Subgroups — #30861802

Order 17152.

1282 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22574920	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569557	C₂	1	1	yes	maximal, characteristic
3	#22569110	C₂	1	2	yes	maximal, complemented
4	#22560593	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560574	C₂	1	2	yes	maximal, complemented
6	#22560448	C₂	1	2	yes	maximal, complemented
7	#22560428	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560172	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560053	C₂	1	2	yes	maximal, complemented
10	#22560009	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2932	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864232	C₂²	1	1	yes	characteristic
13	#16864194	C₂²	1	2	yes	—
14	#16864163	C₂²	1	1	yes	characteristic
15	#16864162	—	2	2	no	—
16	#16864162	C₂²	1	2	yes	—
17	#16864161	—	2	4	no	—
18	C₂×#14524049	—	2	2	no	—
19	C₂×#14524049	C₂²	1	2	yes	—
20	#131×D₆₇	—	2	4	no	—
21	#16863578	C₄	1	2	yes	complemented
22	#16863573	C₄	1	2	yes	complemented
23	#16863568	C₄	1	2	yes	complemented
24	#16863563	C₄	1	2	yes	complemented
25	#16863516	—	2	4	no	—
26	#16863511	—	2	4	no	—
27	#16863444	C₂²	1	2	yes	complemented
28	#16863413	C₂²	1	2	yes	complemented
29	#16863412	C₂²	1	2	yes	—
30	C₂×#14523971	C₂²	1	2	yes	—
31	#16863393	C₂²	1	2	yes	—
32	#16863380	C₂²	1	1	yes	characteristic
33	#16863370	C₂²	1	1	yes	characteristic
34	#16863351	C₂²	1	2	yes	complemented
35	#16863167	C₂²	1	2	yes	complemented
36	#16863161	C₂²	1	2	yes	complemented
37	#16863156	C₂²	1	2	yes	complemented
38	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	#532×C₆₇	C₂²	1	1	yes	characteristic
41	#435×C₆₇	—	2	2	no	—
42	#435×C₆₇	—	2	2	no	—
43	#423×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	#408×C₆₇	C₂²	1	1	yes	characteristic
45	#408×C₆₇	C₂²	1	1	yes	characteristic, complemented
46	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic, complemented
47	#14524091	C₂×C₄	1	2	yes	—
48	#14524090	—	2	2	no	—
49	#14524089	—	2	2	no	—
50	#14524088	—	2	2	no	—
51	#14524088	C₂×C₄	1	2	yes	—
52	C₂×#11761628	—	2	2	no	—
53	C₂×#11761628	C₂³	1	2	yes	—
54	#41×D₆₇	—	2	4	no	—
55	#14524074	C₂×C₄	1	4	yes	—
56	#14524073	—	2	2	no	—
57	#14524069	C₂×C₄	1	4	yes	—
58	#14524068	—	2	2	no	—
59	#14524063	—	2	4	no	—
60	#14524051	—	2	8	no	—
61	#14524051	—	2	4	no	—
62	#14524050	—	2	4	no	—
63	#14524049	—	2	4	no	—
64	#14524049	—	2	4	no	—
65	#14524049	—	2	2	no	—
66	#14524049	—	2	4	no	—
67	#14524049	—	2	2	no	—
68	#14524049	D₄	1	4	yes	complemented
69	#14524049	D₄	1	4	yes	complemented
70	C₂×#11761612	—	2	2	no	—
71	C₂×#11761612	C₂³	1	2	yes	—
72	C₂×#11761612	C₂³	1	2	yes	—
73	C₂×#11761611	—	2	4	no	—
74	C₂×#11761611	—	2	2	no	—
75	M₁₆×D₆₇	—	2	8	no	—
76	M₁₆×D₆₇	—	2	4	no	—
77	C₂×C₈×D₆₇	—	2	4	no	—
78	C₂×C₈×D₆₇	—	2	2	no	—
79	#14524002	C₂³	1	2	yes	—
80	#14523978	—	2	4	no	—
81	#14523971	C₂×C₄	1	4	yes	—
82	#14523971	C₂×C₄	1	4	yes	—
83	#14523970	—	2	4	no	—
84	#14523969	—	2	4	no	—
85	#14523968	—	2	2	no	—
86	#14523965	—	2	2	no	—
87	#14523962	C₂×C₄	1	2	yes	—
88	#14523960	C₂×C₄	1	2	yes	—
89	#14523958	—	2	2	no	—
90	#14523953	—	2	2	no	—
91	C₈×Dic₆₇	—	2	2	no	—
92	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
94	#131×C₆₇	—	2	2	no	—
95	#131×C₆₇	—	2	2	no	—
96	#131×C₆₇	—	2	2	no	—
97	#131×C₆₇	—	2	2	no	—
98	C₂×M₁₆×C₆₇	—	2	1	no	—
99	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
100	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
102	C₂²×C₈×C₆₇	—	2	1	no	—
103	C₂²×C₈×C₆₇	D₄	1	2	yes	—
104	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
105	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
106	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
107	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
108	#124×C₆₇	C₂×C₄	1	1	yes	characteristic
109	#124×C₆₇	C₂×C₄	1	1	yes	characteristic
110	#121×C₆₇	—	2	2	no	—
111	C₄×D₄×C₆₇	—	2	2	no	—
112	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
113	C₄⋊C₈×C₆₇	—	2	2	no	—
114	C₂²⋊C₈×C₆₇	—	2	2	no	—
115	C₂²⋊C₈×C₆₇	—	2	2	no	—
116	#97×C₆₇	—	2	1	no	—
117	#97×C₆₇	—	2	1	no	—
118	#11761629	—	2	4	no	—
119	#11761628	—	2	4	no	—
120	#11761628	—	2	4	no	—
121	#11761628	—	2	4	no	—
122	#11761628	—	2	2	no	—
123	#11761628	—	2	2	no	—
124	#11761628	C₂²×C₄	1	4	yes	—
125	#11761628	C₂×D₄	1	4	yes	—
126	#11761627	—	2	8	no	—
127	#11761627	—	2	4	no	—
128	C₂×Dic₁₃₄	—	2	2	no	—
129	C₂×Dic₁₃₄	#41	1	2	yes	—
130	Q₈×D₆₇	—	2	4	no	—
131	C₂×D₂₆₈	—	2	2	no	—
132	C₂×D₂₆₈	#41	1	2	yes	—
133	C₂×C₂²⋊D₆₇	—	2	4	no	—
134	C₂×C₂²⋊D₆₇	—	2	2	no	—
135	D₄×D₆₇	—	2	8	no	—
136	D₄×D₆₇	—	2	4	no	—
137	C₂×C₄×D₆₇	—	2	4	no	—
138	C₂×C₄×D₆₇	—	2	2	no	—
139	#11761618	—	2	4	no	—
140	#11761616	—	2	4	no	—
141	#11761615	—	2	4	no	—
142	#11761613	—	2	4	no	—
143	#11761612	—	2	4	no	—
144	#11761612	—	2	2	no	—
145	#11761612	—	2	2	no	—
146	#11761612	C₂²×C₄	1	4	yes	—
147	#11761612	C₂×D₄	1	4	yes	—
148	#11761611	—	2	8	no	—
149	#11761611	—	2	4	no	—
150	#11761611	—	2	4	no	—
151	C₂×C₆₇⋊C₈	—	2	4	no	—
152	C₂×C₆₇⋊C₈	—	2	2	no	—
153	C₂×C₆₇⋊C₈	—	2	2	no	—
154	C₈×D₆₇	—	2	8	no	—
155	C₈×D₆₇	—	2	4	no	—
156	C₈×D₆₇	—	2	4	no	—
157	C₄⋊Dic₆₇	—	2	2	no	—
158	C₂²⋊Dic₆₇	—	2	2	no	—
159	C₄×Dic₆₇	—	2	2	no	—
160	#41×C₆₇	—	2	2	no	—
161	#41×C₆₇	—	2	2	no	—
162	#41×C₆₇	—	2	2	no	—
163	#41×C₆₇	—	2	2	no	—
164	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
165	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
166	C₂×D₄×C₆₇	—	2	1	no	—
167	C₂×D₄×C₆₇	—	2	2	no	—
168	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
170	C₂×D₄×C₆₇	#41	1	2	yes	—
171	C₂²×C₄×C₆₇	—	2	1	no	—
172	C₂²×C₄×C₆₇	—	2	1	no	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
175	M₁₆×C₆₇	—	2	2	no	—
176	M₁₆×C₆₇	—	2	4	no	—
177	M₁₆×C₆₇	—	2	2	no	—
178	M₁₆×C₆₇	—	4	1	no	—
179	M₁₆×C₆₇	—	4	1	no	—
180	C₂×C₈×C₆₇	—	2	2	no	—
181	C₂×C₈×C₆₇	—	2	2	no	—
182	C₂×C₈×C₆₇	—	2	1	no	—
183	C₂×C₈×C₆₇	—	2	1	no	—
184	C₂×C₈×C₆₇	—	2	2	no	—
185	C₂×C₈×C₆₇	—	4	2	no	—
186	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
187	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
188	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
189	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
190	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₂²⋊C₄×C₆₇	—	2	2	no	—
193	C₂²⋊C₄×C₆₇	—	2	2	no	—
194	C₄²×C₆₇	—	2	1	no	—
195	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
196	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
197	C₂³×C₆₇	—	2	1	no	—
198	C₂³×C₆₇	C₄×D₄	1	2	yes	—
199	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
200	C₂²×D₆₇	—	2	4	no	—
201	C₂²×D₆₇	—	2	2	no	—
202	Q₈×C₆₇	—	2	2	no	—
203	Q₈×C₆₇	—	2	2	no	—
204	D₄×C₆₇	—	2	4	no	—
205	D₄×C₆₇	—	2	2	no	—
206	D₄×C₆₇	—	2	2	no	—
207	D₄×C₆₇	—	4	2	no	—
208	D₄×C₆₇	—	8	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	1	no	—
212	C₂×C₄×C₆₇	—	2	1	no	—
213	C₂×C₄×C₆₇	—	2	1	no	—
214	C₂×C₄×C₆₇	—	4	1	no	—
215	C₂×C₄×C₆₇	—	4	1	no	—
216	C₂×C₄×C₆₇	—	8	1	no	—
217	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
218	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
222	C₂²⋊D₆₇	—	2	8	no	—
223	C₂²⋊D₆₇	—	2	4	no	—
224	C₂²⋊D₆₇	—	2	4	no	—
225	C₂×Dic₆₇	—	2	4	no	—
226	C₂×Dic₆₇	—	2	2	no	—
227	C₂×Dic₆₇	—	4	2	no	—
228	D₂₆₈	—	2	4	no	—
229	D₂₆₈	—	2	2	no	—
230	D₂₆₈	C₄×D₄	1	4	yes	—
231	C₄×D₆₇	—	2	8	no	—
232	C₄×D₆₇	—	2	4	no	—
233	C₄×D₆₇	—	2	4	no	—
234	Dic₁₃₄	—	2	4	no	—
235	Dic₁₃₄	—	2	2	no	—
236	Dic₁₃₄	C₄×D₄	1	4	yes	—
237	C₈×C₆₇	—	2	2	no	—
238	C₈×C₆₇	—	4	1	no	—
239	C₈×C₆₇	—	4	1	no	—
240	C₈×C₆₇	#120	1	1	yes	characteristic
241	C₈×C₆₇	#120	1	1	yes	characteristic
242	C₈×C₆₇	#120	1	2	yes	—
243	C₆₇⋊C₈	—	2	4	no	—
244	C₆₇⋊C₈	—	2	4	no	—
245	C₂²×C₆₇	—	2	2	no	—
246	C₂²×C₆₇	—	4	1	no	—
247	C₂²×C₆₇	—	4	2	no	—
248	C₂²×C₆₇	—	4	2	no	—
249	C₂²×C₆₇	—	4	1	no	—
250	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
251	C₂²×C₆₇	#525	1	2	yes	—
252	C₂×D₆₇	—	2	4	no	—
253	C₂×D₆₇	—	8	2	no	—
254	C₂×D₆₇	—	8	4	no	—
255	C₄×C₆₇	—	2	2	no	—
256	C₄×C₆₇	—	4	1	no	—
257	C₄×C₆₇	—	4	1	no	—
258	C₄×C₆₇	—	4	1	no	—
259	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
260	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
261	C₄×C₆₇	#525	1	1	yes	characteristic
262	Dic₆₇	—	2	4	no	—
263	Dic₆₇	—	8	2	no	—
264	#21478	—	67	1	no	maximal, Sylow
265	C₂×C₆₇	—	2	2	no	—
266	C₂×C₆₇	—	2	1	no	—
267	C₂×C₆₇	—	4	2	no	—
268	C₂×C₆₇	—	8	1	no	—
269	C₂×C₆₇	C₄×#120	1	1	yes	characteristic
270	D₆₇	—	4	4	no	—
271	#3573	—	67	1	no	—
272	#3018	—	67	1	no	—
273	#2955	—	67	2	no	—
274	#2932	D₆₇	1	1	yes	characteristic, complemented
275	#1884	—	67	1	no	—
276	#1881	—	67	2	no	—
277	#1861	—	67	2	no	—
278	#1858	—	67	1	no	—
279	#1794	—	67	1	no	—
280	#1765	—	67	2	no	—
281	#1757	—	67	1	no	—
282	C₆₇	#21478	1	1	yes	characteristic, complemented, Sylow
283	#585	—	67	1	no	—
284	#575	—	67	2	no	—
285	#568	—	67	1	no	—
286	#568	—	67	2	no	—
287	#568	—	134	4	no	—
288	#568	—	134	2	no	—
289	C₂×#131	—	67	2	no	—
290	C₂×#131	—	134	4	no	—
291	C₂×#131	—	134	2	no	—
292	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
293	C₂×#131	C₂×D₆₇	1	1	yes	characteristic, complemented
294	#532	C₂×D₆₇	1	1	yes	characteristic
295	#456	—	67	2	no	—
296	#455	—	67	2	no	—
297	#454	—	67	2	no	—
298	#453	—	67	2	no	—
299	#444	—	134	4	no	—
300	#443	—	134	4	no	—
301	#435	—	2	2	no	—
302	#435	—	2	2	no	—
303	#430	—	67	2	no	—
304	#423	C₂×D₆₇	1	1	yes	characteristic, complemented
305	#421	—	67	2	no	—
306	#421	—	67	2	no	—
307	C₂×#104	—	67	2	no	—
308	#416	—	67	2	no	—
309	#413	—	67	1	no	—
310	#410	—	67	1	no	—
311	#408	C₂×D₆₇	1	1	yes	characteristic
312	#408	C₂×D₆₇	1	1	yes	characteristic, complemented
313	#405	—	67	2	no	—
314	C₂×#97	C₂×D₆₇	1	1	yes	characteristic, complemented
315	#344	—	67	2	no	—
316	#341	—	67	2	no	—
317	#339	—	67	2	no	—
318	2₋¹⁺⁴	—	67	2	no	—
319	2₋¹⁺⁴	—	134	2	no	—
320	H(5,𝔽₂)	—	67	2	no	—
321	H(5,𝔽₂)	—	134	2	no	—
322	H(5,𝔽₂)	—	134	2	no	—
323	C₂×#41	—	67	2	no	—
324	C₂×#41	—	134	4	no	—
325	C₂×#41	—	134	2	no	—
326	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
327	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
328	#137	—	67	4	no	—
329	#137	—	134	2	no	—
330	AGL(1,ℤ/8)	—	67	4	no	—
331	AGL(1,ℤ/8)	—	134	2	no	—
332	#135	—	134	4	no	—
333	#131	—	2	2	no	—
334	#131	—	2	2	no	—
335	#131	—	2	2	no	—
336	#131	—	2	2	no	—
337	#131	—	67	4	no	—
338	#131	—	67	4	no	—
339	#131	—	134	4	no	—
340	#131	—	134	4	no	—
341	#131	—	134	4	no	—
342	#131	—	134	8	no	—
343	#131	—	134	4	no	—
344	#131	—	134	2	no	—
345	#131	—	134	2	no	—
346	#131	—	134	4	no	—
347	C₂×M₁₆	—	2	1	no	—
348	C₂×M₁₆	—	67	2	no	—
349	C₂×M₁₆	—	67	2	no	—
350	C₂×M₁₆	—	134	4	no	—
351	C₂×M₁₆	—	134	4	no	—
352	C₂×M₁₆	—	134	8	no	—
353	C₂×M₁₆	—	134	2	no	—
354	C₂×M₁₆	—	134	2	no	—
355	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
356	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
357	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
358	C₂²×C₈	—	2	1	no	—
359	C₂²×C₈	—	134	4	no	—
360	C₂²×C₈	—	134	2	no	—
361	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
362	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
363	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
364	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
365	#127	C₄×D₆₇	1	1	yes	characteristic
366	#124	C₄×D₆₇	1	1	yes	characteristic
367	#124	C₄×D₆₇	1	1	yes	characteristic
368	#121	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	#117	—	67	2	no	—
371	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
372	#108	—	134	4	no	—
373	C₄⋊C₈	—	2	2	no	—
374	#104	—	67	4	no	—
375	#104	—	67	4	no	—
376	#104	—	134	4	no	—
377	#104	—	134	4	no	—
378	#103	—	134	2	no	—
379	#102	—	134	2	no	—
380	#101	—	67	2	no	—
381	#100	—	67	2	no	—
382	#99	—	134	2	no	—
383	C₂²⋊C₈	—	2	2	no	—
384	C₂²⋊C₈	—	2	2	no	—
385	#97	—	2	1	no	—
386	#97	—	2	1	no	—
387	#97	—	134	2	no	—
388	C₄×C₈	—	134	2	no	—
389	#41	—	2	2	no	—
390	#41	—	2	2	no	—
391	#41	—	2	2	no	—
392	#41	—	2	2	no	—
393	#41	—	67	4	no	—
394	#41	—	67	4	no	—
395	#41	—	134	4	no	—
396	#41	—	134	4	no	—
397	#41	—	134	4	no	—
398	#41	—	134	8	no	—
399	#41	—	134	4	no	—
400	#41	—	134	4	no	—
401	#41	—	134	2	no	—
402	#41	—	134	2	no	—
403	C₂×Q₈	—	67	2	no	—
404	C₂×Q₈	—	134	2	no	—
405	C₂×Q₈	—	134	4	no	—
406	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
407	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	67	2	no	—
411	C₂×D₄	—	134	2	no	—
412	C₂×D₄	—	134	4	no	—
413	C₂×D₄	—	134	8	no	—
414	C₂×D₄	—	134	4	no	—
415	C₂×D₄	—	134	2	no	—
416	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
417	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
418	C₂×D₄	#11761628	1	2	yes	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	134	4	no	—
422	C₂²×C₄	—	134	2	no	—
423	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
424	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
425	Q₁₆	—	134	4	no	—
426	SD₁₆	—	134	4	no	—
427	SD₁₆	—	134	4	no	—
428	D₈	—	134	4	no	—
429	M₁₆	—	2	2	no	—
430	M₁₆	—	2	4	no	—
431	M₁₆	—	2	2	no	—
432	M₁₆	—	4	1	no	—
433	M₁₆	—	4	1	no	—
434	M₁₆	—	67	4	no	—
435	M₁₆	—	67	4	no	—
436	M₁₆	—	134	8	no	—
437	M₁₆	—	134	4	no	—
438	M₁₆	—	134	2	no	—
439	M₁₆	—	134	2	no	—
440	M₁₆	—	134	4	no	—
441	M₁₆	—	134	4	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	2	no	—
447	C₂×C₈	—	4	2	no	—
448	C₂×C₈	—	134	8	no	—
449	C₂×C₈	—	134	4	no	—
450	C₂×C₈	—	134	2	no	—
451	C₂×C₈	—	134	2	no	—
452	C₂×C₈	—	134	4	no	—
453	C₂×C₈	—	134	4	no	—
454	C₂×C₈	D₄×D₆₇	1	2	yes	—
455	C₂×C₈	D₄×D₆₇	1	2	yes	—
456	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
457	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
458	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	134	2	no	—
461	C₂²⋊C₄	—	2	2	no	—
462	C₂²⋊C₄	—	2	2	no	—
463	C₂²⋊C₄	—	134	2	no	—
464	C₄²	—	2	1	no	—
465	C₄²	—	134	2	no	—
466	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
467	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
468	C₂³	—	2	1	no	—
469	C₂³	—	134	4	no	—
470	C₂³	—	134	2	no	—
471	C₂³	C₄×C₂²⋊D₆₇	1	2	yes	—
472	C₂³	C₂×#11761628	1	1	yes	characteristic
473	Q₈	—	2	2	no	—
474	Q₈	—	2	2	no	—
475	Q₈	—	67	4	no	—
476	Q₈	—	134	4	no	—
477	Q₈	—	134	2	no	—
478	D₄	—	2	4	no	—
479	D₄	—	2	2	no	—
480	D₄	—	2	2	no	—
481	D₄	—	4	2	no	—
482	D₄	—	8	2	no	—
483	D₄	—	67	4	no	—
484	D₄	—	134	4	no	—
485	D₄	—	134	8	no	—
486	D₄	—	134	4	no	—
487	D₄	—	134	4	no	—
488	D₄	—	134	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	4	1	no	—
496	C₂×C₄	—	8	1	no	—
497	C₂×C₄	—	268	2	no	—
498	C₂×C₄	—	134	8	no	—
499	C₂×C₄	—	134	4	no	—
500	C₂×C₄	—	134	2	no	—
501	C₂×C₄	—	134	4	no	—
502	C₂×C₄	—	134	4	no	—
503	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
504	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
505	C₂×C₄	C₂×D₄×D₆₇	1	2	yes	—
506	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
507	C₂×C₄	#41×D₆₇	1	2	yes	—
508	C₈	—	2	2	no	—
509	C₈	—	4	1	no	—
510	C₈	—	4	1	no	—
511	C₈	—	134	4	no	—
512	C₈	—	134	4	no	—
513	C₈	#14524011	1	1	yes	characteristic
514	C₈	#14524011	1	1	yes	characteristic
515	C₈	#14524011	1	2	yes	—
516	C₂²	—	2	2	no	—
517	C₂²	—	4	1	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	1	no	—
521	C₂²	—	536	2	no	—
522	C₂²	—	536	4	no	—
523	C₂²	—	134	4	no	—
524	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
525	C₂²	#16863905	1	2	yes	—
526	C₄	—	2	2	no	—
527	C₄	—	4	1	no	—
528	C₄	—	4	1	no	—
529	C₄	—	4	1	no	—
530	C₄	—	536	2	no	—
531	C₄	—	134	4	no	—
532	C₄	C₄×D₄×D₆₇	1	2	yes	—
533	C₄	C₂×#14524011	1	1	yes	characteristic
534	C₄	#16863903	1	1	yes	characteristic
535	C₂	—	2	1	no	—
536	C₂	—	2	2	no	—
537	C₂	—	4	2	no	—
538	C₂	—	8	1	no	—
539	C₂	—	268	4	no	—
540	C₂	C₄×#14524011	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14524011
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2932	C₂
socle	C₂×C₆₇	C₄×#120
prime radical	#22569557	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2932	D₆₇
hypocenter	C₆₇	#21478
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #2932

Complements

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

N	H ≅ G/N	type
1. #22574920	C₂	semidirect
3. #22569110	C₂	semidirect
4. #22560593	C₂	semidirect
5. #22560574	C₂	semidirect
6. #22560448	C₂	semidirect
7. #22560428	C₂	semidirect
8. #22560172	C₂	semidirect
9. #22560053	C₂	semidirect
10. #22560009	C₂	semidirect
11. C₆₇×#2932	C₂	semidirect
21. #16863578	C₄	semidirect
22. #16863573	C₄	semidirect
23. #16863568	C₄	semidirect
24. #16863563	C₄	semidirect
27. #16863444	C₂²	semidirect
28. #16863413	C₂²	semidirect
34. #16863351	C₂²	semidirect
35. #16863167	C₂²	semidirect
36. #16863161	C₂²	semidirect
37. #16863156	C₂²	semidirect
39. C₂×#131×C₆₇	C₂²	semidirect
43. #423×C₆₇	C₂²	semidirect
45. #408×C₆₇	C₂²	semidirect
46. C₂×#97×C₆₇	C₂²	semidirect
68. #14524049	D₄	semidirect
69. #14524049	D₄	semidirect
274. #2932	D₆₇	semidirect
282. C₆₇	#21478	semidirect
293. C₂×#131	C₂×D₆₇	semidirect
304. #423	C₂×D₆₇	semidirect
312. #408	C₂×D₆₇	semidirect
314. C₂×#97	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22574920	2
#22569557	2
#22569110	2
#22560593	2
#22560574	2
#22560448	2
#22560428	2
#22560172	2
#22560053	2
#22560009	2
C₆₇×#2932	2
#21478	67