Subgroups — #31980662

Order 18176.

646 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22599484	C₂	1	2	yes	maximal
2	#22597274	C₂	1	1	yes	maximal, characteristic
3	#22587831	C₂	1	4	yes	maximal, complemented
4	#22587707	C₂	1	4	yes	maximal, complemented
5	#22587088	C₂	1	1	yes	maximal, characteristic
6	#22587012	C₂	1	2	yes	maximal
7	C₂×#494×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#17986556	C₂²	1	1	yes	characteristic
9	#17986395	—	2	2	no	—
10	#17986351	—	2	2	no	—
11	#17986329	—	2	4	no	—
12	#17986281	C₂²	1	8	yes	—
13	#17986129	—	2	1	no	—
14	#17986129	C₂²	1	2	yes	—
15	C₈×Dic₁₄₂	—	2	1	no	—
16	C₈×Dic₁₄₂	C₂²	1	2	yes	—
17	#17986052	C₂²	1	2	yes	—
18	#17986025	—	2	1	no	—
19	#17986023	C₂²	1	1	yes	characteristic
20	#17985954	—	2	1	no	—
21	#17985752	C₂²	1	2	yes	—
22	#17985750	C₂²	1	2	yes	—
23	#17985745	C₂²	1	8	yes	—
24	C₂²×Q₁₆×C₇₁	—	2	1	no	—
25	C₂²×Q₁₆×C₇₁	—	2	1	no	—
26	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
27	#494×C₇₁	—	2	1	no	—
28	#494×C₇₁	—	2	1	no	—
29	#494×C₇₁	C₂²	1	2	yes	—
30	#494×C₇₁	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
32	C₂×#11765849	—	2	1	no	—
33	#14556128	D₄	1	1	yes	characteristic
34	#14556128	D₄	1	1	yes	characteristic
35	#14556120	—	2	1	no	—
36	#14556107	—	2	1	no	—
37	#14556107	D₄	1	2	yes	—
38	C₄×Dic₁₄₂	—	2	1	no	—
39	C₄×Dic₁₄₂	C₂³	1	2	yes	—
40	#14556087	—	2	1	no	—
41	#14556059	—	2	8	no	—
42	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
43	C₄⋊(C₇₁⋊C₈)	C₂³	1	4	yes	—
44	C₄⋊(C₇₁⋊C₈)	C₂³	1	2	yes	—
45	Dic₇₁⋊C₈	—	2	2	no	—
46	Dic₇₁⋊C₈	—	2	2	no	—
47	#14556051	—	2	8	no	—
48	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
49	#14556038	—	2	2	no	—
50	C₄×C₇₁⋊C₈	—	2	2	no	—
51	C₈×Dic₇₁	—	2	2	no	—
52	C₂²×Q₈×C₇₁	—	2	2	no	—
53	C₂×Q₁₆×C₇₁	—	2	2	no	—
54	C₂×Q₁₆×C₇₁	—	2	2	no	—
55	C₂×Q₁₆×C₇₁	—	2	2	no	—
56	C₂×Q₁₆×C₇₁	—	2	2	no	—
57	C₂×Q₁₆×C₇₁	—	4	4	no	—
58	C₂×Q₁₆×C₇₁	—	4	4	no	—
59	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₇₁	—	2	2	no	—
62	C₄⋊Q₈×C₇₁	—	2	2	no	—
63	C₄⋊Q₈×C₇₁	—	2	2	no	—
64	C₄⋊Q₈×C₇₁	C₂³	1	4	yes	—
65	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
66	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
67	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₄₂	—	2	1	no	—
73	C₂×Dic₁₄₂	C₂×D₄	1	2	yes	—
74	#11765849	—	4	2	no	—
75	C₂×C₇₁⋊C₈	—	2	2	no	—
76	C₂×C₇₁⋊C₈	—	2	4	no	—
77	C₄⋊Dic₇₁	—	2	1	no	—
78	C₄⋊Dic₇₁	—	2	1	no	—
79	C₄⋊Dic₇₁	—	2	1	no	—
80	C₄⋊Dic₇₁	C₂×D₄	1	2	yes	—
81	Dic₇₁⋊C₄	—	2	2	no	—
82	Dic₇₁⋊C₄	—	2	2	no	—
83	C₂²⋊Dic₇₁	—	2	2	no	—
84	C₄×Dic₇₁	—	2	2	no	—
85	C₂×Q₈×C₇₁	—	2	4	no	—
86	C₂×Q₈×C₇₁	—	4	2	no	—
87	C₂×Q₈×C₇₁	—	4	4	no	—
88	C₂×Q₈×C₇₁	—	4	4	no	—
89	C₂²×C₄×C₇₁	—	4	2	no	—
90	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
93	Q₁₆×C₇₁	—	4	4	no	—
94	Q₁₆×C₇₁	—	4	4	no	—
95	Q₁₆×C₇₁	—	4	4	no	—
96	Q₁₆×C₇₁	—	4	4	no	—
97	C₂×C₈×C₇₁	—	2	2	no	—
98	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
99	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₇₁	—	2	4	no	—
104	C₄⋊C₄×C₇₁	—	2	4	no	—
105	C₄⋊C₄×C₇₁	—	4	2	no	—
106	C₄⋊C₄×C₇₁	—	4	2	no	—
107	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₇₁	#41	1	1	yes	characteristic
110	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
111	C₂³×C₇₁	#121	1	1	yes	characteristic
112	Q₈×C₇₁	—	4	4	no	—
113	Q₈×C₇₁	—	4	4	no	—
114	Q₈×C₇₁	—	8	4	no	—
115	C₂×C₄×C₇₁	—	2	2	no	—
116	C₂×C₄×C₇₁	—	2	2	no	—
117	C₂×C₄×C₇₁	—	4	4	no	—
118	C₂×C₄×C₇₁	—	8	4	no	—
119	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
120	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
127	C₂×Dic₇₁	—	2	2	no	—
128	C₂×Dic₇₁	—	2	2	no	—
129	Dic₁₄₂	—	2	4	no	—
130	Dic₁₄₂	—	4	2	no	—
131	C₈×C₇₁	—	2	1	no	—
132	C₈×C₇₁	—	2	1	no	—
133	C₈×C₇₁	C₂×SD₁₆	1	2	yes	—
134	C₈×C₇₁	C₂×SD₁₆	1	2	yes	—
135	C₇₁⋊C₈	—	4	2	no	—
136	C₇₁⋊C₈	—	4	4	no	—
137	C₂²×C₇₁	—	2	1	no	—
138	C₂²×C₇₁	—	2	1	no	—
139	C₂²×C₇₁	#460	1	2	yes	—
140	C₂²×C₇₁	C₂×#121	1	1	yes	characteristic
141	C₄×C₇₁	—	2	1	no	—
142	C₄×C₇₁	—	2	1	no	—
143	C₄×C₇₁	—	8	4	no	—
144	C₄×C₇₁	#460	1	2	yes	—
145	C₄×C₇₁	#473	1	2	yes	—
146	C₄×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
147	C₄×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₄×C₇₁	#578	1	2	yes	—
149	Dic₇₁	—	4	2	no	—
150	Dic₇₁	—	4	2	no	—
151	#19204	—	71	1	no	maximal, Sylow
152	C₂×C₇₁	—	2	2	no	—
153	C₂×C₇₁	C₂×#460	1	1	yes	characteristic
154	C₂×C₇₁	#3059	1	1	yes	characteristic
155	C₂×C₇₁	C₂×#473	1	1	yes	characteristic
156	#3229	—	71	2	no	—
157	C₂×#494	D₇₁	1	1	yes	characteristic
158	#2960	—	71	1	no	—
159	#1690	—	71	4	no	—
160	#1670	—	71	4	no	—
161	#1564	—	71	1	no	—
162	#1546	—	71	2	no	—
163	C₇₁	#19204	1	1	yes	characteristic, complemented, Sylow
164	C₂²×Q₁₆	—	2	1	no	—
165	C₂²×Q₁₆	—	2	1	no	—
166	#533	—	71	1	no	—
167	C₂×C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
168	#498	—	142	2	no	—
169	#494	—	2	1	no	—
170	#494	—	2	1	no	—
171	#494	C₂×D₇₁	1	2	yes	—
172	#494	C₂×D₇₁	1	2	yes	—
173	#487	—	142	2	no	—
174	#483	—	142	4	no	—
175	#475	—	71	8	no	—
176	#446	—	71	2	no	—
177	#446	—	142	1	no	—
178	C₈×Q₈	—	71	2	no	—
179	C₈×Q₈	—	142	1	no	—
180	#431	—	71	2	no	—
181	#424	—	142	1	no	—
182	#423	—	71	1	no	—
183	#405	—	142	1	no	—
184	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
185	#335	—	71	2	no	—
186	#334	—	71	2	no	—
187	#332	—	71	8	no	—
188	C₂²×Q₈	—	2	2	no	—
189	C₂×Q₁₆	—	2	2	no	—
190	C₂×Q₁₆	—	2	2	no	—
191	C₂×Q₁₆	—	2	2	no	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×Q₁₆	—	4	4	no	—
194	C₂×Q₁₆	—	4	4	no	—
195	C₂×M₁₆	—	142	1	no	—
196	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	71	1	no	—
202	C₄⋊Q₈	—	71	1	no	—
203	C₄⋊Q₈	C₂²×D₇₁	1	4	yes	—
204	#125	—	142	1	no	—
205	C₂²⋊Q₈	—	71	2	no	—
206	C₂²⋊Q₈	—	142	1	no	—
207	C₄×Q₈	—	71	2	no	—
208	C₄×Q₈	—	142	1	no	—
209	#117	—	142	1	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
213	#106	—	142	8	no	—
214	C₄⋊C₈	—	71	2	no	—
215	C₄⋊C₈	—	71	4	no	—
216	C₄⋊C₈	—	142	2	no	—
217	C₄⋊C₈	—	142	2	no	—
218	C₄⋊C₈	—	142	1	no	—
219	#103	—	142	8	no	—
220	C₂²⋊C₈	—	142	2	no	—
221	#97	—	142	2	no	—
222	C₄×C₈	—	142	2	no	—
223	C₄×C₈	—	142	2	no	—
224	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
225	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
227	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	—	4	4	no	—
231	C₂×Q₈	—	4	4	no	—
232	C₂×Q₈	—	71	2	no	—
233	C₂×Q₈	—	142	1	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
237	C₂²×C₄	#11765864	1	1	yes	characteristic
238	Q₁₆	—	4	4	no	—
239	Q₁₆	—	4	4	no	—
240	Q₁₆	—	4	4	no	—
241	Q₁₆	—	4	4	no	—
242	M₁₆	—	284	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	142	4	no	—
245	C₂×C₈	—	142	2	no	—
246	C₂×C₈	#11765853	1	4	yes	—
247	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	71	2	no	—
256	C₄⋊C₄	—	142	2	no	—
257	C₄⋊C₄	—	142	2	no	—
258	C₄⋊C₄	—	142	1	no	—
259	C₄⋊C₄	—	142	1	no	—
260	C₄⋊C₄	—	142	1	no	—
261	C₂²⋊C₄	—	142	2	no	—
262	C₄²	—	142	2	no	—
263	C₄²	D₄×D₇₁	1	1	yes	characteristic
264	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
265	C₄²	#11765864	1	1	yes	characteristic
266	C₄²	C₂³×D₇₁	1	1	yes	characteristic
267	C₂³	#14556099	1	1	yes	characteristic
268	Q₈	—	4	4	no	—
269	Q₈	—	4	4	no	—
270	Q₈	—	8	4	no	—
271	Q₈	—	284	2	no	—
272	Q₈	—	142	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	—	142	2	no	—
278	C₂×C₄	—	142	2	no	—
279	C₂×C₄	#14556099	1	1	yes	characteristic
280	C₂×C₄	#14556099	1	1	yes	characteristic
281	C₂×C₄	#14556099	1	1	yes	characteristic
282	C₂×C₄	C₂×#11765853	1	2	yes	—
283	C₂×C₄	#14556154	1	2	yes	—
284	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
286	C₂×C₄	C₂×#11765864	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	—	284	2	no	—
290	C₈	—	284	4	no	—
291	C₈	C₂×#11765853	1	2	yes	—
292	C₈	C₂×#11765853	1	2	yes	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	#17986196	1	2	yes	—
296	C₂²	C₂×#14556099	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	8	4	no	—
300	C₄	—	284	2	no	—
301	C₄	—	284	2	no	—
302	C₄	#17986196	1	2	yes	—
303	C₄	#17986271	1	2	yes	—
304	C₄	C₂²×#11765853	1	1	yes	characteristic
305	C₄	C₂×#14556154	1	1	yes	characteristic
306	C₄	#17986812	1	2	yes	—
307	C₂	—	2	2	no	—
308	C₂	C₂×#17986196	1	1	yes	characteristic
309	C₂	#22597991	1	1	yes	characteristic
310	C₂	C₂×#17986271	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556099
commutator, cosocle	C₄²×C₇₁	C₂⁴
Frattini	C₄²	C₂³×D₇₁
Fitting	C₂×#494×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#121
prime radical	C₂³×C₇₁	#121
hypercenter	C₂×#494	D₇₁
hypocenter	C₇₁	#19204
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #22587831	C₂	semidirect
4. #22587707	C₂	semidirect
163. C₇₁	#19204	semidirect

Maximal subgroups

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

subgroup	index
#22599484	2
#22597274	2
#22587831	2
#22587707	2
#22587088	2
#22587012	2
C₂×#494×C₇₁	2
#19204	71