Subgroups — #30839776

Order 17152.

608 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570723	C₂	1	1	yes	maximal, characteristic, complemented
2	#22570371	C₂	1	2	yes	maximal
3	#22559613	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559608	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559603	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559598	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559591	C₂	1	2	yes	maximal, complemented
8	#22559584	C₂	1	2	yes	maximal, complemented
9	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558917	C₂	1	2	yes	maximal, complemented
11	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
12	#16863953	C₂²	1	2	yes	—
13	#16863801	C₂²	1	1	yes	characteristic
14	#16863801	C₂²	1	1	yes	characteristic
15	#16863798	—	2	1	no	—
16	#16863798	—	2	1	no	—
17	#16863794	C₂²	1	2	yes	—
18	#16863737	—	2	2	no	—
19	#16863691	C₂²	1	2	yes	—
20	#16863686	C₂²	1	2	yes	—
21	#16863606	C₂²	1	2	yes	—
22	#16863601	C₂²	1	2	yes	—
23	#16863558	—	2	2	no	—
24	#16863501	—	2	2	no	—
25	#16863496	—	2	2	no	—
26	#16863453	C₂²	1	2	yes	—
27	#16863434	—	2	1	no	—
28	#16863434	—	2	1	no	—
29	#16863149	C₂²	1	1	yes	characteristic
30	#16863149	C₂²	1	1	yes	characteristic
31	#16863147	C₂²	1	2	yes	—
32	#16863144	C₂²	1	2	yes	—
33	#16863144	C₂²	1	2	yes	—
34	#16863141	C₂²	1	2	yes	—
35	#16863141	C₂²	1	2	yes	—
36	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
37	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₆₇	—	2	1	no	—
40	C₈×D₄×C₆₇	—	2	1	no	—
41	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
42	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
43	#431×C₆₇	—	2	1	no	—
44	#431×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄×C₈×C₆₇	—	2	1	no	—
46	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#11761618	—	2	2	no	—
48	C₂×#11761616	—	2	2	no	—
49	#14524043	C₂³	1	2	yes	—
50	#14524043	C₂³	1	2	yes	—
51	#14524035	—	2	2	no	—
52	#14524022	—	2	2	no	—
53	#14524022	—	2	2	no	—
54	C₄×Dic₁₃₄	—	2	2	no	—
55	C₄×Dic₁₃₄	—	2	2	no	—
56	#14524002	—	2	2	no	—
57	#14523977	—	2	1	no	—
58	#14523977	—	2	1	no	—
59	#14523977	—	2	1	no	—
60	#14523977	—	2	1	no	—
61	#14523976	—	2	2	no	—
62	#14523975	—	2	2	no	—
63	#14523975	—	2	2	no	—
64	#14523974	—	2	2	no	—
65	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
66	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
67	#14523967	—	2	2	no	—
68	#14523966	—	2	2	no	—
69	#14523966	—	2	2	no	—
70	#14523964	—	2	2	no	—
71	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
72	#14523954	—	2	2	no	—
73	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	—	2	1	no	—
75	C₂²×C₈×C₆₇	—	2	1	no	—
76	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
78	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
79	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×D₄×C₆₇	—	2	1	no	—
81	C₄×D₄×C₆₇	—	2	1	no	—
82	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
83	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
84	#117×C₆₇	—	2	1	no	—
85	#117×C₆₇	D₄	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	—	2	1	no	—
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₆₇	—	2	1	no	—
89	C₄⋊C₈×C₆₇	—	2	1	no	—
90	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂²⋊C₈×C₆₇	—	2	1	no	—
93	C₂²⋊C₈×C₆₇	—	2	1	no	—
94	C₂²⋊C₈×C₆₇	—	2	1	no	—
95	C₄×C₈×C₆₇	—	2	1	no	—
96	C₄×C₈×C₆₇	—	2	1	no	—
97	C₄×C₈×C₆₇	—	2	1	no	—
98	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
99	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
100	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×Dic₁₃₄	—	2	2	no	—
103	C₂×Dic₁₃₄	—	2	2	no	—
104	#11761618	—	4	4	no	—
105	#11761616	—	4	4	no	—
106	C₂×C₆₇⋊C₈	—	2	2	no	—
107	C₂×C₆₇⋊C₈	—	2	2	no	—
108	C₄⋊Dic₆₇	—	2	2	no	—
109	C₄⋊Dic₆₇	—	2	2	no	—
110	C₄⋊Dic₆₇	—	2	2	no	—
111	C₄⋊Dic₆₇	—	2	2	no	—
112	Dic₆₇⋊C₄	—	4	2	no	—
113	Dic₆₇⋊C₄	—	4	2	no	—
114	C₂²⋊Dic₆₇	—	4	2	no	—
115	C₄×Dic₆₇	—	4	2	no	—
116	#41×C₆₇	—	2	2	no	—
117	#41×C₆₇	—	2	2	no	—
118	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×D₄×C₆₇	—	2	1	no	—
120	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	—	2	1	no	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	—	2	1	no	—
131	C₂×C₈×C₆₇	—	4	2	no	—
132	C₂×C₈×C₆₇	—	4	2	no	—
133	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₆₇	—	2	1	no	—
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂²⋊C₄×C₆₇	—	2	1	no	—
142	C₂²⋊C₄×C₆₇	—	2	1	no	—
143	C₂²⋊C₄×C₆₇	—	2	1	no	—
144	C₄²×C₆₇	—	2	1	no	—
145	C₄²×C₆₇	—	2	1	no	—
146	C₄²×C₆₇	—	2	1	no	—
147	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₆₇	#41	1	1	yes	characteristic
150	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
151	C₂³×C₆₇	—	2	1	no	—
152	C₂³×C₆₇	#121	1	1	yes	characteristic
153	Q₈×C₆₇	—	2	1	no	—
154	Q₈×C₆₇	#135	1	2	yes	—
155	D₄×C₆₇	—	2	4	no	—
156	D₄×C₆₇	—	2	1	no	—
157	D₄×C₆₇	#135	1	2	yes	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
171	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
172	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
176	C₂×Dic₆₇	—	4	2	no	—
177	C₂×Dic₆₇	—	4	2	no	—
178	Dic₁₃₄	—	4	2	no	—
179	Dic₁₃₄	—	4	2	no	—
180	Dic₁₃₄	—	8	2	no	—
181	C₈×C₆₇	—	2	1	no	—
182	C₈×C₆₇	—	2	1	no	—
183	C₈×C₆₇	—	4	1	no	—
184	C₈×C₆₇	—	4	1	no	—
185	C₈×C₆₇	#137	1	2	yes	—
186	C₈×C₆₇	#137	1	2	yes	—
187	C₆₇⋊C₈	—	4	2	no	—
188	C₆₇⋊C₈	—	4	2	no	—
189	C₂²×C₆₇	—	2	1	no	—
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	2	2	no	—
192	C₂²×C₆₇	—	2	1	no	—
193	C₂²×C₆₇	—	4	1	no	—
194	C₂²×C₆₇	—	4	1	no	—
195	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
196	C₄×C₆₇	—	2	1	no	—
197	C₄×C₆₇	—	2	1	no	—
198	C₄×C₆₇	—	2	2	no	—
199	C₄×C₆₇	—	2	1	no	—
200	C₄×C₆₇	—	2	1	no	—
201	C₄×C₆₇	—	4	1	no	—
202	C₄×C₆₇	—	4	1	no	—
203	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
204	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
205	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
206	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
207	Dic₆₇	—	8	2	no	—
208	Dic₆₇	—	8	2	no	—
209	#19273	—	67	1	no	maximal, Sylow
210	C₂×C₆₇	—	4	1	no	—
211	C₂×C₆₇	—	4	2	no	—
212	C₂×C₆₇	#3039	1	1	yes	characteristic
213	C₂×C₆₇	#3050	1	1	yes	characteristic
214	C₂×C₆₇	#3067	1	1	yes	characteristic
215	#3160	—	67	1	no	—
216	#3116	—	67	2	no	—
217	C₈×#41	D₆₇	1	1	yes	characteristic
218	#1676	—	67	1	no	—
219	#1675	—	67	1	no	—
220	#1674	—	67	1	no	—
221	#1673	—	67	1	no	—
222	#1672	—	67	2	no	—
223	#1671	—	67	2	no	—
224	#1568	—	67	1	no	—
225	#1554	—	67	2	no	—
226	C₆₇	#19273	1	1	yes	characteristic, complemented, Sylow
227	#533	—	67	2	no	—
228	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
229	#500	—	67	1	no	—
230	#500	—	67	1	no	—
231	#499	—	134	1	no	—
232	#499	—	134	1	no	—
233	#498	—	67	2	no	—
234	#485	—	134	2	no	—
235	#477	—	67	2	no	—
236	#476	—	67	2	no	—
237	#462	—	67	2	no	—
238	#461	—	67	2	no	—
239	#452	—	134	2	no	—
240	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
241	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
242	#441	—	134	2	no	—
243	#440	—	134	2	no	—
244	C₈×D₄	—	2	1	no	—
245	C₈×D₄	—	2	1	no	—
246	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
247	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
248	#432	—	67	2	no	—
249	#431	—	2	1	no	—
250	#431	C₂×D₆₇	1	1	yes	characteristic
251	#427	—	134	1	no	—
252	#427	—	134	1	no	—
253	C₂×C₄×C₈	—	2	1	no	—
254	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
255	#335	—	67	1	no	—
256	#335	—	67	1	no	—
257	#334	—	67	2	no	—
258	#333	—	67	2	no	—
259	#333	—	67	2	no	—
260	#332	—	67	2	no	—
261	#332	—	67	2	no	—
262	C₂×#41	D₂₆₈	1	1	yes	characteristic
263	C₂×Q₁₆	—	134	2	no	—
264	C₂×SD₁₆	—	134	2	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
268	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
269	C₄⋊Q₈	—	67	2	no	—
270	C₄⋊Q₈	—	67	2	no	—
271	#125	—	134	2	no	—
272	C₂²⋊Q₈	—	134	2	no	—
273	C₂²⋊Q₈	—	134	2	no	—
274	C₄×Q₈	—	134	2	no	—
275	C₄×Q₈	—	134	2	no	—
276	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
277	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
278	C₄×D₄	—	2	1	no	—
279	C₄×D₄	—	2	1	no	—
280	C₄×D₄	D₂₆₈	1	1	yes	characteristic
281	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
282	#117	—	2	1	no	—
283	#117	—	134	2	no	—
284	#117	D₂₆₈	1	1	yes	characteristic
285	C₂×C₄²	—	2	1	no	—
286	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
287	#107	—	134	1	no	—
288	#107	—	134	1	no	—
289	#107	—	134	1	no	—
290	#107	—	134	1	no	—
291	#107	—	134	2	no	—
292	#106	—	134	2	no	—
293	#106	—	134	2	no	—
294	#106	—	134	2	no	—
295	C₄⋊C₈	—	2	1	no	—
296	C₄⋊C₈	—	2	1	no	—
297	C₄⋊C₈	—	67	2	no	—
298	C₄⋊C₈	—	67	2	no	—
299	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
300	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
301	#103	—	134	2	no	—
302	#103	—	134	2	no	—
303	#103	—	134	2	no	—
304	#102	—	134	2	no	—
305	C₂²⋊C₈	—	2	1	no	—
306	C₂²⋊C₈	—	2	1	no	—
307	C₂²⋊C₈	—	2	1	no	—
308	C₂²⋊C₈	—	134	2	no	—
309	#97	—	134	2	no	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	—	2	1	no	—
312	C₄×C₈	—	2	1	no	—
313	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
314	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
315	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
316	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
317	#41	—	2	2	no	—
318	#41	—	2	2	no	—
319	C₂×Q₈	—	134	2	no	—
320	C₂×Q₈	—	134	2	no	—
321	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	C₂×D₂₆₈	1	1	yes	characteristic
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
328	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
329	C₂²×C₄	#11761628	1	1	yes	characteristic
330	Q₁₆	—	268	4	no	—
331	SD₁₆	—	268	4	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	4	2	no	—
337	C₂×C₈	—	4	2	no	—
338	C₂×C₈	—	134	2	no	—
339	C₂×C₈	—	134	2	no	—
340	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
341	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
342	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
343	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	268	2	no	—
347	C₄⋊C₄	—	268	2	no	—
348	C₄⋊C₄	—	134	2	no	—
349	C₄⋊C₄	—	134	2	no	—
350	C₄⋊C₄	—	134	2	no	—
351	C₄⋊C₄	—	134	2	no	—
352	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
353	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
354	C₂²⋊C₄	—	2	1	no	—
355	C₂²⋊C₄	—	2	1	no	—
356	C₂²⋊C₄	—	2	1	no	—
357	C₂²⋊C₄	—	268	2	no	—
358	C₄²	—	2	1	no	—
359	C₄²	—	2	1	no	—
360	C₄²	—	2	1	no	—
361	C₄²	—	268	2	no	—
362	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
363	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
364	C₄²	#11761628	1	1	yes	characteristic
365	C₄²	C₂³×D₆₇	1	1	yes	characteristic
366	C₂³	—	2	1	no	—
367	C₂³	#14524017	1	1	yes	characteristic
368	Q₈	—	2	1	no	—
369	Q₈	—	268	2	no	—
370	Q₈	—	268	2	no	—
371	Q₈	—	536	2	no	—
372	Q₈	#14524065	1	2	yes	—
373	D₄	—	2	4	no	—
374	D₄	—	2	1	no	—
375	D₄	#14524065	1	2	yes	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	268	2	no	—
389	C₂×C₄	—	268	2	no	—
390	C₂×C₄	#14524017	1	1	yes	characteristic
391	C₂×C₄	#14524017	1	1	yes	characteristic
392	C₂×C₄	#14524017	1	1	yes	characteristic
393	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
394	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
395	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
396	C₈	—	2	1	no	—
397	C₈	—	2	1	no	—
398	C₈	—	4	1	no	—
399	C₈	—	4	1	no	—
400	C₈	—	268	2	no	—
401	C₈	—	268	2	no	—
402	C₈	#14524073	1	2	yes	—
403	C₈	#14524073	1	2	yes	—
404	C₂²	—	2	1	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	4	1	no	—
409	C₂²	—	4	1	no	—
410	C₂²	C₂×#14524017	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	2	no	—
413	C₄	—	2	1	no	—
414	C₄	—	2	1	no	—
415	C₄	—	2	1	no	—
416	C₄	—	4	1	no	—
417	C₄	—	4	1	no	—
418	C₄	—	536	2	no	—
419	C₄	—	536	2	no	—
420	C₄	C₂×#14524065	1	1	yes	characteristic
421	C₄	C₂×#14524065	1	1	yes	characteristic
422	C₄	C₂×#14524073	1	1	yes	characteristic
423	C₄	C₂×#14524073	1	1	yes	characteristic
424	C₂	—	4	1	no	—
425	C₂	—	4	2	no	—
426	C₂	#22569696	1	1	yes	characteristic
427	C₂	#22569790	1	1	yes	characteristic
428	C₂	#22569929	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524017
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×#41×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂×#41×C₆₇	D₄
hypercenter	C₈×#41	D₆₇
hypocenter	C₆₇	#19273
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #22570723	C₂	semidirect
3. #22559613	C₂	semidirect
4. #22559608	C₂	semidirect
5. #22559603	C₂	semidirect
6. #22559598	C₂	semidirect
7. #22559591	C₂	semidirect
8. #22559584	C₂	semidirect
9. #22558972	C₂	semidirect
10. #22558917	C₂	semidirect
226. C₆₇	#19273	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
#22570371	2
#22559613	2
#22559608	2
#22559603	2
#22559598	2
#22559591	2
#22559584	2
#22558972	2
#22558917	2
C₈×#41×C₆₇	2
#19273	67