Subgroups — #30878695

Order 17152.

820 conjugacy classes of subgroups.

The subgroup lattice has 460 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	#22569127	C₂	1	1	yes	maximal, characteristic, complemented
3	#22569115	C₂	1	2	yes	maximal, complemented
4	#22568820	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561102	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560999	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560879	C₂	1	2	yes	maximal, complemented
8	#22560569	C₂	1	2	yes	maximal, complemented
9	#22560430	C₂	1	2	yes	maximal, complemented
10	#22560115	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1765	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864194	C₂²	1	2	yes	—
13	C₂×#14524065	C₂²	1	1	yes	characteristic
14	#16864163	C₂²	1	2	yes	—
15	#16864162	C₂²	1	2	yes	—
16	#16863671	C₄	1	2	yes	complemented
17	#16863666	C₄	1	2	yes	complemented
18	#16863659	C₄	1	4	yes	complemented
19	#16863519	—	2	2	no	—
20	#16863515	—	2	4	no	—
21	#16863514	—	2	2	no	—
22	#16863510	—	2	4	no	—
23	#16863445	—	2	2	no	—
24	#16863445	C₂²	1	2	yes	—
25	#16863444	C₂²	1	2	yes	—
26	C₂×#14523979	—	2	2	no	—
27	C₂×#14523979	C₂²	1	1	yes	characteristic
28	C₂×#14523979	C₂²	1	1	yes	characteristic
29	C₂×#14523979	C₂²	1	1	yes	characteristic
30	#16863414	C₂²	1	2	yes	—
31	C₂×#14523971	C₂²	1	2	yes	—
32	#16863378	C₂²	1	2	yes	—
33	#16863362	C₂²	1	2	yes	—
34	C₄×#11761612	C₂²	1	2	yes	—
35	#16863166	C₂²	1	2	yes	—
36	#16863161	C₂²	1	2	yes	—
37	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
39	#416×C₆₇	C₂²	1	1	yes	characteristic
40	#405×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
42	#341×C₆₇	C₂²	1	1	yes	characteristic
43	#339×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	#14524050	—	2	4	no	—
59	#14524050	—	4	2	no	—
60	#14524049	—	2	4	no	—
61	C₂×#11761612	—	2	4	no	—
62	C₂×#11761612	C₂³	1	2	yes	—
63	C₂×#11761612	C₂³	1	2	yes	—
64	C₂×#11761612	C₂³	1	2	yes	—
65	M₁₆×D₆₇	—	4	2	no	—
66	#14523979	—	2	4	no	—
67	#14523979	—	2	1	no	—
68	#14523979	—	2	1	no	—
69	#14523979	C₂×C₄	1	2	yes	—
70	#14523979	C₂×C₄	1	2	yes	—
71	#14523979	D₄	1	2	yes	—
72	#14523979	D₄	1	2	yes	—
73	#14523978	—	2	4	no	—
74	#14523971	—	2	4	no	—
75	#14523971	—	2	2	no	—
76	#14523971	—	2	2	no	—
77	#14523961	C₂×C₄	1	4	yes	—
78	#14523959	C₂×C₄	1	4	yes	—
79	(C₂×D₆₇)⋊C₈	—	4	2	no	—
80	#14523954	—	2	2	no	—
81	C₄×C₆₇⋊C₈	—	2	2	no	—
82	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
83	#131×C₆₇	—	2	1	no	—
84	#131×C₆₇	—	2	1	no	—
85	#131×C₆₇	C₂×C₄	1	2	yes	—
86	#131×C₆₇	C₂×C₄	1	2	yes	—
87	C₂×M₁₆×C₆₇	—	2	1	no	—
88	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
90	C₂²×C₈×C₆₇	—	2	1	no	—
91	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	#117×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	#104×C₆₇	D₄	1	4	yes	—
96	#103×C₆₇	—	2	1	no	—
97	#102×C₆₇	—	2	1	no	—
98	#97×C₆₇	—	2	1	no	—
99	#97×C₆₇	—	2	1	no	—
100	#97×C₆₇	—	2	1	no	—
101	C₄×C₈×C₆₇	—	2	1	no	—
102	#11761628	—	2	2	no	—
103	#11761628	—	2	2	no	—
104	#11761628	—	2	2	no	—
105	#11761628	—	2	2	no	—
106	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
107	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
108	C₂×C₂²⋊D₆₇	—	2	2	no	—
109	C₂×C₄×D₆₇	—	2	2	no	—
110	Dic₂₆₈	—	2	1	no	—
111	Dic₂₆₈	—	2	1	no	—
112	#11761618	—	2	4	no	—
113	#11761617	—	2	2	no	—
114	#11761617	—	2	2	no	—
115	#11761616	—	2	4	no	—
116	#11761615	—	2	4	no	—
117	D₅₃₆	—	2	1	no	—
118	D₅₃₆	—	2	1	no	—
119	#11761613	—	2	4	no	—
120	#11761612	—	2	2	no	—
121	#11761612	—	2	2	no	—
122	#11761612	—	2	2	no	—
123	#11761612	—	2	2	no	—
124	#11761612	—	4	4	no	—
125	#11761612	C₂²×C₄	1	4	yes	—
126	#11761612	C₂×D₄	1	4	yes	—
127	#11761611	—	4	4	no	—
128	C₂×C₆₇⋊C₈	—	2	2	no	—
129	C₂×C₆₇⋊C₈	—	2	2	no	—
130	C₂×C₆₇⋊C₈	—	2	2	no	—
131	C₂×C₆₇⋊C₈	—	4	4	no	—
132	C₈×D₆₇	—	4	4	no	—
133	#41×C₆₇	—	2	1	no	—
134	#41×C₆₇	—	2	1	no	—
135	#41×C₆₇	C₂²×C₄	1	2	yes	—
136	#41×C₆₇	C₂×D₄	1	2	yes	—
137	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
138	C₂×D₄×C₆₇	—	2	1	no	—
139	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
143	M₁₆×C₆₇	—	2	1	no	—
144	M₁₆×C₆₇	—	2	1	no	—
145	M₁₆×C₆₇	—	2	2	no	—
146	M₁₆×C₆₇	—	2	2	no	—
147	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
148	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
149	C₂×C₈×C₆₇	—	2	1	no	—
150	C₂×C₈×C₆₇	—	2	1	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	—	2	1	no	—
155	C₂×C₈×C₆₇	—	2	1	no	—
156	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
159	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
161	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
162	C₄⋊C₄×C₆₇	—	2	1	no	—
163	C₂²⋊C₄×C₆₇	—	2	1	no	—
164	C₄²×C₆₇	—	2	1	no	—
165	C₄²×C₆₇	—	2	1	no	—
166	C₄²×C₆₇	C₂×D₄	1	2	yes	—
167	C₂³×C₆₇	—	2	1	no	—
168	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
169	C₂²×D₆₇	—	2	2	no	—
170	Q₈×C₆₇	—	2	1	no	—
171	Q₈×C₆₇	C₄×D₄	1	2	yes	—
172	D₄×C₆₇	—	2	2	no	—
173	D₄×C₆₇	—	2	2	no	—
174	D₄×C₆₇	—	2	1	no	—
175	D₄×C₆₇	C₄×D₄	1	2	yes	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	8	1	no	—
182	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂×#41	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	C₂²⋊D₆₇	—	4	2	no	—
189	C₂²⋊D₆₇	—	4	2	no	—
190	C₂×Dic₆₇	—	2	2	no	—
191	D₂₆₈	—	2	2	no	—
192	D₂₆₈	—	2	2	no	—
193	C₄×D₆₇	—	4	2	no	—
194	C₄×D₆₇	—	4	2	no	—
195	Dic₁₃₄	—	2	2	no	—
196	Dic₁₃₄	—	2	2	no	—
197	C₈×C₆₇	—	2	2	no	—
198	C₈×C₆₇	—	2	2	no	—
199	C₈×C₆₇	—	4	1	no	—
200	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
201	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
202	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
203	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
204	C₆₇⋊C₈	—	2	4	no	—
205	C₆₇⋊C₈	—	2	4	no	—
206	C₆₇⋊C₈	—	4	2	no	—
207	C₂²×C₆₇	—	2	2	no	—
208	C₂²×C₆₇	—	4	1	no	—
209	C₂²×C₆₇	—	8	1	no	—
210	C₂²×C₆₇	#520	1	1	yes	characteristic
211	C₂²×C₆₇	#547	1	1	yes	characteristic
212	C₂²×C₆₇	#548	1	1	yes	characteristic
213	C₂×D₆₇	—	4	2	no	—
214	C₂×D₆₇	—	8	2	no	—
215	C₄×C₆₇	—	2	2	no	—
216	C₄×C₆₇	—	4	2	no	—
217	C₄×C₆₇	—	8	1	no	—
218	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
219	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
220	C₄×C₆₇	D₄²	1	1	yes	characteristic
221	C₄×C₆₇	#553	1	1	yes	characteristic
222	Dic₆₇	—	4	2	no	—
223	#23043	—	67	1	no	maximal, Sylow
224	C₂×C₆₇	—	2	1	no	—
225	C₂×C₆₇	—	2	1	no	—
226	C₂×C₆₇	—	2	1	no	—
227	C₂×C₆₇	—	4	2	no	—
228	C₂×C₆₇	#2361	1	1	yes	characteristic
229	D₆₇	—	8	2	no	—
230	#3067	—	67	1	no	—
231	#2957	—	67	1	no	—
232	#2956	—	67	2	no	—
233	#2916	—	67	1	no	—
234	#1975	—	67	1	no	—
235	#1953	—	67	1	no	—
236	#1932	—	67	2	no	—
237	#1881	—	67	2	no	—
238	#1858	—	67	2	no	—
239	#1778	—	67	1	no	—
240	#1765	D₆₇	1	1	yes	characteristic, complemented
241	C₆₇	#23043	1	1	yes	characteristic, complemented, Sylow
242	#575	—	67	2	no	—
243	C₂×#135	—	67	1	no	—
244	#568	—	67	2	no	—
245	#568	—	67	2	no	—
246	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
247	#473	—	67	2	no	—
248	#472	—	67	2	no	—
249	#471	—	67	4	no	—
250	#444	—	134	2	no	—
251	#444	—	134	4	no	—
252	#443	—	134	2	no	—
253	#443	—	134	4	no	—
254	#430	—	67	2	no	—
255	#430	—	67	2	no	—
256	#430	—	134	2	no	—
257	C₂×#108	—	67	1	no	—
258	C₂×#108	—	67	1	no	—
259	C₂×#108	—	67	1	no	—
260	C₂×#108	—	134	2	no	—
261	#421	—	67	2	no	—
262	C₂×#104	—	67	2	no	—
263	C₂×#104	C₂×D₆₇	1	1	yes	characteristic
264	#416	C₂×D₆₇	1	1	yes	characteristic
265	#413	—	67	2	no	—
266	#408	—	67	2	no	—
267	#405	C₂×D₆₇	1	1	yes	characteristic
268	C₄×M₁₆	—	67	2	no	—
269	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
270	#344	—	67	2	no	—
271	#341	—	67	2	no	—
272	#341	C₂×D₆₇	1	1	yes	characteristic
273	#339	C₂×D₆₇	1	1	yes	characteristic
274	C₂×#41	—	67	2	no	—
275	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
276	#137	—	67	4	no	—
277	#137	—	134	2	no	—
278	AGL(1,ℤ/8)	—	67	4	no	—
279	AGL(1,ℤ/8)	—	134	2	no	—
280	#135	—	134	4	no	—
281	#135	—	134	1	no	—
282	#135	—	134	1	no	—
283	#135	—	134	1	no	—
284	#135	—	134	1	no	—
285	C₂×Q₁₆	—	67	1	no	—
286	C₂×SD₁₆	—	67	2	no	—
287	C₂×D₈	—	67	1	no	—
288	#131	—	2	1	no	—
289	#131	—	2	1	no	—
290	#131	—	268	2	no	—
291	#131	—	134	4	no	—
292	#131	—	134	4	no	—
293	#131	—	134	4	no	—
294	#131	C₄×D₆₇	1	2	yes	—
295	#131	C₄×D₆₇	1	2	yes	—
296	C₂×M₁₆	—	2	1	no	—
297	C₂×M₁₆	—	268	2	no	—
298	C₂×M₁₆	—	67	2	no	—
299	C₂×M₁₆	—	67	2	no	—
300	C₂×M₁₆	—	67	2	no	—
301	C₂×M₁₆	—	134	4	no	—
302	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
303	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
304	C₂²×C₈	—	2	1	no	—
305	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
306	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
307	#117	C₂²×D₆₇	1	1	yes	characteristic
308	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
309	#108	—	67	2	no	—
310	#108	—	67	2	no	—
311	#108	—	67	2	no	—
312	#108	—	67	2	no	—
313	#108	—	134	4	no	—
314	#108	—	134	4	no	—
315	#108	—	134	1	no	—
316	#108	—	134	1	no	—
317	#104	—	134	4	no	—
318	#104	—	134	2	no	—
319	#104	—	134	2	no	—
320	#104	D₂₆₈	1	4	yes	—
321	#103	—	2	1	no	—
322	#102	—	2	1	no	—
323	#101	—	67	4	no	—
324	#100	—	67	4	no	—
325	C₂²⋊C₈	—	268	2	no	—
326	#97	—	2	1	no	—
327	#97	—	2	1	no	—
328	#97	—	2	1	no	—
329	#97	—	134	2	no	—
330	C₄×C₈	—	2	1	no	—
331	C₄×C₈	—	134	2	no	—
332	#41	—	2	1	no	—
333	#41	—	2	1	no	—
334	#41	—	134	2	no	—
335	#41	—	134	2	no	—
336	#41	—	134	2	no	—
337	#41	—	134	2	no	—
338	#41	C₂×C₄×D₆₇	1	2	yes	—
339	#41	C₂×D₂₆₈	1	2	yes	—
340	C₂×Q₈	—	67	2	no	—
341	C₂×Q₈	#11761628	1	1	yes	characteristic
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	67	2	no	—
344	C₂×D₄	—	134	2	no	—
345	C₂×D₄	#11761628	1	1	yes	characteristic
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	134	2	no	—
349	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
350	Q₁₆	—	134	4	no	—
351	Q₁₆	—	134	1	no	—
352	Q₁₆	—	134	1	no	—
353	SD₁₆	—	134	4	no	—
354	SD₁₆	—	134	4	no	—
355	SD₁₆	—	134	2	no	—
356	SD₁₆	—	134	2	no	—
357	D₈	—	134	4	no	—
358	D₈	—	134	1	no	—
359	D₈	—	134	1	no	—
360	M₁₆	—	2	1	no	—
361	M₁₆	—	2	1	no	—
362	M₁₆	—	2	2	no	—
363	M₁₆	—	2	2	no	—
364	M₁₆	—	268	4	no	—
365	M₁₆	—	268	4	no	—
366	M₁₆	—	67	4	no	—
367	M₁₆	—	67	4	no	—
368	M₁₆	—	134	2	no	—
369	M₁₆	—	134	2	no	—
370	M₁₆	—	134	2	no	—
371	M₁₆	—	134	2	no	—
372	M₁₆	C₂×C₄×D₆₇	1	2	yes	—
373	M₁₆	C₂×D₂₆₈	1	2	yes	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	268	4	no	—
382	C₂×C₈	—	268	4	no	—
383	C₂×C₈	—	134	2	no	—
384	C₂×C₈	—	134	2	no	—
385	C₂×C₈	—	134	2	no	—
386	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
387	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
388	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
389	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
390	C₂×C₈	#11761627	1	1	yes	characteristic
391	C₂×C₈	#11761627	1	1	yes	characteristic
392	C₄⋊C₄	—	2	1	no	—
393	C₂²⋊C₄	—	2	1	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	C₂×D₂₆₈	1	2	yes	—
397	C₂³	—	2	1	no	—
398	C₂³	—	134	2	no	—
399	C₂³	#14524090	1	1	yes	characteristic
400	Q₈	—	2	1	no	—
401	Q₈	—	134	2	no	—
402	Q₈	—	134	2	no	—
403	Q₈	C₄×D₂₆₈	1	2	yes	—
404	D₄	—	2	2	no	—
405	D₄	—	2	2	no	—
406	D₄	—	2	1	no	—
407	D₄	—	268	2	no	—
408	D₄	—	268	2	no	—
409	D₄	—	134	2	no	—
410	D₄	—	134	2	no	—
411	D₄	C₄×D₂₆₈	1	2	yes	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	8	1	no	—
418	C₂×C₄	—	268	2	no	—
419	C₂×C₄	—	268	2	no	—
420	C₂×C₄	—	134	2	no	—
421	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
422	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
423	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
424	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
425	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
426	C₂×C₄	#14524089	1	1	yes	characteristic
427	C₈	—	2	2	no	—
428	C₈	—	2	2	no	—
429	C₈	—	4	1	no	—
430	C₈	—	268	2	no	—
431	C₈	—	134	4	no	—
432	C₈	—	134	4	no	—
433	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
434	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
435	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
436	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
437	C₂²	—	2	2	no	—
438	C₂²	—	4	1	no	—
439	C₂²	—	8	1	no	—
440	C₂²	—	268	2	no	—
441	C₂²	—	536	2	no	—
442	C₂²	#16863876	1	1	yes	characteristic
443	C₂²	#16864050	1	1	yes	characteristic
444	C₂²	#16864055	1	1	yes	characteristic
445	C₄	—	2	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	8	1	no	—
448	C₄	—	268	2	no	—
449	C₄	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
450	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
451	C₄	D₄×D₂₆₈	1	1	yes	characteristic
452	C₄	#16864092	1	1	yes	characteristic
453	C₂	—	2	1	no	—
454	C₂	—	2	1	no	—
455	C₂	—	2	1	no	—
456	C₂	—	4	2	no	—
457	C₂	—	536	2	no	—
458	C₂	#22563223	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1765	C₂
socle	C₂×C₆₇	#2361
prime radical	#22569929	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1765	D₆₇
hypocenter	C₆₇	#23043
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1765

Complements

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

N	H ≅ G/N	type
2. #22569127	C₂	semidirect
3. #22569115	C₂	semidirect
4. #22568820	C₂	semidirect
5. #22561102	C₂	semidirect
6. #22560999	C₂	semidirect
7. #22560879	C₂	semidirect
8. #22560569	C₂	semidirect
9. #22560430	C₂	semidirect
10. #22560115	C₂	semidirect
11. C₆₇×#1765	C₂	semidirect
16. #16863671	C₄	semidirect
17. #16863666	C₄	semidirect
18. #16863659	C₄	semidirect
240. #1765	D₆₇	semidirect
241. C₆₇	#23043	semidirect

Maximal subgroups

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

subgroup	index
#22569929	2
#22569127	2
#22569115	2
#22568820	2
#22561102	2
#22560999	2
#22560879	2
#22560569	2
#22560430	2
#22560115	2
C₆₇×#1765	2
#23043	67