₄	—	67	32	no	—
378	C₂²×D₄	—	67	4	no	—
379	C₂²×D₄	—	67	2	no	—
380	C₂²×D₄	C₂²×D₆₇	1	4	yes	—
381	C₂²×D₄	C₂²×D₆₇	1	4	yes	—
382	C₂²×D₄	C₂²×D₆₇	1	2	yes	—
383	C₂²×D₄	C₂²×D₆₇	1	4	yes	—
384	C₂²×D₄	C₂²×D₆₇	1	8	yes	—
385	C₂²×D₄	C₂²×D₆₇	1	2	yes	—
386	C₂²×D₄	C₂²×D₆₇	1	16	yes	complemented
387	C₂²×D₄	C₂²×D₆₇	1	32	yes	complemented
388	C₂³×C₄	—	67	8	no	—
389	C₂³×C₄	—	67	4	no	—
390	C₂³×C₄	—	67	4	no	—
391	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
392	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
393	#127	—	67	32	no	—
394	#127	C₂²×D₆₇	1	8	yes	—
395	#127	C₂²×D₆₇	1	32	yes	complemented
396	#127	C₂²×D₆₇	1	8	yes	complemented
397	#127	C₂²×D₆₇	1	16	yes	complemented
398	#126	—	67	32	no	—
399	#125	—	67	16	no	—
400	#124	—	67	32	no	—
401	#124	—	67	16	no	—
402	#123	—	67	32	no	—
403	#123	—	67	32	no	—
404	#123	—	67	64	no	—
405	#123	—	67	32	no	—
406	C₂²⋊Q₈	—	67	16	no	—
407	#121	—	67	64	no	—
408	#121	—	67	64	no	—
409	#121	—	67	64	no	—
410	#121	—	67	32	no	—
411	#121	—	67	32	no	—
412	#121	—	67	64	no	—
413	#121	—	67	16	no	—
414	#120	—	67	64	no	—
415	#120	—	67	32	no	—
416	#120	—	67	32	no	—
417	#120	—	67	64	no	—
418	C₄×Q₈	—	67	16	no	—
419	C₄×D₄	—	67	32	no	—
420	C₄×D₄	—	67	32	no	—
421	C₄×D₄	—	67	32	no	—
422	C₄×D₄	—	67	16	no	—
423	#117	—	67	32	no	—
424	C₂×C₄⋊C₄	—	67	4	no	—
425	C₂×C₄⋊C₄	—	67	4	no	—
426	C₂×C₄⋊C₄	—	67	2	no	—
427	C₂×C₄⋊C₄	—	67	2	no	—
428	C₂×C₄⋊C₄	—	67	4	no	—
429	C₂×C₄⋊C₄	—	67	4	no	—
430	C₂×C₂²⋊C₄	—	67	4	no	—
431	C₂×C₂²⋊C₄	—	67	4	no	—
432	C₂×C₂²⋊C₄	—	67	4	no	—
433	C₂×C₂²⋊C₄	—	67	32	no	—
434	C₂×C₂²⋊C₄	—	67	8	no	—
435	C₂×C₂²⋊C₄	—	67	8	no	—
436	C₂×C₂²⋊C₄	—	67	4	no	—
437	C₂×C₂²⋊C₄	—	67	4	no	—
438	C₂×C₂²⋊C₄	—	67	4	no	—
439	C₂×C₂²⋊C₄	—	67	4	no	—
440	C₂×C₄²	—	67	4	no	—
441	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
442	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
443	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
444	C₂×C₄²	C₂²×D₆₇	1	8	yes	complemented
445	C₂⁴	—	2	16	no	—
446	C₂⁴	—	4	16	no	—
447	C₂⁴	—	268	8	no	—
448	C₂⁴	—	268	16	no	—
449	C₂⁴	—	67	16	no	—
450	C₂⁴	—	67	4	no	—
451	C₂⁴	C₂³×D₆₇	1	4	yes	—
452	C₂⁴	C₂³×D₆₇	1	4	yes	—
453	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
454	C₂⁴	C₂³×D₆₇	1	16	yes	—
455	#41	—	268	64	no	—
456	C₂×Q₈	—	67	8	no	—
457	C₂×Q₈	—	134	8	no	—
458	C₂×D₄	—	2	16	no	—
459	C₂×D₄	—	2	16	no	—
460	C₂×D₄	—	4	64	no	—
461	C₂×D₄	—	4	16	no	—
462	C₂×D₄	—	4	4	no	—
463	C₂×D₄	—	4	8	no	—
464	C₂×D₄	—	4	4	no	—
465	C₂×D₄	—	4	32	no	—
466	C₂×D₄	—	268	64	no	—
467	C₂×D₄	—	268	64	no	—
468	C₂×D₄	—	268	16	no	—
469	C₂×D₄	—	268	16	no	—
470	C₂×D₄	—	268	16	no	—
471	C₂×D₄	—	67	32	no	—
472	C₂×D₄	—	67	32	no	—
473	C₂×D₄	—	67	32	no	—
474	C₂×D₄	—	67	16	no	—
475	C₂×D₄	—	67	8	no	—
476	C₂×D₄	—	134	16	no	—
477	C₂×D₄	—	134	8	no	—
478	C₂×D₄	C₂³×D₆₇	1	16	yes	—
479	C₂×D₄	C₂³×D₆₇	1	16	yes	—
480	C₂×D₄	C₂³×D₆₇	1	8	yes	—
481	C₂×D₄	C₂³×D₆₇	1	16	yes	—
482	C₂×D₄	C₂³×D₆₇	1	32	yes	—
483	C₂×D₄	C₂³×D₆₇	1	8	yes	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	8	no	—
486	C₂²×C₄	—	268	8	no	—
487	C₂²×C₄	—	268	16	no	—
488	C₂²×C₄	—	67	32	no	—
489	C₂²×C₄	—	67	16	no	—
490	C₂²×C₄	—	67	4	no	—
491	C₂²×C₄	—	67	4	no	—
492	C₂²×C₄	—	67	4	no	—
493	C₂²×C₄	—	67	16	no	—
494	C₂²×C₄	—	134	16	no	—
495	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
496	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
497	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
498	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
499	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
500	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
501	C₄⋊C₄	—	67	16	no	—
502	C₄⋊C₄	—	67	16	no	—
503	C₄⋊C₄	—	67	8	no	—
504	C₄⋊C₄	—	67	8	no	—
505	C₄⋊C₄	—	67	16	no	—
506	C₄⋊C₄	—	67	16	no	—
507	C₂²⋊C₄	—	67	16	no	—
508	C₂²⋊C₄	—	67	16	no	—
509	C₂²⋊C₄	—	67	16	no	—
510	C₂²⋊C₄	—	67	32	no	—
511	C₂²⋊C₄	—	67	32	no	—
512	C₂²⋊C₄	—	67	16	no	—
513	C₂²⋊C₄	—	67	16	no	—
514	C₂²⋊C₄	—	67	16	no	—
515	C₂²⋊C₄	—	67	16	no	—
516	C₄²	—	67	16	no	—
517	C₄²	C₂³×D₆₇	1	8	yes	—
518	C₄²	C₂³×D₆₇	1	4	yes	—
519	C₄²	C₂³×D₆₇	1	4	yes	—
520	C₂³	—	2	4	no	—
521	C₂³	—	2	16	no	—
522	C₂³	—	2	8	no	—
523	C₂³	—	2	16	no	—
524	C₂³	—	2	8	no	—
525	C₂³	—	2	32	no	—
526	C₂³	—	4	32	no	—
527	C₂³	—	8	32	no	—
528	C₂³	—	268	16	no	—
529	C₂³	—	268	32	no	—
530	C₂³	—	536	32	no	—
531	C₂³	—	67	8	no	—
532	C₂³	—	134	8	no	—
533	C₂³	—	134	8	no	—
534	C₂³	—	134	8	no	—
535	C₂³	#14524088	1	4	yes	—
536	C₂³	C₂⁴×D₆₇	1	8	yes	—
537	C₂³	C₂⁴×D₆₇	1	8	yes	—
538	C₂³	C₂⁴×D₆₇	1	2	yes	—
539	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
540	Q₈	—	134	16	no	—
541	D₄	—	2	32	no	—
542	D₄	—	2	32	no	—
543	D₄	—	4	32	no	—
544	D₄	—	4	8	no	—
545	D₄	—	4	16	no	—
546	D₄	—	4	8	no	—
547	D₄	—	268	32	no	—
548	D₄	—	268	32	no	—
549	D₄	—	268	32	no	—
550	D₄	—	134	32	no	—
551	D₄	—	134	16	no	—
552	C₂×C₄	—	2	8	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	16	no	—
557	C₂×C₄	—	268	16	no	—
558	C₂×C₄	—	268	32	no	—
559	C₂×C₄	—	67	8	no	—
560	C₂×C₄	—	67	8	no	—
561	C₂×C₄	—	67	8	no	—
562	C₂×C₄	—	134	8	no	—
563	C₂×C₄	—	134	8	no	—
564	C₂×C₄	—	134	8	no	—
565	C₂×C₄	—	134	32	no	—
566	C₂×C₄	#14524088	1	8	yes	—
567	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
568	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
569	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
570	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
571	C₂²	—	2	4	no	—
572	C₂²	—	2	8	no	—
573	C₂²	—	2	16	no	—
574	C₂²	—	2	8	no	—
575	C₂²	—	2	16	no	—
576	C₂²	—	2	8	no	—
577	C₂²	—	4	16	no	—
578	C₂²	—	4	16	no	—
579	C₂²	—	8	32	no	—
580	C₂²	—	268	16	no	—
581	C₂²	—	536	32	no	—
582	C₂²	—	134	8	no	—
583	C₂²	—	134	8	no	—
584	C₂²	—	134	8	no	—
585	C₂²	C₂×#14524088	1	4	yes	—
586	C₂²	C₂×#14524088	1	2	yes	—
587	C₂²	C₂×#14524088	1	4	yes	—
588	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	2	4	no	—
592	C₄	—	134	8	no	—
593	C₄	—	134	8	no	—
594	C₄	—	134	8	no	—
595	C₄	C₂×#14524088	1	8	yes	—
596	C₂	—	2	4	no	—
597	C₂	—	4	8	no	—
598	C₂	—	4	8	no	—
599	C₂	—	268	8	no	—
600	C₂	#22574542	1	4	yes	direct product factor
601	C₂	C₂²×#14524088	1	1	yes	characteristic
602	C₂	C₂²×#14524088	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂²×#127×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂²×#127	D₆₇
hypocenter	C₆₇	C₂×#3543
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#127

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22574542	C₂	direct (N × H)
2. C₂×#16864133	C₂	semidirect
3. C₂×#16864132	C₂	semidirect
4. C₂×#16864066	C₂	semidirect
5. C₂×#16864040	C₂	semidirect
6. C₂×#16864031	C₂	semidirect
7. C₂×D₄×C₂²⋊D₆₇	C₂	semidirect
8. C₂×#16864000	C₂	semidirect
9. C₂×#16863959	C₂	semidirect
10. C₂×#16863955	C₂	semidirect
11. C₂×#16863930	C₂	semidirect
12. C₂²×#127×C₆₇	C₂	semidirect
13. C₂²×#11761627	C₂²	semidirect
14. C₂³×C₂²⋊D₆₇	C₂²	semidirect
15. C₂²×D₄×D₆₇	C₂²	semidirect
16. #16864133	C₂²	semidirect
17. #16864132	C₂²	semidirect
18. #16864066	C₂²	semidirect
19. #16864040	C₂²	semidirect
20. #16864031	C₂²	semidirect
21. D₄×C₂²⋊D₆₇	C₂²	semidirect
22. #16864000	C₂²	semidirect
23. #16863959	C₂²	semidirect
24. #16863955	C₂²	semidirect
25. C₂×#14524039	C₂²	semidirect
26. #16863930	C₂²	semidirect
27. C₂×#14524037	C₂²	semidirect
28. C₂×#14524035	C₂²	semidirect
29. C₂×#14524031	C₂²	semidirect
30. C₂×#14524030	C₂²	semidirect
31. C₂×#14524026	C₂²	semidirect
32. C₂×#14524024	C₂²	semidirect
33. C₂×#14524024	C₂²	semidirect
34. C₂×#14524024	C₂²	semidirect
35. C₂×#14524022	C₂²	semidirect
37. C₂×#14524015	C₂²	semidirect
38. C₂×#14524015	C₂²	semidirect
39. C₂×#14524015	C₂²	semidirect
40. C₂×#14524015	C₂²	semidirect
41. C₂×#14524014	C₂²	semidirect
42. C₂×#14524014	C₂²	semidirect
43. C₂×#14524012	C₂²	semidirect
44. C₂×#14524011	C₂²	semidirect
45. C₂×#14524011	C₂²	semidirect
46. C₂×#14524011	C₂²	semidirect
47. C₂×C₄×Dic₁₃₄	C₂²	semidirect
48. C₂×C₄×D₂₆₈	C₂²	semidirect
49. C₂×D₄×Dic₆₇	C₂²	semidirect
50. C₂×D₄×Dic₆₇	C₂²	semidirect
51. C₂×C₄×C₂²⋊D₆₇	C₂²	semidirect
52. C₂×#14524000	C₂²	semidirect
53. C₂²×C₂²⋊Dic₆₇	C₂²	semidirect
54. C₂³×D₄×C₆₇	C₂²	semidirect
55. C₂³×D₄×C₆₇	C₂²	semidirect
57. C₂×#127×C₆₇	C₂²	semidirect
58. C₂×#127×C₆₇	C₂²	semidirect
59. C₂×#127×C₆₇	C₂²	semidirect
60. C₂×#127×C₆₇	C₂²	semidirect
61. C₂²×C₄²×C₆₇	C₂²	semidirect
64. C₂×#11761627	C₂³	semidirect
72. C₂²×C₂²⋊D₆₇	C₂³	semidirect
74. C₂×D₄×D₆₇	C₂³	semidirect
78. #14524039	C₂³	semidirect
79. #14524037	C₂³	semidirect
80. #14524035	C₂³	semidirect
81. #14524031	C₂³	semidirect
82. #14524030	C₂³	semidirect
83. #14524026	C₂³	semidirect
84. #14524024	C₂³	semidirect
85. #14524024	C₂³	semidirect
86. #14524024	C₂³	semidirect
87. #14524022	C₂³	semidirect
89. #14524015	C₂³	semidirect
90. #14524015	C₂³	semidirect
91. #14524015	C₂³	semidirect
92. #14524015	C₂³	semidirect
93. #14524014	C₂³	semidirect
94. #14524014	C₂³	semidirect
95. #14524012	C₂³	semidirect
96. #14524011	C₂³	semidirect
97. #14524011	C₂³	semidirect
98. #14524011	C₂³	semidirect
99. C₄×Dic₁₃₄	C₂³	semidirect
100. C₄×D₂₆₈	C₂³	semidirect
101. D₄×Dic₆₇	C₂³	semidirect
102. D₄×Dic₆₇	C₂³	semidirect
103. C₄×C₂²⋊D₆₇	C₂³	semidirect
104. #14524000	C₂³	semidirect
116. C₂×C₂²⋊Dic₆₇	C₂³	semidirect
131. C₂²×D₄×C₆₇	C₂³	semidirect
132. C₂²×D₄×C₆₇	C₂³	semidirect
136. #127×C₆₇	C₂³	semidirect
137. #127×C₆₇	C₂³	semidirect
138. #127×C₆₇	C₂³	semidirect
142. C₂×C₄²×C₆₇	C₂³	semidirect
298. C₂×C₆₇	#3543	semidirect
312. C₂²×#127	D₆₇	semidirect
314. C₆₇	C₂×#3543	semidirect
318. C₂³×D₄	C₂×D₆₇	semidirect
319. C₂³×D₄	C₂×D₆₇	semidirect
331. C₂×#127	C₂×D₆₇	semidirect
332. C₂×#127	C₂×D₆₇	semidirect
333. C₂×#127	C₂×D₆₇	semidirect
334. C₂×#127	C₂×D₆₇	semidirect
363. C₂²×C₄²	C₂×D₆₇	semidirect
386. C₂²×D₄	C₂²×D₆₇	semidirect
387. C₂²×D₄	C₂²×D₆₇	semidirect
395. #127	C₂²×D₆₇	semidirect
396. #127	C₂²×D₆₇	semidirect
397. #127	C₂²×D₆₇	semidirect
444. C₂×C₄²	C₂²×D₆₇	semidirect
600. C₂	#22574542	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22574542	2
C₂×#16864133	2
C₂×#16864132	2
C₂×#16864066	2
C₂×#16864040	2
C₂×#16864031	2
C₂×D₄×C₂²⋊D₆₇	2
C₂×#16864000	2
C₂×#16863959	2
C₂×#16863955	2
C₂×#16863930	2
C₂²×#127×C₆₇	2
C₂×#3543	67

Subgroups — C2×#22574542 #31763424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22574542 #31763424