Subgroups — C₇×#4517 #13051299

Order 1344.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₇×#819	C₂	1	1	yes	maximal, characteristic
4	C₇×#362	—	3	1	no	maximal, Hall
5	C₂×C₇×Dic₆	—	2	1	no	—
6	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₇×#208	—	2	1	no	—
8	C₇×Dic₁₂	—	2	1	no	—
9	C₇×Dic₁₂	C₄	1	2	yes	—
10	C₃×C₇×C₁₆	—	2	1	no	—
11	C₂×C₇×Q₁₆	—	3	1	no	—
12	C₇×M₃₂	D₃	1	1	yes	characteristic
13	C₇×#108	—	3	1	no	—
14	#4517	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
16	C₂×C₇×Dic₃	—	4	1	no	—
17	C₇×Dic₆	—	2	2	no	—
18	C₇×Dic₆	—	4	1	no	—
19	C₃×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₃×C₇×C₈	D₄	1	1	yes	characteristic
21	C₇×C₃⋊C₈	—	4	1	no	—
22	C₂×C₇×Q₈	—	6	1	no	—
23	C₇×Q₁₆	—	3	2	no	—
24	C₇×Q₁₆	—	6	1	no	—
25	C₇×M₁₆	—	6	1	no	—
26	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
27	C₇×C₁₆	—	2	1	no	—
28	C₂×Dic₁₂	C₂×C₇	1	1	yes	characteristic
29	C₃×M₃₂	C₂×C₇	1	1	yes	characteristic
30	#819	C₂×C₇	1	1	yes	characteristic
31	C₂²×C₃×C₇	SD₁₆	1	1	yes	characteristic
32	C₃×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₃×C₄×C₇	D₈	1	1	yes	characteristic
34	C₇×Dic₃	—	4	2	no	—
35	#362	—	3	1	no	Sylow
36	C₇×Q₈	—	6	2	no	—
37	C₇×Q₈	—	12	1	no	—
38	C₂×C₄×C₇	—	12	1	no	—
39	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
40	C₇×C₈	—	12	1	no	—
41	C₇×C₈	C₄×D₃	1	1	yes	characteristic
42	C₇×C₈	#67	1	1	yes	characteristic
43	C₂×Dic₆	—	2	1	no	—
44	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
45	#208	—	2	1	no	—
46	Dic₁₂	—	2	1	no	—
47	Dic₁₂	C₄×C₇	1	2	yes	—
48	C₃×C₁₆	—	2	1	no	—
49	C₂×C₃×C₇	—	2	1	no	—
50	C₂×C₃×C₇	#102	1	1	yes	characteristic
51	C₂×Q₁₆	—	3	1	no	—
52	M₃₂	D₃×C₇	1	1	yes	characteristic
53	#108	—	3	1	no	—
54	C₂²×C₇	#204	1	1	yes	characteristic
55	C₄×C₇	—	12	2	no	—
56	C₄×C₇	D₂₄	1	1	yes	characteristic
57	C₄×C₇	(C₂×D₃)⋊C₄	1	1	yes	characteristic
58	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
59	C₂×Dic₃	—	4	1	no	—
60	Dic₆	—	2	2	no	—
61	Dic₆	—	4	1	no	—
62	C₃×C₈	C₂×C₄×C₇	1	1	yes	characteristic
63	C₃×C₈	C₇×D₄	1	1	yes	characteristic
64	C₃⋊C₈	—	4	1	no	—
65	C₃×C₇	#362	1	1	yes	characteristic, complemented, Hall
66	C₂×Q₈	—	6	1	no	—
67	Q₁₆	—	3	2	no	—
68	Q₁₆	—	6	1	no	—
69	M₁₆	—	6	1	no	—
70	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
71	C₁₆	—	2	1	no	—
72	C₂×C₇	—	2	1	no	—
73	C₂×C₇	#821	1	1	yes	characteristic
74	C₂²×C₃	C₇×SD₁₆	1	1	yes	characteristic
75	C₃×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
76	C₃×C₄	C₇×D₈	1	1	yes	characteristic
77	Dic₃	—	4	2	no	—
78	Q₈	—	6	2	no	—
79	Q₈	—	12	1	no	—
80	C₂×C₄	—	12	1	no	—
81	C₂×C₄	C₇×D₁₂	1	1	yes	characteristic
82	C₈	—	12	1	no	—
83	C₈	C₄×D₃×C₇	1	1	yes	characteristic
84	C₈	C₇×#67	1	1	yes	characteristic
85	C₇	#4517	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×C₃	—	2	1	no	—
87	C₂×C₃	C₇×#102	1	1	yes	characteristic
88	C₂²	C₇×#204	1	1	yes	characteristic
89	C₄	—	12	2	no	—
90	C₄	C₇×D₂₄	1	1	yes	characteristic
91	C₄	C₇×(C₂×D₃)⋊C₄	1	1	yes	characteristic
92	C₃	C₇×#362	1	1	yes	characteristic, complemented, Sylow
93	C₂	—	2	1	no	—
94	C₂	C₇×#821	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	#821
commutator	C₃×C₈	C₂×C₄×C₇
Frattini	C₂×C₈	C₂×D₃×C₇
Fitting	C₃×C₇×M₃₂	C₂
socle	C₂×C₃×C₇	#102
prime radical	C₂²×C₃×C₇	SD₁₆
cosocle	C₂×C₃×C₈	C₂²×C₇
hypercenter	C₇×M₃₂	D₃
hypocenter	C₃	C₇×#362
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₈ ◁ C₇×M₃₂

Complements

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

N	H ≅ G/N	type
14. #4517	C₇	direct (N × H)
65. C₃×C₇	#362	semidirect
85. C₇	#4517	direct (N × H)
92. C₃	C₇×#362	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×Dic₁₂	2
C₃×C₇×M₃₂	2
C₇×#819	2
C₇×#362	3
#4517	7

Subgroups — C7×#4517 #13051299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4517 #13051299