Subgroups — C₂×C₄×C₈×Dic₇₁ #31994771

Order 18176.

2046 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₇₁	C₂	1	4	yes	maximal
3	C₂×C₄²×Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	C₄×C₈×Dic₇₁	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₇₁⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₇₁	C₄	1	4	yes	—
8	C₂²×C₄×Dic₇₁	C₂²	1	2	yes	—
9	C₂×C₄×C₇₁⋊C₈	C₄	1	8	yes	complemented
10	C₂×C₄×C₇₁⋊C₈	C₂²	1	4	yes	—
11	C₂×C₈×Dic₇₁	C₄	1	32	yes	direct product factor
12	C₂×C₈×Dic₇₁	C₂²	1	16	yes	—
13	C₄²×Dic₇₁	C₄	1	4	yes	—
14	C₄²×Dic₇₁	C₂²	1	4	yes	—
15	C₂³×C₈×C₇₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₁	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₇₁	C₂²	1	4	yes	—
19	C₂³×Dic₇₁	C₂×C₄	1	4	yes	—
20	C₂²×C₇₁⋊C₈	C₂×C₄	1	16	yes	—
21	C₂²×C₇₁⋊C₈	C₂×C₄	1	8	yes	—
22	C₂²×C₇₁⋊C₈	C₂³	1	4	yes	—
23	C₂×C₄×Dic₇₁	C₈	1	32	yes	direct product factor
24	C₂×C₄×Dic₇₁	C₂×C₄	1	4	yes	—
25	C₂×C₄×Dic₇₁	C₂×C₄	1	16	yes	—
26	C₂×C₄×Dic₇₁	C₂×C₄	1	16	yes	—
27	C₂×C₄×Dic₇₁	C₂³	1	4	yes	—
28	C₄×C₇₁⋊C₈	C₂×C₄	1	16	yes	complemented
29	C₈×Dic₇₁	C₂×C₄	1	64	yes	direct product factor
30	C₂³×C₄×C₇₁	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂×C₄	1	16	yes	—
33	C₂²×C₈×C₇₁	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₇₁	C₂³	1	4	yes	—
35	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₇₁	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₇₁	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
39	C₄×C₈×C₇₁	C₂×C₄	1	16	yes	complemented
40	C₂²×Dic₇₁	C₄²	1	16	yes	—
41	C₂²×Dic₇₁	C₂×C₈	1	32	yes	—
42	C₂²×Dic₇₁	C₂²×C₄	1	8	yes	—
43	C₂×C₇₁⋊C₈	C₄²	1	32	yes	complemented
44	C₂×C₇₁⋊C₈	C₂²×C₄	1	16	yes	—
45	C₂×C₇₁⋊C₈	C₂²×C₄	1	8	yes	—
46	C₄×Dic₇₁	C₄²	1	16	yes	—
47	C₄×Dic₇₁	C₂×C₈	1	64	yes	direct product factor
48	C₄×Dic₇₁	C₂²×C₄	1	16	yes	—
49	C₂⁴×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₄²	1	8	yes	—
51	C₂²×C₄×C₇₁	C₂×C₈	1	16	yes	—
52	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₇₁	C₂²×C₄	1	8	yes	—
55	C₂²×C₄×C₇₁	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₄²	1	32	yes	complemented
59	C₂×C₈×C₇₁	C₂²×C₄	1	16	yes	—
60	C₂×C₈×C₇₁	C₂²×C₄	1	8	yes	—
61	C₄²×C₇₁	C₄²	1	8	yes	—
62	C₄²×C₇₁	C₂²×C₄	1	4	yes	—
63	C₄²×C₇₁	C₂²×C₄	1	4	yes	—
64	C₂³×C₇₁	C₂×C₄²	1	4	yes	—
65	C₂³×C₇₁	C₂×C₄²	1	2	yes	—
66	C₂³×C₇₁	C₂²×C₈	1	8	yes	—
67	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₄×C₈	1	32	yes	complemented
69	C₂×C₄×C₇₁	C₂×C₄²	1	4	yes	—
70	C₂×C₄×C₇₁	C₂×C₄²	1	8	yes	—
71	C₂×C₄×C₇₁	C₂×C₄²	1	2	yes	—
72	C₂×C₄×C₇₁	C₂×C₄²	1	8	yes	—
73	C₂×C₄×C₇₁	C₂×C₄²	1	8	yes	—
74	C₂×C₄×C₇₁	C₂²×C₈	1	16	yes	—
75	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂³×C₄	1	4	yes	—
77	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
78	C₂×Dic₇₁	C₄×C₈	1	64	yes	direct product factor
79	C₂×Dic₇₁	C₂×C₄²	1	16	yes	—
80	C₂×Dic₇₁	C₂²×C₈	1	32	yes	—
81	C₈×C₇₁	C₂×C₄²	1	16	yes	complemented
82	C₇₁⋊C₈	C₂×C₄²	1	16	yes	complemented
83	C₂²×C₇₁	C₄³	1	4	yes	—
84	C₂²×C₇₁	C₂×C₄×C₈	1	16	yes	—
85	C₂²×C₇₁	C₂×C₄×C₈	1	8	yes	—
86	C₂²×C₇₁	C₂²×C₄²	1	2	yes	—
87	C₂²×C₇₁	C₂²×C₄²	1	1	yes	characteristic
88	C₂²×C₇₁	C₂³×C₈	1	4	yes	—
89	C₄×C₇₁	C₄³	1	4	yes	—
90	C₄×C₇₁	C₂×C₄×C₈	1	16	yes	complemented
91	C₄×C₇₁	C₂²×C₄²	1	2	yes	—
92	C₄×C₇₁	C₂²×C₄²	1	2	yes	—
93	Dic₇₁	C₂×C₄×C₈	1	32	yes	direct product factor
94	C₂×C₄²×C₈	—	71	1	no	maximal, Sylow
95	C₂×C₇₁	C₄²×C₈	1	8	yes	complemented
96	C₂×C₇₁	C₂×C₄³	1	1	yes	characteristic
97	C₂×C₇₁	C₂²×C₄×C₈	1	4	yes	—
98	C₂×C₇₁	C₂²×C₄×C₈	1	2	yes	—
99	C₂²×C₄×C₈	—	71	1	no	—
100	C₂²×C₄×C₈	—	71	4	no	—
101	C₂²×C₄×C₈	D₇₁	1	1	yes	characteristic
102	C₂×C₄³	—	71	1	no	—
103	C₄²×C₈	—	71	8	no	—
104	C₇₁	C₂×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₈	—	71	2	no	—
106	C₂³×C₈	C₂×D₇₁	1	2	yes	—
107	C₂²×C₄²	—	71	4	no	—
108	C₂²×C₄²	—	71	2	no	—
109	C₂²×C₄²	C₂×D₇₁	1	1	yes	characteristic
110	C₂×C₄×C₈	—	71	32	no	—
111	C₂×C₄×C₈	—	71	16	no	—
112	C₂×C₄×C₈	—	71	8	no	—
113	C₂×C₄×C₈	—	71	4	no	—
114	C₂×C₄×C₈	Dic₇₁	1	8	yes	direct product factor
115	C₂×C₄×C₈	C₂×D₇₁	1	4	yes	—
116	C₄³	—	71	4	no	—
117	C₄³	—	71	4	no	—
118	C₂³×C₄	—	71	4	no	—
119	C₂³×C₄	C₄×D₇₁	1	2	yes	—
120	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
121	C₂²×C₈	—	71	8	no	—
122	C₂²×C₈	—	71	16	no	—
123	C₂²×C₈	—	71	4	no	—
124	C₂²×C₈	C₄×D₇₁	1	16	yes	—
125	C₂²×C₈	C₂×Dic₇₁	1	8	yes	—
126	C₂²×C₈	C₂²×D₇₁	1	4	yes	—
127	C₂×C₄²	—	71	32	no	—
128	C₂×C₄²	—	71	4	no	—
129	C₂×C₄²	—	71	16	no	—
130	C₂×C₄²	—	71	16	no	—
131	C₂×C₄²	—	71	4	no	—
132	C₂×C₄²	C₄×D₇₁	1	2	yes	—
133	C₂×C₄²	C₄×D₇₁	1	4	yes	—
134	C₂×C₄²	C₂×Dic₇₁	1	4	yes	—
135	C₂×C₄²	C₂²×D₇₁	1	2	yes	—
136	C₄×C₈	—	71	64	no	—
137	C₄×C₈	—	71	16	no	—
138	C₄×C₈	C₂×Dic₇₁	1	16	yes	direct product factor
139	C₂⁴	C₂×C₄×D₇₁	1	1	yes	characteristic
140	C₂²×C₄	—	71	8	no	—
141	C₂²×C₄	—	71	16	no	—
142	C₂²×C₄	—	71	32	no	—
143	C₂²×C₄	C₄×Dic₇₁	1	8	yes	—
144	C₂²×C₄	C₈×D₇₁	1	16	yes	—
145	C₂²×C₄	C₂×C₄×D₇₁	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₇₁	1	8	yes	—
147	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₄×D₇₁	1	4	yes	—
149	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
150	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
151	C₂×C₈	—	71	8	no	—
152	C₂×C₈	—	71	16	no	—
153	C₂×C₈	—	71	32	no	—
154	C₂×C₈	C₄×Dic₇₁	1	32	yes	direct product factor
155	C₂×C₈	C₂×C₄×D₇₁	1	16	yes	—
156	C₂×C₈	C₂²×Dic₇₁	1	8	yes	—
157	C₄²	—	71	16	no	—
158	C₄²	—	71	16	no	—
159	C₄²	—	71	64	no	—
160	C₄²	C₄×Dic₇₁	1	8	yes	—
161	C₄²	C₂×C₄×D₇₁	1	4	yes	—
162	C₄²	C₂²×Dic₇₁	1	4	yes	—
163	C₂³	C₄²×D₇₁	1	4	yes	—
164	C₂³	C₂×C₄×Dic₇₁	1	2	yes	—
165	C₂³	C₂×C₈×D₇₁	1	8	yes	—
166	C₂³	C₂²×C₄×D₇₁	1	1	yes	characteristic
167	C₂×C₄	—	71	16	no	—
168	C₂×C₄	—	71	64	no	—
169	C₂×C₄	—	71	32	no	—
170	C₂×C₄	C₈×Dic₇₁	1	32	yes	direct product factor
171	C₂×C₄	C₄²×D₇₁	1	4	yes	—
172	C₂×C₄	C₄²×D₇₁	1	8	yes	—
173	C₂×C₄	C₂×C₄×Dic₇₁	1	2	yes	—
174	C₂×C₄	C₂×C₄×Dic₇₁	1	8	yes	—
175	C₂×C₄	C₂×C₄×Dic₇₁	1	8	yes	—
176	C₂×C₄	C₂×C₈×D₇₁	1	16	yes	—
177	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
178	C₂×C₄	C₂²×C₄×D₇₁	1	4	yes	—
179	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
180	C₈	—	71	16	no	—
181	C₈	C₂×C₄×Dic₇₁	1	16	yes	direct product factor
182	C₂²	C₄²×Dic₇₁	1	4	yes	—
183	C₂²	C₄×C₈×D₇₁	1	16	yes	—
184	C₂²	C₂×C₈×Dic₇₁	1	8	yes	—
185	C₂²	C₂×C₄²×D₇₁	1	2	yes	—
186	C₂²	C₂²×C₄×Dic₇₁	1	1	yes	characteristic
187	C₂²	C₂²×C₈×D₇₁	1	4	yes	—
188	C₄	—	71	32	no	—
189	C₄	C₄²×Dic₇₁	1	4	yes	—
190	C₄	C₂×C₈×Dic₇₁	1	16	yes	direct product factor
191	C₄	C₂×C₄²×D₇₁	1	2	yes	—
192	C₄	C₂²×C₄×Dic₇₁	1	2	yes	—
193	C₂	C₄×C₈×Dic₇₁	1	8	yes	direct product factor
194	C₂	C₂×C₄²×Dic₇₁	1	1	yes	characteristic
195	C₂	C₂×C₄×C₈×D₇₁	1	4	yes	—
196	C₂	C₂²×C₈×Dic₇₁	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄×C₈	D₇₁
commutator, hypocenter	C₇₁	C₂×C₄²×C₈
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₂²×C₄×C₈×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₂²×C₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₄×C₈×Dic₇₁	C₂	direct (N × H)
9. C₂×C₄×C₇₁⋊C₈	C₄	semidirect
11. C₂×C₈×Dic₇₁	C₄	direct (N × H)
17. C₂×C₄×C₈×C₇₁	C₄	semidirect
23. C₂×C₄×Dic₇₁	C₈	direct (N × H)
28. C₄×C₇₁⋊C₈	C₂×C₄	semidirect
29. C₈×Dic₇₁	C₂×C₄	direct (N × H)
39. C₄×C₈×C₇₁	C₂×C₄	semidirect
43. C₂×C₇₁⋊C₈	C₄²	semidirect
47. C₄×Dic₇₁	C₂×C₈	direct (N × H)
58. C₂×C₈×C₇₁	C₄²	semidirect
68. C₂×C₄×C₇₁	C₄×C₈	semidirect
78. C₂×Dic₇₁	C₄×C₈	direct (N × H)
81. C₈×C₇₁	C₂×C₄²	semidirect
82. C₇₁⋊C₈	C₂×C₄²	semidirect
90. C₄×C₇₁	C₂×C₄×C₈	semidirect
93. Dic₇₁	C₂×C₄×C₈	direct (N × H)
95. C₂×C₇₁	C₄²×C₈	semidirect
104. C₇₁	C₂×C₄²×C₈	semidirect
114. C₂×C₄×C₈	Dic₇₁	direct (N × H)
138. C₄×C₈	C₂×Dic₇₁	direct (N × H)
154. C₂×C₈	C₄×Dic₇₁	direct (N × H)
170. C₂×C₄	C₈×Dic₇₁	direct (N × H)
181. C₈	C₂×C₄×Dic₇₁	direct (N × H)
190. C₄	C₂×C₈×Dic₇₁	direct (N × H)
193. C₂	C₄×C₈×Dic₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇₁⋊C₈	2
C₂²×C₈×Dic₇₁	2
C₂×C₄²×Dic₇₁	2
C₄×C₈×Dic₇₁	2
C₂²×C₄×C₈×C₇₁	2
C₂×C₄²×C₈	71

Subgroups — C2×C4×C8×Dic71 #31994771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₈×Dic₇₁ #31994771