Subgroups — C₂×C₃×C₇×C₂²⋊C₈ #13056917

Order 1344.

404 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₇×C₈	C₂	1	2	yes	maximal, complemented
3	C₃×C₇×C₂²⋊C₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×C₂²⋊C₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂⁴×C₃×C₇	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₄×C₇	—	2	4	no	—
7	C₂²×C₃×C₄×C₇	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₇	C₄	1	2	yes	—
9	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₇×C₈	—	2	4	no	—
12	C₂×C₃×C₇×C₈	C₂²	1	4	yes	complemented
13	C₂³×C₄×C₇	C₂×C₃	1	1	yes	characteristic
14	C₂²×C₇×C₈	C₂×C₃	1	2	yes	complemented
15	C₇×C₂²⋊C₈	C₂×C₃	1	4	yes	direct product factor
16	C₂×C₃×C₂²⋊C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂³×C₃×C₇	—	2	2	no	—
18	C₂³×C₃×C₇	—	2	2	no	—
19	C₂³×C₃×C₇	C₈	1	4	yes	complemented
20	C₂³×C₃×C₇	C₂×C₄	1	2	yes	—
21	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄×C₇	—	2	2	no	—
23	C₂×C₃×C₄×C₇	—	2	4	no	—
24	C₂×C₃×C₄×C₇	—	2	4	no	—
25	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₇	D₄	1	4	yes	—
28	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₃×C₇×C₈	—	2	4	no	—
30	C₂⁴×C₇	C₃×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇	—	2	4	no	—
32	C₂²×C₄×C₇	C₃×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇	C₃×C₄	1	2	yes	—
34	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
35	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
36	C₂×C₇×C₈	—	2	4	no	—
37	C₂×C₇×C₈	C₂²×C₃	1	4	yes	complemented
38	C₂³×C₃×C₄	C₂×C₇	1	1	yes	characteristic
39	C₂²×C₃×C₈	C₂×C₇	1	2	yes	complemented
40	C₃×C₂²⋊C₈	C₂×C₇	1	4	yes	direct product factor
41	C₂²×C₃×C₇	—	2	8	no	—
42	C₂²×C₃×C₇	—	2	2	no	—
43	C₂²×C₃×C₇	—	2	2	no	—
44	C₂²×C₃×C₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₃×C₇	C₂×C₈	1	2	yes	—
46	C₂²×C₃×C₇	C₂×C₈	1	4	yes	complemented
47	C₂²×C₃×C₇	M₁₆	1	2	yes	—
48	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
49	C₃×C₄×C₇	—	2	2	no	—
50	C₃×C₄×C₇	C₂²⋊C₄	1	2	yes	—
51	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
52	C₂×C₂²⋊C₈	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³×C₇	—	2	2	no	—
54	C₂³×C₇	—	2	2	no	—
55	C₂³×C₇	C₃×C₈	1	4	yes	complemented
56	C₂³×C₇	C₂×C₃×C₄	1	2	yes	—
57	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
58	C₂×C₄×C₇	—	2	2	no	—
59	C₂×C₄×C₇	—	2	4	no	—
60	C₂×C₄×C₇	—	2	4	no	—
61	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
62	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
63	C₂×C₄×C₇	C₃×D₄	1	4	yes	—
64	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
65	C₇×C₈	—	2	4	no	—
66	C₂⁴×C₃	C₄×C₇	1	1	yes	characteristic
67	C₂²×C₃×C₄	—	2	4	no	—
68	C₂²×C₃×C₄	C₄×C₇	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₄×C₇	1	2	yes	—
70	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
71	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
72	C₂×C₃×C₈	—	2	4	no	—
73	C₂×C₃×C₈	C₂²×C₇	1	4	yes	complemented
74	C₂×C₃×C₇	—	2	4	no	—
75	C₂×C₃×C₇	C₂²⋊C₈	1	4	yes	direct product factor
76	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₃×C₇	C₂×M₁₆	1	1	yes	characteristic
79	C₂³×C₄	C₂×C₃×C₇	1	1	yes	characteristic
80	C₂²×C₈	C₂×C₃×C₇	1	2	yes	complemented
81	C₂²⋊C₈	C₂×C₃×C₇	1	4	yes	direct product factor
82	C₂²×C₇	—	2	8	no	—
83	C₂²×C₇	—	2	2	no	—
84	C₂²×C₇	—	2	2	no	—
85	C₂²×C₇	C₃×C₂²⋊C₄	1	2	yes	—
86	C₂²×C₇	C₂×C₃×C₈	1	2	yes	—
87	C₂²×C₇	C₂×C₃×C₈	1	4	yes	complemented
88	C₂²×C₇	C₃×M₁₆	1	2	yes	—
89	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
90	C₄×C₇	—	2	2	no	—
91	C₄×C₇	C₃×C₂²⋊C₄	1	2	yes	—
92	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
93	C₂³×C₃	—	2	2	no	—
94	C₂³×C₃	—	2	2	no	—
95	C₂³×C₃	C₇×C₈	1	4	yes	complemented
96	C₂³×C₃	C₂×C₄×C₇	1	2	yes	—
97	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	4	no	—
100	C₂×C₃×C₄	—	2	4	no	—
101	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
102	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
103	C₂×C₃×C₄	C₇×D₄	1	4	yes	—
104	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
105	C₃×C₈	—	2	4	no	—
106	C₃×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic, direct product factor, Hall
107	C₂⁴	C₃×C₄×C₇	1	1	yes	characteristic
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	C₃×C₄×C₇	1	1	yes	characteristic
110	C₂²×C₄	C₃×C₄×C₇	1	2	yes	—
111	C₂²×C₄	C₂²×C₃×C₇	1	2	yes	—
112	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	C₂²×C₃×C₇	1	4	yes	complemented
115	C₂×C₇	—	2	4	no	—
116	C₂×C₇	C₃×C₂²⋊C₈	1	4	yes	direct product factor
117	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₇	C₂²×C₃×C₈	1	1	yes	characteristic
119	C₂×C₇	C₂×C₃×M₁₆	1	1	yes	characteristic
120	C₂²×C₃	—	2	8	no	—
121	C₂²×C₃	—	2	2	no	—
122	C₂²×C₃	—	2	2	no	—
123	C₂²×C₃	C₇×C₂²⋊C₄	1	2	yes	—
124	C₂²×C₃	C₂×C₇×C₈	1	2	yes	—
125	C₂²×C₃	C₂×C₇×C₈	1	4	yes	complemented
126	C₂²×C₃	C₇×M₁₆	1	2	yes	—
127	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
128	C₃×C₄	—	2	2	no	—
129	C₃×C₄	C₇×C₂²⋊C₄	1	2	yes	—
130	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
131	C₂³	—	2	2	no	—
132	C₂³	—	2	2	no	—
133	C₂³	C₃×C₇×C₈	1	4	yes	complemented
134	C₂³	C₂×C₃×C₄×C₇	1	2	yes	—
135	C₂³	C₂×C₃×C₄×C₇	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂×C₃×C₄×C₇	1	2	yes	—
140	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
141	C₂×C₄	C₃×C₇×D₄	1	4	yes	—
142	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
143	C₈	—	2	4	no	—
144	C₇	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×C₃	—	2	4	no	—
146	C₂×C₃	C₇×C₂²⋊C₈	1	4	yes	direct product factor
147	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₃	C₂²×C₇×C₈	1	1	yes	characteristic
149	C₂×C₃	C₂×C₇×M₁₆	1	1	yes	characteristic
150	C₂²	—	2	8	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	2	no	—
153	C₂²	C₃×C₇×C₂²⋊C₄	1	2	yes	—
154	C₂²	C₂×C₃×C₇×C₈	1	2	yes	—
155	C₂²	C₂×C₃×C₇×C₈	1	4	yes	complemented
156	C₂²	C₃×C₇×M₁₆	1	2	yes	—
157	C₂²	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	C₃×C₇×C₂²⋊C₄	1	2	yes	—
160	C₄	C₂×C₃×C₇×D₄	1	2	yes	—
161	C₃	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
162	C₂	—	2	4	no	—
163	C₂	C₃×C₇×C₂²⋊C₈	1	4	yes	direct product factor
164	C₂	C₂×C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
165	C₂	C₂²×C₃×C₇×C₈	1	1	yes	characteristic
166	C₂	C₂×C₃×C₇×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄×C₇	C₂²
commutator	C₂	C₂²×C₃×C₇×C₈
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle	C₂³×C₃×C₇	C₂×C₄
prime radical	C₂⁴×C₃×C₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₇×C₈	C₂	semidirect
3. C₃×C₇×C₂²⋊C₈	C₂	direct (N × H)
4. C₂×C₇×C₂²⋊C₈	C₃	direct (N × H)
12. C₂×C₃×C₇×C₈	C₂²	semidirect
14. C₂²×C₇×C₈	C₂×C₃	semidirect
15. C₇×C₂²⋊C₈	C₂×C₃	direct (N × H)
16. C₂×C₃×C₂²⋊C₈	C₇	direct (N × H)
19. C₂³×C₃×C₇	C₈	semidirect
37. C₂×C₇×C₈	C₂²×C₃	semidirect
39. C₂²×C₃×C₈	C₂×C₇	semidirect
40. C₃×C₂²⋊C₈	C₂×C₇	direct (N × H)
46. C₂²×C₃×C₇	C₂×C₈	semidirect
52. C₂×C₂²⋊C₈	C₃×C₇	direct (N × H)
55. C₂³×C₇	C₃×C₈	semidirect
73. C₂×C₃×C₈	C₂²×C₇	semidirect
75. C₂×C₃×C₇	C₂²⋊C₈	direct (N × H)
80. C₂²×C₈	C₂×C₃×C₇	semidirect
81. C₂²⋊C₈	C₂×C₃×C₇	direct (N × H)
87. C₂²×C₇	C₂×C₃×C₈	semidirect
95. C₂³×C₃	C₇×C₈	semidirect
106. C₃×C₇	C₂×C₂²⋊C₈	direct (N × H)
114. C₂×C₈	C₂²×C₃×C₇	semidirect
116. C₂×C₇	C₃×C₂²⋊C₈	direct (N × H)
125. C₂²×C₃	C₂×C₇×C₈	semidirect
133. C₂³	C₃×C₇×C₈	semidirect
144. C₇	C₂×C₃×C₂²⋊C₈	direct (N × H)
146. C₂×C₃	C₇×C₂²⋊C₈	direct (N × H)
155. C₂²	C₂×C₃×C₇×C₈	semidirect
161. C₃	C₂×C₇×C₂²⋊C₈	direct (N × H)
163. C₂	C₃×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
C₂³×C₃×C₄×C₇	2
C₂²×C₃×C₇×C₈	2
C₃×C₇×C₂²⋊C₈	2
C₂×C₇×C₂²⋊C₈	3
C₂×C₃×C₂²⋊C₈	7

Subgroups — C2×C3×C7×C22⋊C8 #13056917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×C₂²⋊C₈ #13056917