Subgroups — #30878686

Order 17152.

824 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569929	C₂	1	1	yes	maximal, characteristic
2	#22569125	C₂	1	1	yes	maximal, characteristic, complemented
3	#22569116	C₂	1	2	yes	maximal, complemented
4	#22561095	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560997	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560877	C₂	1	2	yes	maximal, complemented
7	#22560571	C₂	1	2	yes	maximal, complemented
8	#22560427	C₂	1	2	yes	maximal, complemented
9	#108×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560044	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2916	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864194	C₂²	1	2	yes	—
13	C₂×#14524065	C₂²	1	1	yes	characteristic
14	#16864162	C₂²	1	2	yes	—
15	#16863671	C₄	1	2	yes	complemented
16	#16863666	C₄	1	2	yes	complemented
17	#16863659	C₄	1	4	yes	complemented
18	#16863517	—	2	2	no	—
19	#16863516	—	2	4	no	—
20	#16863512	—	2	2	no	—
21	#16863511	—	2	4	no	—
22	#16863444	C₂²	1	2	yes	—
23	C₂×#14523979	C₂²	1	1	yes	characteristic
24	#16863412	C₂²	1	2	yes	—
25	C₂×#14523970	C₂²	1	1	yes	characteristic
26	C₂×#14523969	C₂²	1	2	yes	—
27	#16863392	C₂²	1	1	yes	characteristic
28	#16863378	C₂²	1	2	yes	—
29	#16863364	C₂²	1	2	yes	—
30	#16863351	C₂²	1	1	yes	characteristic
31	M₁₆×Dic₆₇	C₂²	1	2	yes	—
32	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
33	#16863167	C₂²	1	2	yes	—
34	#16863160	C₂²	1	1	yes	characteristic
35	#16863160	C₂²	1	2	yes	—
36	#16863157	C₂²	1	1	yes	characteristic
37	#568×C₆₇	C₂²	1	2	yes	—
38	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
39	#430×C₆₇	—	2	2	no	—
40	#430×C₆₇	C₂²	1	2	yes	—
41	C₂×#108×C₆₇	—	2	2	no	—
42	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#11761628	C₂³	1	2	yes	—
45	#14524074	—	2	2	no	—
46	#14524073	C₂×C₄	1	4	yes	—
47	#14524069	—	2	2	no	—
48	#14524068	C₂×C₄	1	4	yes	—
49	#14524065	—	2	1	no	—
50	#14524065	—	2	1	no	—
51	#14524065	—	2	1	no	—
52	#14524065	—	2	1	no	—
53	#14524063	—	2	4	no	—
54	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
55	C₂×#11761617	C₂×C₄	1	2	yes	—
56	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
57	#14524051	—	2	4	no	—
58	#14524049	—	2	4	no	—
59	C₂×#11761612	C₂³	1	2	yes	—
60	C₂×#11761612	C₂³	1	1	yes	characteristic
61	M₁₆×D₆₇	—	4	2	no	—
62	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
63	#14524002	C₂³	1	1	yes	characteristic
64	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
65	#14523979	C₂×C₄	1	2	yes	—
66	#14523979	C₂×C₄	1	2	yes	—
67	#14523978	—	2	4	no	—
68	#14523970	D₄	1	2	yes	—
69	#14523970	D₄	1	2	yes	—
70	#14523969	—	2	2	no	—
71	#14523969	—	2	2	no	—
72	#14523969	—	2	4	no	—
73	#14523967	—	2	1	no	—
74	#14523964	—	2	1	no	—
75	#14523961	C₂×C₄	1	4	yes	—
76	#14523959	C₂×C₄	1	4	yes	—
77	(C₂×D₆₇)⋊C₈	—	4	2	no	—
78	#14523953	—	2	2	no	—
79	#14523953	—	2	1	no	—
80	C₈×Dic₆₇	—	2	2	no	—
81	C₈×Dic₆₇	—	2	1	no	—
82	C₈×Dic₆₇	—	2	1	no	—
83	C₈×Dic₆₇	—	2	1	no	—
84	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
85	#131×C₆₇	—	2	1	no	—
86	#131×C₆₇	—	2	1	no	—
87	#131×C₆₇	—	2	4	no	—
88	#131×C₆₇	—	4	2	no	—
89	#131×C₆₇	C₂×C₄	1	2	yes	—
90	#131×C₆₇	C₂×C₄	1	2	yes	—
91	C₂×M₁₆×C₆₇	—	2	1	no	—
92	C₂×M₁₆×C₆₇	—	2	4	no	—
93	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
94	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
96	C₂²×C₈×C₆₇	—	2	1	no	—
97	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
98	#108×C₆₇	—	2	4	no	—
99	#108×C₆₇	—	2	1	no	—
100	#108×C₆₇	—	2	1	no	—
101	#108×C₆₇	D₄	1	2	yes	—
102	#108×C₆₇	D₄	1	2	yes	—
103	#11761628	—	2	2	no	—
104	#11761628	—	2	2	no	—
105	#11761628	—	2	2	no	—
106	#11761628	—	2	2	no	—
107	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
108	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
109	C₂×C₂²⋊D₆₇	—	2	2	no	—
110	C₂²×Dic₆₇	—	2	1	no	—
111	C₂×C₄×D₆₇	—	2	2	no	—
112	Dic₂₆₈	—	2	1	no	—
113	Dic₂₆₈	—	2	1	no	—
114	#11761618	—	2	4	no	—
115	#11761617	—	2	2	no	—
116	#11761617	—	2	2	no	—
117	#11761616	—	2	4	no	—
118	#11761615	—	2	4	no	—
119	D₅₃₆	—	2	1	no	—
120	D₅₃₆	—	2	1	no	—
121	#11761613	—	2	4	no	—
122	#11761612	—	2	2	no	—
123	#11761612	—	2	1	no	—
124	#11761612	C₂²×C₄	1	4	yes	—
125	#11761612	C₂×D₄	1	2	yes	—
126	#11761611	—	4	4	no	—
127	C₂×C₆₇⋊C₈	—	2	2	no	—
128	C₂×C₆₇⋊C₈	—	2	1	no	—
129	C₂×C₆₇⋊C₈	—	2	1	no	—
130	C₂×C₆₇⋊C₈	—	2	1	no	—
131	C₂×C₆₇⋊C₈	—	2	1	no	—
132	C₈×D₆₇	—	4	4	no	—
133	C₄⋊Dic₆₇	—	2	1	no	—
134	C₂²⋊Dic₆₇	—	2	1	no	—
135	C₄×Dic₆₇	—	2	1	no	—
136	C₄×Dic₆₇	—	2	1	no	—
137	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
138	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
139	#41×C₆₇	—	2	1	no	—
140	#41×C₆₇	—	2	1	no	—
141	#41×C₆₇	C₂²×C₄	1	2	yes	—
142	#41×C₆₇	C₂×D₄	1	2	yes	—
143	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
144	C₂×D₄×C₆₇	—	2	1	no	—
145	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
148	M₁₆×C₆₇	—	2	1	no	—
149	M₁₆×C₆₇	—	2	2	no	—
150	M₁₆×C₆₇	—	2	2	no	—
151	M₁₆×C₆₇	—	2	2	no	—
152	M₁₆×C₆₇	—	2	2	no	—
153	M₁₆×C₆₇	—	2	2	no	—
154	M₁₆×C₆₇	—	4	4	no	—
155	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
156	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	4	4	no	—
163	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
165	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
166	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
167	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
168	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
169	C₂³×C₆₇	—	2	1	no	—
170	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
171	C₂²×D₆₇	—	2	2	no	—
172	Q₈×C₆₇	—	2	1	no	—
173	Q₈×C₆₇	C₄×D₄	1	2	yes	—
174	D₄×C₆₇	—	2	2	no	—
175	D₄×C₆₇	—	2	2	no	—
176	D₄×C₆₇	—	2	1	no	—
177	D₄×C₆₇	C₄×D₄	1	2	yes	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂²⋊D₆₇	—	4	2	no	—
188	C₂²⋊D₆₇	—	4	2	no	—
189	C₂×Dic₆₇	—	2	1	no	—
190	C₂×Dic₆₇	—	2	2	no	—
191	C₂×Dic₆₇	—	2	1	no	—
192	C₂×Dic₆₇	—	4	2	no	—
193	C₂×Dic₆₇	—	4	1	no	—
194	D₂₆₈	—	2	2	no	—
195	D₂₆₈	—	2	2	no	—
196	C₄×D₆₇	—	4	2	no	—
197	C₄×D₆₇	—	4	2	no	—
198	Dic₁₃₄	—	2	2	no	—
199	Dic₁₃₄	—	2	2	no	—
200	C₈×C₆₇	—	2	2	no	—
201	C₈×C₆₇	—	2	4	no	—
202	C₈×C₆₇	—	4	2	no	—
203	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
204	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
205	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
206	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
207	C₆₇⋊C₈	—	2	4	no	—
208	C₆₇⋊C₈	—	2	2	no	—
209	C₆₇⋊C₈	—	4	1	no	—
210	C₂²×C₆₇	—	2	2	no	—
211	C₂²×C₆₇	—	4	1	no	—
212	C₂²×C₆₇	—	8	1	no	—
213	C₂²×C₆₇	#520	1	1	yes	characteristic
214	C₂²×C₆₇	#547	1	1	yes	characteristic
215	C₂²×C₆₇	#548	1	1	yes	characteristic
216	C₂×D₆₇	—	4	2	no	—
217	C₂×D₆₇	—	8	2	no	—
218	C₄×C₆₇	—	2	2	no	—
219	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
220	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
221	C₄×C₆₇	D₄²	1	1	yes	characteristic
222	C₄×C₆₇	#553	1	1	yes	characteristic
223	Dic₆₇	—	2	2	no	—
224	Dic₆₇	—	4	1	no	—
225	Dic₆₇	—	4	2	no	—
226	Dic₆₇	—	8	1	no	—
227	#23042	—	67	1	no	maximal, Sylow
228	C₂×C₆₇	—	2	1	no	—
229	C₂×C₆₇	—	2	1	no	—
230	C₂×C₆₇	—	2	1	no	—
231	C₂×C₆₇	—	4	2	no	—
232	C₂×C₆₇	#2361	1	1	yes	characteristic
233	D₆₇	—	8	2	no	—
234	#3067	—	67	1	no	—
235	#2957	—	67	1	no	—
236	#2956	—	67	2	no	—
237	#2916	D₆₇	1	1	yes	characteristic, complemented
238	#1974	—	67	1	no	—
239	#1953	—	67	1	no	—
240	#1932	—	67	2	no	—
241	#1881	—	67	2	no	—
242	#1858	—	67	2	no	—
243	C₄×#108	—	67	1	no	—
244	#1764	—	67	1	no	—
245	C₆₇	#23042	1	1	yes	characteristic, complemented, Sylow
246	#575	—	67	2	no	—
247	C₂×#135	—	67	1	no	—
248	#568	—	67	2	no	—
249	#568	C₂×D₆₇	1	2	yes	—
250	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
251	#473	—	67	2	no	—
252	#472	—	67	2	no	—
253	#471	—	67	4	no	—
254	#444	—	134	4	no	—
255	#444	—	134	2	no	—
256	#443	—	134	4	no	—
257	#443	—	134	2	no	—
258	#430	—	2	2	no	—
259	#430	—	67	2	no	—
260	#430	C₂×D₆₇	1	2	yes	—
261	C₂×#108	—	2	2	no	—
262	C₂×#108	—	67	1	no	—
263	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
264	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
265	#421	—	67	2	no	—
266	C₂×#104	—	67	1	no	—
267	C₂×#104	—	67	2	no	—
268	#416	—	67	1	no	—
269	#413	—	67	2	no	—
270	#408	—	67	2	no	—
271	#405	—	67	1	no	—
272	C₄×M₁₆	—	67	2	no	—
273	C₂×C₄×C₈	—	67	1	no	—
274	#344	—	67	2	no	—
275	#341	—	67	1	no	—
276	#341	—	67	2	no	—
277	#339	—	67	1	no	—
278	C₂×#41	—	67	2	no	—
279	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
280	#137	—	67	4	no	—
281	#137	—	134	2	no	—
282	AGL(1,ℤ/8)	—	67	4	no	—
283	AGL(1,ℤ/8)	—	134	2	no	—
284	#135	—	134	4	no	—
285	#135	—	134	1	no	—
286	#135	—	134	1	no	—
287	#135	—	134	1	no	—
288	#135	—	134	1	no	—
289	C₂×Q₁₆	—	67	1	no	—
290	C₂×SD₁₆	—	67	2	no	—
291	C₂×D₈	—	67	1	no	—
292	#131	—	2	1	no	—
293	#131	—	2	1	no	—
294	#131	—	2	4	no	—
295	#131	—	4	2	no	—
296	#131	—	134	4	no	—
297	#131	—	134	4	no	—
298	#131	C₄×D₆₇	1	2	yes	—
299	#131	C₄×D₆₇	1	2	yes	—
300	C₂×M₁₆	—	2	1	no	—
301	C₂×M₁₆	—	2	4	no	—
302	C₂×M₁₆	—	268	2	no	—
303	C₂×M₁₆	—	67	1	no	—
304	C₂×M₁₆	—	67	2	no	—
305	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
306	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
307	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
308	C₂²×C₈	—	2	1	no	—
309	C₂²×C₈	—	67	1	no	—
310	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
311	#117	—	67	1	no	—
312	C₂×C₄²	—	67	1	no	—
313	#108	—	2	4	no	—
314	#108	—	2	1	no	—
315	#108	—	2	1	no	—
316	#108	—	67	2	no	—
317	#108	—	67	2	no	—
318	#108	—	134	4	no	—
319	#108	C₂²⋊D₆₇	1	2	yes	—
320	#108	C₂²⋊D₆₇	1	2	yes	—
321	#104	—	67	2	no	—
322	#104	—	67	2	no	—
323	#104	—	134	2	no	—
324	#104	—	134	2	no	—
325	#104	—	134	4	no	—
326	#103	—	134	1	no	—
327	#102	—	134	1	no	—
328	#101	—	67	4	no	—
329	#100	—	67	4	no	—
330	C₂²⋊C₈	—	268	2	no	—
331	#97	—	134	2	no	—
332	#97	—	134	1	no	—
333	C₄×C₈	—	134	2	no	—
334	C₄×C₈	—	134	1	no	—
335	C₄×C₈	—	134	1	no	—
336	C₄×C₈	—	134	1	no	—
337	#41	—	2	1	no	—
338	#41	—	2	1	no	—
339	#41	—	134	2	no	—
340	#41	—	134	2	no	—
341	#41	—	134	2	no	—
342	#41	—	134	2	no	—
343	#41	C₂×C₄×D₆₇	1	2	yes	—
344	#41	D₄×D₆₇	1	2	yes	—
345	C₂×Q₈	—	67	2	no	—
346	C₂×Q₈	#11761629	1	1	yes	characteristic
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	67	2	no	—
349	C₂×D₄	—	134	2	no	—
350	C₂×D₄	#11761629	1	1	yes	characteristic
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	134	1	no	—
353	C₂²×C₄	—	134	2	no	—
354	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
355	Q₁₆	—	134	4	no	—
356	Q₁₆	—	134	1	no	—
357	Q₁₆	—	134	1	no	—
358	SD₁₆	—	134	4	no	—
359	SD₁₆	—	134	4	no	—
360	SD₁₆	—	134	2	no	—
361	SD₁₆	—	134	2	no	—
362	D₈	—	134	4	no	—
363	D₈	—	134	1	no	—
364	D₈	—	134	1	no	—
365	M₁₆	—	2	1	no	—
366	M₁₆	—	2	2	no	—
367	M₁₆	—	2	2	no	—
368	M₁₆	—	2	2	no	—
369	M₁₆	—	2	2	no	—
370	M₁₆	—	2	2	no	—
371	M₁₆	—	4	4	no	—
372	M₁₆	—	268	4	no	—
373	M₁₆	—	67	4	no	—
374	M₁₆	—	67	2	no	—
375	M₁₆	—	134	2	no	—
376	M₁₆	—	134	1	no	—
377	M₁₆	C₂×C₄×D₆₇	1	2	yes	—
378	M₁₆	C₂×C₂²⋊D₆₇	1	4	yes	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	4	4	no	—
385	C₂×C₈	—	268	4	no	—
386	C₂×C₈	—	134	1	no	—
387	C₂×C₈	—	134	1	no	—
388	C₂×C₈	—	134	1	no	—
389	C₂×C₈	—	134	2	no	—
390	C₂×C₈	—	134	1	no	—
391	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
392	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
393	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
394	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
395	C₂×C₈	#11761628	1	1	yes	characteristic
396	C₂×C₈	#11761628	1	1	yes	characteristic
397	C₄⋊C₄	—	134	1	no	—
398	C₂²⋊C₄	—	134	1	no	—
399	C₄²	—	67	1	no	—
400	C₄²	—	67	1	no	—
401	C₄²	—	134	1	no	—
402	C₄²	—	134	1	no	—
403	C₂³	—	2	1	no	—
404	C₂³	—	134	2	no	—
405	C₂³	#14524091	1	1	yes	characteristic
406	Q₈	—	2	1	no	—
407	Q₈	—	134	2	no	—
408	Q₈	—	134	2	no	—
409	Q₈	#14524005	1	2	yes	—
410	D₄	—	2	2	no	—
411	D₄	—	2	2	no	—
412	D₄	—	2	1	no	—
413	D₄	—	268	2	no	—
414	D₄	—	268	2	no	—
415	D₄	—	134	2	no	—
416	D₄	—	134	2	no	—
417	D₄	#14524005	1	2	yes	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	268	2	no	—
422	C₂×C₄	—	268	2	no	—
423	C₂×C₄	—	268	2	no	—
424	C₂×C₄	—	268	1	no	—
425	C₂×C₄	—	134	1	no	—
426	C₂×C₄	—	134	2	no	—
427	C₂×C₄	—	134	1	no	—
428	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
429	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
430	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
431	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
432	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
433	C₂×C₄	#14524088	1	1	yes	characteristic
434	C₈	—	2	4	no	—
435	C₈	—	2	2	no	—
436	C₈	—	4	2	no	—
437	C₈	—	268	1	no	—
438	C₈	—	134	2	no	—
439	C₈	—	134	4	no	—
440	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
441	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
442	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
443	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
444	C₂²	—	2	2	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	8	1	no	—
447	C₂²	—	268	2	no	—
448	C₂²	—	536	2	no	—
449	C₂²	#16863877	1	1	yes	characteristic
450	C₂²	#16864047	1	1	yes	characteristic
451	C₂²	#16864057	1	1	yes	characteristic
452	C₄	—	2	2	no	—
453	C₄	—	268	1	no	—
454	C₄	—	268	2	no	—
455	C₄	—	536	1	no	—
456	C₄	—	134	2	no	—
457	C₄	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
458	C₄	C₂×#14524005	1	1	yes	characteristic
459	C₄	D₄×C₂²⋊D₆₇	1	1	yes	characteristic
460	C₄	#16864094	1	1	yes	characteristic
461	C₂	—	2	1	no	—
462	C₂	—	2	1	no	—
463	C₂	—	2	1	no	—
464	C₂	—	4	2	no	—
465	C₂	—	536	2	no	—
466	C₂	#22563225	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2916	C₂
socle	C₂×C₆₇	#2361
prime radical	#22569929	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2916	D₆₇
hypocenter	C₆₇	#23042
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2916

Complements

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

N	H ≅ G/N	type
2. #22569125	C₂	semidirect
3. #22569116	C₂	semidirect
4. #22561095	C₂	semidirect
5. #22560997	C₂	semidirect
6. #22560877	C₂	semidirect
7. #22560571	C₂	semidirect
8. #22560427	C₂	semidirect
9. #108×Dic₆₇	C₂	semidirect
10. #22560044	C₂	semidirect
11. C₆₇×#2916	C₂	semidirect
15. #16863671	C₄	semidirect
16. #16863666	C₄	semidirect
17. #16863659	C₄	semidirect
237. #2916	D₆₇	semidirect
245. C₆₇	#23042	semidirect

Maximal subgroups

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

subgroup	index
#22569929	2
#22569125	2
#22569116	2
#22561095	2
#22560997	2
#22560877	2
#22560571	2
#22560427	2
#108×Dic₆₇	2
#22560044	2
C₆₇×#2916	2
#23042	67