Subgroups — #31908718

Order 18176.

306 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#22587088	C₂	1	1	yes	maximal, characteristic
3	#22586392	C₂	1	4	yes	maximal, complemented
4	C₇₁×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
6	#17986052	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
9	#17985752	—	2	1	no	—
10	#17985750	C₂²	1	2	yes	—
11	#533×C₇₁	—	2	1	no	—
12	#500×C₇₁	C₄	1	1	yes	characteristic
13	#500×C₇₁	C₄	1	1	yes	characteristic
14	#499×C₇₁	—	2	1	no	—
15	#499×C₇₁	C₂²	1	2	yes	—
16	#498×C₇₁	—	2	1	no	—
17	#427×C₇₁	—	2	1	no	—
18	#427×C₇₁	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₇₁⋊C₁₆	—	2	2	no	—
21	C₂×C₇₁⋊C₁₆	—	4	2	no	—
22	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
23	C₂²⋊(C₇₁⋊C₈)	—	4	1	no	—
24	C₄×C₇₁⋊C₈	—	4	1	no	—
25	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₄⋊Q₈×C₇₁	—	2	1	no	—
28	C₄⋊Q₈×C₇₁	—	2	1	no	—
29	#125×C₇₁	—	2	2	no	—
30	C₂²⋊Q₈×C₇₁	—	4	1	no	—
31	C₂²⋊Q₈×C₇₁	—	4	1	no	—
32	C₄×Q₈×C₇₁	—	4	1	no	—
33	C₄×Q₈×C₇₁	—	4	1	no	—
34	#117×C₇₁	—	2	2	no	—
35	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
36	#107×C₇₁	—	2	1	no	—
37	#107×C₇₁	—	2	1	no	—
38	#107×C₇₁	C₂×C₄	1	2	yes	—
39	#107×C₇₁	C₂×C₄	1	2	yes	—
40	#106×C₇₁	—	2	2	no	—
41	#106×C₇₁	—	2	2	no	—
42	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₂×C₇₁⋊C₈	—	4	2	no	—
47	C₇₁⋊C₁₆	—	4	2	no	—
48	C₂×Q₈×C₇₁	—	4	1	no	—
49	C₂×Q₈×C₇₁	—	4	1	no	—
50	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₇₁	—	2	2	no	—
54	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
57	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
58	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₇₁	—	2	2	no	—
61	C₄⋊C₄×C₇₁	—	2	2	no	—
62	C₄⋊C₄×C₇₁	—	4	1	no	—
63	C₄⋊C₄×C₇₁	—	4	1	no	—
64	C₄⋊C₄×C₇₁	—	4	2	no	—
65	C₄⋊C₄×C₇₁	—	4	2	no	—
66	C₂²⋊C₄×C₇₁	—	4	2	no	—
67	C₄²×C₇₁	—	4	2	no	—
68	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₇₁	D₈	1	1	yes	characteristic
70	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
71	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₂³×C₇₁	#102	1	1	yes	characteristic
73	Q₈×C₇₁	—	8	2	no	—
74	Q₈×C₇₁	—	8	2	no	—
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	—	4	1	no	—
77	C₂×C₄×C₇₁	—	4	2	no	—
78	C₂×C₄×C₇₁	—	4	2	no	—
79	C₂×C₄×C₇₁	#101	1	2	yes	—
80	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
82	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
83	C₂×C₄×C₇₁	#103	1	2	yes	—
84	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
87	C₈×C₇₁	—	2	1	no	—
88	C₈×C₇₁	—	2	1	no	—
89	C₈×C₇₁	#103	1	2	yes	—
90	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
91	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
92	C₇₁⋊C₈	—	8	2	no	—
93	C₂²×C₇₁	—	2	1	no	—
94	C₂²×C₇₁	—	2	1	no	—
95	C₂²×C₇₁	#332	1	2	yes	—
96	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
97	C₄×C₇₁	—	2	1	no	—
98	C₄×C₇₁	—	2	1	no	—
99	C₄×C₇₁	—	2	1	no	—
100	C₄×C₇₁	—	8	2	no	—
101	C₄×C₇₁	—	8	2	no	—
102	C₄×C₇₁	#332	1	2	yes	—
103	C₄×C₇₁	#359	1	2	yes	—
104	C₄×C₇₁	C₂×#101	1	1	yes	characteristic
105	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
106	#11934	—	71	1	no	maximal, Sylow
107	C₂×C₇₁	—	2	2	no	—
108	C₂×C₇₁	C₂×#332	1	1	yes	characteristic
109	C₂×C₇₁	C₂×#359	1	1	yes	characteristic
110	C₂×C₇₁	#2145	1	1	yes	characteristic
111	#3160	D₇₁	1	1	yes	characteristic
112	C₂×C₄⋊C₁₆	—	71	1	no	—
113	#1564	—	71	1	no	—
114	#1363	—	71	4	no	—
115	C₇₁	#11934	1	1	yes	characteristic, complemented, Sylow
116	#533	—	2	1	no	—
117	C₂²×C₁₆	—	142	1	no	—
118	#500	Dic₇₁	1	1	yes	characteristic
119	#500	Dic₇₁	1	1	yes	characteristic
120	#499	—	2	1	no	—
121	#499	C₂×D₇₁	1	2	yes	—
122	#498	—	2	1	no	—
123	#431	—	142	1	no	—
124	#427	—	2	1	no	—
125	#427	Dic₇₁	1	2	yes	—
126	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
127	C₄⋊C₁₆	—	71	2	no	—
128	C₄⋊C₁₆	—	142	1	no	—
129	#335	—	142	1	no	—
130	#334	—	71	2	no	—
131	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
132	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
133	C₄⋊Q₈	—	2	1	no	—
134	C₄⋊Q₈	—	2	1	no	—
135	#125	—	2	2	no	—
136	C₂²⋊Q₈	—	4	1	no	—
137	C₂²⋊Q₈	—	4	1	no	—
138	C₄×Q₈	—	4	1	no	—
139	C₄×Q₈	—	4	1	no	—
140	#117	—	2	2	no	—
141	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
142	C₂×C₁₆	—	284	2	no	—
143	C₂×C₁₆	—	142	2	no	—
144	#107	—	2	1	no	—
145	#107	—	2	1	no	—
146	#107	C₂×Dic₇₁	1	2	yes	—
147	#107	C₂×Dic₇₁	1	2	yes	—
148	#106	—	2	2	no	—
149	#106	—	2	2	no	—
150	C₄⋊C₈	—	142	2	no	—
151	C₂²⋊C₈	—	284	1	no	—
152	C₄×C₈	—	284	1	no	—
153	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
156	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
157	C₂×Q₈	—	4	1	no	—
158	C₂×Q₈	—	4	1	no	—
159	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
160	C₂²×C₄	#11765850	1	1	yes	characteristic
161	C₂²×C₄	#11765853	1	1	yes	characteristic
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	284	2	no	—
164	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
165	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
166	C₂×C₈	#11765852	1	2	yes	—
167	C₂×C₈	#11765855	1	2	yes	—
168	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₂²⋊C₄	—	4	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
179	C₄²	#11765850	1	1	yes	characteristic
180	C₄²	#11765853	1	1	yes	characteristic
181	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
182	C₁₆	—	284	2	no	—
183	C₂³	#14556049	1	1	yes	characteristic
184	Q₈	—	8	2	no	—
185	Q₈	—	8	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	#14556047	1	2	yes	—
191	C₂×C₄	#14556049	1	1	yes	characteristic
192	C₂×C₄	#14556049	1	1	yes	characteristic
193	C₂×C₄	#14556049	1	1	yes	characteristic
194	C₂×C₄	#14556052	1	2	yes	—
195	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
196	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
197	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	568	2	no	—
201	C₈	#14556052	1	2	yes	—
202	C₈	C₂×#11765852	1	1	yes	characteristic
203	C₈	C₂×#11765855	1	1	yes	characteristic
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	#17985745	1	2	yes	—
207	C₂²	C₂×#14556049	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	8	2	no	—
212	C₄	—	8	2	no	—
213	C₄	#17985745	1	2	yes	—
214	C₄	#17985806	1	2	yes	—
215	C₄	C₂×#14556047	1	1	yes	characteristic
216	C₄	C₂×#14556052	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	C₂×#17985745	1	1	yes	characteristic
219	C₂	C₂×#17985806	1	1	yes	characteristic
220	C₂	#22590076	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556049
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#3160	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	#102
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#3160	D₇₁
hypocenter	C₇₁	#11934
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #22586392	C₂	semidirect
115. C₇₁	#11934	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
#22587088	2
#22586392	2
C₇₁×#3160	2
#11934	71