Subgroups — C₇×#13627 #13190843

Order 1792.

218 conjugacy classes of subgroups.

The subgroup lattice has 124 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₆₄	C₂	1	2	yes	maximal
2	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×C₄⋊C₃₂	C₂	1	4	yes	maximal, complemented
4	C₂²×C₇×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
7	C₇×M₆₄	—	2	4	no	—
8	C₂×C₇×C₃₂	C₂²	1	4	yes	—
9	C₄×C₇×C₁₆	C₄	1	2	yes	—
10	C₄×C₇×C₁₆	C₂²	1	2	yes	—
11	#13627	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂²×C₇×C₈	C₈	1	1	yes	characteristic
13	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₇	C₈	1	1	yes	characteristic
15	C₂×C₇×C₁₆	—	2	2	no	—
16	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
17	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₇×C₁₆	D₄	1	2	yes	—
19	C₂×C₇×C₁₆	Q₈	1	2	yes	—
20	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
21	C₄×C₇×C₈	C₈	1	2	yes	—
22	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
23	C₇×C₃₂	—	2	4	no	—
24	C₂×M₆₄	C₂×C₇	1	2	yes	—
25	C₂×C₄×C₁₆	C₂×C₇	1	1	yes	characteristic
26	C₄⋊C₃₂	C₂×C₇	1	4	yes	complemented
27	C₂²×C₄×C₇	C₁₆	1	2	yes	—
28	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₇×C₈	—	2	2	no	—
30	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
31	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
32	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
33	C₂×C₇×C₈	M₁₆	1	2	yes	—
34	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₇	C₁₆	1	1	yes	characteristic
36	C₄²×C₇	C₁₆	1	1	yes	characteristic
37	C₄²×C₇	C₂×C₈	1	2	yes	—
38	C₇×C₁₆	—	2	2	no	—
39	C₇×C₁₆	C₄⋊C₄	1	2	yes	—
40	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
41	C₇×C₁₆	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₁₆	C₄×C₇	1	1	yes	characteristic
43	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
44	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
45	M₆₄	—	2	4	no	—
46	C₂×C₃₂	C₂²×C₇	1	4	yes	—
47	C₄×C₁₆	C₄×C₇	1	2	yes	—
48	C₄×C₁₆	C₂²×C₇	1	2	yes	—
49	C₂³×C₇	C₂×C₁₆	1	1	yes	characteristic
50	C₂×C₄×C₇	—	2	4	no	—
51	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
52	C₂×C₄×C₇	C₂×C₁₆	1	2	yes	—
53	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₇	C₂×C₁₆	1	2	yes	—
55	C₂×C₄×C₇	M₃₂	1	2	yes	—
56	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
57	C₇×C₈	—	2	2	no	—
58	C₇×C₈	C₄⋊C₈	1	2	yes	—
59	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
61	C₂²×C₈	C₇×C₈	1	1	yes	characteristic
62	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
63	C₂×C₄²	C₇×C₈	1	1	yes	characteristic
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
66	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
67	C₂×C₁₆	C₇×D₄	1	2	yes	—
68	C₂×C₁₆	C₇×Q₈	1	2	yes	—
69	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
70	C₄×C₈	C₇×C₈	1	2	yes	—
71	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
72	C₃₂	—	2	4	no	—
73	C₂²×C₇	—	2	2	no	—
74	C₂²×C₇	C₄⋊C₁₆	1	2	yes	—
75	C₂²×C₇	C₂²×C₁₆	1	1	yes	characteristic
76	C₄×C₇	—	2	2	no	—
77	C₄×C₇	C₄⋊C₁₆	1	2	yes	—
78	C₄×C₇	M₆₄	1	4	yes	complemented
79	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
81	C₂²×C₄	C₇×C₁₆	1	2	yes	—
82	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
85	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
86	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
87	C₂×C₈	C₇×M₁₆	1	2	yes	—
88	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
89	C₄²	C₇×C₁₆	1	1	yes	characteristic
90	C₄²	C₇×C₁₆	1	1	yes	characteristic
91	C₄²	C₂×C₇×C₈	1	2	yes	—
92	C₁₆	—	2	2	no	—
93	C₁₆	C₇×C₄⋊C₄	1	2	yes	—
94	C₁₆	C₂×C₇×D₄	1	1	yes	characteristic
95	C₁₆	C₂×C₇×Q₈	1	1	yes	characteristic
96	C₂×C₇	—	2	2	no	—
97	C₂×C₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
98	C₂×C₇	C₂×M₆₄	1	2	yes	—
99	C₂³	C₂×C₇×C₁₆	1	1	yes	characteristic
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	C₇×C₄⋊C₈	1	2	yes	—
102	C₂×C₄	C₂×C₇×C₁₆	1	2	yes	—
103	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
104	C₂×C₄	C₂×C₇×C₁₆	1	2	yes	—
105	C₂×C₄	C₇×M₃₂	1	2	yes	—
106	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	C₇×C₄⋊C₈	1	2	yes	—
109	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
110	C₈	C₂×C₇×M₁₆	1	1	yes	characteristic
111	C₇	#13627	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²	—	2	2	no	—
113	C₂²	C₇×C₄⋊C₁₆	1	2	yes	—
114	C₂²	C₂²×C₇×C₁₆	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	C₇×C₄⋊C₁₆	1	2	yes	—
117	C₄	C₇×M₆₄	1	4	yes	complemented
118	C₄	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
119	C₄	C₂×C₇×M₃₂	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₇×C₄⋊C₁₆	1	1	yes	characteristic
122	C₂	C₂×C₇×M₆₄	1	2	yes	—

	subgroup	G/H
center	C₂×C₇×C₁₆	C₂³
commutator	C₂²	C₂²×C₇×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂²×C₇	C₂²×C₁₆
prime radical	C₂³×C₇	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×C₄⋊C₃₂	C₂	semidirect
11. #13627	C₇	direct (N × H)
26. C₄⋊C₃₂	C₂×C₇	semidirect
78. C₄×C₇	M₆₄	semidirect
111. C₇	#13627	direct (N × H)
117. C₄	C₇×M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×M₆₄	2
C₂×C₄×C₇×C₁₆	2
C₇×C₄⋊C₃₂	2
#13627	7

Subgroups — C7×#13627 #13190843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13627 #13190843