Subgroups — C₂×#22573973 #31763725

Order 17152.

12378 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524088	C₂	1	2	yes	maximal, complemented
2	#22573973	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16864050	C₂	1	4	yes	maximal, complemented
4	C₂×#16864041	C₂	1	8	yes	maximal, complemented
5	C₂×D₄×D₂₆₈	C₂	1	4	yes	maximal, complemented
6	C₂×#16863977	C₂	1	2	yes	maximal, complemented
7	C₂×#16863958	C₂	1	4	yes	maximal, complemented
8	C₂×#16863946	C₂	1	2	yes	maximal, complemented
9	C₂²×#14524017	C₂	1	4	yes	maximal, complemented
10	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#14524088	—	2	32	no	—
12	C₂×#14524088	C₂²	1	32	yes	complemented
13	C₂²×#11761628	C₂²	1	4	yes	complemented
14	C₂²×#11761627	C₂²	1	8	yes	complemented
15	C₂³×D₂₆₈	C₂²	1	4	yes	complemented
16	C₂³×C₂²⋊D₆₇	C₂²	1	8	yes	complemented
17	C₂²×D₄×D₆₇	C₂²	1	8	yes	complemented
18	#16864050	C₂²	1	64	yes	complemented
19	#16864041	C₂²	1	128	yes	complemented
20	D₄×D₂₆₈	C₂²	1	64	yes	complemented
21	#16863977	C₂²	1	32	yes	complemented
22	#16863958	C₂²	1	64	yes	complemented
23	#16863946	C₂²	1	32	yes	complemented
24	C₂×#14524043	C₂²	1	2	yes	complemented
25	C₂×#14524040	C₂²	1	2	yes	complemented
26	C₂×#14524031	C₂²	1	4	yes	complemented
27	C₂×#14524027	C₂²	1	16	yes	complemented
28	C₂×#14524023	C₂²	1	8	yes	complemented
29	C₂×#14524018	C₂²	1	8	yes	complemented
30	C₂×#14524017	C₂²	1	8	yes	complemented
31	C₂×#14524017	C₂²	1	8	yes	complemented
32	C₂×#14524017	C₂²	1	4	yes	complemented
33	C₂×#14524017	C₂²	1	8	yes	complemented
34	C₂×#14524017	C₂²	1	4	yes	complemented
35	C₂×#14524017	C₂²	1	64	yes	complemented
36	C₂×#14524013	C₂²	1	16	yes	complemented
37	C₂×C₄×D₂₆₈	C₂²	1	4	yes	complemented
38	C₂×C₄×D₂₆₈	C₂²	1	4	yes	complemented
39	C₂²×C₄⋊Dic₆₇	C₂²	1	4	yes	complemented
40	C₂²×(C₂×D₆₇)⋊C₄	C₂²	1	8	yes	complemented
41	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
42	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	complemented
43	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
44	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
45	C₂×C₄×D₄×C₆₇	C₂²	1	16	yes	complemented
46	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
47	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	complemented
48	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic, complemented
49	C₂⁴×D₆₇	C₂³	1	8	yes	—
50	#14524088	—	2	256	no	—
51	C₂×#11761628	—	2	32	no	—
52	C₂×#11761628	C₂³	1	32	yes	complemented
53	C₂×#11761627	—	2	64	no	—
54	C₂×#11761627	C₂³	1	64	yes	complemented
55	C₂²×Dic₁₃₄	C₂³	1	4	yes	—
56	C₂²×D₂₆₈	—	4	16	no	—
57	C₂²×D₂₆₈	C₂³	1	4	yes	—
58	C₂²×D₂₆₈	C₂³	1	4	yes	—
59	C₂²×D₂₆₈	C₂³	1	4	yes	—
60	C₂²×D₂₆₈	C₂³	1	8	yes	—
61	C₂²×D₂₆₈	C₂³	1	32	yes	complemented
62	C₂²×C₂²⋊D₆₇	—	2	64	no	—
63	C₂²×C₂²⋊D₆₇	C₂³	1	16	yes	—
64	C₂²×C₂²⋊D₆₇	C₂³	1	8	yes	—
65	C₂²×C₂²⋊D₆₇	C₂³	1	64	yes	complemented
66	C₂×D₄×D₆₇	—	2	64	no	—
67	C₂×D₄×D₆₇	C₂³	1	64	yes	complemented
68	C₂³×Dic₆₇	C₂³	1	8	yes	—
69	C₂²×C₄×D₆₇	C₂³	1	8	yes	—
70	#14524043	C₂³	1	16	yes	complemented
71	#14524040	C₂³	1	16	yes	complemented
72	#14524031	C₂³	1	32	yes	complemented
73	#14524027	C₂³	1	128	yes	complemented
74	#14524023	C₂³	1	64	yes	complemented
75	#14524018	C₂³	1	64	yes	complemented
76	#14524017	C₂³	1	64	yes	complemented
77	#14524017	C₂³	1	64	yes	complemented
78	#14524017	C₂³	1	32	yes	complemented
79	#14524017	C₂³	1	64	yes	complemented
80	#14524017	C₂³	1	32	yes	complemented
81	#14524013	C₂³	1	128	yes	complemented
82	C₄×D₂₆₈	C₂³	1	32	yes	complemented
83	C₄×D₂₆₈	C₂³	1	32	yes	complemented
84	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
85	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
86	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
87	C₂×C₄⋊Dic₆₇	C₂³	1	32	yes	complemented
88	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
89	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	16	yes	—
90	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
91	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	64	yes	complemented
92	C₂⁵×C₆₇	C₂³	1	2	yes	—
93	C₂²×D₄×C₆₇	D₄	1	16	yes	complemented
94	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
95	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
96	C₂³×C₄×C₆₇	—	4	8	no	—
97	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
98	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
99	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
100	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
101	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
102	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
103	C₂³×C₄×C₆₇	C₂³	1	16	yes	complemented
104	C₄×D₄×C₆₇	C₂³	1	32	yes	complemented
105	C₄×D₄×C₆₇	C₂³	1	32	yes	complemented
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	complemented
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
109	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	complemented
110	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
112	C₂×C₄²×C₆₇	C₂³	1	8	yes	complemented
113	C₂³×D₆₇	—	2	32	no	—
114	C₂³×D₆₇	—	4	16	no	—
115	C₂³×D₆₇	—	4	16	no	—
116	C₂³×D₆₇	C₂⁴	1	8	yes	—
117	C₂³×D₆₇	C₂⁴	1	32	yes	—
118	#11761628	—	2	128	no	—
119	#11761627	—	2	256	no	—
120	C₂×Dic₁₃₄	—	2	16	no	—
121	C₂×Dic₁₃₄	C₂⁴	1	16	yes	—
122	C₂×D₂₆₈	—	2	16	no	—
123	C₂×D₂₆₈	—	2	16	no	—
124	C₂×D₂₆₈	—	4	64	no	—
125	C₂×D₂₆₈	—	4	16	no	—
126	C₂×D₂₆₈	—	4	8	no	—
127	C₂×D₂₆₈	C₂⁴	1	16	yes	—
128	C₂×D₂₆₈	C₂⁴	1	16	yes	—
129	C₂×D₂₆₈	C₂⁴	1	16	yes	—
130	C₂×D₂₆₈	C₂⁴	1	32	yes	—
131	C₂×C₂²⋊D₆₇	—	2	64	no	—
132	C₂×C₂²⋊D₆₇	—	2	32	no	—
133	C₂×C₂²⋊D₆₇	—	2	256	no	—
134	C₂×C₂²⋊D₆₇	C₂⁴	1	64	yes	—
135	C₂×C₂²⋊D₆₇	C₂⁴	1	32	yes	—
136	D₄×D₆₇	—	2	256	no	—
137	C₂²×Dic₆₇	—	2	32	no	—
138	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
139	C₂²×Dic₆₇	C₂⁴	1	32	yes	—
140	C₂×C₄×D₆₇	—	2	32	no	—
141	C₂×C₄×D₆₇	C₂⁴	1	32	yes	—
142	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
143	C₄⋊Dic₆₇	C₂⁴	1	32	yes	—
144	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
145	(C₂×D₆₇)⋊C₄	C₂⁴	1	32	yes	—
146	(C₂×D₆₇)⋊C₄	C₂⁴	1	64	yes	—
147	(C₂×D₆₇)⋊C₄	C₂⁴	1	32	yes	—
148	C₂⁴×C₆₇	—	4	4	no	—
149	C₂⁴×C₆₇	—	4	4	no	—
150	C₂⁴×C₆₇	C₂×D₄	1	16	yes	complemented
151	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
152	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
153	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
154	C₂×D₄×C₆₇	C₂×D₄	1	32	yes	complemented
155	C₂×D₄×C₆₇	C₂×D₄	1	64	yes	complemented
156	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
157	C₂²×C₄×C₆₇	—	2	4	no	—
158	C₂²×C₄×C₆₇	—	2	16	no	—
159	C₂²×C₄×C₆₇	—	4	32	no	—
160	C₂²×C₄×C₆₇	—	4	8	no	—
161	C₂²×C₄×C₆₇	—	4	4	no	—
162	C₂²×C₄×C₆₇	—	4	2	no	—
163	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	complemented
164	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
175	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
176	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
177	C₄²×C₆₇	C₂⁴	1	8	yes	—
178	C₄²×C₆₇	C₂⁴	1	8	yes	—
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	2	8	no	—
181	C₂³×C₆₇	—	2	2	no	—
182	C₂³×C₆₇	—	2	8	no	—
183	C₂³×C₆₇	—	4	8	no	—
184	C₂³×C₆₇	—	4	32	no	—
185	C₂³×C₆₇	—	4	8	no	—
186	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
187	C₂³×C₆₇	C₂²×D₄	1	16	yes	—
188	C₂³×C₆₇	C₂²×D₄	1	32	yes	complemented
189	C₂³×C₆₇	C₂⁵	1	8	yes	—
190	C₂³×C₆₇	C₂⁵	1	2	yes	—
191	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
192	C₂²×D₆₇	—	2	16	no	—
193	C₂²×D₆₇	—	2	16	no	—
194	C₂²×D₆₇	—	2	16	no	—
195	C₂²×D₆₇	—	2	64	no	—
196	C₂²×D₆₇	—	4	32	no	—
197	C₂²×D₆₇	—	4	32	no	—
198	C₂²×D₆₇	—	8	64	no	—
199	C₂²×D₆₇	C₂⁵	1	16	yes	—
200	D₄×C₆₇	C₂²×D₄	1	64	yes	complemented
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	8	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	8	no	—
205	C₂×C₄×C₆₇	—	2	32	no	—
206	C₂×C₄×C₆₇	—	4	8	no	—
207	C₂×C₄×C₆₇	—	4	16	no	—
208	C₂×C₄×C₆₇	—	4	4	no	—
209	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
210	C₂×C₄×C₆₇	C₂²×D₄	1	16	yes	complemented
211	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	complemented
212	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
213	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
214	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
215	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
216	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
217	C₂²⋊D₆₇	—	2	128	no	—
218	C₂²⋊D₆₇	—	2	64	no	—
219	C₂×Dic₆₇	—	2	16	no	—
220	C₂×Dic₆₇	—	2	64	no	—
221	C₂×Dic₆₇	C₂⁵	1	16	yes	—
222	D₂₆₈	—	2	32	no	—
223	D₂₆₈	—	2	32	no	—
224	D₂₆₈	—	4	32	no	—
225	D₂₆₈	—	4	16	no	—
226	C₄×D₆₇	—	2	64	no	—
227	Dic₁₃₄	—	2	32	no	—
228	C₂²×C₆₇	—	2	2	no	—
229	C₂²×C₆₇	—	2	8	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	2	8	no	—
232	C₂²×C₆₇	—	2	32	no	—
233	C₂²×C₆₇	—	2	8	no	—
234	C₂²×C₆₇	—	4	32	no	—
235	C₂²×C₆₇	C₂³×D₄	1	16	yes	—
236	C₂²×C₆₇	C₂³×D₄	1	4	yes	—
237	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
238	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
239	C₂²×C₆₇	#585	1	2	yes	—
240	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
241	C₂×D₆₇	—	2	16	no	—
242	C₂×D₆₇	—	2	16	no	—
243	C₂×D₆₇	—	2	16	no	—
244	C₂×D₆₇	—	4	32	no	—
245	C₂×D₆₇	—	8	64	no	—
246	C₄×C₆₇	—	2	8	no	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	2	4	no	—
249	C₄×C₆₇	C₂³×D₄	1	8	yes	—
250	C₄×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	complemented
251	Dic₆₇	—	2	16	no	—
252	C₂×#3496	—	67	1	no	maximal, Sylow
253	C₂×C₆₇	—	2	16	no	—
254	C₂×C₆₇	—	2	4	no	—
255	C₂×C₆₇	#3496	1	4	yes	complemented
256	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
257	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
258	C₂×C₆₇	C₂×#585	1	1	yes	characteristic
259	D₆₇	—	4	16	no	—
260	C₂²×H(5,𝔽₂)	—	67	2	no	—
261	#3496	—	67	32	no	—
262	C₂×#547	—	67	4	no	—
263	C₂×#546	—	67	8	no	—
264	C₂×D₄²	—	67	4	no	—
265	C₂×#537	—	67	2	no	—
266	C₂×#534	—	67	4	no	—
267	C₂×#532	—	67	2	no	—
268	C₂²×#121	—	67	4	no	—
269	C₂²×C₄×D₄	D₆₇	1	1	yes	characteristic, complemented
270	C₆₇	C₂×#3496	1	1	yes	characteristic, complemented, Sylow
271	C₂×H(5,𝔽₂)	—	67	32	no	—
272	C₂×H(5,𝔽₂)	—	134	32	no	—
273	C₂²×#41	—	67	8	no	—
274	C₂²×#41	—	67	4	no	—
275	C₂³×D₄	—	67	8	no	—
276	C₂³×D₄	—	67	8	no	—
277	C₂³×D₄	—	67	4	no	—
278	C₂³×D₄	C₂×D₆₇	1	1	yes	characteristic
279	C₂⁴×C₄	C₂×D₆₇	1	2	yes	complemented
280	#547	—	67	64	no	—
281	#546	—	67	128	no	—
282	D₄²	—	67	64	no	—
283	#537	—	67	32	no	—
284	#534	—	67	64	no	—
285	#532	—	67	32	no	—
286	C₂×C₄⋊Q₈	—	67	2	no	—
287	C₂×#127	—	67	2	no	—
288	C₂×#124	—	67	4	no	—
289	C₂×#123	—	67	16	no	—
290	C₂×C₂²⋊Q₈	—	67	8	no	—
291	C₂×#121	—	67	4	no	—
292	C₂×#121	—	67	8	no	—
293	C₂×#121	—	67	8	no	—
294	C₂×#121	—	67	8	no	—
295	C₂×#121	—	67	4	no	—
296	C₂×#121	—	67	64	no	—
297	C₂×#121	—	67	8	no	—
298	C₂×#120	—	67	16	no	—
299	C₂×C₄×D₄	—	67	4	no	—
300	C₂×C₄×D₄	—	67	4	no	—
301	C₂×C₄×D₄	C₂×D₆₇	1	4	yes	complemented
302	C₂×C₄×D₄	C₂×D₆₇	1	4	yes	complemented
303	C₂×C₄×D₄	C₂×D₆₇	1	16	yes	complemented
304	C₂²×C₄⋊C₄	—	67	4	no	—
305	C₂²×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic, complemented
306	C₂²×C₂²⋊C₄	—	67	8	no	—
307	C₂²×C₂²⋊C₄	C₂×D₆₇	1	2	yes	complemented
308	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic, complemented
309	C₂⁵	—	67	8	no	—
310	C₂⁵	C₂²×D₆₇	1	2	yes	—
311	H(5,𝔽₂)	—	134	256	no	—
312	C₂×#41	—	67	64	no	—
313	C₂×#41	—	67	32	no	—
314	C₂×#41	—	134	64	no	—
315	C₂×#41	—	134	32	no	—
316	C₂²×Q₈	—	67	4	no	—
317	C₂²×D₄	—	268	16	no	—
318	C₂²×D₄	—	67	16	no	—
319	C₂²×D₄	—	67	8	no	—
320	C₂²×D₄	—	67	64	no	—
321	C₂²×D₄	—	67	64	no	—
322	C₂²×D₄	—	67	4	no	—
323	C₂²×D₄	—	67	8	no	—
324	C₂²×D₄	—	67	32	no	—
325	C₂²×D₄	—	67	4	no	—
326	C₂²×D₄	—	67	4	no	—
327	C₂²×D₄	—	134	64	no	—
328	C₂²×D₄	—	134	64	no	—
329	C₂²×D₄	D₂₆₈	1	16	yes	complemented
330	C₂²×D₄	C₂²×D₆₇	1	4	yes	—
331	C₂²×D₄	C₂²×D₆₇	1	8	yes	—
332	C₂³×C₄	—	4	8	no	—
333	C₂³×C₄	—	67	8	no	—
334	C₂³×C₄	—	67	8	no	—
335	C₂³×C₄	C₂²×D₆₇	1	4	yes	—
336	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
337	C₂³×C₄	C₂²×D₆₇	1	4	yes	—
338	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
339	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
340	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
341	C₂³×C₄	C₂²×D₆₇	1	16	yes	complemented
342	C₄⋊Q₈	—	67	16	no	—
343	#127	—	67	16	no	—
344	#124	—	67	32	no	—
345	#123	—	67	128	no	—
346	C₂²⋊Q₈	—	67	64	no	—
347	#121	—	67	32	no	—
348	#121	—	67	64	no	—
349	#121	—	67	64	no	—
350	#121	—	67	64	no	—
351	#121	—	67	32	no	—
352	#121	—	67	64	no	—
353	#120	—	67	128	no	—
354	C₄×D₄	—	67	32	no	—
355	C₄×D₄	—	67	32	no	—
356	C₄×D₄	C₂²×D₆₇	1	32	yes	complemented
357	C₄×D₄	C₂²×D₆₇	1	32	yes	complemented
358	C₂×C₄⋊C₄	—	67	4	no	—
359	C₂×C₄⋊C₄	—	67	8	no	—
360	C₂×C₄⋊C₄	—	67	32	no	—
361	C₂×C₄⋊C₄	—	67	4	no	—
362	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
363	C₂×C₄⋊C₄	C₂²×D₆₇	1	8	yes	complemented
364	C₂×C₂²⋊C₄	—	67	8	no	—
365	C₂×C₂²⋊C₄	—	67	16	no	—
366	C₂×C₂²⋊C₄	—	67	64	no	—
367	C₂×C₂²⋊C₄	—	67	8	no	—
368	C₂×C₂²⋊C₄	C₂²×D₆₇	1	8	yes	—
369	C₂×C₂²⋊C₄	C₂²×D₆₇	1	16	yes	complemented
370	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
371	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
372	C₂×C₄²	C₂²×D₆₇	1	8	yes	complemented
373	C₂⁴	—	4	4	no	—
374	C₂⁴	—	4	4	no	—
375	C₂⁴	—	268	16	no	—
376	C₂⁴	—	268	16	no	—
377	C₂⁴	—	67	32	no	—
378	C₂⁴	—	67	8	no	—
379	C₂⁴	—	134	32	no	—
380	C₂⁴	C₂×D₂₆₈	1	16	yes	complemented
381	C₂⁴	C₂³×D₆₇	1	4	yes	—
382	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
383	C₂⁴	C₂³×D₆₇	1	8	yes	—
384	#41	—	134	256	no	—
385	#41	—	134	128	no	—
386	C₂×Q₈	—	67	16	no	—
387	C₂×Q₈	—	134	16	no	—
388	C₂×D₄	—	268	16	no	—
389	C₂×D₄	—	268	8	no	—
390	C₂×D₄	—	268	64	no	—
391	C₂×D₄	—	67	64	no	—
392	C₂×D₄	—	67	32	no	—
393	C₂×D₄	—	67	16	no	—
394	C₂×D₄	—	67	32	no	—
395	C₂×D₄	—	67	16	no	—
396	C₂×D₄	—	67	16	no	—
397	C₂×D₄	—	134	64	no	—
398	C₂×D₄	—	134	32	no	—
399	C₂×D₄	—	134	256	no	—
400	C₂×D₄	—	134	256	no	—
401	C₂×D₄	—	134	16	no	—
402	C₂×D₄	—	134	16	no	—
403	C₂×D₄	C₂×D₂₆₈	1	32	yes	complemented
404	C₂×D₄	C₂×D₂₆₈	1	64	yes	complemented
405	C₂×D₄	C₂³×D₆₇	1	16	yes	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	2	16	no	—
408	C₂²×C₄	—	4	32	no	—
409	C₂²×C₄	—	4	8	no	—
410	C₂²×C₄	—	4	4	no	—
411	C₂²×C₄	—	4	2	no	—
412	C₂²×C₄	—	67	32	no	—
413	C₂²×C₄	—	67	8	no	—
414	C₂²×C₄	—	67	32	no	—
415	C₂²×C₄	—	134	32	no	—
416	C₂²×C₄	—	134	32	no	—
417	C₂²×C₄	C₂×D₂₆₈	1	8	yes	complemented
418	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
419	C₂²×C₄	C₂³×D₆₇	1	16	yes	—
420	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
421	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
422	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
423	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
424	C₂²×C₄	C₂³×D₆₇	1	16	yes	—
425	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
426	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
427	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
428	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
429	C₄⋊C₄	—	67	16	no	—
430	C₄⋊C₄	—	67	32	no	—
431	C₄⋊C₄	—	67	16	no	—
432	C₄⋊C₄	C₂³×D₆₇	1	16	yes	—
433	C₂²⋊C₄	—	67	32	no	—
434	C₂²⋊C₄	—	67	64	no	—
435	C₂²⋊C₄	—	67	32	no	—
436	C₂²⋊C₄	C₂³×D₆₇	1	32	yes	—
437	C₄²	C₂³×D₆₇	1	8	yes	—
438	C₄²	C₂³×D₆₇	1	8	yes	—
439	C₂³	—	2	2	no	—
440	C₂³	—	2	8	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	8	no	—
443	C₂³	—	4	8	no	—
444	C₂³	—	4	32	no	—
445	C₂³	—	4	8	no	—
446	C₂³	—	268	32	no	—
447	C₂³	—	268	32	no	—
448	C₂³	—	536	64	no	—
449	C₂³	—	67	16	no	—
450	C₂³	—	134	16	no	—
451	C₂³	—	134	16	no	—
452	C₂³	—	134	16	no	—
453	C₂³	—	134	64	no	—
454	C₂³	C₂²×D₂₆₈	1	16	yes	—
455	C₂³	C₂²×D₂₆₈	1	4	yes	—
456	C₂³	C₂²×D₂₆₈	1	32	yes	complemented
457	C₂³	C₂⁴×D₆₇	1	8	yes	—
458	C₂³	C₂⁴×D₆₇	1	2	yes	—
459	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
460	Q₈	—	134	32	no	—
461	D₄	—	268	32	no	—
462	D₄	—	268	16	no	—
463	D₄	—	134	128	no	—
464	D₄	—	134	64	no	—
465	D₄	—	134	32	no	—
466	D₄	—	134	32	no	—
467	D₄	C₂²×D₂₆₈	1	64	yes	complemented
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	8	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	32	no	—
473	C₂×C₄	—	4	16	no	—
474	C₂×C₄	—	4	8	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	67	16	no	—
477	C₂×C₄	—	134	16	no	—
478	C₂×C₄	—	134	64	no	—
479	C₂×C₄	—	134	64	no	—
480	C₂×C₄	C₂²×D₂₆₈	1	8	yes	—
481	C₂×C₄	C₂²×D₂₆₈	1	16	yes	complemented
482	C₂×C₄	#14524088	1	4	yes	complemented
483	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
484	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
485	C₂×C₄	C₂⁴×D₆₇	1	8	yes	—
486	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
487	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
488	C₂²	—	2	8	no	—
489	C₂²	—	2	32	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	8	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	8	no	—
494	C₂²	—	4	32	no	—
495	C₂²	—	268	32	no	—
496	C₂²	—	536	64	no	—
497	C₂²	—	134	16	no	—
498	C₂²	—	134	16	no	—
499	C₂²	—	134	16	no	—
500	C₂²	C₂³×D₂₆₈	1	16	yes	—
501	C₂²	C₂³×D₂₆₈	1	4	yes	—
502	C₂²	C₂³×D₂₆₈	1	2	yes	—
503	C₂²	C₂×#14524088	1	2	yes	—
504	C₂²	#16864232	1	2	yes	—
505	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
506	C₄	—	2	2	no	—
507	C₄	—	2	8	no	—
508	C₄	—	2	4	no	—
509	C₄	—	134	16	no	—
510	C₄	C₂³×D₂₆₈	1	8	yes	—
511	C₄	C₂×#14524088	1	4	yes	complemented
512	C₂	—	2	4	no	—
513	C₂	—	2	16	no	—
514	C₂	—	268	16	no	—
515	C₂	#22573973	1	4	yes	direct product factor
516	C₂	C₂⁴×D₂₆₈	1	1	yes	characteristic
517	C₂	C₂²×#14524088	1	1	yes	characteristic
518	C₂	C₂×#16864232	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂²×C₄×D₄×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂²×C₄×D₄	D₆₇
hypocenter	C₆₇	C₂×#3496
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14524088	C₂	semidirect
2. #22573973	C₂	direct (N × H)
3. C₂×#16864050	C₂	semidirect
4. C₂×#16864041	C₂	semidirect
5. C₂×D₄×D₂₆₈	C₂	semidirect
6. C₂×#16863977	C₂	semidirect
7. C₂×#16863958	C₂	semidirect
8. C₂×#16863946	C₂	semidirect
9. C₂²×#14524017	C₂	semidirect
10. C₂²×C₄×D₄×C₆₇	C₂	semidirect
12. C₂×#14524088	C₂²	semidirect
13. C₂²×#11761628	C₂²	semidirect
14. C₂²×#11761627	C₂²	semidirect
15. C₂³×D₂₆₈	C₂²	semidirect
16. C₂³×C₂²⋊D₆₇	C₂²	semidirect
17. C₂²×D₄×D₆₇	C₂²	semidirect
18. #16864050	C₂²	semidirect
19. #16864041	C₂²	semidirect
20. D₄×D₂₆₈	C₂²	semidirect
21. #16863977	C₂²	semidirect
22. #16863958	C₂²	semidirect
23. #16863946	C₂²	semidirect
24. C₂×#14524043	C₂²	semidirect
25. C₂×#14524040	C₂²	semidirect
26. C₂×#14524031	C₂²	semidirect
27. C₂×#14524027	C₂²	semidirect
28. C₂×#14524023	C₂²	semidirect
29. C₂×#14524018	C₂²	semidirect
30. C₂×#14524017	C₂²	semidirect
31. C₂×#14524017	C₂²	semidirect
32. C₂×#14524017	C₂²	semidirect
33. C₂×#14524017	C₂²	semidirect
34. C₂×#14524017	C₂²	semidirect
35. C₂×#14524017	C₂²	semidirect
36. C₂×#14524013	C₂²	semidirect
37. C₂×C₄×D₂₆₈	C₂²	semidirect
38. C₂×C₄×D₂₆₈	C₂²	semidirect
39. C₂²×C₄⋊Dic₆₇	C₂²	semidirect
40. C₂²×(C₂×D₆₇)⋊C₄	C₂²	semidirect
42. C₂⁴×C₄×C₆₇	C₂²	semidirect
43. C₂×C₄×D₄×C₆₇	C₂²	semidirect
44. C₂×C₄×D₄×C₆₇	C₂²	semidirect
45. C₂×C₄×D₄×C₆₇	C₂²	semidirect
46. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
47. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
48. C₂²×C₄²×C₆₇	C₂²	semidirect
52. C₂×#11761628	C₂³	semidirect
54. C₂×#11761627	C₂³	semidirect
61. C₂²×D₂₆₈	C₂³	semidirect
65. C₂²×C₂²⋊D₆₇	C₂³	semidirect
67. C₂×D₄×D₆₇	C₂³	semidirect
70. #14524043	C₂³	semidirect
71. #14524040	C₂³	semidirect
72. #14524031	C₂³	semidirect
73. #14524027	C₂³	semidirect
74. #14524023	C₂³	semidirect
75. #14524018	C₂³	semidirect
76. #14524017	C₂³	semidirect
77. #14524017	C₂³	semidirect
78. #14524017	C₂³	semidirect
79. #14524017	C₂³	semidirect
80. #14524017	C₂³	semidirect
81. #14524013	C₂³	semidirect
82. C₄×D₂₆₈	C₂³	semidirect
83. C₄×D₂₆₈	C₂³	semidirect
87. C₂×C₄⋊Dic₆₇	C₂³	semidirect
91. C₂×(C₂×D₆₇)⋊C₄	C₂³	semidirect
93. C₂²×D₄×C₆₇	D₄	semidirect
103. C₂³×C₄×C₆₇	C₂³	semidirect
104. C₄×D₄×C₆₇	C₂³	semidirect
105. C₄×D₄×C₆₇	C₂³	semidirect
107. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
109. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
112. C₂×C₄²×C₆₇	C₂³	semidirect
150. C₂⁴×C₆₇	C₂×D₄	semidirect
154. C₂×D₄×C₆₇	C₂×D₄	semidirect
155. C₂×D₄×C₆₇	C₂×D₄	semidirect
163. C₂²×C₄×C₆₇	C₂×D₄	semidirect
188. C₂³×C₆₇	C₂²×D₄	semidirect
200. D₄×C₆₇	C₂²×D₄	semidirect
210. C₂×C₄×C₆₇	C₂²×D₄	semidirect
211. C₂×C₄×C₆₇	H(5,𝔽₂)	semidirect
250. C₄×C₆₇	C₂×H(5,𝔽₂)	semidirect
255. C₂×C₆₇	#3496	semidirect
269. C₂²×C₄×D₄	D₆₇	semidirect
270. C₆₇	C₂×#3496	semidirect
279. C₂⁴×C₄	C₂×D₆₇	semidirect
301. C₂×C₄×D₄	C₂×D₆₇	semidirect
302. C₂×C₄×D₄	C₂×D₆₇	semidirect
303. C₂×C₄×D₄	C₂×D₆₇	semidirect
305. C₂²×C₄⋊C₄	C₂×D₆₇	semidirect
307. C₂²×C₂²⋊C₄	C₂×D₆₇	semidirect
308. C₂²×C₄²	C₂×D₆₇	semidirect
329. C₂²×D₄	D₂₆₈	semidirect
341. C₂³×C₄	C₂²×D₆₇	semidirect
356. C₄×D₄	C₂²×D₆₇	semidirect
357. C₄×D₄	C₂²×D₆₇	semidirect
363. C₂×C₄⋊C₄	C₂²×D₆₇	semidirect
369. C₂×C₂²⋊C₄	C₂²×D₆₇	semidirect
372. C₂×C₄²	C₂²×D₆₇	semidirect
380. C₂⁴	C₂×D₂₆₈	semidirect
403. C₂×D₄	C₂×D₂₆₈	semidirect
404. C₂×D₄	C₂×D₂₆₈	semidirect
417. C₂²×C₄	C₂×D₂₆₈	semidirect
456. C₂³	C₂²×D₂₆₈	semidirect
467. D₄	C₂²×D₂₆₈	semidirect
481. C₂×C₄	C₂²×D₂₆₈	semidirect
482. C₂×C₄	#14524088	semidirect
511. C₄	C₂×#14524088	semidirect
515. C₂	#22573973	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14524088	2
#22573973	2
C₂×#16864050	2
C₂×#16864041	2
C₂×D₄×D₂₆₈	2
C₂×#16863977	2
C₂×#16863958	2
C₂×#16863946	2
C₂²×#14524017	2
C₂²×C₄×D₄×C₆₇	2
C₂×#3496	67

Subgroups — C2×#22573973 #31763725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22573973 #31763725