Subgroups — C₇×#7476 #13184692

Order 1792.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#368	C₂	1	2	yes	maximal
3	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
4	C₂×C₇×#108	—	2	2	no	—
5	C₇×M₆₄	—	2	2	no	—
6	C₂×C₇×C₃₂	C₄	1	2	yes	—
7	C₇×#368	—	2	2	no	—
8	C₇×#368	C₄	1	4	yes	—
9	#7476	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₇×M₁₆	—	4	2	no	—
11	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
12	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
13	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
14	C₂×C₇×C₁₆	D₄	1	2	yes	—
15	C₂×C₇×C₁₆	Q₈	1	1	yes	characteristic
16	C₇×#108	—	2	4	no	—
17	C₇×#108	—	4	2	no	—
18	C₇×C₃₂	—	2	2	no	—
19	C₂×M₆₄	C₂×C₇	1	1	yes	characteristic
20	C₂×#368	C₂×C₇	1	2	yes	—
21	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
22	C₇×M₁₆	—	4	4	no	—
23	C₇×M₁₆	—	8	2	no	—
24	C₂×C₇×C₈	—	8	2	no	—
25	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
26	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₇×C₈	SD₁₆	1	2	yes	—
28	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
29	C₇×C₁₆	C₄²	1	1	yes	characteristic
30	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
31	C₇×C₁₆	C₄⋊C₄	1	2	yes	—
32	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
33	C₂×#108	—	2	2	no	—
34	M₆₄	—	2	2	no	—
35	C₂×C₃₂	C₄×C₇	1	2	yes	—
36	#368	—	2	2	no	—
37	#368	C₄×C₇	1	4	yes	—
38	C₂³×C₇	Q₃₂	1	1	yes	characteristic
39	C₂×C₄×C₇	#102	1	2	yes	—
40	C₂×C₄×C₇	#107	1	1	yes	characteristic
41	C₂×C₄×C₇	D₁₆	1	1	yes	characteristic
42	C₂×C₄×C₇	SD₃₂	1	2	yes	—
43	C₇×C₈	—	8	4	no	—
44	C₇×C₈	#95	1	1	yes	characteristic
45	C₇×C₈	#103	1	2	yes	—
46	C₇×C₈	#106	1	1	yes	characteristic
47	C₂×M₁₆	—	4	2	no	—
48	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
49	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
50	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
51	C₂×C₁₆	C₇×D₄	1	2	yes	—
52	C₂×C₁₆	C₇×Q₈	1	1	yes	characteristic
53	#108	—	2	4	no	—
54	#108	—	4	2	no	—
55	C₃₂	—	2	2	no	—
56	C₂²×C₇	—	4	1	no	—
57	C₂²×C₇	#358	1	2	yes	—
58	C₂²×C₇	#366	1	1	yes	characteristic
59	C₄×C₇	#340	1	1	yes	characteristic
60	C₄×C₇	#357	1	2	yes	—
61	C₄×C₇	#367	1	1	yes	characteristic
62	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
63	M₁₆	—	4	4	no	—
64	M₁₆	—	8	2	no	—
65	C₂×C₈	—	8	2	no	—
66	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
67	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₈	C₇×SD₁₆	1	2	yes	—
69	C₂×C₈	C₇×Q₁₆	1	1	yes	characteristic
70	C₁₆	C₄²×C₇	1	1	yes	characteristic
71	C₁₆	C₇×C₂²⋊C₄	1	1	yes	characteristic
72	C₁₆	C₇×C₄⋊C₄	1	2	yes	—
73	C₂×C₇	—	2	2	no	—
74	C₂×C₇	—	2	1	no	—
75	C₂×C₇	#1380	1	1	yes	characteristic
76	C₂³	C₇×Q₃₂	1	1	yes	characteristic
77	C₂×C₄	C₇×#102	1	2	yes	—
78	C₂×C₄	C₇×#107	1	1	yes	characteristic
79	C₂×C₄	C₇×D₁₆	1	1	yes	characteristic
80	C₂×C₄	C₇×SD₃₂	1	2	yes	—
81	C₈	—	8	4	no	—
82	C₈	C₇×#95	1	1	yes	characteristic
83	C₈	C₇×#103	1	2	yes	—
84	C₈	C₇×#106	1	1	yes	characteristic
85	C₇	#7476	1	1	yes	characteristic, direct product factor, Sylow
86	C₂²	—	4	1	no	—
87	C₂²	C₇×#358	1	2	yes	—
88	C₂²	C₇×#366	1	1	yes	characteristic
89	C₄	C₇×#340	1	1	yes	characteristic
90	C₄	C₇×#357	1	2	yes	—
91	C₄	C₇×#367	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	—	2	1	no	—
94	C₂	C₇×#1380	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	#367
commutator	C₁₆	C₄²×C₇
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇
socle	C₂×C₇	#1380
prime radical	C₂³×C₇	Q₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
9. #7476	C₇	direct (N × H)
85. C₇	#7476	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
C₂×C₇×#368	2
#7476	7

Subgroups — C7×#7476 #13184692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7476 #13184692