Subgroups — C₇×#6399 #13096102

Order 1568.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1157	C₂	1	1	yes	maximal, characteristic
2	C₇²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
5	C₇×C₂²⋊D₇	—	2	1	no	—
6	C₂×C₇×Dic₇	—	2	1	no	—
7	C₇×D₂₈	C₄	1	1	yes	characteristic, complemented
8	C₄×C₇×D₇	—	2	1	no	—
9	C₇×Dic₁₄	C₄	1	1	yes	characteristic, complemented
10	C₇²×C₈	—	2	1	no	—
11	C₇×#104	—	7	1	no	maximal
12	#6399	C₇	1	1	yes	maximal, characteristic, direct product factor
13	C₂²×C₇²	D₄	1	1	yes	characteristic
14	C₂×C₇×D₇	—	2	1	no	—
15	C₄×C₇²	C₂×C₄	1	1	yes	characteristic
16	C₄×C₇²	D₄	1	1	yes	characteristic
17	C₇×Dic₇	—	2	2	no	—
18	C₇×Dic₇	—	2	1	no	—
19	C₇×#41	—	7	1	no	—
20	#1157	C₂×C₇	1	1	yes	characteristic
21	C₇×M₁₆	—	2	3	no	—
22	C₇×M₁₆	D₇	1	1	yes	characteristic, complemented
23	C₇×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
24	C₄²×C₇	—	7	1	no	—
25	C₄×Dic₇	C₂×C₇	1	1	yes	characteristic, complemented
26	C₂×C₇²	—	2	1	no	—
27	C₂×C₇²	C₂²⋊C₄	1	1	yes	characteristic
28	C₇×D₇	—	4	1	no	—
29	C₇×Q₈	—	7	1	no	—
30	C₇×D₄	—	7	1	no	—
31	C₇×D₄	—	14	1	no	—
32	C₂×C₄×C₇	—	2	3	no	—
33	C₂×C₄×C₇	—	14	1	no	—
34	C₂×C₄×C₇	—	14	1	no	—
35	C₂×C₄×C₇	C₂×D₇	1	1	yes	characteristic
36	C₂×C₄×C₇	C₂²×C₇	1	1	yes	characteristic
37	C₂²⋊D₇	—	2	1	no	—
38	C₂×Dic₇	—	2	1	no	—
39	D₂₈	C₄×C₇	1	1	yes	characteristic, complemented
40	C₄×D₇	—	2	1	no	—
41	Dic₁₄	C₄×C₇	1	1	yes	characteristic, complemented
42	C₇×C₈	—	2	6	no	—
43	C₇×C₈	—	2	1	no	—
44	C₇×C₈	—	2	1	no	—
45	C₇²	#104	1	1	yes	characteristic, complemented, Sylow
46	#104	—	7	1	no	Sylow
47	C₂²×C₇	—	2	3	no	—
48	C₂²×C₇	—	14	1	no	—
49	C₂²×C₇	D₂₈	1	1	yes	characteristic
50	C₂²×C₇	C₇×D₄	1	1	yes	characteristic
51	C₂×D₇	—	2	1	no	—
52	C₄×C₇	—	2	3	no	—
53	C₄×C₇	—	2	3	no	—
54	C₄×C₇	—	14	2	no	—
55	C₄×C₇	—	14	1	no	—
56	C₄×C₇	C₄×D₇	1	1	yes	characteristic
57	C₄×C₇	C₂²⋊D₇	1	1	yes	characteristic
58	C₄×C₇	C₂×C₄×C₇	1	1	yes	characteristic
59	C₄×C₇	C₇×D₄	1	1	yes	characteristic
60	Dic₇	—	2	2	no	—
61	Dic₇	—	2	1	no	—
62	#41	—	7	1	no	—
63	M₁₆	C₇×D₇	1	1	yes	characteristic, complemented
64	C₄²	—	7	1	no	—
65	C₂×C₇	—	2	1	no	—
66	C₂×C₇	—	2	1	no	—
67	C₂×C₇	—	2	3	no	—
68	C₂×C₇	—	4	3	no	—
69	C₂×C₇	—	28	1	no	—
70	C₂×C₇	(C₂×D₇)⋊C₄	1	1	yes	characteristic
71	C₂×C₇	C₇×C₂²⋊C₄	1	1	yes	characteristic
72	D₇	—	4	1	no	—
73	Q₈	—	7	1	no	—
74	D₄	—	7	1	no	—
75	D₄	—	14	1	no	—
76	C₂×C₄	—	14	1	no	—
77	C₂×C₄	—	14	1	no	—
78	C₂×C₄	C₂×C₇×D₇	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₇	—	2	3	no	—
81	C₇	#6399	1	1	yes	characteristic, direct product factor
82	C₇	C₇×#104	1	1	yes	characteristic, complemented
83	C₂²	—	14	1	no	—
84	C₂²	C₇×D₂₈	1	1	yes	characteristic
85	C₄	—	14	2	no	—
86	C₄	—	14	1	no	—
87	C₄	C₄×C₇×D₇	1	1	yes	characteristic
88	C₄	C₇×C₂²⋊D₇	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	28	1	no	—
91	C₂	C₇×(C₂×D₇)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	C₂²⋊D₇
commutator	C₄×C₇	C₂×C₄×C₇
Frattini	C₂×C₄	C₂×C₇×D₇
Fitting	C₇²×M₁₆	C₂
socle	C₂×C₇²	C₂²⋊C₄
prime radical	C₇×#1157	C₂
cosocle	C₂×C₄×C₇	C₂²×C₇
hypercenter	C₇×M₁₆	D₇
hypocenter	C₇	C₇×#104
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₇×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₇²×M₁₆	C₂	semidirect
3. C₄×C₇×Dic₇	C₂	semidirect
7. C₇×D₂₈	C₄	semidirect
9. C₇×Dic₁₄	C₄	semidirect
12. #6399	C₇	direct (N × H)
22. C₇×M₁₆	D₇	semidirect
23. C₇×M₁₆	C₂×C₇	semidirect
25. C₄×Dic₇	C₂×C₇	semidirect
39. D₂₈	C₄×C₇	semidirect
41. Dic₁₄	C₄×C₇	semidirect
45. C₇²	#104	semidirect
63. M₁₆	C₇×D₇	semidirect
81. C₇	#6399	direct (N × H)
82. C₇	C₇×#104	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1157	2
C₇²×M₁₆	2
C₄×C₇×Dic₇	2
C₇×#104	7
#6399	7

Subgroups — C7×#6399 #13096102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6399 #13096102