Subgroups — C₂³×C₇×C₂²⋊C₈ #13237275

Order 1792.

7646 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₇×C₈	C₂	1	2	yes	maximal, complemented
3	C₂²×C₇×C₂²⋊C₈	C₂	1	28	yes	maximal, direct product factor
4	C₂⁶×C₇	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₇	—	2	16	no	—
6	C₂⁴×C₄×C₇	C₄	1	1	yes	characteristic
7	C₂⁴×C₄×C₇	C₄	1	14	yes	—
8	C₂⁴×C₄×C₇	C₂²	1	14	yes	—
9	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂³×C₇×C₈	—	2	16	no	—
11	C₂³×C₇×C₈	C₂²	1	28	yes	complemented
12	C₂×C₇×C₂²⋊C₈	C₂²	1	112	yes	direct product factor
13	C₂³×C₂²⋊C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂⁵×C₇	—	2	8	no	—
15	C₂⁵×C₇	—	2	8	no	—
16	C₂⁵×C₇	C₈	1	16	yes	complemented
17	C₂⁵×C₇	C₂×C₄	1	14	yes	—
18	C₂⁵×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₇	—	2	8	no	—
20	C₂³×C₄×C₇	—	2	112	no	—
21	C₂³×C₄×C₇	—	2	112	no	—
22	C₂³×C₄×C₇	C₂×C₄	1	14	yes	—
23	C₂³×C₄×C₇	C₂×C₄	1	84	yes	—
24	C₂³×C₄×C₇	C₂×C₄	1	7	yes	—
25	C₂³×C₄×C₇	D₄	1	16	yes	—
26	C₂³×C₄×C₇	C₂³	1	28	yes	—
27	C₂³×C₄×C₇	C₂³	1	7	yes	—
28	C₂²×C₇×C₈	—	2	112	no	—
29	C₂²×C₇×C₈	C₂³	1	56	yes	complemented
30	C₇×C₂²⋊C₈	C₂³	1	64	yes	direct product factor
31	C₂⁵×C₄	C₂×C₇	1	1	yes	characteristic
32	C₂⁴×C₈	C₂×C₇	1	2	yes	complemented
33	C₂²×C₂²⋊C₈	C₂×C₇	1	28	yes	direct product factor
34	C₂⁴×C₇	—	2	128	no	—
35	C₂⁴×C₇	—	2	56	no	—
36	C₂⁴×C₇	—	2	56	no	—
37	C₂⁴×C₇	C₂²⋊C₄	1	8	yes	—
38	C₂⁴×C₇	C₂×C₈	1	8	yes	—
39	C₂⁴×C₇	C₂×C₈	1	112	yes	complemented
40	C₂⁴×C₇	M₁₆	1	8	yes	—
41	C₂⁴×C₇	C₂²×C₄	1	28	yes	—
42	C₂⁴×C₇	C₂²×C₄	1	7	yes	—
43	C₂²×C₄×C₇	—	2	56	no	—
44	C₂²×C₄×C₇	—	2	336	no	—
45	C₂²×C₄×C₇	—	2	112	no	—
46	C₂²×C₄×C₇	C₂²⋊C₄	1	56	yes	—
47	C₂²×C₄×C₇	C₂²×C₄	1	28	yes	—
48	C₂²×C₄×C₇	C₂²×C₄	1	56	yes	—
49	C₂²×C₄×C₇	C₂²×C₄	1	21	yes	—
50	C₂²×C₄×C₇	C₂×D₄	1	56	yes	—
51	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
52	C₂²×C₄×C₇	C₂⁴	1	7	yes	—
53	C₂×C₇×C₈	—	2	112	no	—
54	C₂×C₇×C₈	C₂⁴	1	16	yes	complemented
55	C₂⁶	C₄×C₇	1	1	yes	characteristic
56	C₂⁴×C₄	—	2	16	no	—
57	C₂⁴×C₄	C₄×C₇	1	1	yes	characteristic
58	C₂⁴×C₄	C₄×C₇	1	14	yes	—
59	C₂⁴×C₄	C₂²×C₇	1	14	yes	—
60	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
61	C₂³×C₈	—	2	16	no	—
62	C₂³×C₈	C₂²×C₇	1	28	yes	complemented
63	C₂×C₂²⋊C₈	C₂²×C₇	1	112	yes	direct product factor
64	C₂³×C₇	—	2	56	no	—
65	C₂³×C₇	—	2	56	no	—
66	C₂³×C₇	—	2	448	no	—
67	C₂³×C₇	C₂²⋊C₈	1	64	yes	direct product factor
68	C₂³×C₇	C₂×C₂²⋊C₄	1	28	yes	—
69	C₂³×C₇	C₂²×C₈	1	28	yes	—
70	C₂³×C₇	C₂²×C₈	1	112	yes	complemented
71	C₂³×C₇	C₂×M₁₆	1	28	yes	—
72	C₂³×C₇	C₂³×C₄	1	8	yes	—
73	C₂³×C₇	C₂³×C₄	1	7	yes	—
74	C₂×C₄×C₇	—	2	56	no	—
75	C₂×C₄×C₇	—	2	112	no	—
76	C₂×C₄×C₇	—	2	16	no	—
77	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	84	yes	—
78	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
79	C₂×C₄×C₇	C₂³×C₄	1	7	yes	—
80	C₂×C₄×C₇	C₂²×D₄	1	28	yes	—
81	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
82	C₇×C₈	—	2	16	no	—
83	C₂⁵	—	2	8	no	—
84	C₂⁵	—	2	8	no	—
85	C₂⁵	C₇×C₈	1	16	yes	complemented
86	C₂⁵	C₂×C₄×C₇	1	14	yes	—
87	C₂⁵	C₂×C₄×C₇	1	1	yes	characteristic
88	C₂³×C₄	—	2	8	no	—
89	C₂³×C₄	—	2	112	no	—
90	C₂³×C₄	—	2	112	no	—
91	C₂³×C₄	C₂×C₄×C₇	1	14	yes	—
92	C₂³×C₄	C₂×C₄×C₇	1	84	yes	—
93	C₂³×C₄	C₂×C₄×C₇	1	7	yes	—
94	C₂³×C₄	C₇×D₄	1	16	yes	—
95	C₂³×C₄	C₂³×C₇	1	28	yes	—
96	C₂³×C₄	C₂³×C₇	1	7	yes	—
97	C₂²×C₈	—	2	112	no	—
98	C₂²×C₈	C₂³×C₇	1	56	yes	complemented
99	C₂²⋊C₈	C₂³×C₇	1	64	yes	direct product factor
100	C₂²×C₇	—	2	224	no	—
101	C₂²×C₇	—	2	8	no	—
102	C₂²×C₇	—	2	8	no	—
103	C₂²×C₇	C₂×C₂²⋊C₈	1	112	yes	direct product factor
104	C₂²×C₇	C₂²×C₂²⋊C₄	1	14	yes	—
105	C₂²×C₇	C₂³×C₈	1	14	yes	—
106	C₂²×C₇	C₂³×C₈	1	16	yes	complemented
107	C₂²×C₇	C₂²×M₁₆	1	14	yes	—
108	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
109	C₄×C₇	—	2	8	no	—
110	C₄×C₇	C₂²×C₂²⋊C₄	1	14	yes	—
111	C₄×C₇	C₂³×D₄	1	2	yes	—
112	C₂⁴	—	2	128	no	—
113	C₂⁴	—	2	56	no	—
114	C₂⁴	—	2	56	no	—
115	C₂⁴	C₇×C₂²⋊C₄	1	8	yes	—
116	C₂⁴	C₂×C₇×C₈	1	8	yes	—
117	C₂⁴	C₂×C₇×C₈	1	112	yes	complemented
118	C₂⁴	C₇×M₁₆	1	8	yes	—
119	C₂⁴	C₂²×C₄×C₇	1	28	yes	—
120	C₂⁴	C₂²×C₄×C₇	1	7	yes	—
121	C₂²×C₄	—	2	56	no	—
122	C₂²×C₄	—	2	336	no	—
123	C₂²×C₄	—	2	112	no	—
124	C₂²×C₄	C₇×C₂²⋊C₄	1	56	yes	—
125	C₂²×C₄	C₂²×C₄×C₇	1	28	yes	—
126	C₂²×C₄	C₂²×C₄×C₇	1	56	yes	—
127	C₂²×C₄	C₂²×C₄×C₇	1	21	yes	—
128	C₂²×C₄	C₂×C₇×D₄	1	56	yes	—
129	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
130	C₂²×C₄	C₂⁴×C₇	1	7	yes	—
131	C₂×C₈	—	2	112	no	—
132	C₂×C₈	C₂⁴×C₇	1	16	yes	complemented
133	C₂×C₇	—	2	16	no	—
134	C₂×C₇	C₂²×C₂²⋊C₈	1	28	yes	direct product factor
135	C₂×C₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₇	C₂⁴×C₈	1	1	yes	characteristic
137	C₂×C₇	C₂³×M₁₆	1	1	yes	characteristic
138	C₂³	—	2	56	no	—
139	C₂³	—	2	448	no	—
140	C₂³	—	2	56	no	—
141	C₂³	C₇×C₂²⋊C₈	1	64	yes	direct product factor
142	C₂³	C₂×C₇×C₂²⋊C₄	1	28	yes	—
143	C₂³	C₂²×C₇×C₈	1	28	yes	—
144	C₂³	C₂²×C₇×C₈	1	112	yes	complemented
145	C₂³	C₂×C₇×M₁₆	1	28	yes	—
146	C₂³	C₂³×C₄×C₇	1	8	yes	—
147	C₂³	C₂³×C₄×C₇	1	7	yes	—
148	C₂×C₄	—	2	56	no	—
149	C₂×C₄	—	2	16	no	—
150	C₂×C₄	—	2	112	no	—
151	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	84	yes	—
152	C₂×C₄	C₂³×C₄×C₇	1	8	yes	—
153	C₂×C₄	C₂³×C₄×C₇	1	7	yes	—
154	C₂×C₄	C₂²×C₇×D₄	1	28	yes	—
155	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
156	C₈	—	2	16	no	—
157	C₇	C₂³×C₂²⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²	—	2	8	no	—
159	C₂²	—	2	8	no	—
160	C₂²	—	2	224	no	—
161	C₂²	C₂×C₇×C₂²⋊C₈	1	112	yes	direct product factor
162	C₂²	C₂²×C₇×C₂²⋊C₄	1	14	yes	—
163	C₂²	C₂³×C₇×C₈	1	14	yes	—
164	C₂²	C₂³×C₇×C₈	1	16	yes	complemented
165	C₂²	C₂²×C₇×M₁₆	1	14	yes	—
166	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
167	C₄	—	2	8	no	—
168	C₄	C₂²×C₇×C₂²⋊C₄	1	14	yes	—
169	C₄	C₂³×C₇×D₄	1	2	yes	—
170	C₂	—	2	16	no	—
171	C₂	C₂²×C₇×C₂²⋊C₈	1	28	yes	direct product factor
172	C₂	C₂³×C₇×C₂²⋊C₄	1	1	yes	characteristic
173	C₂	C₂⁴×C₇×C₈	1	1	yes	characteristic
174	C₂	C₂³×C₇×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄×C₇	C₂²
commutator	C₂	C₂⁴×C₇×C₈
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂⁵×C₇	C₂×C₄
prime radical	C₂⁶×C₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₇×C₈	C₂	semidirect
3. C₂²×C₇×C₂²⋊C₈	C₂	direct (N × H)
11. C₂³×C₇×C₈	C₂²	semidirect
12. C₂×C₇×C₂²⋊C₈	C₂²	direct (N × H)
13. C₂³×C₂²⋊C₈	C₇	direct (N × H)
16. C₂⁵×C₇	C₈	semidirect
29. C₂²×C₇×C₈	C₂³	semidirect
30. C₇×C₂²⋊C₈	C₂³	direct (N × H)
32. C₂⁴×C₈	C₂×C₇	semidirect
33. C₂²×C₂²⋊C₈	C₂×C₇	direct (N × H)
39. C₂⁴×C₇	C₂×C₈	semidirect
54. C₂×C₇×C₈	C₂⁴	semidirect
62. C₂³×C₈	C₂²×C₇	semidirect
63. C₂×C₂²⋊C₈	C₂²×C₇	direct (N × H)
67. C₂³×C₇	C₂²⋊C₈	direct (N × H)
70. C₂³×C₇	C₂²×C₈	semidirect
85. C₂⁵	C₇×C₈	semidirect
98. C₂²×C₈	C₂³×C₇	semidirect
99. C₂²⋊C₈	C₂³×C₇	direct (N × H)
103. C₂²×C₇	C₂×C₂²⋊C₈	direct (N × H)
106. C₂²×C₇	C₂³×C₈	semidirect
117. C₂⁴	C₂×C₇×C₈	semidirect
132. C₂×C₈	C₂⁴×C₇	semidirect
134. C₂×C₇	C₂²×C₂²⋊C₈	direct (N × H)
141. C₂³	C₇×C₂²⋊C₈	direct (N × H)
144. C₂³	C₂²×C₇×C₈	semidirect
157. C₇	C₂³×C₂²⋊C₈	direct (N × H)
161. C₂²	C₂×C₇×C₂²⋊C₈	direct (N × H)
164. C₂²	C₂³×C₇×C₈	semidirect
171. C₂	C₂²×C₇×C₂²⋊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
C₂⁵×C₄×C₇	2
C₂⁴×C₇×C₈	2
C₂²×C₇×C₂²⋊C₈	2
C₂³×C₂²⋊C₈	7

Subgroups — C23×C7×C22⋊C8 #13237275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×C₂²⋊C₈ #13237275