Subgroups — #30874295

Order 17152.

870 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863511	C₂	1	2	yes	maximal
2	C₄×#14524063	C₂	1	1	yes	maximal, characteristic
3	#22568774	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560954	C₂	1	2	yes	maximal, complemented
5	#22560484	C₂	1	2	yes	maximal, complemented
6	#22560483	C₂	1	2	yes	maximal, complemented
7	#22560380	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560229	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560226	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560074	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2906	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₂²	1	1	yes	characteristic
13	C₂×#14524049	C₂²	1	2	yes	—
14	C₄×#11761628	C₂²	1	1	yes	characteristic
15	#16863511	—	2	8	no	—
16	C₄×#11761618	C₄	1	1	yes	characteristic, complemented
17	C₄×#11761618	C₄	1	1	yes	characteristic, complemented
18	C₄×#11761615	C₄	1	1	yes	characteristic, complemented
19	C₄×#11761615	C₄	1	1	yes	characteristic, complemented
20	C₄×#11761616	C₄	1	1	yes	characteristic, complemented
21	C₄×#11761616	C₄	1	1	yes	characteristic, complemented
22	C₄×#11761613	C₄	1	1	yes	characteristic, complemented
23	C₄×#11761613	C₄	1	1	yes	characteristic, complemented
24	C₂×#14523978	C₂²	1	2	yes	—
25	C₂×#14523978	C₂²	1	2	yes	—
26	#16863432	C₂²	1	1	yes	characteristic
27	#16863422	C₂²	1	2	yes	—
28	#16863421	—	2	2	no	—
29	#16863420	C₂²	1	1	yes	characteristic
30	#16863418	—	2	1	no	—
31	C₄⋊C₈×D₆₇	—	2	1	no	—
32	C₂×#14523970	C₂²	1	2	yes	—
33	C₂×#14523969	C₂²	1	2	yes	—
34	#16863392	C₂²	1	1	yes	characteristic
35	#16863390	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
38	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
39	#16863157	C₂²	1	2	yes	—
40	#16863155	C₂²	1	2	yes	—
41	#16863150	C₂²	1	2	yes	—
42	C₂×#131×C₆₇	C₂²	1	2	yes	—
43	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
44	#424×C₆₇	—	2	2	no	—
45	#424×C₆₇	C₂²	1	2	yes	—
46	#423×C₆₇	—	2	1	no	—
47	#423×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
49	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
50	C₂×#11761628	C₂³	1	1	yes	characteristic
51	#14524063	D₄	1	4	yes	—
52	#14524063	Q₈	1	4	yes	—
53	C₂×#11761618	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761616	C₂×C₄	1	1	yes	characteristic
55	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
56	C₂×#11761613	C₂×C₄	1	1	yes	characteristic
57	#14524049	—	2	4	no	—
58	#14524049	—	4	2	no	—
59	C₂×#11761612	C₂³	1	2	yes	—
60	C₂×#11761611	—	2	2	no	—
61	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
62	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
63	C₂×C₈×D₆₇	—	2	2	no	—
64	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
66	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
67	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₂²⋊D₆₇	—	2	1	no	—
69	C₄×C₂²⋊D₆₇	—	2	1	no	—
70	#14524002	C₂³	1	1	yes	characteristic
71	#14524000	—	2	1	no	—
72	C₂×C₄⋊Dic₆₇	—	2	1	no	—
73	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
74	C₄²×D₆₇	—	2	1	no	—
75	#14523978	—	2	4	no	—
76	#14523978	—	4	2	no	—
77	#14523976	C₂×C₄	1	1	yes	characteristic
78	#14523976	C₂×C₄	1	1	yes	characteristic
79	#14523974	C₂×C₄	1	1	yes	characteristic
80	#14523974	C₂×C₄	1	1	yes	characteristic
81	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
82	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
83	Dic₆₇⋊C₈	—	2	2	no	—
84	Dic₆₇⋊C₈	—	2	2	no	—
85	#14523970	—	2	4	no	—
86	#14523969	—	2	4	no	—
87	#14523967	C₂×C₄	1	1	yes	characteristic
88	#14523967	C₂×C₄	1	1	yes	characteristic
89	#14523966	C₂×C₄	1	1	yes	characteristic
90	#14523966	C₂×C₄	1	1	yes	characteristic
91	#14523964	C₂×C₄	1	1	yes	characteristic
92	#14523964	C₂×C₄	1	1	yes	characteristic
93	#14523963	C₂×C₄	1	1	yes	characteristic
94	#14523963	C₂×C₄	1	1	yes	characteristic
95	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
96	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
97	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
98	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
99	C₈×Dic₆₇	—	2	4	no	—
100	C₆₇⋊#95	—	2	1	no	—
101	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
102	#131×C₆₇	—	2	4	no	—
103	#131×C₆₇	—	4	2	no	—
104	C₂×M₁₆×C₆₇	—	2	2	no	—
105	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
106	C₂²×C₈×C₆₇	—	2	2	no	—
107	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
108	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
109	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
110	C₄×D₄×C₆₇	—	2	1	no	—
111	C₄×D₄×C₆₇	—	2	1	no	—
112	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
113	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
114	#117×C₆₇	—	2	1	no	—
115	#117×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	—	2	1	no	—
117	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	C₄⋊C₈×C₆₇	—	2	2	no	—
119	C₄⋊C₈×C₆₇	—	2	2	no	—
120	C₄⋊C₈×C₆₇	—	2	1	no	—
121	C₄⋊C₈×C₆₇	—	2	1	no	—
122	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
123	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
124	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
125	#11761628	—	2	1	no	—
126	#11761628	—	2	1	no	—
127	#11761628	C₂×D₄	1	2	yes	—
128	#11761628	C₂×Q₈	1	2	yes	—
129	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
130	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₂²⋊D₆₇	—	2	1	no	—
132	C₂²×Dic₆₇	—	2	2	no	—
133	C₂×C₄×D₆₇	—	2	1	no	—
134	C₂×C₄×D₆₇	—	2	1	no	—
135	C₂×C₄×D₆₇	—	2	1	no	—
136	#11761618	C₄⋊C₄	1	4	yes	complemented
137	#11761616	C₄⋊C₄	1	4	yes	complemented
138	#11761615	C₄⋊C₄	1	4	yes	complemented
139	#11761613	C₄⋊C₄	1	4	yes	complemented
140	#11761612	—	2	2	no	—
141	#11761612	—	2	2	no	—
142	#11761611	—	4	4	no	—
143	C₂×C₆₇⋊C₈	—	2	1	no	—
144	C₂×C₆₇⋊C₈	—	2	1	no	—
145	C₂×C₆₇⋊C₈	—	2	2	no	—
146	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
151	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
152	C₈×D₆₇	—	4	4	no	—
153	C₄⋊Dic₆₇	—	2	4	no	—
154	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
155	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
156	Dic₆₇⋊C₄	—	2	1	no	—
157	Dic₆₇⋊C₄	—	2	1	no	—
158	C₂²⋊Dic₆₇	—	2	1	no	—
159	(C₂×D₆₇)⋊C₄	—	2	1	no	—
160	(C₂×D₆₇)⋊C₄	—	2	1	no	—
161	C₄×Dic₆₇	—	2	1	no	—
162	C₄×Dic₆₇	—	2	1	no	—
163	C₄×Dic₆₇	—	2	1	no	—
164	C₄×Dic₆₇	—	2	2	no	—
165	C₄×Dic₆₇	—	2	2	no	—
166	#41×C₆₇	—	2	1	no	—
167	#41×C₆₇	—	2	1	no	—
168	#41×C₆₇	C₂×D₄	1	2	yes	—
169	#41×C₆₇	C₂×Q₈	1	2	yes	—
170	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
171	C₂×D₄×C₆₇	—	2	1	no	—
172	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	—	2	1	no	—
174	C₂²×C₄×C₆₇	—	2	1	no	—
175	C₂²×C₄×C₆₇	—	2	1	no	—
176	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
179	M₁₆×C₆₇	—	2	2	no	—
180	M₁₆×C₆₇	—	2	2	no	—
181	M₁₆×C₆₇	—	4	4	no	—
182	C₂×C₈×C₆₇	—	2	2	no	—
183	C₂×C₈×C₆₇	—	2	2	no	—
184	C₂×C₈×C₆₇	—	2	2	no	—
185	C₂×C₈×C₆₇	—	4	4	no	—
186	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
187	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
188	C₄⋊C₄×C₆₇	—	2	1	no	—
189	C₄⋊C₄×C₆₇	—	2	1	no	—
190	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
192	C₂²⋊C₄×C₆₇	—	2	1	no	—
193	C₂²⋊C₄×C₆₇	—	2	1	no	—
194	C₂²⋊C₄×C₆₇	—	2	1	no	—
195	C₄²×C₆₇	—	2	1	no	—
196	C₄²×C₆₇	—	2	1	no	—
197	C₄²×C₆₇	—	2	1	no	—
198	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
199	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
200	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
202	C₂³×C₆₇	—	2	1	no	—
203	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
204	C₂²×D₆₇	—	2	1	no	—
205	Q₈×C₆₇	—	2	1	no	—
206	Q₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
207	D₄×C₆₇	—	2	2	no	—
208	D₄×C₆₇	—	2	2	no	—
209	D₄×C₆₇	—	2	1	no	—
210	D₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
211	C₂×C₄×C₆₇	—	2	1	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	1	no	—
216	C₂×C₄×C₆₇	—	2	1	no	—
217	C₂×C₄×C₆₇	—	4	1	no	—
218	C₂×C₄×C₆₇	—	4	2	no	—
219	C₂×C₄×C₆₇	—	8	1	no	—
220	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
225	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
228	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
230	C₂²⋊D₆₇	—	2	2	no	—
231	C₂²⋊D₆₇	—	2	2	no	—
232	C₂×Dic₆₇	—	2	1	no	—
233	C₂×Dic₆₇	—	2	4	no	—
234	C₂×Dic₆₇	—	2	1	no	—
235	C₂×Dic₆₇	—	2	1	no	—
236	C₂×Dic₆₇	—	4	2	no	—
237	C₂×Dic₆₇	—	4	2	no	—
238	D₂₆₈	—	2	1	no	—
239	D₂₆₈	C₂×C₄⋊C₄	1	2	yes	—
240	C₄×D₆₇	—	2	2	no	—
241	C₄×D₆₇	—	2	2	no	—
242	C₄×D₆₇	—	4	2	no	—
243	C₄×D₆₇	—	8	1	no	—
244	Dic₁₃₄	—	2	1	no	—
245	Dic₁₃₄	C₂×C₄⋊C₄	1	2	yes	—
246	C₈×C₆₇	—	2	2	no	—
247	C₈×C₆₇	—	2	2	no	—
248	C₈×C₆₇	—	4	2	no	—
249	C₆₇⋊C₈	—	2	1	no	—
250	C₆₇⋊C₈	—	2	1	no	—
251	C₆₇⋊C₈	—	4	2	no	—
252	C₆₇⋊C₈	C₂×C₄⋊C₄	1	2	yes	—
253	C₆₇⋊C₈	C₂×C₄⋊C₄	1	2	yes	—
254	C₂²×C₆₇	—	2	1	no	—
255	C₂²×C₆₇	—	2	1	no	—
256	C₂²×C₆₇	—	2	2	no	—
257	C₂²×C₆₇	—	4	1	no	—
258	C₂²×C₆₇	—	4	1	no	—
259	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
260	C₂²×C₆₇	#547	1	1	yes	characteristic
261	C₂²×C₆₇	#554	1	1	yes	characteristic
262	C₂×D₆₇	—	2	2	no	—
263	C₂×D₆₇	—	4	1	no	—
264	C₂×D₆₇	—	4	1	no	—
265	C₄×C₆₇	—	2	2	no	—
266	C₄×C₆₇	—	2	1	no	—
267	C₄×C₆₇	—	2	1	no	—
268	C₄×C₆₇	—	2	1	no	—
269	C₄×C₆₇	—	4	1	no	—
270	C₄×C₆₇	—	4	1	no	—
271	C₄×C₆₇	#443	1	2	yes	complemented
272	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
273	C₄×C₆₇	#520	1	1	yes	characteristic
274	C₄×C₆₇	D₄²	1	1	yes	characteristic
275	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
276	Dic₆₇	—	2	2	no	—
277	Dic₆₇	—	4	4	no	—
278	Dic₆₇	—	4	1	no	—
279	Dic₆₇	—	4	1	no	—
280	#22688	—	67	1	no	maximal, Sylow
281	C₂×C₆₇	—	2	1	no	—
282	C₂×C₆₇	—	2	1	no	—
283	C₂×C₆₇	—	4	2	no	—
284	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
285	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
286	C₂×C₆₇	#2957	1	1	yes	characteristic
287	D₆₇	—	4	2	no	—
288	C₂×#443	—	67	2	no	—
289	C₄×#135	—	67	1	no	—
290	#2906	—	67	1	no	—
291	#2906	D₆₇	1	1	yes	characteristic, complemented
292	#1945	—	67	2	no	—
293	#1866	—	67	2	no	—
294	#1866	—	67	2	no	—
295	#1848	—	67	1	no	—
296	#1806	—	67	1	no	—
297	#1806	—	67	1	no	—
298	C₈×C₄⋊C₄	—	67	1	no	—
299	C₆₇	#22688	1	1	yes	characteristic, complemented, Sylow
300	C₂×#135	—	67	1	no	—
301	C₂×#131	—	67	2	no	—
302	C₂×#131	C₂×D₆₇	1	2	yes	—
303	C₄×#41	—	67	1	no	—
304	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
305	#443	—	134	8	no	—
306	C₄×Q₁₆	—	67	1	no	—
307	C₄×Q₁₆	—	67	1	no	—
308	C₄×SD₁₆	—	67	1	no	—
309	C₄×SD₁₆	—	67	1	no	—
310	C₄×SD₁₆	—	67	1	no	—
311	C₄×SD₁₆	—	67	1	no	—
312	C₄×D₈	—	67	1	no	—
313	C₄×D₈	—	67	1	no	—
314	C₂×#108	—	67	2	no	—
315	C₂×#108	—	67	2	no	—
316	#427	—	67	1	no	—
317	#424	—	2	2	no	—
318	#424	—	67	2	no	—
319	#424	—	134	2	no	—
320	#424	C₂×D₆₇	1	2	yes	—
321	#423	—	2	1	no	—
322	#423	—	67	1	no	—
323	#423	—	134	1	no	—
324	#423	C₂×D₆₇	1	1	yes	characteristic
325	C₂×C₄⋊C₈	—	2	1	no	—
326	C₂×C₄⋊C₈	—	134	1	no	—
327	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
328	C₂×#104	—	67	2	no	—
329	C₂×#104	—	67	2	no	—
330	#416	—	67	1	no	—
331	#416	—	67	1	no	—
332	C₂×C₄×C₈	—	67	1	no	—
333	C₂×C₄×C₈	—	67	2	no	—
334	C₄×C₄⋊C₄	—	67	1	no	—
335	#339	—	67	2	no	—
336	#339	—	67	2	no	—
337	#336	—	67	2	no	—
338	C₂×#41	—	67	1	no	—
339	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
340	#135	—	67	4	no	—
341	#135	—	67	4	no	—
342	C₂×Q₁₆	—	67	1	no	—
343	C₂×SD₁₆	—	67	1	no	—
344	C₂×SD₁₆	—	67	1	no	—
345	C₂×D₈	—	67	1	no	—
346	#131	—	2	4	no	—
347	#131	—	4	2	no	—
348	#131	—	268	2	no	—
349	#131	—	134	4	no	—
350	C₂×M₁₆	—	2	2	no	—
351	C₂×M₁₆	—	67	2	no	—
352	C₂×M₁₆	—	134	2	no	—
353	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
354	C₂²×C₈	—	2	2	no	—
355	C₂²×C₈	—	67	1	no	—
356	C₂²×C₈	—	67	1	no	—
357	C₂²×C₈	—	134	2	no	—
358	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
359	C₄×Q₈	—	67	1	no	—
360	C₄×Q₈	—	67	1	no	—
361	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
362	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
363	C₄×D₄	—	2	1	no	—
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	—	67	1	no	—
366	C₄×D₄	—	67	1	no	—
367	C₄×D₄	—	134	1	no	—
368	C₄×D₄	—	134	1	no	—
369	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
370	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
371	#117	—	2	1	no	—
372	#117	—	67	1	no	—
373	#117	—	134	1	no	—
374	#117	C₂²×D₆₇	1	1	yes	characteristic
375	C₂×C₄⋊C₄	—	134	1	no	—
376	C₂×C₄²	—	2	1	no	—
377	C₂×C₄²	—	67	2	no	—
378	C₂×C₄²	—	134	1	no	—
379	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
380	#108	—	268	2	no	—
381	#108	—	134	4	no	—
382	#107	—	67	1	no	—
383	#107	—	67	1	no	—
384	#106	—	67	1	no	—
385	#106	—	67	1	no	—
386	C₄⋊C₈	—	2	2	no	—
387	C₄⋊C₈	—	2	2	no	—
388	C₄⋊C₈	—	2	1	no	—
389	C₄⋊C₈	—	2	1	no	—
390	C₄⋊C₈	—	134	2	no	—
391	C₄⋊C₈	—	134	2	no	—
392	C₄⋊C₈	—	134	1	no	—
393	C₄⋊C₈	—	134	1	no	—
394	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
395	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
396	C₄⋊C₈	C₂²⋊D₆₇	1	2	yes	—
397	#104	—	134	4	no	—
398	#104	—	134	4	no	—
399	#103	—	67	1	no	—
400	#103	—	67	1	no	—
401	#103	—	67	1	no	—
402	#103	—	67	1	no	—
403	#102	—	67	1	no	—
404	#102	—	67	1	no	—
405	#102	—	67	1	no	—
406	#102	—	67	1	no	—
407	C₄×C₈	—	67	1	no	—
408	C₄×C₈	—	67	1	no	—
409	C₄×C₈	—	67	1	no	—
410	C₄×C₈	—	67	1	no	—
411	C₄×C₈	—	134	4	no	—
412	#95	—	134	1	no	—
413	#41	—	2	1	no	—
414	#41	—	2	1	no	—
415	#41	—	67	2	no	—
416	#41	—	67	2	no	—
417	#41	—	134	1	no	—
418	#41	—	134	1	no	—
419	#41	D₄×D₆₇	1	2	yes	—
420	#41	Q₈×D₆₇	1	2	yes	—
421	C₂×Q₈	—	67	1	no	—
422	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	—	67	1	no	—
425	C₂×D₄	—	134	1	no	—
426	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	134	2	no	—
431	C₂²×C₄	—	134	1	no	—
432	C₂²×C₄	—	134	1	no	—
433	C₂²×C₄	—	134	1	no	—
434	C₂²×C₄	#11761628	1	1	yes	characteristic
435	C₂²×C₄	#11761628	1	1	yes	characteristic
436	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
437	Q₁₆	—	67	4	no	—
438	SD₁₆	—	67	4	no	—
439	SD₁₆	—	67	4	no	—
440	D₈	—	67	4	no	—
441	M₁₆	—	2	2	no	—
442	M₁₆	—	2	2	no	—
443	M₁₆	—	4	4	no	—
444	M₁₆	—	268	4	no	—
445	M₁₆	—	134	2	no	—
446	M₁₆	—	134	2	no	—
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	4	4	no	—
451	C₂×C₈	—	268	4	no	—
452	C₂×C₈	—	67	1	no	—
453	C₂×C₈	—	67	1	no	—
454	C₂×C₈	—	67	2	no	—
455	C₂×C₈	—	67	2	no	—
456	C₂×C₈	—	67	1	no	—
457	C₂×C₈	—	67	1	no	—
458	C₂×C₈	—	134	2	no	—
459	C₂×C₈	—	134	1	no	—
460	C₂×C₈	—	134	1	no	—
461	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
462	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	67	1	no	—
466	C₄⋊C₄	—	67	1	no	—
467	C₄⋊C₄	—	134	4	no	—
468	C₄⋊C₄	—	134	1	no	—
469	C₄⋊C₄	—	134	1	no	—
470	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
471	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
472	C₂²⋊C₄	—	2	1	no	—
473	C₂²⋊C₄	—	2	1	no	—
474	C₂²⋊C₄	—	2	1	no	—
475	C₂²⋊C₄	—	134	1	no	—
476	C₂²⋊C₄	—	134	1	no	—
477	C₂²⋊C₄	—	134	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	—	2	1	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	134	2	no	—
482	C₄²	—	134	2	no	—
483	C₄²	—	134	1	no	—
484	C₄²	—	134	1	no	—
485	C₄²	—	134	1	no	—
486	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
487	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
488	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
489	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
490	C₂³	—	2	1	no	—
491	C₂³	—	134	1	no	—
492	C₂³	C₂×#11761628	1	1	yes	characteristic
493	Q₈	—	2	1	no	—
494	Q₈	—	67	2	no	—
495	Q₈	—	134	1	no	—
496	Q₈	C₄⋊C₄×D₆₇	1	2	yes	—
497	D₄	—	2	2	no	—
498	D₄	—	2	2	no	—
499	D₄	—	2	1	no	—
500	D₄	—	67	2	no	—
501	D₄	—	134	2	no	—
502	D₄	—	134	2	no	—
503	D₄	—	134	1	no	—
504	D₄	C₄⋊C₄×D₆₇	1	2	yes	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	8	1	no	—
514	C₂×C₄	—	268	2	no	—
515	C₂×C₄	—	268	2	no	—
516	C₂×C₄	—	268	2	no	—
517	C₂×C₄	—	536	1	no	—
518	C₂×C₄	—	134	4	no	—
519	C₂×C₄	—	134	1	no	—
520	C₂×C₄	—	134	1	no	—
521	C₂×C₄	—	134	1	no	—
522	C₂×C₄	—	134	2	no	—
523	C₂×C₄	—	134	2	no	—
524	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
525	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
526	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
527	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
528	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
529	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
530	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
531	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
532	C₂×C₄	#14524088	1	1	yes	characteristic
533	C₂×C₄	#14524091	1	1	yes	characteristic
534	C₈	—	2	2	no	—
535	C₈	—	2	2	no	—
536	C₈	—	4	2	no	—
537	C₈	—	268	2	no	—
538	C₈	—	67	2	no	—
539	C₈	—	67	2	no	—
540	C₈	—	134	1	no	—
541	C₈	—	134	1	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	2	1	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	268	1	no	—
548	C₂²	—	268	1	no	—
549	C₂²	—	134	2	no	—
550	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
551	C₂²	#16864047	1	1	yes	characteristic
552	C₂²	#16864101	1	1	yes	characteristic
553	C₄	—	2	2	no	—
554	C₄	—	2	1	no	—
555	C₄	—	2	1	no	—
556	C₄	—	2	1	no	—
557	C₄	—	4	1	no	—
558	C₄	—	4	1	no	—
559	C₄	—	268	4	no	—
560	C₄	—	268	1	no	—
561	C₄	—	268	1	no	—
562	C₄	—	134	2	no	—
563	C₄	#16863511	1	2	yes	complemented
564	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
565	C₄	#16863877	1	1	yes	characteristic
566	C₄	D₄×C₂²⋊D₆₇	1	1	yes	characteristic
567	C₄	Q₈×C₂²⋊D₆₇	1	1	yes	characteristic
568	C₂	—	2	1	no	—
569	C₂	—	2	1	no	—
570	C₂	—	4	2	no	—
571	C₂	—	268	2	no	—
572	C₂	C₄⋊C₄×C₂²⋊D₆₇	1	1	yes	characteristic
573	C₂	C₂×#16863511	1	1	yes	characteristic
574	C₂	#22569124	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2906	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524063	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2906	D₆₇
hypocenter	C₆₇	#22688
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #2906

Complements

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

N	H ≅ G/N	type
3. #22568774	C₂	semidirect
4. #22560954	C₂	semidirect
5. #22560484	C₂	semidirect
6. #22560483	C₂	semidirect
7. #22560380	C₂	semidirect
8. #22560229	C₂	semidirect
9. #22560226	C₂	semidirect
10. #22560074	C₂	semidirect
11. C₆₇×#2906	C₂	semidirect
16. C₄×#11761618	C₄	semidirect
17. C₄×#11761618	C₄	semidirect
18. C₄×#11761615	C₄	semidirect
19. C₄×#11761615	C₄	semidirect
20. C₄×#11761616	C₄	semidirect
21. C₄×#11761616	C₄	semidirect
22. C₄×#11761613	C₄	semidirect
23. C₄×#11761613	C₄	semidirect
136. #11761618	C₄⋊C₄	semidirect
137. #11761616	C₄⋊C₄	semidirect
138. #11761615	C₄⋊C₄	semidirect
139. #11761613	C₄⋊C₄	semidirect
271. C₄×C₆₇	#443	semidirect
291. #2906	D₆₇	semidirect
299. C₆₇	#22688	semidirect
563. C₄	#16863511	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863511	2
C₄×#14524063	2
#22568774	2
#22560954	2
#22560484	2
#22560483	2
#22560380	2
#22560229	2
#22560226	2
#22560074	2
C₆₇×#2906	2
#22688	67