Subgroups — #30877905

Order 17152.

708 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569991	C₂	1	1	yes	maximal, characteristic
2	#22568774	C₂	1	1	yes	maximal, characteristic
3	#22561187	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560984	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560959	C₂	1	2	yes	maximal, complemented
6	#22560808	C₂	1	2	yes	maximal, complemented
7	#22560483	C₂	1	2	yes	maximal
8	#22560064	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560055	C₂	1	2	yes	maximal
10	C₄×#14523969	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2962	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524049	C₂²	1	2	yes	—
13	C₄×#11761628	C₂²	1	1	yes	characteristic
14	#16863700	C₄	1	2	yes	complemented
15	#16863693	C₄	1	2	yes	complemented
16	#16863688	C₄	1	1	yes	characteristic, complemented
17	#16863683	C₄	1	1	yes	characteristic, complemented
18	#16863678	C₄	1	1	yes	characteristic, complemented
19	#16863673	C₄	1	1	yes	characteristic, complemented
20	C₂×#14523978	C₂²	1	2	yes	—
21	#16863432	C₂²	1	2	yes	—
22	#16863422	C₂²	1	2	yes	—
23	#16863421	—	2	2	no	—
24	#16863420	C₂²	1	1	yes	characteristic
25	#16863418	—	2	1	no	—
26	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×D₆₇	—	2	1	no	—
28	C₂×#14523969	C₂²	1	1	yes	characteristic
29	C₂×#14523969	C₂²	1	1	yes	characteristic
30	#16863390	C₂²	1	2	yes	—
31	C₂×#14523953	C₂²	1	2	yes	—
32	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
33	#16863160	C₂²	1	2	yes	—
34	#16863157	C₂²	1	1	yes	characteristic
35	#16863157	C₂²	1	1	yes	characteristic
36	#16863155	C₂²	1	2	yes	—
37	#16863155	C₂²	1	2	yes	—
38	#16863150	C₂²	1	2	yes	—
39	#533×C₆₇	C₂²	1	1	yes	characteristic
40	#446×C₆₇	—	2	2	no	—
41	#446×C₆₇	—	2	1	no	—
42	C₈×Q₈×C₆₇	—	2	1	no	—
43	#424×C₆₇	C₂²	1	2	yes	—
44	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
45	#405×C₆₇	C₂²	1	2	yes	—
46	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#11761628	C₂³	1	1	yes	characteristic
48	#14524049	—	2	4	no	—
49	#14524049	—	4	2	no	—
50	C₂×#11761612	C₂³	1	2	yes	—
51	C₂×#11761611	—	2	2	no	—
52	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
53	C₂×C₈×D₆₇	—	2	2	no	—
54	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
56	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
57	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
58	C₄×C₂²⋊D₆₇	—	2	1	no	—
59	C₄×C₂²⋊D₆₇	—	2	1	no	—
60	#14524002	C₂³	1	1	yes	characteristic
61	#14524000	—	2	1	no	—
62	C₂×C₄×Dic₆₇	—	2	1	no	—
63	C₂×C₄×Dic₆₇	—	2	1	no	—
64	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
66	C₄²×D₆₇	—	2	1	no	—
67	#14523978	—	4	2	no	—
68	#14523976	C₂×C₄	1	2	yes	—
69	#14523976	C₂×C₄	1	2	yes	—
70	#14523974	C₂×C₄	1	2	yes	—
71	#14523974	C₂×C₄	1	2	yes	—
72	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
73	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
74	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
75	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
76	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
77	Dic₆₇⋊C₈	—	2	2	no	—
78	Dic₆₇⋊C₈	—	2	2	no	—
79	#14523969	—	2	2	no	—
80	#14523969	Q₈	1	4	yes	complemented
81	#14523966	C₂×C₄	1	2	yes	—
82	#14523966	C₂×C₄	1	2	yes	—
83	#14523963	C₂×C₄	1	2	yes	—
84	#14523963	C₂×C₄	1	2	yes	—
85	#14523953	—	4	2	no	—
86	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
89	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
91	#125×C₆₇	C₂×C₄	1	2	yes	—
92	C₂²⋊Q₈×C₆₇	—	2	1	no	—
93	C₂²⋊Q₈×C₆₇	—	2	1	no	—
94	C₄×Q₈×C₆₇	—	2	1	no	—
95	C₄×Q₈×C₆₇	—	2	1	no	—
96	#117×C₆₇	C₂³	1	2	yes	—
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₆₇	—	2	2	no	—
99	C₄⋊C₈×C₆₇	—	2	2	no	—
100	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
101	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
102	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
103	#97×C₆₇	—	2	2	no	—
104	#97×C₆₇	—	2	1	no	—
105	C₄×C₈×C₆₇	—	2	2	no	—
106	C₄×C₈×C₆₇	—	2	1	no	—
107	#11761628	—	2	1	no	—
108	#11761628	—	2	1	no	—
109	#11761628	C₂×Q₈	1	2	yes	—
110	#11761628	#41	1	2	yes	—
111	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
112	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₂²⋊D₆₇	—	2	1	no	—
114	C₂²×Dic₆₇	—	2	1	no	—
115	C₂²×Dic₆₇	—	2	1	no	—
116	C₂×C₄×D₆₇	—	2	1	no	—
117	C₂×C₄×D₆₇	—	2	1	no	—
118	C₂×C₄×D₆₇	—	2	1	no	—
119	#11761612	—	2	2	no	—
120	#11761612	—	2	2	no	—
121	#11761611	—	4	4	no	—
122	C₂×C₆₇⋊C₈	—	2	2	no	—
123	C₂×C₆₇⋊C₈	—	2	2	no	—
124	C₂×C₆₇⋊C₈	—	2	2	no	—
125	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
126	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
127	C₈×D₆₇	—	4	4	no	—
128	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
130	Dic₆₇⋊C₄	—	2	1	no	—
131	Dic₆₇⋊C₄	—	2	1	no	—
132	C₂²⋊Dic₆₇	—	2	1	no	—
133	(C₂×D₆₇)⋊C₄	—	2	1	no	—
134	(C₂×D₆₇)⋊C₄	—	2	1	no	—
135	C₄×Dic₆₇	—	2	1	no	—
136	C₄×Dic₆₇	—	2	1	no	—
137	C₄×Dic₆₇	—	2	1	no	—
138	C₄×Dic₆₇	—	2	1	no	—
139	C₄×Dic₆₇	—	2	1	no	—
140	C₄×Dic₆₇	—	2	1	no	—
141	C₄×Dic₆₇	—	4	2	no	—
142	C₄×Dic₆₇	—	4	2	no	—
143	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
144	C₂×Q₈×C₆₇	—	2	1	no	—
145	C₂×Q₈×C₆₇	—	2	1	no	—
146	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	M₁₆×C₆₇	—	2	1	no	—
150	M₁₆×C₆₇	—	2	1	no	—
151	M₁₆×C₆₇	—	2	1	no	—
152	M₁₆×C₆₇	C₂×Q₈	1	2	yes	—
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	1	no	—
157	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
159	C₄⋊C₄×C₆₇	—	2	2	no	—
160	C₄⋊C₄×C₆₇	—	2	2	no	—
161	C₄⋊C₄×C₆₇	—	2	1	no	—
162	C₄⋊C₄×C₆₇	—	2	1	no	—
163	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
164	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
165	C₂²⋊C₄×C₆₇	—	2	2	no	—
166	C₄²×C₆₇	—	2	2	no	—
167	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
172	C₂²×D₆₇	—	2	1	no	—
173	Q₈×C₆₇	—	4	2	no	—
174	Q₈×C₆₇	—	8	1	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	4	1	no	—
179	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
189	C₂²⋊D₆₇	—	2	2	no	—
190	C₂²⋊D₆₇	—	2	2	no	—
191	C₂×Dic₆₇	—	2	1	no	—
192	C₂×Dic₆₇	—	2	2	no	—
193	C₂×Dic₆₇	—	2	1	no	—
194	C₂×Dic₆₇	—	2	1	no	—
195	C₂×Dic₆₇	—	2	2	no	—
196	C₂×Dic₆₇	—	4	1	no	—
197	C₂×Dic₆₇	—	4	1	no	—
198	C₂×Dic₆₇	—	4	1	no	—
199	C₂×Dic₆₇	—	4	1	no	—
200	D₂₆₈	—	2	1	no	—
201	D₂₆₈	C₄×Q₈	1	2	yes	—
202	C₄×D₆₇	—	2	2	no	—
203	C₄×D₆₇	—	2	2	no	—
204	C₄×D₆₇	—	4	2	no	—
205	C₄×D₆₇	—	8	1	no	—
206	Dic₁₃₄	—	2	1	no	—
207	Dic₁₃₄	C₄×Q₈	1	2	yes	—
208	C₈×C₆₇	—	2	2	no	—
209	C₈×C₆₇	—	2	2	no	—
210	C₈×C₆₇	—	2	2	no	—
211	C₈×C₆₇	—	4	1	no	—
212	C₆₇⋊C₈	—	2	2	no	—
213	C₆₇⋊C₈	—	2	2	no	—
214	C₆₇⋊C₈	—	4	2	no	—
215	C₂²×C₆₇	—	2	1	no	—
216	C₂²×C₆₇	—	2	1	no	—
217	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂²×C₆₇	#552	1	1	yes	characteristic
219	C₂²×C₆₇	#554	1	1	yes	characteristic
220	C₂×D₆₇	—	2	2	no	—
221	C₂×D₆₇	—	4	1	no	—
222	C₂×D₆₇	—	4	1	no	—
223	C₄×C₆₇	—	2	1	no	—
224	C₄×C₆₇	—	2	1	no	—
225	C₄×C₆₇	—	2	1	no	—
226	C₄×C₆₇	—	4	2	no	—
227	C₄×C₆₇	—	4	2	no	—
228	C₄×C₆₇	#444	1	2	yes	—
229	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
230	C₄×C₆₇	#520	1	1	yes	characteristic
231	C₄×C₆₇	#546	1	1	yes	characteristic
232	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
233	Dic₆₇	—	2	2	no	—
234	Dic₆₇	—	4	2	no	—
235	Dic₆₇	—	4	2	no	—
236	Dic₆₇	—	4	1	no	—
237	Dic₆₇	—	4	1	no	—
238	#22983	—	67	1	no	maximal, Sylow
239	C₂×C₆₇	—	2	1	no	—
240	C₂×C₆₇	—	2	1	no	—
241	C₂×C₆₇	#2360	1	1	yes	characteristic
242	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
243	C₂×C₆₇	#2956	1	1	yes	characteristic
244	D₆₇	—	4	2	no	—
245	#3076	—	67	1	no	—
246	#2962	D₆₇	1	1	yes	characteristic, complemented
247	#2906	—	67	1	no	—
248	#1994	—	67	1	no	—
249	#1950	—	67	1	no	—
250	#1946	—	67	2	no	—
251	#1922	—	67	2	no	—
252	#1866	—	67	2	no	—
253	#1767	—	67	1	no	—
254	#1765	—	67	2	no	—
255	C₄×#104	—	67	1	no	—
256	C₆₇	#22983	1	1	yes	characteristic, complemented, Sylow
257	C₂×#131	—	67	2	no	—
258	#533	C₂×D₆₇	1	1	yes	characteristic
259	C₄×#41	—	67	1	no	—
260	#479	—	67	2	no	—
261	#478	—	67	2	no	—
262	#477	—	67	1	no	—
263	#476	—	67	1	no	—
264	#475	—	67	1	no	—
265	#474	—	67	1	no	—
266	#446	—	2	2	no	—
267	#446	—	2	1	no	—
268	C₈×Q₈	—	2	1	no	—
269	C₂×#108	—	67	2	no	—
270	#427	—	67	2	no	—
271	#424	—	67	2	no	—
272	#424	—	134	2	no	—
273	#424	C₂×D₆₇	1	2	yes	—
274	#423	—	67	1	no	—
275	#423	—	134	1	no	—
276	C₂×C₄⋊C₈	—	67	1	no	—
277	C₂×C₄⋊C₈	—	134	1	no	—
278	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
279	C₂×#104	—	67	1	no	—
280	C₂×#104	—	67	1	no	—
281	#416	—	67	2	no	—
282	#405	C₂×D₆₇	1	2	yes	—
283	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
284	C₂×#97	—	67	2	no	—
285	C₄³	—	67	1	no	—
286	#341	—	67	2	no	—
287	#339	—	67	1	no	—
288	#339	—	67	1	no	—
289	#339	—	67	2	no	—
290	#339	—	67	2	no	—
291	#336	—	67	2	no	—
292	C₂×#41	—	67	1	no	—
293	#131	—	268	2	no	—
294	#131	—	134	4	no	—
295	C₂×M₁₆	—	67	2	no	—
296	C₂×M₁₆	—	134	2	no	—
297	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
298	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
299	C₂²×C₈	—	67	2	no	—
300	C₂²×C₈	—	134	2	no	—
301	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
302	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
303	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
304	#125	C₄×D₆₇	1	2	yes	—
305	C₂²⋊Q₈	—	2	1	no	—
306	C₂²⋊Q₈	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×Q₈	—	2	1	no	—
309	C₄×Q₈	—	67	1	no	—
310	C₄×Q₈	—	67	1	no	—
311	C₄×D₄	—	67	1	no	—
312	C₄×D₄	—	67	1	no	—
313	C₄×D₄	—	134	1	no	—
314	C₄×D₄	—	134	1	no	—
315	#117	—	67	1	no	—
316	#117	—	134	1	no	—
317	#117	C₂²×D₆₇	1	2	yes	—
318	C₂×C₄²	—	67	1	no	—
319	C₂×C₄²	—	67	1	no	—
320	C₂×C₄²	—	134	1	no	—
321	C₂×C₄²	—	134	1	no	—
322	C₂×C₄²	—	134	1	no	—
323	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
324	#108	—	268	2	no	—
325	#107	—	67	2	no	—
326	#107	—	67	2	no	—
327	#106	—	67	2	no	—
328	#106	—	67	2	no	—
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	—	67	1	no	—
332	C₄⋊C₈	—	67	1	no	—
333	C₄⋊C₈	—	67	2	no	—
334	C₄⋊C₈	—	134	2	no	—
335	C₄⋊C₈	—	134	2	no	—
336	C₄⋊C₈	—	134	1	no	—
337	C₄⋊C₈	—	134	1	no	—
338	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
339	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
340	C₄⋊C₈	C₂²⋊D₆₇	1	2	yes	—
341	#104	—	67	4	no	—
342	#104	—	134	2	no	—
343	#103	—	67	2	no	—
344	#103	—	67	2	no	—
345	#102	—	67	2	no	—
346	#102	—	67	2	no	—
347	#97	—	2	2	no	—
348	#97	—	2	1	no	—
349	#97	—	268	2	no	—
350	C₄×C₈	—	2	2	no	—
351	C₄×C₈	—	2	1	no	—
352	#41	—	67	2	no	—
353	#41	—	67	2	no	—
354	#41	—	134	1	no	—
355	#41	—	134	1	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	67	1	no	—
359	C₂×D₄	—	67	1	no	—
360	C₂×D₄	—	134	1	no	—
361	C₂²×C₄	—	134	1	no	—
362	C₂²×C₄	—	134	1	no	—
363	C₂²×C₄	—	134	1	no	—
364	C₂²×C₄	—	134	1	no	—
365	C₂²×C₄	—	134	1	no	—
366	C₂²×C₄	#11761628	1	1	yes	characteristic
367	C₂²×C₄	#11761628	1	1	yes	characteristic
368	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
369	M₁₆	—	2	1	no	—
370	M₁₆	—	2	1	no	—
371	M₁₆	—	2	1	no	—
372	M₁₆	—	268	4	no	—
373	M₁₆	—	134	2	no	—
374	M₁₆	—	134	2	no	—
375	M₁₆	Q₈×D₆₇	1	2	yes	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	268	4	no	—
381	C₂×C₈	—	67	2	no	—
382	C₂×C₈	—	67	2	no	—
383	C₂×C₈	—	134	2	no	—
384	C₂×C₈	—	134	2	no	—
385	C₂×C₈	—	134	2	no	—
386	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
387	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	67	1	no	—
393	C₄⋊C₄	—	67	1	no	—
394	C₄⋊C₄	—	134	1	no	—
395	C₄⋊C₄	—	134	1	no	—
396	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
397	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
398	C₂²⋊C₄	—	2	2	no	—
399	C₂²⋊C₄	—	134	1	no	—
400	C₂²⋊C₄	—	134	1	no	—
401	C₂²⋊C₄	—	134	1	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	268	2	no	—
404	C₄²	—	268	2	no	—
405	C₄²	—	67	2	no	—
406	C₄²	—	134	1	no	—
407	C₄²	—	134	1	no	—
408	C₄²	—	134	1	no	—
409	C₄²	—	134	1	no	—
410	C₄²	—	134	1	no	—
411	C₄²	—	134	1	no	—
412	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
413	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
414	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
415	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
416	C₂³	—	134	1	no	—
417	C₂³	C₂×#11761628	1	1	yes	characteristic
418	Q₈	—	4	2	no	—
419	Q₈	—	8	1	no	—
420	Q₈	—	67	2	no	—
421	Q₈	—	134	1	no	—
422	D₄	—	67	2	no	—
423	D₄	—	134	2	no	—
424	D₄	—	134	2	no	—
425	D₄	—	134	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	268	1	no	—
431	C₂×C₄	—	268	1	no	—
432	C₂×C₄	—	268	1	no	—
433	C₂×C₄	—	268	1	no	—
434	C₂×C₄	—	268	2	no	—
435	C₂×C₄	—	536	1	no	—
436	C₂×C₄	—	134	2	no	—
437	C₂×C₄	—	134	2	no	—
438	C₂×C₄	—	134	1	no	—
439	C₂×C₄	—	134	1	no	—
440	C₂×C₄	—	134	1	no	—
441	C₂×C₄	—	134	2	no	—
442	C₂×C₄	—	134	2	no	—
443	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
444	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
445	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
446	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
447	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
448	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
449	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
450	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
451	C₂×C₄	#14524089	1	1	yes	characteristic
452	C₂×C₄	#14524091	1	1	yes	characteristic
453	C₈	—	2	2	no	—
454	C₈	—	2	2	no	—
455	C₈	—	2	2	no	—
456	C₈	—	4	1	no	—
457	C₈	—	268	2	no	—
458	C₈	—	134	2	no	—
459	C₈	—	134	2	no	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	268	1	no	—
463	C₂²	—	268	1	no	—
464	C₂²	—	134	2	no	—
465	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
466	C₂²	#16864087	1	1	yes	characteristic
467	C₂²	#16864106	1	1	yes	characteristic
468	C₄	—	2	1	no	—
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	4	2	no	—
472	C₄	—	4	2	no	—
473	C₄	—	268	2	no	—
474	C₄	—	268	2	no	—
475	C₄	—	268	1	no	—
476	C₄	—	268	1	no	—
477	C₄	—	134	2	no	—
478	C₄	#16863516	1	2	yes	—
479	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
480	C₄	#16863878	1	1	yes	characteristic
481	C₄	#16864034	1	1	yes	characteristic
482	C₄	Q₈×C₂²⋊D₆₇	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	268	2	no	—
486	C₂	#22563210	1	1	yes	characteristic
487	C₂	C₂×#16863516	1	1	yes	characteristic
488	C₂	#22569116	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×#2962	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#11761628	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2962	D₆₇
hypocenter	C₆₇	#22983
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #2962

Complements

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

N	H ≅ G/N	type
3. #22561187	C₂	semidirect
4. #22560984	C₂	semidirect
5. #22560959	C₂	semidirect
6. #22560808	C₂	semidirect
8. #22560064	C₂	semidirect
11. C₆₇×#2962	C₂	semidirect
14. #16863700	C₄	semidirect
15. #16863693	C₄	semidirect
16. #16863688	C₄	semidirect
17. #16863683	C₄	semidirect
18. #16863678	C₄	semidirect
19. #16863673	C₄	semidirect
80. #14523969	Q₈	semidirect
246. #2962	D₆₇	semidirect
256. C₆₇	#22983	semidirect

Maximal subgroups

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

subgroup	index
#22569991	2
#22568774	2
#22561187	2
#22560984	2
#22560959	2
#22560808	2
#22560483	2
#22560064	2
#22560055	2
C₄×#14523969	2
C₆₇×#2962	2
#22983	67