Subgroups — #31897456

Order 18176.

512 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588193	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22586181	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14556039	—	2	1	no	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
9	#17985754	—	2	1	no	—
10	#17985754	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₇₁	—	2	1	no	—
13	#391×C₇₁	—	2	1	no	—
14	#389×C₇₁	—	2	1	no	—
15	#389×C₇₁	C₄	1	2	yes	—
16	#384×C₇₁	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
18	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
19	C₄³×C₇₁	C₂²	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	—	2	2	no	—
21	C₂²×C₇₁⋊C₈	—	2	2	no	—
22	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	Q₈	1	4	yes	complemented
25	#14556039	—	4	2	no	—
26	C₄×C₇₁⋊C₈	—	2	4	no	—
27	C₂²×Q₈×C₇₁	—	2	1	no	—
28	C₄⋊Q₈×C₇₁	—	2	4	no	—
29	C₄×Q₈×C₇₁	—	4	2	no	—
30	C₄×Q₈×C₇₁	—	4	2	no	—
31	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
35	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇₁	—	2	2	no	—
38	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
45	#95×C₇₁	—	2	2	no	—
46	#95×C₇₁	—	2	2	no	—
47	C₂×C₇₁⋊C₈	—	2	4	no	—
48	C₂×C₇₁⋊C₈	—	2	4	no	—
49	C₂×C₇₁⋊C₈	—	4	2	no	—
50	C₂×C₇₁⋊C₈	—	4	2	no	—
51	C₂×C₇₁⋊C₈	—	4	2	no	—
52	C₂×C₇₁⋊C₈	—	4	2	no	—
53	C₂×Q₈×C₇₁	—	2	4	no	—
54	C₂×Q₈×C₇₁	—	4	2	no	—
55	C₂×Q₈×C₇₁	—	4	2	no	—
56	C₂²×C₄×C₇₁	—	2	2	no	—
57	C₂²×C₄×C₇₁	—	2	2	no	—
58	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₇₁	—	2	4	no	—
66	C₄⋊C₄×C₇₁	—	2	2	no	—
67	C₄⋊C₄×C₇₁	—	2	4	no	—
68	C₄⋊C₄×C₇₁	—	4	4	no	—
69	C₄⋊C₄×C₇₁	—	4	2	no	—
70	C₄⋊C₄×C₇₁	—	4	2	no	—
71	C₄⋊C₄×C₇₁	M₁₆	1	4	yes	—
72	C₄²×C₇₁	—	2	2	no	—
73	C₄²×C₇₁	—	2	2	no	—
74	C₄²×C₇₁	—	2	2	no	—
75	C₄²×C₇₁	—	2	2	no	—
76	C₄²×C₇₁	—	4	1	no	—
77	C₄²×C₇₁	—	4	2	no	—
78	C₄²×C₇₁	—	4	2	no	—
79	C₄²×C₇₁	D₈	1	2	yes	—
80	C₄²×C₇₁	SD₁₆	1	2	yes	—
81	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
82	C₄²×C₇₁	#41	1	2	yes	—
83	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈×C₇₁	—	4	8	no	—
85	C₂×C₄×C₇₁	—	2	1	no	—
86	C₂×C₄×C₇₁	—	2	1	no	—
87	C₂×C₄×C₇₁	—	2	1	no	—
88	C₂×C₄×C₇₁	—	2	1	no	—
89	C₂×C₄×C₇₁	—	2	1	no	—
90	C₂×C₄×C₇₁	—	2	1	no	—
91	C₂×C₄×C₇₁	—	2	1	no	—
92	C₂×C₄×C₇₁	—	2	1	no	—
93	C₂×C₄×C₇₁	—	2	4	no	—
94	C₂×C₄×C₇₁	—	2	1	no	—
95	C₂×C₄×C₇₁	—	2	1	no	—
96	C₂×C₄×C₇₁	—	2	1	no	—
97	C₂×C₄×C₇₁	—	4	2	no	—
98	C₂×C₄×C₇₁	—	4	2	no	—
99	C₂×C₄×C₇₁	—	4	2	no	—
100	C₂×C₄×C₇₁	—	4	2	no	—
101	C₂×C₄×C₇₁	—	4	2	no	—
102	C₂×C₄×C₇₁	#102	1	2	yes	—
103	C₂×C₄×C₇₁	#102	1	2	yes	—
104	C₂×C₄×C₇₁	#104	1	2	yes	—
105	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₇₁	#131	1	2	yes	—
111	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₇₁	#137	1	2	yes	—
114	C₇₁⋊C₈	—	4	4	no	—
115	C₇₁⋊C₈	—	4	4	no	—
116	C₂²×C₇₁	#391	1	1	yes	characteristic
117	C₂²×C₇₁	#407	1	1	yes	characteristic
118	C₂²×C₇₁	#408	1	1	yes	characteristic
119	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
120	C₂²×C₇₁	#419	1	1	yes	characteristic
121	C₂²×C₇₁	C₂×#104	1	1	yes	characteristic
122	C₂²×C₇₁	#421	1	1	yes	characteristic
123	C₄×C₇₁	—	2	1	no	—
124	C₄×C₇₁	—	2	1	no	—
125	C₄×C₇₁	—	2	2	no	—
126	C₄×C₇₁	—	2	2	no	—
127	C₄×C₇₁	—	2	2	no	—
128	C₄×C₇₁	—	4	4	no	—
129	C₄×C₇₁	—	4	1	no	—
130	C₄×C₇₁	—	8	2	no	—
131	C₄×C₇₁	#446	1	4	yes	—
132	C₄×C₇₁	#474	1	2	yes	—
133	C₄×C₇₁	#475	1	2	yes	—
134	#9869	—	71	1	no	maximal, Sylow
135	C₂×C₇₁	#1486	1	1	yes	characteristic
136	C₂×C₇₁	#1491	1	1	yes	characteristic
137	C₂×C₇₁	#1492	1	1	yes	characteristic
138	C₂×C₇₁	#1497	1	1	yes	characteristic
139	C₂×C₇₁	#1985	1	1	yes	characteristic
140	C₂×C₇₁	#1986	1	1	yes	characteristic
141	C₂×C₇₁	#1988	1	1	yes	characteristic
142	C₄×C₄⋊Q₈	D₇₁	1	1	yes	characteristic
143	#1767	—	71	1	no	—
144	C₄×C₄⋊C₈	—	71	1	no	—
145	#1279	—	71	4	no	—
146	C₇₁	#9869	1	1	yes	characteristic, complemented, Sylow
147	C₂×C₄⋊Q₈	Dic₇₁	1	1	yes	characteristic
148	C₂×C₄×Q₈	—	2	1	no	—
149	C₂×C₄⋊C₈	—	71	2	no	—
150	C₂×C₄⋊C₈	—	142	1	no	—
151	C₂×#97	—	142	1	no	—
152	C₂×C₄×C₈	—	71	2	no	—
153	#391	—	2	1	no	—
154	#389	—	2	1	no	—
155	#389	Dic₇₁	1	2	yes	—
156	#384	Dic₇₁	1	1	yes	characteristic
157	C₄×C₄⋊C₄	—	2	1	no	—
158	C₄×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
159	C₄³	C₂×D₇₁	1	1	yes	characteristic
160	#336	—	142	1	no	—
161	#336	—	142	1	no	—
162	C₂²×Q₈	—	2	1	no	—
163	C₂²×C₈	—	142	2	no	—
164	C₂²×C₈	—	142	2	no	—
165	C₄⋊Q₈	—	2	4	no	—
166	C₄×Q₈	—	4	2	no	—
167	C₄×Q₈	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
176	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
182	C₄⋊C₈	—	71	4	no	—
183	C₄⋊C₈	—	142	4	no	—
184	C₄⋊C₈	—	142	2	no	—
185	#97	—	284	2	no	—
186	C₄×C₈	—	142	4	no	—
187	#95	—	2	2	no	—
188	#95	—	2	2	no	—
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
196	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
201	C₂×C₈	—	284	2	no	—
202	C₂×C₈	—	284	2	no	—
203	C₂×C₈	—	284	2	no	—
204	C₂×C₈	—	284	2	no	—
205	C₂×C₈	—	142	4	no	—
206	C₂×C₈	—	142	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	#11765849	1	4	yes	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	4	2	no	—
220	C₄²	—	4	2	no	—
221	C₄²	#11765850	1	2	yes	—
222	C₄²	#11765853	1	2	yes	—
223	C₄²	Q₈×D₇₁	1	2	yes	—
224	C₄²	#11765866	1	2	yes	—
225	C₂³	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
226	Q₈	—	4	8	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	#14556049	1	2	yes	—
245	C₂×C₄	#14556049	1	2	yes	—
246	C₂×C₄	#14556055	1	2	yes	—
247	C₂×C₄	Q₈×Dic₇₁	1	2	yes	—
248	C₂×C₄	#14556104	1	2	yes	—
249	C₂×C₄	#14556117	1	1	yes	characteristic
250	C₂×C₄	#14556126	1	1	yes	characteristic
251	C₂×C₄	C₂×#11765849	1	2	yes	—
252	C₂×C₄	#14556135	1	2	yes	—
253	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
254	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
255	C₂×C₄	#14556159	1	2	yes	—
256	C₈	—	284	4	no	—
257	C₈	—	284	4	no	—
258	C₂²	#17985906	1	1	yes	characteristic
259	C₂²	#17985963	1	1	yes	characteristic
260	C₂²	#17985966	1	1	yes	characteristic
261	C₂²	C₂×#14556049	1	1	yes	characteristic
262	C₂²	#17986008	1	1	yes	characteristic
263	C₂²	C₂×#14556055	1	1	yes	characteristic
264	C₂²	#17986016	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	4	no	—
271	C₄	—	4	1	no	—
272	C₄	—	8	2	no	—
273	C₄	#17986130	1	4	yes	—
274	C₄	#17986276	1	2	yes	—
275	C₄	#17986281	1	2	yes	—
276	C₂	#22586765	1	1	yes	characteristic
277	C₂	#22586787	1	1	yes	characteristic
278	C₂	#22586791	1	1	yes	characteristic
279	C₂	#22586813	1	1	yes	characteristic
280	C₂	#22589281	1	1	yes	characteristic
281	C₂	#22589285	1	1	yes	characteristic
282	C₂	#22589293	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₁
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×C₄⋊Q₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×C₄⋊Q₈	D₇₁
hypocenter	C₇₁	#9869
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₄⋊(C₇₁⋊C₈)	Q₈	semidirect
146. C₇₁	#9869	semidirect

Maximal subgroups

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

subgroup	index
#22588193	2
C₄×C₄⋊(C₇₁⋊C₈)	2
#22586181	2
C₄×C₄⋊Q₈×C₇₁	2
#9869	71