Subgroups — #30821556

Order 17152.

814 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569175	C₂	1	1	yes	maximal, characteristic
2	#22569170	C₂	1	1	yes	maximal, characteristic
3	C₄×#14524051	C₂	1	1	yes	maximal, characteristic
4	#22558513	C₂	1	2	yes	maximal, complemented
5	#22558505	C₂	1	2	yes	maximal, complemented
6	#22558496	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558493	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558488	C₂	1	2	yes	maximal, complemented
9	#22558488	C₂	1	2	yes	maximal, complemented
10	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524051	C₄	1	2	yes	—
13	C₄×#11761628	C₂²	1	1	yes	characteristic
14	#16863526	C₄	1	2	yes	—
15	#16863523	C₄	1	2	yes	—
16	#16863475	C₄	1	2	yes	—
17	#16863473	C₄	1	2	yes	—
18	#16863467	C₄	1	4	yes	—
19	#16863464	C₄	1	4	yes	—
20	#16863449	C₂²	1	1	yes	characteristic
21	#16863447	C₂²	1	2	yes	—
22	#16863447	C₂²	1	2	yes	—
23	#16863354	C₄	1	2	yes	—
24	#16863353	C₄	1	4	yes	—
25	C₄×#11761611	C₂²	1	2	yes	—
26	C₄×#11761611	C₂²	1	2	yes	—
27	C₄×#11761611	C₂²	1	2	yes	—
28	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₆₇	C₂²	1	2	yes	—
31	#16863118	C₂²	1	2	yes	—
32	#16863118	C₂²	1	2	yes	—
33	#16863118	C₂²	1	2	yes	—
34	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
36	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
37	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
39	#431×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
42	#322×C₆₇	C₂²	1	2	yes	—
43	#322×C₆₇	C₂²	1	2	yes	—
44	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
45	#14524051	—	2	8	no	—
46	C₂×#11761611	C₂×C₄	1	4	yes	—
47	C₂×#11761611	C₂×C₄	1	4	yes	—
48	C₂×#11761611	C₂×C₄	1	4	yes	—
49	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
50	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
52	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
54	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
55	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
57	C₄×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
58	#14524002	C₂×C₄	1	1	yes	characteristic
59	#14524000	C₂×C₄	1	2	yes	—
60	C₄²×D₆₇	C₂³	1	2	yes	—
61	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
62	Dic₆₇⋊C₈	C₂×C₄	1	4	yes	—
63	Dic₆₇⋊C₈	C₂×C₄	1	4	yes	—
64	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
65	(C₂×D₆₇)⋊C₈	C₂×C₄	1	4	yes	—
66	(C₂×D₆₇)⋊C₈	C₂×C₄	1	4	yes	—
67	#14523953	C₂×C₄	1	4	yes	—
68	#14523953	C₂×C₄	1	4	yes	—
69	#14523953	C₂×C₄	1	4	yes	—
70	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
71	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
72	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
77	C₈×Dic₆₇	C₂×C₄	1	4	yes	—
78	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
79	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
82	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
83	#97×C₆₇	C₂×C₄	1	2	yes	—
84	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₆₇	C₂³	1	2	yes	—
86	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₆₇	C₂³	1	2	yes	—
88	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	#11761628	—	2	4	no	—
90	C₂×Dic₁₃₄	C₄²	1	1	yes	characteristic
91	C₂×D₂₆₈	C₄²	1	1	yes	characteristic
92	C₂×C₂²⋊D₆₇	C₄²	1	2	yes	—
93	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
96	#11761611	—	2	8	no	—
97	#11761611	—	2	8	no	—
98	#11761611	—	2	8	no	—
99	C₂×C₆₇⋊C₈	—	2	4	no	—
100	C₂×C₆₇⋊C₈	M₁₆	1	8	yes	complemented
101	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
103	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
106	C₈×D₆₇	—	2	8	no	—
107	C₄⋊Dic₆₇	C₄²	1	1	yes	characteristic
108	C₄⋊Dic₆₇	C₄²	1	1	yes	characteristic
109	Dic₆₇⋊C₄	C₄²	1	2	yes	—
110	Dic₆₇⋊C₄	C₄²	1	2	yes	—
111	C₂²⋊Dic₆₇	C₄²	1	2	yes	—
112	(C₂×D₆₇)⋊C₄	C₄²	1	2	yes	—
113	(C₂×D₆₇)⋊C₄	C₄²	1	2	yes	—
114	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
115	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
116	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	M₁₆×C₆₇	—	2	4	no	—
121	C₂×C₈×C₆₇	—	2	4	no	—
122	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
123	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
125	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
126	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
130	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
131	C₂²×D₆₇	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
138	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
141	C₂²⋊D₆₇	—	2	4	no	—
142	C₂×Dic₆₇	C₂×C₄²	1	2	yes	—
143	C₂×Dic₆₇	C₂×C₄²	1	2	yes	—
144	C₂×Dic₆₇	C₂×C₄²	1	2	yes	—
145	D₂₆₈	—	2	2	no	—
146	C₄×D₆₇	—	2	4	no	—
147	C₄×D₆₇	—	2	4	no	—
148	C₄×D₆₇	—	2	4	no	—
149	Dic₁₃₄	—	2	2	no	—
150	C₈×C₆₇	—	2	4	no	—
151	C₈×C₆₇	—	2	2	no	—
152	C₈×C₆₇	—	2	4	no	—
153	C₈×C₆₇	#131	1	4	yes	complemented
154	C₆₇⋊C₈	—	2	2	no	—
155	C₆₇⋊C₈	—	2	4	no	—
156	C₆₇⋊C₈	C₂×M₁₆	1	4	yes	—
157	C₆₇⋊C₈	C₂×M₁₆	1	4	yes	—
158	C₆₇⋊C₈	#131	1	4	yes	complemented
159	C₂²×C₆₇	—	2	1	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
162	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
163	C₂×D₆₇	—	2	2	no	—
164	C₂×D₆₇	—	2	2	no	—
165	C₂×D₆₇	—	2	2	no	—
166	C₄×C₆₇	—	2	1	no	—
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
169	C₄×C₆₇	C₂²×M₁₆	1	2	yes	—
170	C₄×C₆₇	C₂×#131	1	2	yes	—
171	C₄×C₆₇	C₂×#131	1	2	yes	—
172	Dic₆₇	—	2	2	no	—
173	Dic₆₇	—	2	2	no	—
174	Dic₆₇	—	2	2	no	—
175	#17335	—	67	1	no	maximal, Sylow
176	C₂×C₆₇	—	2	2	no	—
177	C₂×C₆₇	C₂×C₄×M₁₆	1	1	yes	characteristic
178	C₂×C₆₇	C₄×#131	1	1	yes	characteristic
179	C₂×C₆₇	C₄×#131	1	1	yes	characteristic
180	D₆₇	—	4	2	no	—
181	#2965	—	67	1	no	—
182	#2965	—	67	1	no	—
183	C₄×#131	—	67	1	no	—
184	#1455	—	67	2	no	—
185	#1453	—	67	2	no	—
186	#1451	—	67	1	no	—
187	#1451	D₆₇	1	1	yes	characteristic, complemented
188	#1450	—	67	1	no	—
189	C₈×M₁₆	—	67	2	no	—
190	C₈×M₁₆	—	67	2	no	—
191	C₂×C₈²	—	67	1	no	—
192	C₆₇	#17335	1	1	yes	characteristic, complemented, Sylow
193	C₂×#131	—	67	2	no	—
194	C₄×#41	—	67	1	no	—
195	#446	—	67	2	no	—
196	#446	—	67	2	no	—
197	#436	—	67	2	no	—
198	#436	—	67	2	no	—
199	#435	—	67	4	no	—
200	#435	—	67	4	no	—
201	#431	—	67	1	no	—
202	#431	—	67	2	no	—
203	#431	—	67	2	no	—
204	#431	C₂×D₆₇	1	1	yes	characteristic
205	#405	—	67	2	no	—
206	#405	—	67	4	no	—
207	C₄×M₁₆	—	67	2	no	—
208	C₄×M₁₆	—	67	2	no	—
209	C₄×M₁₆	—	67	2	no	—
210	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
211	C₂×C₄×C₈	—	67	1	no	—
212	C₂×C₄×C₈	—	67	1	no	—
213	C₂×C₄×C₈	—	67	2	no	—
214	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
215	#322	—	67	2	no	—
216	#322	—	67	2	no	—
217	#322	—	67	2	no	—
218	#322	C₂×D₆₇	1	2	yes	—
219	#322	C₂×D₆₇	1	2	yes	—
220	C₈²	—	67	1	no	—
221	C₈²	—	67	1	no	—
222	C₈²	—	67	1	no	—
223	C₈²	—	67	1	no	—
224	C₈²	—	67	2	no	—
225	C₂×#41	—	67	1	no	—
226	#131	—	134	8	no	—
227	C₂×M₁₆	—	67	4	no	—
228	C₂×M₁₆	—	67	4	no	—
229	C₂×M₁₆	—	67	4	no	—
230	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
231	C₂²×C₈	—	67	2	no	—
232	C₂²×C₈	—	67	2	no	—
233	C₂²×C₈	—	67	4	no	—
234	C₂²×C₈	C₄×D₆₇	1	2	yes	—
235	C₄×Q₈	—	67	1	no	—
236	C₄×Q₈	—	67	1	no	—
237	C₄×D₄	—	67	2	no	—
238	C₄×D₄	—	67	2	no	—
239	C₄×D₄	—	67	1	no	—
240	C₄×D₄	—	67	1	no	—
241	#117	—	67	1	no	—
242	#117	—	67	2	no	—
243	C₂×C₄²	—	67	2	no	—
244	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
245	C₄⋊C₈	—	67	4	no	—
246	C₄⋊C₈	—	67	4	no	—
247	C₄⋊C₈	—	67	2	no	—
248	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
249	C₂²⋊C₈	—	67	2	no	—
250	C₂²⋊C₈	—	67	4	no	—
251	C₂²⋊C₈	—	67	4	no	—
252	C₂²⋊C₈	C₄×D₆₇	1	2	yes	—
253	#97	—	67	4	no	—
254	#97	—	67	4	no	—
255	#97	—	67	4	no	—
256	#97	C₄×D₆₇	1	2	yes	—
257	C₄×C₈	—	67	4	no	—
258	C₄×C₈	—	67	2	no	—
259	C₄×C₈	—	67	2	no	—
260	C₄×C₈	—	67	1	no	—
261	C₄×C₈	—	67	1	no	—
262	C₄×C₈	—	67	1	no	—
263	C₄×C₈	—	67	1	no	—
264	C₄×C₈	—	67	2	no	—
265	C₄×C₈	C₄×D₆₇	1	2	yes	—
266	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₆₇	1	2	yes	—
269	C₄×C₈	C₂²×D₆₇	1	2	yes	—
270	#41	—	134	4	no	—
271	C₂×Q₈	—	67	1	no	—
272	C₂×D₄	—	67	2	no	—
273	C₂×D₄	—	67	1	no	—
274	C₂²×C₄	—	67	2	no	—
275	C₂²×C₄	—	67	2	no	—
276	C₂²×C₄	—	67	2	no	—
277	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
280	M₁₆	—	2	4	no	—
281	M₁₆	—	134	8	no	—
282	M₁₆	—	134	8	no	—
283	M₁₆	—	134	8	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	67	8	no	—
286	C₂×C₈	—	67	2	no	—
287	C₂×C₈	—	67	2	no	—
288	C₂×C₈	—	67	2	no	—
289	C₂×C₈	—	67	2	no	—
290	C₂×C₈	—	67	4	no	—
291	C₂×C₈	—	134	8	no	—
292	C₂×C₈	—	134	4	no	—
293	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
294	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
295	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
296	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
297	C₄⋊C₄	—	67	2	no	—
298	C₄⋊C₄	—	67	2	no	—
299	C₄⋊C₄	—	67	1	no	—
300	C₄⋊C₄	—	67	1	no	—
301	C₂²⋊C₄	—	67	2	no	—
302	C₂²⋊C₄	—	67	2	no	—
303	C₂²⋊C₄	—	67	2	no	—
304	C₄²	—	67	2	no	—
305	C₄²	—	67	2	no	—
306	C₄²	—	67	2	no	—
307	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
308	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
309	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
310	C₄²	C₂³×D₆₇	1	1	yes	characteristic
311	C₂³	—	67	2	no	—
312	C₂³	C₄²×D₆₇	1	1	yes	characteristic
313	Q₈	—	134	2	no	—
314	D₄	—	134	4	no	—
315	D₄	—	134	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	67	2	no	—
319	C₂×C₄	—	67	2	no	—
320	C₂×C₄	—	67	2	no	—
321	C₂×C₄	—	134	4	no	—
322	C₂×C₄	—	134	4	no	—
323	C₂×C₄	—	134	4	no	—
324	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
325	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
326	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
327	C₂×C₄	M₁₆×D₆₇	1	4	yes	—
328	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
331	C₈	—	2	4	no	—
332	C₈	—	2	4	no	—
333	C₈	—	2	2	no	—
334	C₈	—	67	4	no	—
335	C₈	—	67	4	no	—
336	C₈	—	67	4	no	—
337	C₈	—	134	2	no	—
338	C₈	—	134	4	no	—
339	C₈	#14524051	1	4	yes	complemented
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	134	2	no	—
343	C₂²	—	134	2	no	—
344	C₂²	—	134	2	no	—
345	C₂²	#16863348	1	2	yes	—
346	C₂²	C₂×C₄²×D₆₇	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	134	2	no	—
350	C₄	—	134	2	no	—
351	C₄	—	134	2	no	—
352	C₄	#16863348	1	2	yes	—
353	C₄	C₂×M₁₆×D₆₇	1	2	yes	—
354	C₄	#131×D₆₇	1	2	yes	—
355	C₄	C₂×#14524051	1	2	yes	—
356	C₂	—	2	2	no	—
357	C₂	—	268	2	no	—
358	C₂	C₂×#16863348	1	1	yes	characteristic
359	C₂	#22568631	1	1	yes	characteristic
360	C₂	C₄×#14524051	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₄²
Fitting	C₆₇×#1451	C₂
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1451	D₆₇
hypocenter	C₆₇	#17335
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ #1451

Complements

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

N	H ≅ G/N	type
4. #22558513	C₂	semidirect
5. #22558505	C₂	semidirect
6. #22558496	C₂	semidirect
7. #22558493	C₂	semidirect
8. #22558488	C₂	semidirect
9. #22558488	C₂	semidirect
10. C₂×C₈×C₆₇⋊C₈	C₂	semidirect
11. C₆₇×#1451	C₂	semidirect
100. C₂×C₆₇⋊C₈	M₁₆	semidirect
153. C₈×C₆₇	#131	semidirect
158. C₆₇⋊C₈	#131	semidirect
187. #1451	D₆₇	semidirect
192. C₆₇	#17335	semidirect
339. C₈	#14524051	semidirect

Maximal subgroups

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

subgroup	index
#22569175	2
#22569170	2
C₄×#14524051	2
#22558513	2
#22558505	2
#22558496	2
#22558493	2
#22558488	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#1451	2
#17335	67