Subgroups — C₂³×D₈×D₇₁ #32911225

Order 18176.

15454 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	C₂³×D₄×D₇₁	C₂	1	2	yes	maximal, complemented
2	C₂³×D₅₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#11765850	C₂	1	2	yes	maximal, complemented
4	C₂²×D₈×D₇₁	C₂	1	56	yes	maximal, direct product factor
5	C₂³×C₈×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₇₁	—	2	2	no	—
8	C₂³×D₂₈₄	C₂²	1	2	yes	complemented
9	C₂³×C₂²⋊D₇₁	—	2	2	no	—
10	C₂²×D₄×D₇₁	—	2	28	no	—
11	C₂²×D₄×D₇₁	C₂²	1	56	yes	complemented
12	C₂³×C₄×D₇₁	C₂²	1	1	yes	characteristic
13	C₂²×D₅₆₈	C₂²	1	28	yes	complemented
14	C₂²×#11765850	C₂²	1	56	yes	complemented
15	C₂×D₈×D₇₁	C₂²	1	448	yes	direct product factor
16	C₂³×C₇₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₈×D₇₁	C₂²	1	28	yes	complemented
18	C₂³×D₄×C₇₁	C₂²	1	2	yes	complemented
19	C₂²×D₈×C₇₁	C₂²	1	28	yes	complemented
20	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic, complemented
21	C₂⁴×D₇₁	—	2	56	no	—
22	C₂⁴×D₇₁	—	2	2	no	—
23	C₂⁴×D₇₁	—	4	32	no	—
24	C₂⁴×D₇₁	D₄	1	1	yes	characteristic
25	C₂²×D₂₈₄	—	2	14	no	—
26	C₂²×D₂₈₄	C₂³	1	28	yes	complemented
27	C₂²×C₂²⋊D₇₁	—	2	56	no	—
28	C₂×D₄×D₇₁	—	2	336	no	—
29	C₂×D₄×D₇₁	C₂³	1	224	yes	complemented
30	C₂³×Dic₇₁	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₇₁	D₄	1	14	yes	—
32	C₂²×C₄×D₇₁	C₂³	1	14	yes	—
33	C₂×D₅₆₈	C₂³	1	112	yes	complemented
34	C₂×#11765850	C₂³	1	224	yes	complemented
35	D₈×D₇₁	C₂³	1	512	yes	direct product factor
36	C₂²×C₇₁⋊C₈	C₂³	1	14	yes	complemented
37	C₂×C₈×D₇₁	C₂³	1	112	yes	complemented
38	C₂⁵×C₇₁	—	2	2	no	—
39	C₂²×D₄×C₇₁	—	2	14	no	—
40	C₂²×D₄×C₇₁	C₂³	1	28	yes	complemented
41	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
42	C₂×D₈×C₇₁	C₂³	1	112	yes	complemented
43	C₂²×C₈×C₇₁	C₂³	1	14	yes	complemented
44	C₂³×D₇₁	—	2	28	no	—
45	C₂³×D₇₁	—	2	224	no	—
46	C₂³×D₇₁	—	4	16	no	—
47	C₂³×D₇₁	—	4	448	no	—
48	C₂³×D₇₁	D₈	1	16	yes	direct product factor
49	C₂³×D₇₁	C₂×D₄	1	14	yes	—
50	C₂×D₂₈₄	—	2	84	no	—
51	C₂×D₂₈₄	C₂⁴	1	56	yes	complemented
52	C₂×C₂²⋊D₇₁	—	2	224	no	—
53	D₄×D₇₁	—	2	448	no	—
54	D₄×D₇₁	C₂⁴	1	128	yes	complemented
55	C₂²×Dic₇₁	C₂×D₄	1	14	yes	—
56	C₂×C₄×D₇₁	C₂×D₄	1	84	yes	—
57	C₂×C₄×D₇₁	C₂⁴	1	28	yes	—
58	D₅₆₈	C₂⁴	1	64	yes	complemented
59	#11765850	C₂⁴	1	128	yes	complemented
60	C₂×C₇₁⋊C₈	C₂⁴	1	28	yes	complemented
61	C₈×D₇₁	C₂⁴	1	64	yes	complemented
62	C₂⁴×C₇₁	—	2	28	no	—
63	C₂⁴×C₇₁	—	4	16	no	—
64	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₂×D₄×C₇₁	—	2	84	no	—
66	C₂×D₄×C₇₁	C₂⁴	1	56	yes	complemented
67	C₂²×C₄×C₇₁	C₂×D₄	1	7	yes	—
68	C₂²×C₄×C₇₁	C₂⁴	1	7	yes	—
69	D₈×C₇₁	C₂⁴	1	64	yes	complemented
70	C₂×C₈×C₇₁	C₂⁴	1	28	yes	complemented
71	C₂³×C₇₁	—	2	56	no	—
72	C₂³×C₇₁	—	4	112	no	—
73	C₂³×C₇₁	C₂×D₈	1	8	yes	complemented
74	C₂³×C₇₁	C₂²×D₄	1	7	yes	—
75	C₂²×D₇₁	—	2	56	no	—
76	C₂²×D₇₁	—	2	128	no	—
77	C₂²×D₇₁	—	4	112	no	—
78	C₂²×D₇₁	—	4	896	no	—
79	C₂²×D₇₁	C₂×D₈	1	112	yes	direct product factor
80	C₂²×D₇₁	C₂²×D₄	1	28	yes	—
81	D₄×C₇₁	—	2	56	no	—
82	D₄×C₇₁	C₂⁵	1	16	yes	complemented
83	C₂×C₄×C₇₁	C₂²×D₄	1	21	yes	—
84	C₂×C₄×C₇₁	C₂⁵	1	7	yes	—
85	C₂²⋊D₇₁	—	2	128	no	—
86	C₂×Dic₇₁	C₂²×D₄	1	28	yes	—
87	D₂₈₄	—	2	56	no	—
88	D₂₈₄	C₂⁵	1	16	yes	complemented
89	C₄×D₇₁	C₂²×D₄	1	56	yes	—
90	C₄×D₇₁	C₂⁵	1	8	yes	—
91	C₈×C₇₁	C₂⁵	1	8	yes	complemented
92	C₇₁⋊C₈	C₂⁵	1	8	yes	complemented
93	C₂²×C₇₁	—	2	16	no	—
94	C₂²×C₇₁	—	4	112	no	—
95	C₂²×C₇₁	C₂²×D₈	1	28	yes	complemented
96	C₂²×C₇₁	C₂³×D₄	1	7	yes	—
97	C₂×D₇₁	—	2	16	no	—
98	C₂×D₇₁	—	4	112	no	—
99	C₂×D₇₁	—	4	256	no	—
100	C₂×D₇₁	C₂²×D₈	1	112	yes	direct product factor
101	C₂×D₇₁	C₂³×D₄	1	8	yes	—
102	C₄×C₇₁	C₂³×D₄	1	7	yes	—
103	C₄×C₇₁	C₂⁶	1	1	yes	characteristic
104	Dic₇₁	C₂³×D₄	1	8	yes	—
105	C₂⁴×D₈	—	71	1	no	maximal, Sylow
106	C₂×C₇₁	—	4	16	no	—
107	C₂×C₇₁	C₂³×D₈	1	14	yes	complemented
108	C₂×C₇₁	C₂⁴×D₄	1	1	yes	characteristic
109	D₇₁	—	4	16	no	—
110	D₇₁	C₂³×D₈	1	16	yes	direct product factor
111	C₂⁴×D₄	—	71	2	no	—
112	C₂³×D₈	—	71	2	no	—
113	C₂³×D₈	—	71	56	no	—
114	C₂³×D₈	—	71	1	no	—
115	C₂³×D₈	D₇₁	1	1	yes	characteristic, direct product factor
116	C₂⁴×C₈	—	71	1	no	—
117	C₇₁	C₂⁴×D₈	1	1	yes	characteristic, complemented, Sylow
118	C₂⁶	—	142	2	no	—
119	C₂³×D₄	—	71	56	no	—
120	C₂³×D₄	—	71	2	no	—
121	C₂³×D₄	—	142	28	no	—
122	C₂³×D₄	—	142	2	no	—
123	C₂³×D₄	C₂×D₇₁	1	2	yes	complemented
124	C₂⁴×C₄	—	71	1	no	—
125	C₂²×D₈	—	71	56	no	—
126	C₂²×D₈	—	71	448	no	—
127	C₂²×D₈	—	71	28	no	—
128	C₂²×D₈	C₂×D₇₁	1	28	yes	direct product factor
129	C₂³×C₈	—	71	1	no	—
130	C₂³×C₈	—	71	28	no	—
131	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic, complemented
132	C₂⁵	—	2	2	no	—
133	C₂⁵	—	284	32	no	—
134	C₂⁵	—	71	1	no	—
135	C₂⁵	—	142	56	no	—
136	C₂⁵	—	142	2	no	—
137	C₂²×D₄	—	2	14	no	—
138	C₂²×D₄	—	71	224	no	—
139	C₂²×D₄	—	71	28	no	—
140	C₂²×D₄	—	142	336	no	—
141	C₂²×D₄	—	142	56	no	—
142	C₂²×D₄	—	142	14	no	—
143	C₂²×D₄	C₂²×D₇₁	1	28	yes	complemented
144	C₂³×C₄	—	71	1	no	—
145	C₂³×C₄	—	71	14	no	—
146	C₂³×C₄	—	71	14	no	—
147	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
148	C₂×D₈	—	71	224	no	—
149	C₂×D₈	—	71	512	no	—
150	C₂×D₈	—	71	112	no	—
151	C₂×D₈	C₂²×D₇₁	1	112	yes	direct product factor
152	C₂²×C₈	—	71	14	no	—
153	C₂²×C₈	—	71	112	no	—
154	C₂²×C₈	C₂²×D₇₁	1	14	yes	complemented
155	C₂⁴	—	2	28	no	—
156	C₂⁴	—	4	16	no	—
157	C₂⁴	—	284	448	no	—
158	C₂⁴	—	284	16	no	—
159	C₂⁴	—	71	14	no	—
160	C₂⁴	—	71	16	no	—
161	C₂⁴	—	142	224	no	—
162	C₂⁴	—	142	28	no	—
163	C₂⁴	D₄×D₇₁	1	1	yes	characteristic
164	C₂×D₄	—	2	84	no	—
165	C₂×D₄	—	71	128	no	—
166	C₂×D₄	—	71	56	no	—
167	C₂×D₄	—	142	448	no	—
168	C₂×D₄	—	142	224	no	—
169	C₂×D₄	—	142	84	no	—
170	C₂×D₄	C₂³×D₇₁	1	56	yes	complemented
171	C₂²×C₄	—	71	14	no	—
172	C₂²×C₄	—	71	84	no	—
173	C₂²×C₄	—	71	28	no	—
174	C₂²×C₄	D₄×D₇₁	1	7	yes	—
175	C₂²×C₄	C₂³×D₇₁	1	7	yes	—
176	D₈	—	71	128	no	—
177	D₈	—	71	64	no	—
178	D₈	C₂³×D₇₁	1	64	yes	direct product factor
179	C₂×C₈	—	71	28	no	—
180	C₂×C₈	—	71	64	no	—
181	C₂×C₈	C₂³×D₇₁	1	28	yes	complemented
182	C₂³	—	2	56	no	—
183	C₂³	—	4	112	no	—
184	C₂³	—	284	896	no	—
185	C₂³	—	284	112	no	—
186	C₂³	—	71	28	no	—
187	C₂³	—	71	112	no	—
188	C₂³	—	142	128	no	—
189	C₂³	—	142	56	no	—
190	C₂³	D₈×D₇₁	1	8	yes	direct product factor
191	C₂³	C₂×D₄×D₇₁	1	7	yes	—
192	D₄	—	2	56	no	—
193	D₄	—	71	16	no	—
194	D₄	—	142	128	no	—
195	D₄	—	142	56	no	—
196	D₄	C₂⁴×D₇₁	1	16	yes	complemented
197	C₂×C₄	—	71	28	no	—
198	C₂×C₄	—	71	56	no	—
199	C₂×C₄	—	71	8	no	—
200	C₂×C₄	C₂×D₄×D₇₁	1	21	yes	—
201	C₂×C₄	C₂⁴×D₇₁	1	7	yes	—
202	C₈	—	71	8	no	—
203	C₈	C₂⁴×D₇₁	1	8	yes	complemented
204	C₂²	—	2	16	no	—
205	C₂²	—	4	112	no	—
206	C₂²	—	284	256	no	—
207	C₂²	—	284	112	no	—
208	C₂²	—	71	112	no	—
209	C₂²	—	71	8	no	—
210	C₂²	—	142	16	no	—
211	C₂²	C₂×D₈×D₇₁	1	28	yes	direct product factor
212	C₂²	C₂²×D₄×D₇₁	1	7	yes	—
213	C₄	—	71	8	no	—
214	C₄	C₂²×D₄×D₇₁	1	7	yes	—
215	C₄	C₂⁵×D₇₁	1	1	yes	characteristic
216	C₂	—	4	16	no	—
217	C₂	—	284	16	no	—
218	C₂	—	71	16	no	—
219	C₂	C₂²×D₈×D₇₁	1	14	yes	direct product factor
220	C₂	C₂³×D₄×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₇₁
commutator, cosocle	C₄×C₇₁	C₂⁶
Frattini	C₄	C₂⁵×D₇₁
Fitting	C₂³×D₈×C₇₁	C₂
socle	C₂⁴×C₇₁	C₂×D₄
hypercenter	C₂³×D₈	D₇₁
hypocenter	C₇₁	C₂⁴×D₈
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×D₇₁	C₂	semidirect
2. C₂³×D₅₆₈	C₂	semidirect
3. C₂³×#11765850	C₂	semidirect
4. C₂²×D₈×D₇₁	C₂	direct (N × H)
5. C₂³×C₈×D₇₁	C₂	semidirect
6. C₂³×D₈×C₇₁	C₂	semidirect
8. C₂³×D₂₈₄	C₂²	semidirect
11. C₂²×D₄×D₇₁	C₂²	semidirect
13. C₂²×D₅₆₈	C₂²	semidirect
14. C₂²×#11765850	C₂²	semidirect
15. C₂×D₈×D₇₁	C₂²	direct (N × H)
16. C₂³×C₇₁⋊C₈	C₂²	semidirect
17. C₂²×C₈×D₇₁	C₂²	semidirect
18. C₂³×D₄×C₇₁	C₂²	semidirect
19. C₂²×D₈×C₇₁	C₂²	semidirect
20. C₂³×C₈×C₇₁	C₂²	semidirect
26. C₂²×D₂₈₄	C₂³	semidirect
29. C₂×D₄×D₇₁	C₂³	semidirect
33. C₂×D₅₆₈	C₂³	semidirect
34. C₂×#11765850	C₂³	semidirect
35. D₈×D₇₁	C₂³	direct (N × H)
36. C₂²×C₇₁⋊C₈	C₂³	semidirect
37. C₂×C₈×D₇₁	C₂³	semidirect
40. C₂²×D₄×C₇₁	C₂³	semidirect
42. C₂×D₈×C₇₁	C₂³	semidirect
43. C₂²×C₈×C₇₁	C₂³	semidirect
48. C₂³×D₇₁	D₈	direct (N × H)
51. C₂×D₂₈₄	C₂⁴	semidirect
54. D₄×D₇₁	C₂⁴	semidirect
58. D₅₆₈	C₂⁴	semidirect
59. #11765850	C₂⁴	semidirect
60. C₂×C₇₁⋊C₈	C₂⁴	semidirect
61. C₈×D₇₁	C₂⁴	semidirect
66. C₂×D₄×C₇₁	C₂⁴	semidirect
69. D₈×C₇₁	C₂⁴	semidirect
70. C₂×C₈×C₇₁	C₂⁴	semidirect
73. C₂³×C₇₁	C₂×D₈	semidirect
79. C₂²×D₇₁	C₂×D₈	direct (N × H)
82. D₄×C₇₁	C₂⁵	semidirect
88. D₂₈₄	C₂⁵	semidirect
91. C₈×C₇₁	C₂⁵	semidirect
92. C₇₁⋊C₈	C₂⁵	semidirect
95. C₂²×C₇₁	C₂²×D₈	semidirect
100. C₂×D₇₁	C₂²×D₈	direct (N × H)
107. C₂×C₇₁	C₂³×D₈	semidirect
110. D₇₁	C₂³×D₈	direct (N × H)
115. C₂³×D₈	D₇₁	direct (N × H)
117. C₇₁	C₂⁴×D₈	semidirect
123. C₂³×D₄	C₂×D₇₁	semidirect
128. C₂²×D₈	C₂×D₇₁	direct (N × H)
131. C₂³×C₈	C₂×D₇₁	semidirect
143. C₂²×D₄	C₂²×D₇₁	semidirect
151. C₂×D₈	C₂²×D₇₁	direct (N × H)
154. C₂²×C₈	C₂²×D₇₁	semidirect
170. C₂×D₄	C₂³×D₇₁	semidirect
178. D₈	C₂³×D₇₁	direct (N × H)
181. C₂×C₈	C₂³×D₇₁	semidirect
190. C₂³	D₈×D₇₁	direct (N × H)
196. D₄	C₂⁴×D₇₁	semidirect
203. C₈	C₂⁴×D₇₁	semidirect
211. C₂²	C₂×D₈×D₇₁	direct (N × H)
219. C₂	C₂²×D₈×D₇₁	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×D₄×D₇₁	2
C₂³×D₅₆₈	2
C₂³×#11765850	2
C₂²×D₈×D₇₁	2
C₂³×C₈×D₇₁	2
C₂³×D₈×C₇₁	2
C₂⁴×D₈	71

Subgroups — C23×D8×D71 #32911225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₈×D₇₁ #32911225