Subgroups — #31009356

Order 17152.

558 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11761629	C₂	1	1	yes	maximal, characteristic
2	#22562118	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562042	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562040	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562036	C₂	1	2	yes	maximal, complemented
6	#22562033	C₂	1	2	yes	maximal, complemented
7	#22562007	C₂	1	1	yes	maximal, characteristic, complemented
8	#22562007	C₂	1	2	yes	maximal, complemented
9	#22561802	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11761629	C₄	1	1	yes	characteristic
13	C₂×#14523982	C₂²	1	2	yes	—
14	C₂×#14523982	C₂²	1	2	yes	—
15	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
17	Q₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
18	C₈×D₂₆₈	C₂²	1	2	yes	—
19	C₈×D₂₆₈	C₂²	1	1	yes	characteristic
20	#16863461	C₄	1	1	yes	characteristic
21	#16863461	C₄	1	2	yes	—
22	#16863448	C₄	1	2	yes	—
23	#16863448	C₄	1	1	yes	characteristic
24	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₆₇	C₂²	1	2	yes	—
26	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
27	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
28	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	—
29	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	—
30	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	—
31	#16863172	C₂²	1	2	yes	—
32	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₆₇	C₂²	1	2	yes	—
36	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
38	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	#346×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#11761629	C₈	1	1	yes	characteristic
42	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₆₇	C₂³	1	2	yes	—
46	Q₈×Dic₆₇	C₈	1	1	yes	characteristic
47	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
48	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
49	#14524005	C₈	1	2	yes	—
50	#14524005	C₈	1	1	yes	characteristic
51	#14524001	C₈	1	2	yes	—
52	#14524001	C₈	1	1	yes	characteristic
53	C₄²×D₆₇	C₂×C₄	1	2	yes	—
54	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
55	#14523982	—	2	4	no	—
56	#14523982	—	2	4	no	—
57	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₁₆×D₆₇	—	2	2	no	—
61	C₁₆×D₆₇	—	2	2	no	—
62	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
63	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
64	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
65	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
66	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
67	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
69	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
71	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
75	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
77	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₆₇	C₂³	1	2	yes	—
80	#11761629	—	2	4	no	—
81	C₂×D₂₆₈	C₂×C₈	1	2	yes	—
82	C₂×D₂₆₈	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
92	C₈×D₆₇	—	2	1	no	—
93	C₈×D₆₇	—	2	1	no	—
94	C₈×D₆₇	—	2	2	no	—
95	C₈×D₆₇	—	2	2	no	—
96	C₈×D₆₇	—	2	2	no	—
97	C₈×D₆₇	—	2	4	no	—
98	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₆₇	C₂×C₈	1	1	yes	characteristic
100	(C₂×D₆₇)⋊C₄	C₂×C₈	1	2	yes	—
101	(C₂×D₆₇)⋊C₄	C₂×C₈	1	2	yes	—
102	(C₂×D₆₇)⋊C₄	C₂×C₈	1	2	yes	—
103	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
104	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
106	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
107	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₆₇⋊C₁₆	—	2	1	no	—
109	C₆₇⋊C₁₆	—	2	2	no	—
110	C₆₇⋊C₁₆	#41	1	2	yes	complemented
111	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
116	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
117	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
118	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₁₆×C₆₇	—	2	1	no	—
120	C₁₆×C₆₇	—	2	2	no	—
121	C₁₆×C₆₇	#41	1	2	yes	complemented
122	C₂²×D₆₇	C₂²×C₈	1	2	yes	—
123	C₂²×D₆₇	C₂²×C₈	1	1	yes	characteristic
124	Q₈×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
131	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
133	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
134	D₂₆₈	—	2	4	no	—
135	D₂₆₈	—	2	2	no	—
136	C₄×D₆₇	—	2	1	no	—
137	C₄×D₆₇	—	2	2	no	—
138	C₄×D₆₇	—	2	1	no	—
139	C₄×D₆₇	—	2	2	no	—
140	C₄×D₆₇	—	2	4	no	—
141	C₄×D₆₇	—	2	2	no	—
142	C₄×D₆₇	—	2	4	no	—
143	C₄×D₆₇	M₃₂	1	4	yes	complemented
144	C₈×C₆₇	—	2	1	no	—
145	C₈×C₆₇	—	2	2	no	—
146	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
147	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
148	C₆₇⋊C₈	—	2	1	no	—
149	C₆₇⋊C₈	—	2	2	no	—
150	C₆₇⋊C₈	#131	1	2	yes	—
151	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
152	C₂×D₆₇	—	2	1	no	—
153	C₂×D₆₇	—	2	1	no	—
154	C₂×D₆₇	—	2	2	no	—
155	C₂×D₆₇	—	2	2	no	—
156	C₂×D₆₇	—	2	2	no	—
157	C₂×D₆₇	C₂×M₃₂	1	2	yes	—
158	C₄×C₆₇	—	2	2	no	—
159	C₄×C₆₇	—	2	1	no	—
160	C₄×C₆₇	—	2	2	no	—
161	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
162	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
163	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
164	Dic₆₇	—	2	1	no	—
165	Dic₆₇	—	2	2	no	—
166	Dic₆₇	C₂×M₃₂	1	2	yes	—
167	Dic₆₇	#504	1	2	yes	—
168	#33479	—	67	1	no	maximal, Sylow
169	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
170	C₂×C₆₇	C₂²×M₃₂	1	1	yes	characteristic
171	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
172	D₆₇	—	2	2	no	—
173	D₆₇	—	4	2	no	—
174	C₈×#41	—	67	1	no	—
175	#2183	—	67	1	no	—
176	#2183	D₆₇	1	1	yes	characteristic, complemented
177	#2169	—	67	1	no	—
178	#2169	—	67	1	no	—
179	#2168	—	67	2	no	—
180	#2168	—	67	2	no	—
181	#2162	—	67	2	no	—
182	#2162	—	67	1	no	—
183	C₄×M₃₂	—	67	2	no	—
184	C₂×C₄×C₁₆	—	67	1	no	—
185	C₆₇	#33479	1	1	yes	characteristic, complemented, Sylow
186	C₄×#41	—	67	1	no	—
187	C₂×M₃₂	—	67	2	no	—
188	C₂×M₃₂	—	67	2	no	—
189	C₂²×C₁₆	—	67	1	no	—
190	C₂²×C₁₆	—	67	1	no	—
191	C₈×Q₈	—	67	1	no	—
192	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
193	C₈×D₄	—	67	1	no	—
194	C₈×D₄	—	67	2	no	—
195	C₈×D₄	—	67	2	no	—
196	C₈×D₄	—	67	1	no	—
197	#431	—	67	2	no	—
198	#431	—	67	1	no	—
199	C₂×C₄×C₈	—	67	1	no	—
200	C₂×C₄×C₈	—	67	2	no	—
201	C₄⋊C₁₆	—	67	2	no	—
202	C₄⋊C₁₆	—	67	1	no	—
203	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
204	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
205	C₂²⋊C₁₆	—	67	2	no	—
206	C₂²⋊C₁₆	—	67	2	no	—
207	C₂²⋊C₁₆	—	67	2	no	—
208	#346	—	67	2	no	—
209	#346	C₂×D₆₇	1	2	yes	—
210	C₄×C₁₆	—	67	1	no	—
211	C₄×C₁₆	—	67	1	no	—
212	C₄×C₁₆	—	67	2	no	—
213	C₄×C₁₆	—	67	1	no	—
214	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
215	C₂×#41	—	67	1	no	—
216	C₂²×C₈	—	67	1	no	—
217	C₂²×C₈	—	67	2	no	—
218	C₂²×C₈	—	67	1	no	—
219	C₂²×C₈	—	67	2	no	—
220	C₄×Q₈	—	67	1	no	—
221	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
222	C₄×D₄	—	67	2	no	—
223	C₄×D₄	—	67	1	no	—
224	C₄×D₄	—	67	2	no	—
225	C₄×D₄	—	67	1	no	—
226	#117	—	67	1	no	—
227	#117	—	67	2	no	—
228	C₂×C₄²	—	67	1	no	—
229	C₂×C₄²	—	67	2	no	—
230	M₃₂	—	134	4	no	—
231	M₃₂	—	134	4	no	—
232	C₂×C₁₆	—	67	1	no	—
233	C₂×C₁₆	—	67	1	no	—
234	C₂×C₁₆	—	67	2	no	—
235	C₂×C₁₆	—	134	2	no	—
236	C₂×C₁₆	—	134	2	no	—
237	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
238	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
239	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
240	C₄⋊C₈	—	67	2	no	—
241	C₄⋊C₈	—	67	1	no	—
242	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
243	C₄⋊C₈	C₄×D₆₇	1	1	yes	characteristic
244	C₂²⋊C₈	—	67	2	no	—
245	C₂²⋊C₈	—	67	2	no	—
246	C₂²⋊C₈	—	67	2	no	—
247	C₄×C₈	—	67	1	no	—
248	C₄×C₈	—	67	2	no	—
249	C₄×C₈	—	67	1	no	—
250	C₄×C₈	—	67	1	no	—
251	C₄×C₈	—	67	2	no	—
252	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₆₇	1	2	yes	—
254	#41	—	134	4	no	—
255	C₂×Q₈	C₈×D₆₇	1	1	yes	characteristic
256	C₂×D₄	—	67	2	no	—
257	C₂×D₄	—	67	1	no	—
258	C₂²×C₄	—	67	1	no	—
259	C₂²×C₄	—	67	2	no	—
260	C₂²×C₄	—	67	1	no	—
261	C₂²×C₄	—	67	2	no	—
262	C₂²×C₄	—	67	1	no	—
263	C₂²×C₄	—	67	2	no	—
264	C₂×C₈	—	67	1	no	—
265	C₂×C₈	—	67	1	no	—
266	C₂×C₈	—	67	2	no	—
267	C₂×C₈	—	134	4	no	—
268	C₂×C₈	—	134	2	no	—
269	C₂×C₈	—	134	1	no	—
270	C₂×C₈	—	134	2	no	—
271	C₂×C₈	—	134	1	no	—
272	C₂×C₈	—	134	2	no	—
273	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
275	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
276	C₄⋊C₄	—	67	2	no	—
277	C₄⋊C₄	—	67	1	no	—
278	C₄⋊C₄	C₈×D₆₇	1	2	yes	—
279	C₄⋊C₄	C₈×D₆₇	1	1	yes	characteristic
280	C₂²⋊C₄	—	67	2	no	—
281	C₂²⋊C₄	—	67	2	no	—
282	C₂²⋊C₄	—	67	2	no	—
283	C₄²	—	67	1	no	—
284	C₄²	—	67	2	no	—
285	C₄²	—	67	2	no	—
286	C₄²	—	67	1	no	—
287	C₄²	—	67	1	no	—
288	C₄²	C₂×C₄×D₆₇	1	2	yes	—
289	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
290	C₁₆	—	2	1	no	—
291	C₁₆	—	2	2	no	—
292	C₁₆	—	67	2	no	—
293	C₁₆	—	134	1	no	—
294	C₁₆	—	134	2	no	—
295	C₁₆	#11761629	1	2	yes	complemented
296	C₂³	—	67	1	no	—
297	C₂³	—	67	2	no	—
298	Q₈	—	2	2	no	—
299	D₄	—	134	4	no	—
300	D₄	—	134	2	no	—
301	C₂×C₄	—	67	1	no	—
302	C₂×C₄	—	67	1	no	—
303	C₂×C₄	—	67	4	no	—
304	C₂×C₄	—	67	2	no	—
305	C₂×C₄	—	67	1	no	—
306	C₂×C₄	—	134	4	no	—
307	C₂×C₄	—	134	4	no	—
308	C₂×C₄	—	134	2	no	—
309	C₂×C₄	—	134	2	no	—
310	C₂×C₄	—	134	1	no	—
311	C₂×C₄	—	134	2	no	—
312	C₂×C₄	—	134	1	no	—
313	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
314	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
315	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
316	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	—	67	2	no	—
321	C₈	—	134	1	no	—
322	C₈	—	134	2	no	—
323	C₈	#41×D₆₇	1	1	yes	characteristic
324	C₈	C₂×#11761629	1	1	yes	characteristic
325	C₂²	—	67	2	no	—
326	C₂²	—	134	2	no	—
327	C₂²	—	134	2	no	—
328	C₂²	—	134	2	no	—
329	C₂²	—	134	1	no	—
330	C₂²	—	134	1	no	—
331	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	67	2	no	—
336	C₄	—	67	2	no	—
337	C₄	—	134	2	no	—
338	C₄	—	134	1	no	—
339	C₄	M₃₂×D₆₇	1	2	yes	—
340	C₄	C₄×#11761629	1	1	yes	characteristic
341	C₄	#131×D₆₇	1	1	yes	characteristic
342	C₂	—	268	2	no	—
343	C₂	—	134	2	no	—
344	C₂	C₈×#11761629	1	1	yes	characteristic
345	C₂	C₂×M₃₂×D₆₇	1	1	yes	characteristic
346	C₂	#504×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₆₇×#2183	C₂
prime radical	C₂×#11761629	C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2183	D₆₇
hypocenter	C₆₇	#33479
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #22562118	C₂	semidirect
3. #22562042	C₂	semidirect
4. #22562040	C₂	semidirect
5. #22562036	C₂	semidirect
6. #22562033	C₂	semidirect
7. #22562007	C₂	semidirect
8. #22562007	C₂	semidirect
9. #22561802	C₂	semidirect
10. C₄×C₁₆×D₆₇	C₂	semidirect
11. C₆₇×#2183	C₂	semidirect
110. C₆₇⋊C₁₆	#41	semidirect
121. C₁₆×C₆₇	#41	semidirect
143. C₄×D₆₇	M₃₂	semidirect
176. #2183	D₆₇	semidirect
185. C₆₇	#33479	semidirect
295. C₁₆	#11761629	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11761629	2
#22562118	2
#22562042	2
#22562040	2
#22562036	2
#22562033	2
#22562007	2
#22561802	2
C₄×C₁₆×D₆₇	2
C₆₇×#2183	2
#33479	67