Subgroups — C₇×#13628 #13190844

Order 1792.

206 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	1	yes	maximal, characteristic
2	C₂²×C₇×C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₇×#2108	C₂	1	1	yes	maximal, characteristic
4	C₇×C₄⋊C₃₂	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×M₃₂	C₄	1	1	yes	characteristic
6	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
7	C₇×#405	C₄	1	1	yes	characteristic
8	C₇×M₆₄	—	2	2	no	—
9	C₂×C₇×C₃₂	—	2	2	no	—
10	C₂×C₇×C₃₂	C₂²	1	2	yes	—
11	C₂×C₇×C₃₂	C₂²	1	2	yes	—
12	C₇×#346	C₄	1	2	yes	—
13	C₄×C₇×C₁₆	C₂²	1	2	yes	—
14	#13628	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂×C₇×M₁₆	C₈	1	1	yes	characteristic
16	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₇×#117	C₈	1	1	yes	characteristic
18	C₇×M₃₂	—	2	2	no	—
19	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₇×C₁₆	D₄	1	2	yes	—
22	C₂×C₇×C₁₆	Q₈	1	2	yes	—
23	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
24	C₇×#97	C₈	1	2	yes	—
25	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₇×C₃₂	—	2	2	no	—
27	C₇×C₃₂	—	2	2	no	—
28	C₂×M₆₄	C₂×C₇	1	1	yes	characteristic
29	C₂²×C₃₂	C₂×C₇	1	1	yes	characteristic
30	#2108	C₂×C₇	1	1	yes	characteristic
31	C₄⋊C₃₂	C₂×C₇	1	4	yes	complemented
32	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
33	C₇×M₁₆	—	2	2	no	—
34	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
35	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
36	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
37	C₂×C₇×C₈	M₁₆	1	2	yes	—
38	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₇×C₄⋊C₄	C₁₆	1	2	yes	—
40	C₇×C₂²⋊C₄	C₁₆	1	2	yes	—
41	C₄²×C₇	C₂×C₈	1	2	yes	—
42	C₇×C₁₆	—	2	2	no	—
43	C₇×C₁₆	C₄⋊C₄	1	2	yes	—
44	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
45	C₇×C₁₆	C₂×Q₈	1	1	yes	characteristic
46	C₂×M₃₂	C₄×C₇	1	1	yes	characteristic
47	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
48	#405	C₄×C₇	1	1	yes	characteristic
49	M₆₄	—	2	2	no	—
50	C₂×C₃₂	—	2	2	no	—
51	C₂×C₃₂	C₂²×C₇	1	2	yes	—
52	C₂×C₃₂	C₂²×C₇	1	2	yes	—
53	#346	C₄×C₇	1	2	yes	—
54	C₄×C₁₆	C₂²×C₇	1	2	yes	—
55	C₂³×C₇	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₇	C₂×C₁₆	1	4	yes	—
58	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
59	C₂×C₄×C₇	M₃₂	1	2	yes	—
60	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
61	C₇×C₈	—	2	2	no	—
62	C₇×C₈	C₄⋊C₈	1	2	yes	—
63	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
65	C₂×M₁₆	C₇×C₈	1	1	yes	characteristic
66	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
67	#117	C₇×C₈	1	1	yes	characteristic
68	M₃₂	—	2	2	no	—
69	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
70	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
71	C₂×C₁₆	C₇×D₄	1	2	yes	—
72	C₂×C₁₆	C₇×Q₈	1	2	yes	—
73	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
74	#97	C₇×C₈	1	2	yes	—
75	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
76	C₃₂	—	2	2	no	—
77	C₃₂	—	2	2	no	—
78	C₂²×C₇	—	2	2	no	—
79	C₂²×C₇	C₄⋊C₁₆	1	2	yes	—
80	C₂²×C₇	C₂²×C₁₆	1	1	yes	characteristic
81	C₄×C₇	—	2	4	no	—
82	C₄×C₇	C₄⋊C₁₆	1	2	yes	—
83	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
85	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
88	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
89	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
90	C₂×C₈	C₇×M₁₆	1	2	yes	—
91	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
92	C₄⋊C₄	C₇×C₁₆	1	2	yes	—
93	C₂²⋊C₄	C₇×C₁₆	1	2	yes	—
94	C₄²	C₂×C₇×C₈	1	2	yes	—
95	C₁₆	—	2	2	no	—
96	C₁₆	C₇×C₄⋊C₄	1	2	yes	—
97	C₁₆	C₂×C₇×D₄	1	1	yes	characteristic
98	C₁₆	C₂×C₇×Q₈	1	1	yes	characteristic
99	C₂×C₇	—	2	2	no	—
100	C₂×C₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
101	C₂×C₇	#2258	1	2	yes	—
102	C₂³	C₂×C₇×C₁₆	1	1	yes	characteristic
103	C₂×C₄	C₇×C₄⋊C₈	1	2	yes	—
104	C₂×C₄	C₂×C₇×C₁₆	1	4	yes	—
105	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
106	C₂×C₄	C₇×M₃₂	1	2	yes	—
107	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	C₇×C₄⋊C₈	1	2	yes	—
110	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
111	C₈	C₂×C₇×M₁₆	1	1	yes	characteristic
112	C₇	#13628	1	1	yes	characteristic, direct product factor, Sylow
113	C₂²	—	2	2	no	—
114	C₂²	C₇×C₄⋊C₁₆	1	2	yes	—
115	C₂²	C₂²×C₇×C₁₆	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	C₇×C₄⋊C₁₆	1	2	yes	—
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	C₇×#2258	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
4. C₇×C₄⋊C₃₂	C₂	semidirect
14. #13628	C₇	direct (N × H)
31. C₄⋊C₃₂	C₂×C₇	semidirect
112. C₇	#13628	direct (N × H)

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
C₂²×C₇×C₃₂	2
C₇×#2108	2
C₇×C₄⋊C₃₂	2
#13628	7

Subgroups — C7×#13628 #13190844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13628 #13190844