Subgroups — C₂²×C₄⋊(C₇₁⋊C₁₆) #32150427

Order 18176.

862 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇₁⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₁₆	—	2	8	no	—
5	C₂²×C₇₁⋊C₁₆	C₂²	1	12	yes	—
6	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	16	yes	direct product factor
7	C₂³×C₈×C₇₁	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₇₁	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₁	C₄	1	6	yes	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	6	yes	—
12	C₂×C₇₁⋊C₁₆	—	2	24	no	—
13	C₂×C₇₁⋊C₁₆	C₂³	1	8	yes	—
14	C₂³×C₄×C₇₁	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₇₁	C₈	1	1	yes	characteristic
16	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₇₁	—	2	4	no	—
18	C₂²×C₈×C₇₁	C₂×C₄	1	6	yes	—
19	C₂²×C₈×C₇₁	C₂×C₄	1	3	yes	—
20	C₂²×C₈×C₇₁	D₄	1	4	yes	—
21	C₂²×C₈×C₇₁	Q₈	1	4	yes	—
22	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
23	C₂×C₄²×C₇₁	C₈	1	6	yes	—
24	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
25	C₄×C₈×C₇₁	C₂×C₄	1	12	yes	—
26	C₄×C₈×C₇₁	C₂³	1	4	yes	—
27	C₇₁⋊C₁₆	—	2	8	no	—
28	C₂⁴×C₇₁	C₂×C₈	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	—	2	4	no	—
30	C₂²×C₄×C₇₁	C₁₆	1	8	yes	complemented
31	C₂²×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
32	C₂²×C₄×C₇₁	C₂×C₈	1	6	yes	—
33	C₂²×C₄×C₇₁	C₂×C₈	1	3	yes	—
34	C₂²×C₄×C₇₁	C₂×C₈	1	6	yes	—
35	C₂²×C₄×C₇₁	M₁₆	1	4	yes	—
36	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
37	C₂×C₈×C₇₁	—	2	12	no	—
38	C₂×C₈×C₇₁	C₄⋊C₄	1	12	yes	—
39	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
40	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
41	C₂×C₈×C₇₁	C₂×D₄	1	6	yes	—
42	C₂×C₈×C₇₁	C₂×Q₈	1	6	yes	—
43	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
44	C₄²×C₇₁	C₂×C₈	1	12	yes	—
45	C₄²×C₇₁	C₂²×C₄	1	4	yes	—
46	C₂³×C₇₁	C₄⋊C₈	1	4	yes	—
47	C₂³×C₇₁	C₂×C₁₆	1	4	yes	—
48	C₂³×C₇₁	M₃₂	1	4	yes	—
49	C₂³×C₇₁	C₂²×C₈	1	3	yes	—
50	C₂×C₄×C₇₁	—	2	12	no	—
51	C₂×C₄×C₇₁	C₄⋊C₈	1	12	yes	—
52	C₂×C₄×C₇₁	C₂×C₁₆	1	24	yes	complemented
53	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
54	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
55	C₂×C₄×C₇₁	C₂²×C₈	1	3	yes	—
56	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
57	C₂×C₄×C₇₁	C₂×M₁₆	1	6	yes	—
58	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
59	C₈×C₇₁	—	2	4	no	—
60	C₈×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
61	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
62	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
63	C₂²×C₇₁	C₄⋊C₁₆	1	16	yes	complemented
64	C₂²×C₇₁	C₂×C₄⋊C₈	1	6	yes	—
65	C₂²×C₇₁	C₂²×C₁₆	1	6	yes	—
66	C₂²×C₇₁	C₂×M₃₂	1	6	yes	—
67	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
68	C₄×C₇₁	—	2	4	no	—
69	C₄×C₇₁	C₂×C₄⋊C₈	1	6	yes	—
70	C₄×C₇₁	C₂²×C₁₆	1	8	yes	complemented
71	C₄×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₄×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
73	C₂²×C₄⋊C₁₆	—	71	1	no	maximal, Sylow
74	C₂×C₇₁	C₂×C₄⋊C₁₆	1	12	yes	complemented
75	C₂×C₇₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₇₁	C₂³×C₁₆	1	1	yes	characteristic
77	C₂×C₇₁	C₂²×M₃₂	1	1	yes	characteristic
78	C₂³×C₁₆	—	71	2	no	—
79	C₂²×C₄×C₈	D₇₁	1	1	yes	characteristic
80	C₂×C₄⋊C₁₆	—	71	12	no	—
81	C₇₁	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₈	Dic₇₁	1	1	yes	characteristic
83	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
84	C₂²×C₄²	Dic₇₁	1	1	yes	characteristic
85	C₂²×C₁₆	—	71	12	no	—
86	C₂²×C₁₆	—	142	8	no	—
87	C₂×C₄×C₈	Dic₇₁	1	6	yes	—
88	C₂×C₄×C₈	C₂×D₇₁	1	6	yes	—
89	C₄⋊C₁₆	—	71	16	no	—
90	C₂³×C₄	C₇₁⋊C₈	1	1	yes	characteristic
91	C₂³×C₄	C₇₁⋊C₈	1	1	yes	characteristic
92	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
93	C₂²×C₈	—	2	4	no	—
94	C₂²×C₈	Dic₁₄₂	1	4	yes	—
95	C₂²×C₈	D₂₈₄	1	4	yes	—
96	C₂²×C₈	C₂×Dic₇₁	1	6	yes	—
97	C₂²×C₈	C₂×Dic₇₁	1	3	yes	—
98	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
99	C₂×C₄²	C₇₁⋊C₈	1	6	yes	—
100	C₂×C₄²	C₂×Dic₇₁	1	6	yes	—
101	C₂×C₁₆	—	71	8	no	—
102	C₂×C₁₆	—	142	24	no	—
103	C₄×C₈	C₂×Dic₇₁	1	12	yes	—
104	C₄×C₈	C₂²×D₇₁	1	4	yes	—
105	C₂⁴	C₂×C₇₁⋊C₈	1	1	yes	characteristic
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	C₇₁⋊C₁₆	1	8	yes	complemented
108	C₂²×C₄	C₄⋊Dic₇₁	1	4	yes	—
109	C₂²×C₄	C₂×C₇₁⋊C₈	1	6	yes	—
110	C₂²×C₄	C₂×C₇₁⋊C₈	1	3	yes	—
111	C₂²×C₄	C₂×C₇₁⋊C₈	1	6	yes	—
112	C₂²×C₄	#11765849	1	4	yes	—
113	C₂²×C₄	C₂²×Dic₇₁	1	3	yes	—
114	C₂×C₈	—	2	12	no	—
115	C₂×C₈	C₄⋊Dic₇₁	1	12	yes	—
116	C₂×C₈	C₂²×Dic₇₁	1	4	yes	—
117	C₂×C₈	C₂²×Dic₇₁	1	3	yes	—
118	C₂×C₈	C₂×D₂₈₄	1	6	yes	—
119	C₂×C₈	C₂×Dic₁₄₂	1	6	yes	—
120	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
121	C₄²	C₂×C₇₁⋊C₈	1	12	yes	—
122	C₄²	C₂²×Dic₇₁	1	4	yes	—
123	C₁₆	—	142	8	no	—
124	C₂³	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
125	C₂³	C₂×C₇₁⋊C₁₆	1	4	yes	—
126	C₂³	#14556068	1	4	yes	—
127	C₂³	C₂²×C₇₁⋊C₈	1	3	yes	—
128	C₂×C₄	—	2	12	no	—
129	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	12	yes	—
130	C₂×C₄	C₂×C₇₁⋊C₁₆	1	24	yes	complemented
131	C₂×C₄	C₂×C₄⋊Dic₇₁	1	6	yes	—
132	C₂×C₄	C₂²×C₇₁⋊C₈	1	4	yes	—
133	C₂×C₄	C₂²×C₇₁⋊C₈	1	4	yes	—
134	C₂×C₄	C₂²×C₇₁⋊C₈	1	3	yes	—
135	C₂×C₄	C₂×#11765849	1	6	yes	—
136	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
137	C₈	—	2	4	no	—
138	C₈	C₂×C₄⋊Dic₇₁	1	6	yes	—
139	C₈	C₂²×D₂₈₄	1	1	yes	characteristic
140	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
141	C₂²	C₄⋊(C₇₁⋊C₁₆)	1	16	yes	direct product factor
142	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	6	yes	—
143	C₂²	C₂²×C₇₁⋊C₁₆	1	6	yes	—
144	C₂²	C₂×#14556068	1	6	yes	—
145	C₂²	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	6	yes	—
148	C₄	C₂²×C₇₁⋊C₁₆	1	8	yes	complemented
149	C₄	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
150	C₄	C₂²×#11765849	1	1	yes	characteristic
151	C₂	C₂×C₄⋊(C₇₁⋊C₁₆)	1	12	yes	direct product factor
152	C₂	C₂²×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
153	C₂	C₂³×C₇₁⋊C₁₆	1	1	yes	characteristic
154	C₂	C₂²×#14556068	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₇₁
commutator	C₂×C₇₁	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂²×C₄×C₈×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₂×C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂²×C₄×C₈	D₇₁
hypocenter	C₇₁	C₂²×C₄⋊C₁₆
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	direct (N × H)
6. C₄⋊(C₇₁⋊C₁₆)	C₂²	direct (N × H)
30. C₂²×C₄×C₇₁	C₁₆	semidirect
52. C₂×C₄×C₇₁	C₂×C₁₆	semidirect
63. C₂²×C₇₁	C₄⋊C₁₆	semidirect
70. C₄×C₇₁	C₂²×C₁₆	semidirect
74. C₂×C₇₁	C₂×C₄⋊C₁₆	semidirect
81. C₇₁	C₂²×C₄⋊C₁₆	semidirect
107. C₂²×C₄	C₇₁⋊C₁₆	semidirect
130. C₂×C₄	C₂×C₇₁⋊C₁₆	semidirect
141. C₂²	C₄⋊(C₇₁⋊C₁₆)	direct (N × H)
148. C₄	C₂²×C₇₁⋊C₁₆	semidirect
151. C₂	C₂×C₄⋊(C₇₁⋊C₁₆)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇₁⋊C₁₆	2
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₂²×C₄×C₈×C₇₁	2
C₂²×C₄⋊C₁₆	71

Subgroups — C22×C4⋊(C71⋊C16) #32150427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄⋊(C₇₁⋊C₁₆) #32150427