Subgroups — #31912761

Order 18176.

414 conjugacy classes of subgroups.

The subgroup lattice has 204 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	2	yes	maximal
2	C₂×#17985775	C₂	1	2	yes	maximal
3	#22586418	C₂	1	2	yes	maximal
4	C₄²×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	4	yes	—
6	C₄⋊(C₇₁⋊C₁₆)	C₄	1	8	yes	complemented
7	#17985775	—	2	4	no	—
8	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₄³×C₇₁	C₄	1	1	yes	characteristic
15	C₂×C₇₁⋊C₁₆	—	2	4	no	—
16	C₂×C₇₁⋊C₁₆	—	2	4	no	—
17	C₂×C₇₁⋊C₁₆	C₂×C₄	1	8	yes	—
18	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₇₁	C₈	1	2	yes	—
23	C₂×C₄²×C₇₁	C₈	1	2	yes	—
24	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₁	—	2	2	no	—
28	C₄×C₈×C₇₁	—	2	2	no	—
29	C₄×C₈×C₇₁	—	2	2	no	—
30	C₄×C₈×C₇₁	C₈	1	4	yes	—
31	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₇₁	D₄	1	2	yes	—
33	C₄×C₈×C₇₁	Q₈	1	2	yes	—
34	C₇₁⋊C₁₆	—	2	8	no	—
35	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	—	2	1	no	—
42	C₂×C₈×C₇₁	—	2	1	no	—
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	C₄²	1	2	yes	—
45	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
49	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₁	#41	1	2	yes	—
53	C₄²×C₇₁	—	2	4	no	—
54	C₄²×C₇₁	—	2	4	no	—
55	C₄²×C₇₁	—	2	1	no	—
56	C₄²×C₇₁	—	2	1	no	—
57	C₄²×C₇₁	C₄²	1	2	yes	—
58	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
59	C₄²×C₇₁	C₂×C₈	1	2	yes	—
60	C₄²×C₇₁	M₁₆	1	2	yes	—
61	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	—	2	2	no	—
63	C₂×C₄×C₇₁	—	2	2	no	—
64	C₂×C₄×C₇₁	—	2	2	no	—
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	—	2	2	no	—
67	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
68	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₇₁	C₄⋊C₈	1	4	yes	—
70	C₂×C₄×C₇₁	M₃₂	1	4	yes	—
71	C₂×C₄×C₇₁	M₃₂	1	4	yes	—
72	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
78	C₈×C₇₁	—	2	2	no	—
79	C₈×C₇₁	—	2	2	no	—
80	C₈×C₇₁	C₄⋊C₈	1	4	yes	—
81	C₈×C₇₁	C₄×D₄	1	2	yes	—
82	C₈×C₇₁	C₄×Q₈	1	2	yes	—
83	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
84	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₇₁	#431	1	1	yes	characteristic
86	C₂²×C₇₁	C₂×M₃₂	1	2	yes	—
87	C₂²×C₇₁	C₂×M₃₂	1	2	yes	—
88	C₄×C₇₁	—	2	2	no	—
89	C₄×C₇₁	—	2	4	no	—
90	C₄×C₇₁	—	2	4	no	—
91	C₄×C₇₁	#346	1	4	yes	complemented
92	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
94	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄×C₇₁	#431	1	1	yes	characteristic
96	#12653	—	71	1	no	maximal, Sylow
97	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
98	C₂×C₇₁	C₂×#346	1	1	yes	characteristic
99	C₂×C₇₁	C₄×M₃₂	1	1	yes	characteristic
100	C₂×C₇₁	#2150	1	2	yes	—
101	C₂×C₇₁	#2163	1	2	yes	—
102	C₂×C₄⋊C₁₆	—	71	2	no	—
103	C₂×#346	—	71	2	no	—
104	C₄²×C₈	D₇₁	1	1	yes	characteristic
105	#1374	—	71	2	no	—
106	C₇₁	#12653	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₁₆	—	71	4	no	—
108	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
109	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
110	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
114	C₄³	Dic₇₁	1	1	yes	characteristic
115	C₄⋊C₁₆	—	71	8	no	—
116	#346	—	142	4	no	—
117	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
119	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
121	C₂×C₄²	C₇₁⋊C₈	1	2	yes	—
122	C₂×C₄²	C₇₁⋊C₈	1	2	yes	—
123	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
124	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
125	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
126	C₂×C₁₆	—	71	8	no	—
127	C₂×C₁₆	—	142	4	no	—
128	C₂×C₁₆	—	142	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	C₇₁⋊C₈	1	4	yes	—
133	C₄×C₈	Dic₁₄₂	1	2	yes	—
134	C₄×C₈	C₄×D₇₁	1	4	yes	—
135	C₄×C₈	D₂₈₄	1	2	yes	—
136	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
138	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
139	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
146	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
147	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
148	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
149	C₂×C₈	#11765849	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
151	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
153	C₂×C₈	#11765865	1	2	yes	—
154	C₄²	—	2	4	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	C₄×Dic₇₁	1	2	yes	—
159	C₄²	C₄⋊Dic₇₁	1	2	yes	—
160	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
161	C₄²	#11765849	1	2	yes	—
162	C₁₆	—	142	8	no	—
163	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	C₄×C₇₁⋊C₈	1	2	yes	—
170	C₂×C₄	C₄×C₇₁⋊C₈	1	2	yes	—
171	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
172	C₂×C₄	#14556068	1	4	yes	—
173	C₂×C₄	#14556068	1	4	yes	—
174	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
176	C₂×C₄	#14556087	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
178	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
179	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
183	C₈	C₄×D₂₈₄	1	2	yes	—
184	C₈	C₄×Dic₁₄₂	1	2	yes	—
185	C₂²	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
186	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
187	C₂²	#17986052	1	1	yes	characteristic
188	C₂²	C₂×#14556068	1	2	yes	—
189	C₂²	C₂×#14556068	1	2	yes	—
190	C₄	—	2	4	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	#17985775	1	4	yes	complemented
194	C₄	C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
195	C₄	C₄×#11765849	1	1	yes	characteristic
196	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
197	C₄	#17986052	1	1	yes	characteristic
198	C₂	C₄×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
199	C₂	C₂×#17985775	1	1	yes	characteristic
200	C₂	C₄×#14556068	1	1	yes	characteristic
201	C₂	#22590096	1	2	yes	—
202	C₂	#22590143	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄×C₈
Fitting	C₄²×C₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₄²×C₈	D₇₁
hypocenter	C₇₁	#12653
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
6. C₄⋊(C₇₁⋊C₁₆)	C₄	semidirect
91. C₄×C₇₁	#346	semidirect
106. C₇₁	#12653	semidirect
193. C₄	#17985775	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
C₂×#17985775	2
#22586418	2
C₄²×C₈×C₇₁	2
#12653	71