Subgroups — C₂×#22574012 #31763735

Order 17152.

7738 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22574012	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16864105	C₂	1	8	yes	maximal, complemented
3	C₂×#16864103	C₂	1	4	yes	maximal, complemented
4	C₂×D₄×Dic₁₃₄	C₂	1	4	yes	maximal, complemented
5	C₂×#16864027	C₂	1	2	yes	maximal, complemented
6	C₂×#16864022	C₂	1	4	yes	maximal, complemented
7	C₂×#16863953	C₂	1	2	yes	maximal, complemented
8	C₂²×#14524022	C₂	1	4	yes	maximal, complemented
9	C₂×#16863871	C₂	1	2	yes	maximal
10	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
11	C₂³×Dic₁₃₄	C₂²	1	4	yes	complemented
12	#16864105	C₂²	1	128	yes	complemented
13	#16864103	C₂²	1	64	yes	complemented
14	D₄×Dic₁₃₄	C₂²	1	64	yes	complemented
15	#16864027	C₂²	1	32	yes	complemented
16	#16864022	C₂²	1	64	yes	complemented
17	#16863953	C₂²	1	32	yes	complemented
18	C₂×#14524043	C₂²	1	2	yes	complemented
19	C₂×#14524043	C₂²	1	2	yes	complemented
20	C₂×#14524042	C₂²	1	8	yes	complemented
21	C₂×#14524035	C₂²	1	4	yes	complemented
22	C₂×#14524034	C₂²	1	8	yes	complemented
23	C₂×#14524022	C₂²	1	4	yes	—
24	C₂×#14524022	C₂²	1	8	yes	—
25	C₂×#14524022	C₂²	1	4	yes	—
26	C₂×#14524022	C₂²	1	8	yes	complemented
27	C₂×#14524022	C₂²	1	8	yes	complemented
28	C₂×#14524022	C₂²	1	64	yes	complemented
29	C₂×#14524020	C₂²	1	16	yes	complemented
30	C₂×#14524020	C₂²	1	16	yes	complemented
31	#16863871	C₂²	1	32	yes	—
32	C₂×C₄×Dic₁₃₄	C₂²	1	4	yes	complemented
33	C₂×C₄×Dic₁₃₄	C₂²	1	4	yes	complemented
34	C₂×D₄×Dic₆₇	C₂²	1	8	yes	—
35	C₂×D₄×Dic₆₇	C₂²	1	8	yes	—
36	C₂×#14524002	C₂²	1	4	yes	—
37	C₂²×C₄⋊Dic₆₇	C₂²	1	4	yes	complemented
38	C₂²×Dic₆₇⋊C₄	C₂²	1	8	yes	complemented
39	C₂²×C₂²⋊Dic₆₇	C₂²	1	8	yes	—
40	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
41	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
42	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
43	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
44	C₂×C₄×D₄×C₆₇	C₂²	1	16	yes	—
45	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
46	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
47	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
48	C₂²×Dic₁₃₄	—	4	16	no	—
49	C₂²×Dic₁₃₄	C₂³	1	4	yes	—
50	C₂²×Dic₁₃₄	C₂³	1	4	yes	—
51	C₂²×Dic₁₃₄	C₂³	1	8	yes	—
52	C₂²×Dic₁₃₄	C₂³	1	32	yes	complemented
53	C₂³×Dic₆₇	C₂³	1	8	yes	—
54	C₂³×Dic₆₇	C₂³	1	8	yes	—
55	#14524043	C₂³	1	16	yes	complemented
56	#14524043	C₂³	1	16	yes	complemented
57	#14524042	C₂³	1	64	yes	complemented
58	#14524035	C₂³	1	32	yes	complemented
59	#14524034	C₂³	1	64	yes	complemented
60	#14524022	C₂³	1	32	yes	—
61	#14524022	C₂³	1	64	yes	—
62	#14524022	C₂³	1	32	yes	—
63	#14524022	C₂³	1	64	yes	complemented
64	#14524022	C₂³	1	64	yes	complemented
65	#14524020	C₂³	1	128	yes	complemented
66	#14524020	C₂³	1	128	yes	complemented
67	C₄×Dic₁₃₄	C₂³	1	32	yes	complemented
68	C₄×Dic₁₃₄	C₂³	1	32	yes	complemented
69	D₄×Dic₆₇	C₂³	1	64	yes	—
70	D₄×Dic₆₇	C₂³	1	64	yes	—
71	#14524002	C₂³	1	32	yes	—
72	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
73	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
74	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
75	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
76	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
77	C₂×C₄⋊Dic₆₇	C₂³	1	32	yes	complemented
78	C₂×Dic₆₇⋊C₄	C₂³	1	8	yes	—
79	C₂×Dic₆₇⋊C₄	C₂³	1	16	yes	—
80	C₂×Dic₆₇⋊C₄	C₂³	1	8	yes	—
81	C₂×Dic₆₇⋊C₄	C₂³	1	64	yes	complemented
82	C₂×C₂²⋊Dic₆₇	C₂³	1	64	yes	—
83	C₂×C₂²⋊Dic₆₇	C₂³	1	8	yes	—
84	C₂×C₂²⋊Dic₆₇	C₂³	1	16	yes	—
85	C₂×C₄×Dic₆₇	C₂³	1	8	yes	—
86	C₂⁵×C₆₇	C₂³	1	2	yes	—
87	C₂²×D₄×C₆₇	Q₈	1	16	yes	complemented
88	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
89	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
90	C₂³×C₄×C₆₇	—	4	8	no	—
91	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
92	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
93	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
94	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
95	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
96	C₂³×C₄×C₆₇	C₂³	1	16	yes	—
97	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₆₇	C₂³	1	32	yes	—
99	C₄×D₄×C₆₇	C₂³	1	32	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
104	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
105	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
106	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
107	C₂×Dic₁₃₄	—	2	16	no	—
108	C₂×Dic₁₃₄	—	4	64	no	—
109	C₂×Dic₁₃₄	—	4	16	no	—
110	C₂×Dic₁₃₄	—	4	8	no	—
111	C₂×Dic₁₃₄	C₂⁴	1	16	yes	—
112	C₂×Dic₁₃₄	C₂⁴	1	16	yes	—
113	C₂×Dic₁₃₄	C₂⁴	1	32	yes	—
114	C₂²×Dic₆₇	—	4	16	no	—
115	C₂²×Dic₆₇	—	4	16	no	—
116	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
117	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
118	C₂²×Dic₆₇	C₂⁴	1	32	yes	—
119	C₂²×Dic₆₇	C₂⁴	1	32	yes	—
120	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
121	C₄⋊Dic₆₇	C₂⁴	1	32	yes	—
122	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
123	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
124	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
125	Dic₆₇⋊C₄	C₂⁴	1	32	yes	—
126	Dic₆₇⋊C₄	C₂⁴	1	64	yes	—
127	Dic₆₇⋊C₄	C₂⁴	1	32	yes	—
128	C₂²⋊Dic₆₇	C₂⁴	1	32	yes	—
129	C₂²⋊Dic₆₇	C₂⁴	1	64	yes	—
130	C₄×Dic₆₇	C₂⁴	1	32	yes	—
131	C₂⁴×C₆₇	—	4	4	no	—
132	C₂⁴×C₆₇	—	4	4	no	—
133	C₂⁴×C₆₇	C₂×Q₈	1	16	yes	complemented
134	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
135	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
136	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
137	C₂×D₄×C₆₇	C₂×Q₈	1	32	yes	complemented
138	C₂×D₄×C₆₇	C₂×Q₈	1	64	yes	complemented
139	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
140	C₂²×C₄×C₆₇	—	2	4	no	—
141	C₂²×C₄×C₆₇	—	2	16	no	—
142	C₂²×C₄×C₆₇	—	4	32	no	—
143	C₂²×C₄×C₆₇	—	4	8	no	—
144	C₂²×C₄×C₆₇	—	4	4	no	—
145	C₂²×C₄×C₆₇	—	4	2	no	—
146	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	complemented
147	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
152	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
154	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
155	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
158	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
159	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
160	C₄²×C₆₇	C₂⁴	1	8	yes	—
161	C₄²×C₆₇	C₂⁴	1	8	yes	—
162	C₂³×C₆₇	—	2	2	no	—
163	C₂³×C₆₇	—	2	8	no	—
164	C₂³×C₆₇	—	2	2	no	—
165	C₂³×C₆₇	—	2	8	no	—
166	C₂³×C₆₇	—	4	8	no	—
167	C₂³×C₆₇	—	4	32	no	—
168	C₂³×C₆₇	—	4	8	no	—
169	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
170	C₂³×C₆₇	C₂²×Q₈	1	16	yes	—
171	C₂³×C₆₇	C₂²×Q₈	1	32	yes	complemented
172	C₂³×C₆₇	C₂⁵	1	8	yes	—
173	C₂³×C₆₇	C₂⁵	1	2	yes	—
174	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
175	D₄×C₆₇	C₂²×Q₈	1	64	yes	complemented
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	8	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	8	no	—
180	C₂×C₄×C₆₇	—	2	32	no	—
181	C₂×C₄×C₆₇	—	4	8	no	—
182	C₂×C₄×C₆₇	—	4	16	no	—
183	C₂×C₄×C₆₇	—	4	4	no	—
184	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
185	C₂×C₄×C₆₇	C₂²×Q₈	1	16	yes	complemented
186	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
187	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
188	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
189	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
190	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
191	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
192	C₂×Dic₆₇	—	2	16	no	—
193	C₂×Dic₆₇	—	2	16	no	—
194	C₂×Dic₆₇	—	4	32	no	—
195	C₂×Dic₆₇	—	4	32	no	—
196	C₂×Dic₆₇	C₂⁵	1	16	yes	—
197	C₂×Dic₆₇	C₂⁵	1	16	yes	—
198	Dic₁₃₄	—	2	32	no	—
199	Dic₁₃₄	—	4	32	no	—
200	Dic₁₃₄	—	4	16	no	—
201	C₂²×C₆₇	—	2	8	no	—
202	C₂²×C₆₇	—	2	2	no	—
203	C₂²×C₆₇	—	2	8	no	—
204	C₂²×C₆₇	—	2	2	no	—
205	C₂²×C₆₇	—	2	8	no	—
206	C₂²×C₆₇	—	2	32	no	—
207	C₂²×C₆₇	—	4	32	no	—
208	C₂²×C₆₇	C₂³×Q₈	1	4	yes	—
209	C₂²×C₆₇	C₂³×Q₈	1	16	yes	—
210	C₂²×C₆₇	C₂³×Q₈	1	2	yes	—
211	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
212	C₂²×C₆₇	#585	1	2	yes	—
213	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
214	C₄×C₆₇	—	2	8	no	—
215	C₄×C₆₇	—	2	2	no	—
216	C₄×C₆₇	—	2	4	no	—
217	C₄×C₆₇	C₂³×Q₈	1	8	yes	—
218	C₄×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
219	Dic₆₇	—	2	16	no	—
220	Dic₆₇	—	2	16	no	—
221	C₂×#3501	—	67	1	no	maximal, Sylow
222	C₂×C₆₇	—	2	4	no	—
223	C₂×C₆₇	—	2	16	no	—
224	C₂×C₆₇	#3501	1	4	yes	complemented
225	C₂×C₆₇	C₂⁴×Q₈	1	1	yes	characteristic
226	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
227	C₂×C₆₇	C₂×#585	1	1	yes	characteristic
228	#3501	—	67	32	no	—
229	C₂×#554	—	67	4	no	—
230	C₂×#554	—	67	8	no	—
231	C₂×D₄×Q₈	—	67	4	no	—
232	C₂×#544	—	67	2	no	—
233	C₂×#543	—	67	4	no	—
234	C₂×#533	—	67	2	no	—
235	C₂²×C₂²⋊Q₈	—	67	4	no	—
236	C₂×#518	—	67	2	no	—
237	C₂²×C₄×D₄	D₆₇	1	1	yes	characteristic
238	C₆₇	C₂×#3501	1	1	yes	characteristic, complemented, Sylow
239	C₂³×Q₈	—	67	4	no	—
240	C₂³×D₄	C₂×D₆₇	1	1	yes	characteristic
241	C₂⁴×C₄	C₂×D₆₇	1	2	yes	—
242	#554	—	67	64	no	—
243	#554	—	67	128	no	—
244	D₄×Q₈	—	67	64	no	—
245	#544	—	67	32	no	—
246	#543	—	67	64	no	—
247	#533	—	67	32	no	—
248	C₂×C₄⋊Q₈	—	67	8	no	—
249	C₂×C₄⋊Q₈	—	67	2	no	—
250	C₂×C₄⋊Q₈	—	67	2	no	—
251	C₂×#125	—	67	8	no	—
252	C₂×#125	—	67	4	no	—
253	C₂×C₂²⋊Q₈	—	67	4	no	—
254	C₂×C₂²⋊Q₈	—	67	8	no	—
255	C₂×C₂²⋊Q₈	—	67	8	no	—
256	C₂×C₂²⋊Q₈	—	67	8	no	—
257	C₂×C₂²⋊Q₈	—	67	64	no	—
258	C₂×C₂²⋊Q₈	—	67	4	no	—
259	C₂×C₂²⋊Q₈	—	67	16	no	—
260	C₂×C₂²⋊Q₈	—	67	16	no	—
261	#518	—	67	32	no	—
262	C₂×C₄×Q₈	—	67	4	no	—
263	C₂×C₄×Q₈	—	67	4	no	—
264	C₂×C₄×D₄	—	67	8	no	—
265	C₂×C₄×D₄	—	67	8	no	—
266	C₂×C₄×D₄	C₂×D₆₇	1	4	yes	—
267	C₂×C₄×D₄	C₂×D₆₇	1	4	yes	—
268	C₂×C₄×D₄	C₂×D₆₇	1	16	yes	—
269	C₂×#117	—	67	4	no	—
270	C₂²×C₄⋊C₄	—	67	8	no	—
271	C₂²×C₄⋊C₄	—	67	4	no	—
272	C₂²×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
273	C₂²×C₂²⋊C₄	—	67	8	no	—
274	C₂²×C₂²⋊C₄	C₂×D₆₇	1	2	yes	—
275	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
276	C₂⁵	C₂²×D₆₇	1	2	yes	—
277	C₂²×Q₈	—	268	16	no	—
278	C₂²×Q₈	—	67	4	no	—
279	C₂²×Q₈	—	67	8	no	—
280	C₂²×Q₈	—	67	32	no	—
281	C₂²×Q₈	—	67	4	no	—
282	C₂²×D₄	Dic₁₃₄	1	16	yes	complemented
283	C₂²×D₄	C₂²×D₆₇	1	4	yes	—
284	C₂²×D₄	C₂²×D₆₇	1	8	yes	—
285	C₂³×C₄	—	4	8	no	—
286	C₂³×C₄	—	67	8	no	—
287	C₂³×C₄	—	67	8	no	—
288	C₂³×C₄	C₂²×D₆₇	1	4	yes	—
289	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
290	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
291	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
292	C₂³×C₄	C₂²×D₆₇	1	4	yes	—
293	C₂³×C₄	C₂²×D₆₇	1	16	yes	—
294	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
295	C₄⋊Q₈	—	67	64	no	—
296	C₄⋊Q₈	—	67	16	no	—
297	C₄⋊Q₈	—	67	16	no	—
298	#125	—	67	64	no	—
299	#125	—	67	32	no	—
300	C₂²⋊Q₈	—	67	32	no	—
301	C₂²⋊Q₈	—	67	64	no	—
302	C₂²⋊Q₈	—	67	64	no	—
303	C₂²⋊Q₈	—	67	64	no	—
304	C₂²⋊Q₈	—	67	32	no	—
305	C₂²⋊Q₈	—	67	128	no	—
306	C₂²⋊Q₈	—	67	128	no	—
307	C₄×Q₈	—	67	32	no	—
308	C₄×Q₈	—	67	32	no	—
309	C₄×D₄	—	67	64	no	—
310	C₄×D₄	—	67	64	no	—
311	C₄×D₄	C₂²×D₆₇	1	32	yes	—
312	C₄×D₄	C₂²×D₆₇	1	32	yes	—
313	#117	—	67	32	no	—
314	C₂×C₄⋊C₄	—	67	8	no	—
315	C₂×C₄⋊C₄	—	67	4	no	—
316	C₂×C₄⋊C₄	—	67	8	no	—
317	C₂×C₄⋊C₄	—	67	16	no	—
318	C₂×C₄⋊C₄	—	67	64	no	—
319	C₂×C₄⋊C₄	—	67	32	no	—
320	C₂×C₄⋊C₄	—	67	4	no	—
321	C₂×C₄⋊C₄	—	67	4	no	—
322	C₂×C₄⋊C₄	—	67	8	no	—
323	C₂×C₄⋊C₄	—	67	4	no	—
324	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
325	C₂×C₄⋊C₄	C₂²×D₆₇	1	8	yes	—
326	C₂×C₂²⋊C₄	—	67	8	no	—
327	C₂×C₂²⋊C₄	—	67	16	no	—
328	C₂×C₂²⋊C₄	—	67	64	no	—
329	C₂×C₂²⋊C₄	C₂²×D₆₇	1	8	yes	—
330	C₂×C₂²⋊C₄	C₂²×D₆₇	1	16	yes	—
331	C₂×C₄²	—	67	8	no	—
332	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
333	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
334	C₂×C₄²	C₂²×D₆₇	1	8	yes	—
335	C₂⁴	—	4	4	no	—
336	C₂⁴	—	4	4	no	—
337	C₂⁴	C₂×Dic₁₃₄	1	16	yes	complemented
338	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
339	C₂⁴	C₂³×D₆₇	1	4	yes	—
340	C₂⁴	C₂³×D₆₇	1	8	yes	—
341	C₂×Q₈	—	268	64	no	—
342	C₂×Q₈	—	268	16	no	—
343	C₂×Q₈	—	268	8	no	—
344	C₂×Q₈	—	67	16	no	—
345	C₂×Q₈	—	67	32	no	—
346	C₂×Q₈	—	67	16	no	—
347	C₂×Q₈	—	134	16	no	—
348	C₂×D₄	C₂×Dic₁₃₄	1	32	yes	complemented
349	C₂×D₄	C₂×Dic₁₃₄	1	64	yes	complemented
350	C₂×D₄	C₂³×D₆₇	1	16	yes	—
351	C₂²×C₄	—	2	4	no	—
352	C₂²×C₄	—	2	16	no	—
353	C₂²×C₄	—	4	32	no	—
354	C₂²×C₄	—	4	8	no	—
355	C₂²×C₄	—	4	4	no	—
356	C₂²×C₄	—	4	2	no	—
357	C₂²×C₄	—	268	16	no	—
358	C₂²×C₄	—	268	16	no	—
359	C₂²×C₄	—	67	32	no	—
360	C₂²×C₄	—	67	32	no	—
361	C₂²×C₄	—	67	8	no	—
362	C₂²×C₄	—	67	8	no	—
363	C₂²×C₄	C₂×Dic₁₃₄	1	8	yes	complemented
364	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
365	C₂²×C₄	C₂³×D₆₇	1	16	yes	—
366	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
367	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
368	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
369	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
370	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
371	C₂²×C₄	C₂³×D₆₇	1	16	yes	—
372	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
373	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
374	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
375	C₄⋊C₄	—	67	32	no	—
376	C₄⋊C₄	—	67	16	no	—
377	C₄⋊C₄	—	67	32	no	—
378	C₄⋊C₄	—	67	64	no	—
379	C₄⋊C₄	—	67	16	no	—
380	C₄⋊C₄	—	67	16	no	—
381	C₄⋊C₄	—	67	32	no	—
382	C₄⋊C₄	—	67	16	no	—
383	C₄⋊C₄	C₂³×D₆₇	1	16	yes	—
384	C₂²⋊C₄	—	67	32	no	—
385	C₂²⋊C₄	—	67	64	no	—
386	C₂²⋊C₄	C₂³×D₆₇	1	32	yes	—
387	C₄²	—	67	32	no	—
388	C₄²	C₂³×D₆₇	1	8	yes	—
389	C₄²	C₂³×D₆₇	1	8	yes	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	8	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	8	no	—
394	C₂³	—	4	8	no	—
395	C₂³	—	4	32	no	—
396	C₂³	—	4	8	no	—
397	C₂³	C₂²×Dic₁₃₄	1	4	yes	—
398	C₂³	C₂²×Dic₁₃₄	1	16	yes	—
399	C₂³	C₂²×Dic₁₃₄	1	32	yes	complemented
400	C₂³	C₂⁴×D₆₇	1	2	yes	—
401	C₂³	C₂⁴×D₆₇	1	8	yes	—
402	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
403	Q₈	—	268	32	no	—
404	Q₈	—	268	16	no	—
405	Q₈	—	134	32	no	—
406	D₄	C₂²×Dic₁₃₄	1	64	yes	complemented
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	8	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	8	no	—
411	C₂×C₄	—	2	32	no	—
412	C₂×C₄	—	4	16	no	—
413	C₂×C₄	—	4	8	no	—
414	C₂×C₄	—	4	4	no	—
415	C₂×C₄	—	268	32	no	—
416	C₂×C₄	—	268	32	no	—
417	C₂×C₄	—	67	16	no	—
418	C₂×C₄	—	67	16	no	—
419	C₂×C₄	—	134	16	no	—
420	C₂×C₄	—	134	16	no	—
421	C₂×C₄	C₂²×Dic₁₃₄	1	8	yes	—
422	C₂×C₄	C₂²×Dic₁₃₄	1	16	yes	complemented
423	C₂×C₄	#14524088	1	4	yes	—
424	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
425	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
426	C₂×C₄	C₂⁴×D₆₇	1	8	yes	—
427	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
428	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
429	C₂²	—	2	2	no	—
430	C₂²	—	2	8	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	8	no	—
433	C₂²	—	2	8	no	—
434	C₂²	—	2	32	no	—
435	C₂²	—	4	32	no	—
436	C₂²	C₂³×Dic₁₃₄	1	4	yes	—
437	C₂²	C₂³×Dic₁₃₄	1	16	yes	—
438	C₂²	C₂³×Dic₁₃₄	1	2	yes	—
439	C₂²	C₂×#14524088	1	2	yes	—
440	C₂²	#16864232	1	2	yes	—
441	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
442	C₄	—	2	2	no	—
443	C₄	—	2	8	no	—
444	C₄	—	2	4	no	—
445	C₄	—	134	16	no	—
446	C₄	—	134	16	no	—
447	C₄	C₂³×Dic₁₃₄	1	8	yes	—
448	C₄	C₂×#14524088	1	4	yes	—
449	C₂	—	2	4	no	—
450	C₂	—	2	16	no	—
451	C₂	#22574012	1	4	yes	direct product factor
452	C₂	C₂⁴×Dic₁₃₄	1	1	yes	characteristic
453	C₂	C₂²×#14524088	1	1	yes	characteristic
454	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
prime radical	C₂³×D₄×C₆₇	C₂²
hypercenter	C₂²×C₄×D₄	D₆₇
hypocenter	C₆₇	C₂×#3501
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. #22574012	C₂	direct (N × H)
2. C₂×#16864105	C₂	semidirect
3. C₂×#16864103	C₂	semidirect
4. C₂×D₄×Dic₁₃₄	C₂	semidirect
5. C₂×#16864027	C₂	semidirect
6. C₂×#16864022	C₂	semidirect
7. C₂×#16863953	C₂	semidirect
8. C₂²×#14524022	C₂	semidirect
11. C₂³×Dic₁₃₄	C₂²	semidirect
12. #16864105	C₂²	semidirect
13. #16864103	C₂²	semidirect
14. D₄×Dic₁₃₄	C₂²	semidirect
15. #16864027	C₂²	semidirect
16. #16864022	C₂²	semidirect
17. #16863953	C₂²	semidirect
18. C₂×#14524043	C₂²	semidirect
19. C₂×#14524043	C₂²	semidirect
20. C₂×#14524042	C₂²	semidirect
21. C₂×#14524035	C₂²	semidirect
22. C₂×#14524034	C₂²	semidirect
26. C₂×#14524022	C₂²	semidirect
27. C₂×#14524022	C₂²	semidirect
28. C₂×#14524022	C₂²	semidirect
29. C₂×#14524020	C₂²	semidirect
30. C₂×#14524020	C₂²	semidirect
32. C₂×C₄×Dic₁₃₄	C₂²	semidirect
33. C₂×C₄×Dic₁₃₄	C₂²	semidirect
37. C₂²×C₄⋊Dic₆₇	C₂²	semidirect
38. C₂²×Dic₆₇⋊C₄	C₂²	semidirect
52. C₂²×Dic₁₃₄	C₂³	semidirect
55. #14524043	C₂³	semidirect
56. #14524043	C₂³	semidirect
57. #14524042	C₂³	semidirect
58. #14524035	C₂³	semidirect
59. #14524034	C₂³	semidirect
63. #14524022	C₂³	semidirect
64. #14524022	C₂³	semidirect
65. #14524020	C₂³	semidirect
66. #14524020	C₂³	semidirect
67. C₄×Dic₁₃₄	C₂³	semidirect
68. C₄×Dic₁₃₄	C₂³	semidirect
77. C₂×C₄⋊Dic₆₇	C₂³	semidirect
81. C₂×Dic₆₇⋊C₄	C₂³	semidirect
87. C₂²×D₄×C₆₇	Q₈	semidirect
133. C₂⁴×C₆₇	C₂×Q₈	semidirect
137. C₂×D₄×C₆₇	C₂×Q₈	semidirect
138. C₂×D₄×C₆₇	C₂×Q₈	semidirect
146. C₂²×C₄×C₆₇	C₂×Q₈	semidirect
171. C₂³×C₆₇	C₂²×Q₈	semidirect
175. D₄×C₆₇	C₂²×Q₈	semidirect
185. C₂×C₄×C₆₇	C₂²×Q₈	semidirect
224. C₂×C₆₇	#3501	semidirect
238. C₆₇	C₂×#3501	semidirect
282. C₂²×D₄	Dic₁₃₄	semidirect
337. C₂⁴	C₂×Dic₁₃₄	semidirect
348. C₂×D₄	C₂×Dic₁₃₄	semidirect
349. C₂×D₄	C₂×Dic₁₃₄	semidirect
363. C₂²×C₄	C₂×Dic₁₃₄	semidirect
399. C₂³	C₂²×Dic₁₃₄	semidirect
406. D₄	C₂²×Dic₁₃₄	semidirect
422. C₂×C₄	C₂²×Dic₁₃₄	semidirect
451. C₂	#22574012	direct (N × H)

Maximal subgroups

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

subgroup	index
#22574012	2
C₂×#16864105	2
C₂×#16864103	2
C₂×D₄×Dic₁₃₄	2
C₂×#16864027	2
C₂×#16864022	2
C₂×#16863953	2
C₂²×#14524022	2
C₂×#16863871	2
C₂²×C₄×D₄×C₆₇	2
C₂×#3501	67

Subgroups — C2×#22574012 #31763735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22574012 #31763735