Subgroups — C₇×#398 #90044

Order 448.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×C₄⋊C₄	C₂	1	2	yes	maximal
4	C₇×#95	C₂	1	2	yes	maximal
5	C₇×#95	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₇	C₂²	1	2	yes	—
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₇	C₂²	1	2	yes	—
11	C₇×C₄⋊C₄	—	2	2	no	—
12	C₇×C₄⋊C₄	—	2	4	no	—
13	C₇×C₄⋊C₄	—	2	2	no	—
14	#398	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₇	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₇	—	2	1	no	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	—	2	1	no	—
19	C₂×C₄×C₇	—	2	2	no	—
20	C₂×C₄×C₇	—	2	2	no	—
21	C₂×C₄×C₇	—	2	2	no	—
22	C₂×C₄×C₇	—	2	2	no	—
23	C₂×C₄×C₇	—	2	1	no	—
24	C₂×C₄×C₇	—	2	2	no	—
25	C₂×C₄×C₇	—	2	2	no	—
26	C₂×C₄×C₇	D₄	1	2	yes	—
27	C₂×C₄×C₇	D₄	1	2	yes	—
28	C₂×C₄×C₇	Q₈	1	4	yes	—
29	C₂×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
30	C₂×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
31	C₂×C₄⋊C₄	C₂×C₇	1	2	yes	—
32	#95	C₂×C₇	1	2	yes	—
33	#95	C₂×C₇	1	1	yes	characteristic
34	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₇	C₂×Q₈	1	2	yes	—
37	C₂²×C₇	#41	1	1	yes	characteristic
38	C₂²×C₇	#41	1	2	yes	—
39	C₄×C₇	—	2	2	no	—
40	C₄×C₇	—	2	4	no	—
41	C₄×C₇	—	2	2	no	—
42	C₄×C₇	—	4	2	no	—
43	C₄×C₇	—	4	1	no	—
44	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
45	C₂²×C₄	C₂²×C₇	1	2	yes	—
46	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
47	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
48	C₂²×C₄	C₂²×C₇	1	2	yes	—
49	C₄⋊C₄	—	2	2	no	—
50	C₄⋊C₄	—	2	4	no	—
51	C₄⋊C₄	—	2	2	no	—
52	C₂×C₇	#121	1	1	yes	characteristic
53	C₂×C₇	C₂²⋊Q₈	1	2	yes	—
54	C₂×C₇	#123	1	1	yes	characteristic
55	C₂×C₇	#125	1	2	yes	—
56	C₂×C₇	C₄⋊Q₈	1	1	yes	characteristic
57	C₂³	C₂³×C₇	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	C₇×D₄	1	2	yes	—
69	C₂×C₄	C₇×D₄	1	2	yes	—
70	C₂×C₄	C₇×Q₈	1	4	yes	—
71	C₇	#398	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
73	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
74	C₂²	C₂×C₇×Q₈	1	2	yes	—
75	C₂²	C₇×#41	1	1	yes	characteristic
76	C₂²	C₇×#41	1	2	yes	—
77	C₄	—	2	2	no	—
78	C₄	—	2	4	no	—
79	C₄	—	2	2	no	—
80	C₄	—	4	2	no	—
81	C₄	—	4	1	no	—
82	C₂	C₇×#121	1	1	yes	characteristic
83	C₂	C₇×C₂²⋊Q₈	1	2	yes	—
84	C₂	C₇×#123	1	1	yes	characteristic
85	C₂	C₇×#125	1	2	yes	—
86	C₂	C₇×C₄⋊Q₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂³
commutator, Frattini, cosocle	C₂³	C₂³×C₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
14. #398	C₇	direct (N × H)
71. C₇	#398	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×C₄⋊C₄	2
C₇×#95	2
#398	7

Subgroups — C7×#398 #90044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#398 #90044