Subgroups — C₂³×Dic₇₁⋊C₈ #32861059

Order 18176.

3870 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₇₁⋊C₈	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₇₁	C₄	1	1	yes	characteristic
6	C₂⁴×Dic₇₁	C₄	1	1	yes	characteristic
7	C₂³×C₇₁⋊C₈	—	2	8	no	—
8	C₂³×C₇₁⋊C₈	C₂²	1	14	yes	—
9	C₂²×C₄×Dic₇₁	C₄	1	14	yes	—
10	C₂²×C₄×Dic₇₁	C₂²	1	14	yes	—
11	C₂×Dic₇₁⋊C₈	C₂²	1	112	yes	direct product factor
12	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₇₁	—	2	8	no	—
14	C₂³×C₈×C₇₁	C₂²	1	14	yes	—
15	C₂³×Dic₇₁	—	2	8	no	—
16	C₂³×Dic₇₁	C₈	1	16	yes	complemented
17	C₂³×Dic₇₁	C₂×C₄	1	14	yes	—
18	C₂³×Dic₇₁	C₂×C₄	1	14	yes	—
19	C₂²×C₇₁⋊C₈	—	2	56	no	—
20	C₂²×C₇₁⋊C₈	C₂³	1	28	yes	—
21	C₂×C₄×Dic₇₁	C₂×C₄	1	84	yes	—
22	C₂×C₄×Dic₇₁	C₂³	1	28	yes	—
23	Dic₇₁⋊C₈	C₂³	1	64	yes	direct product factor
24	C₂⁵×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₇₁	C₂×C₄	1	7	yes	—
26	C₂³×C₄×C₇₁	D₄	1	8	yes	—
27	C₂³×C₄×C₇₁	Q₈	1	8	yes	—
28	C₂³×C₄×C₇₁	C₂³	1	7	yes	—
29	C₂²×C₈×C₇₁	—	2	56	no	—
30	C₂²×C₈×C₇₁	C₂³	1	28	yes	—
31	C₂²×Dic₇₁	—	2	56	no	—
32	C₂²×Dic₇₁	C₂×C₈	1	112	yes	complemented
33	C₂²×Dic₇₁	C₂²×C₄	1	28	yes	—
34	C₂²×Dic₇₁	C₂²×C₄	1	28	yes	—
35	C₂×C₇₁⋊C₈	—	2	56	no	—
36	C₂×C₇₁⋊C₈	C₂⁴	1	8	yes	—
37	C₄×Dic₇₁	C₂²×C₄	1	56	yes	—
38	C₄×Dic₇₁	C₂⁴	1	8	yes	—
39	C₂⁴×C₇₁	C₄⋊C₄	1	8	yes	—
40	C₂⁴×C₇₁	C₂×C₈	1	8	yes	—
41	C₂⁴×C₇₁	M₁₆	1	8	yes	—
42	C₂⁴×C₇₁	C₂²×C₄	1	7	yes	—
43	C₂²×C₄×C₇₁	C₄⋊C₄	1	56	yes	—
44	C₂²×C₄×C₇₁	C₂²×C₄	1	21	yes	—
45	C₂²×C₄×C₇₁	C₂×D₄	1	28	yes	—
46	C₂²×C₄×C₇₁	C₂×Q₈	1	28	yes	—
47	C₂²×C₄×C₇₁	C₂⁴	1	7	yes	—
48	C₂×C₈×C₇₁	—	2	56	no	—
49	C₂×C₈×C₇₁	C₂⁴	1	8	yes	—
50	C₂³×C₇₁	C₄⋊C₈	1	64	yes	complemented
51	C₂³×C₇₁	C₂×C₄⋊C₄	1	28	yes	—
52	C₂³×C₇₁	C₂²×C₈	1	28	yes	—
53	C₂³×C₇₁	C₂×M₁₆	1	28	yes	—
54	C₂³×C₇₁	C₂³×C₄	1	7	yes	—
55	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	84	yes	—
56	C₂×C₄×C₇₁	C₂³×C₄	1	7	yes	—
57	C₂×C₄×C₇₁	C₂²×D₄	1	14	yes	—
58	C₂×C₄×C₇₁	C₂²×Q₈	1	14	yes	—
59	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
60	C₂×Dic₇₁	—	2	56	no	—
61	C₂×Dic₇₁	C₂²×C₈	1	112	yes	complemented
62	C₂×Dic₇₁	C₂³×C₄	1	8	yes	—
63	C₂×Dic₇₁	C₂³×C₄	1	8	yes	—
64	C₈×C₇₁	—	2	8	no	—
65	C₇₁⋊C₈	—	2	8	no	—
66	C₂²×C₇₁	C₂×C₄⋊C₈	1	112	yes	complemented
67	C₂²×C₇₁	C₂²×C₄⋊C₄	1	14	yes	—
68	C₂²×C₇₁	C₂³×C₈	1	14	yes	—
69	C₂²×C₇₁	C₂²×M₁₆	1	14	yes	—
70	C₂²×C₇₁	C₂⁴×C₄	1	1	yes	characteristic
71	C₄×C₇₁	C₂²×C₄⋊C₄	1	14	yes	—
72	C₄×C₇₁	C₂³×D₄	1	1	yes	characteristic
73	C₄×C₇₁	C₂³×Q₈	1	1	yes	characteristic
74	Dic₇₁	—	2	8	no	—
75	Dic₇₁	C₂³×C₈	1	16	yes	complemented
76	C₂³×C₄⋊C₈	—	71	1	no	maximal, Sylow
77	C₂×C₇₁	C₂²×C₄⋊C₈	1	28	yes	complemented
78	C₂×C₇₁	C₂³×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₇₁	C₂⁴×C₈	1	1	yes	characteristic
80	C₂×C₇₁	C₂³×M₁₆	1	1	yes	characteristic
81	C₂⁴×C₈	—	71	1	no	—
82	C₂⁴×C₈	D₇₁	1	1	yes	characteristic
83	C₂³×C₄²	—	71	1	no	—
84	C₂²×C₄⋊C₈	—	71	28	no	—
85	C₇₁	C₂³×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
86	C₂⁴×C₄	—	71	1	no	—
87	C₂⁴×C₄	—	71	1	no	—
88	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
89	C₂³×C₈	—	2	8	no	—
90	C₂³×C₈	—	71	14	no	—
91	C₂³×C₈	—	142	8	no	—
92	C₂³×C₈	C₂×D₇₁	1	14	yes	—
93	C₂²×C₄²	—	71	14	no	—
94	C₂²×C₄²	—	71	14	no	—
95	C₂×C₄⋊C₈	—	71	112	no	—
96	C₂⁵	C₄×D₇₁	1	1	yes	characteristic
97	C₂³×C₄	—	71	16	no	—
98	C₂³×C₄	—	71	14	no	—
99	C₂³×C₄	—	71	14	no	—
100	C₂³×C₄	—	142	8	no	—
101	C₂³×C₄	Dic₁₄₂	1	8	yes	—
102	C₂³×C₄	C₄×D₇₁	1	7	yes	—
103	C₂³×C₄	C₂²⋊D₇₁	1	8	yes	—
104	C₂³×C₄	C₂²×D₇₁	1	7	yes	—
105	C₂²×C₈	—	2	56	no	—
106	C₂²×C₈	—	71	28	no	—
107	C₂²×C₈	—	142	56	no	—
108	C₂²×C₈	C₂²×D₇₁	1	28	yes	—
109	C₂×C₄²	—	71	84	no	—
110	C₂×C₄²	—	71	28	no	—
111	C₄⋊C₈	—	71	64	no	—
112	C₂⁴	Dic₇₁⋊C₄	1	8	yes	—
113	C₂⁴	C₈×D₇₁	1	8	yes	—
114	C₂⁴	#11765848	1	8	yes	—
115	C₂⁴	C₂×C₄×D₇₁	1	7	yes	—
116	C₂²×C₄	—	71	28	no	—
117	C₂²×C₄	—	71	112	no	—
118	C₂²×C₄	—	71	28	no	—
119	C₂²×C₄	—	142	56	no	—
120	C₂²×C₄	Dic₇₁⋊C₄	1	56	yes	—
121	C₂²×C₄	C₂×C₄×D₇₁	1	21	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₇₁	1	28	yes	—
123	C₂²×C₄	C₂×Dic₁₄₂	1	28	yes	—
124	C₂²×C₄	C₂³×D₇₁	1	7	yes	—
125	C₂×C₈	—	2	56	no	—
126	C₂×C₈	—	71	8	no	—
127	C₂×C₈	—	142	56	no	—
128	C₂×C₈	C₂³×D₇₁	1	8	yes	—
129	C₄²	—	71	56	no	—
130	C₄²	—	71	8	no	—
131	C₂³	Dic₇₁⋊C₈	1	64	yes	direct product factor
132	C₂³	C₂×Dic₇₁⋊C₄	1	28	yes	—
133	C₂³	C₂×C₈×D₇₁	1	28	yes	—
134	C₂³	C₂×#11765848	1	28	yes	—
135	C₂³	C₂²×C₄×D₇₁	1	7	yes	—
136	C₂×C₄	—	71	8	no	—
137	C₂×C₄	—	71	112	no	—
138	C₂×C₄	—	71	8	no	—
139	C₂×C₄	—	142	56	no	—
140	C₂×C₄	C₂×Dic₇₁⋊C₄	1	84	yes	—
141	C₂×C₄	C₂²×C₄×D₇₁	1	7	yes	—
142	C₂×C₄	C₂²×C₂²⋊D₇₁	1	14	yes	—
143	C₂×C₄	C₂²×Dic₁₄₂	1	14	yes	—
144	C₂×C₄	C₂⁴×D₇₁	1	1	yes	characteristic
145	C₈	—	2	8	no	—
146	C₈	—	142	8	no	—
147	C₂²	C₂×Dic₇₁⋊C₈	1	112	yes	direct product factor
148	C₂²	C₂²×Dic₇₁⋊C₄	1	14	yes	—
149	C₂²	C₂²×C₈×D₇₁	1	14	yes	—
150	C₂²	C₂²×#11765848	1	14	yes	—
151	C₂²	C₂³×C₄×D₇₁	1	1	yes	characteristic
152	C₄	—	71	16	no	—
153	C₄	—	142	8	no	—
154	C₄	C₂²×Dic₇₁⋊C₄	1	14	yes	—
155	C₄	C₂³×C₂²⋊D₇₁	1	1	yes	characteristic
156	C₄	C₂³×Dic₁₄₂	1	1	yes	characteristic
157	C₂	C₂²×Dic₇₁⋊C₈	1	28	yes	direct product factor
158	C₂	C₂³×Dic₇₁⋊C₄	1	1	yes	characteristic
159	C₂	C₂³×C₈×D₇₁	1	1	yes	characteristic
160	C₂	C₂³×#11765848	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₇₁
commutator	C₂×C₇₁	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₇₁
Fitting	C₂⁴×C₈×C₇₁	C₂
socle, prime radical	C₂⁵×C₇₁	C₂×C₄
cosocle	C₂×C₄×C₇₁	C₂⁵
hypercenter	C₂⁴×C₈	D₇₁
hypocenter	C₇₁	C₂³×C₄⋊C₈
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×Dic₇₁⋊C₈	C₂	direct (N × H)
11. C₂×Dic₇₁⋊C₈	C₂²	direct (N × H)
16. C₂³×Dic₇₁	C₈	semidirect
23. Dic₇₁⋊C₈	C₂³	direct (N × H)
32. C₂²×Dic₇₁	C₂×C₈	semidirect
50. C₂³×C₇₁	C₄⋊C₈	semidirect
61. C₂×Dic₇₁	C₂²×C₈	semidirect
66. C₂²×C₇₁	C₂×C₄⋊C₈	semidirect
75. Dic₇₁	C₂³×C₈	semidirect
77. C₂×C₇₁	C₂²×C₄⋊C₈	semidirect
85. C₇₁	C₂³×C₄⋊C₈	semidirect
131. C₂³	Dic₇₁⋊C₈	direct (N × H)
147. C₂²	C₂×Dic₇₁⋊C₈	direct (N × H)
157. C₂	C₂²×Dic₇₁⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇₁⋊C₈	2
C₂³×C₄×Dic₇₁	2
C₂²×Dic₇₁⋊C₈	2
C₂⁴×C₈×C₇₁	2
C₂³×C₄⋊C₈	71

Subgroups — C23×Dic71⋊C8 #32861059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×Dic₇₁⋊C₈ #32861059