Subgroups — C₇×#339 #89381

Order 448.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#117	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₇	—	2	1	no	—
5	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₇×M₁₆	—	2	1	no	—
7	C₇×M₁₆	C₄	1	2	yes	complemented
8	C₂×C₇×C₈	—	2	1	no	—
9	C₇×C₄⋊C₄	C₄	1	2	yes	complemented
10	C₇×C₂²⋊C₄	—	2	1	no	—
11	C₄²×C₇	—	2	1	no	—
12	C₄²×C₇	—	2	1	no	—
13	C₄²×C₇	C₄	1	2	yes	complemented
14	#339	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₇	—	2	4	no	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₇	Q₈	1	1	yes	characteristic
25	C₇×C₈	—	2	2	no	—
26	C₂×M₁₆	C₂×C₇	1	1	yes	characteristic
27	#117	C₂×C₇	1	1	yes	characteristic
28	C₂×C₄²	C₂×C₇	1	1	yes	characteristic
29	C₂²×C₇	—	2	2	no	—
30	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₇	—	2	4	no	—
34	C₄×C₇	—	4	2	no	—
35	C₄×C₇	C₄²	1	1	yes	characteristic
36	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
38	C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄	—	2	1	no	—
40	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
41	M₁₆	—	2	1	no	—
42	M₁₆	C₄×C₇	1	2	yes	complemented
43	C₂×C₈	—	2	1	no	—
44	C₄⋊C₄	C₄×C₇	1	2	yes	complemented
45	C₂²⋊C₄	—	2	1	no	—
46	C₄²	—	2	1	no	—
47	C₄²	—	2	1	no	—
48	C₄²	C₄×C₇	1	2	yes	complemented
49	C₂×C₇	—	2	1	no	—
50	C₂×C₇	—	2	1	no	—
51	C₂×C₇	#95	1	1	yes	characteristic
52	C₂×C₇	#104	1	2	yes	—
53	C₂³	C₇×D₄	1	1	yes	characteristic
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
58	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
59	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
60	C₂×C₄	C₇×D₄	1	1	yes	characteristic
61	C₂×C₄	C₇×D₄	1	1	yes	characteristic
62	C₂×C₄	C₇×Q₈	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₇	#339	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	—	2	2	no	—
66	C₂²	C₇×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²	C₇×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²	C₇×C₄⋊C₄	1	1	yes	characteristic
69	C₄	—	2	4	no	—
70	C₄	—	4	2	no	—
71	C₄	C₄²×C₇	1	1	yes	characteristic
72	C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
73	C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
74	C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	2	1	no	—
77	C₂	C₇×#95	1	1	yes	characteristic
78	C₂	C₇×#104	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	D₄
commutator	C₄	C₄²×C₇
Frattini, cosocle	C₂²×C₄	C₂²×C₇
socle	C₂²×C₇	C₂²⋊C₄
prime radical	C₂³×C₇	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₇×M₁₆	C₄	semidirect
9. C₇×C₄⋊C₄	C₄	semidirect
13. C₄²×C₇	C₄	semidirect
14. #339	C₇	direct (N × H)
42. M₁₆	C₄×C₇	semidirect
44. C₄⋊C₄	C₄×C₇	semidirect
48. C₄²	C₄×C₇	semidirect
64. C₇	#339	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×M₁₆	2
C₇×#117	2
C₂×C₄²×C₇	2
#339	7

Subgroups — C7×#339 #89381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#339 #89381