Subgroups — C₇×#354 #89396

Order 448.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#124	C₂	1	1	yes	maximal, characteristic
2	C₇×#99	C₂	1	2	yes	maximal
3	C₂×C₇×Q₈	C₄	1	1	yes	characteristic, complemented
4	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
5	C₇×C₂²⋊C₄	—	2	2	no	—
6	C₇×C₂²⋊C₄	—	2	2	no	—
7	C₄²×C₇	C₄	1	1	yes	characteristic, complemented
8	#354	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂³×C₇	D₄	1	2	yes	—
10	C₇×Q₈	—	4	1	no	—
11	C₇×D₄	—	4	1	no	—
12	C₂×C₄×C₇	—	2	1	no	—
13	C₂×C₄×C₇	—	2	1	no	—
14	C₂×C₄×C₇	—	4	2	no	—
15	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	#124	C₂×C₇	1	1	yes	characteristic
17	#99	C₂×C₇	1	2	yes	—
18	C₂²×C₇	—	2	2	no	—
19	C₂²×C₇	—	4	2	no	—
20	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₇	—	2	1	no	—
22	C₄×C₇	—	4	1	no	—
23	C₄×C₇	—	4	1	no	—
24	C₄×C₇	—	8	2	no	—
25	C₂×Q₈	C₄×C₇	1	1	yes	characteristic, complemented
26	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
27	C₂²⋊C₄	—	2	2	no	—
28	C₂²⋊C₄	—	2	2	no	—
29	C₄²	C₄×C₇	1	1	yes	characteristic, complemented
30	C₂×C₇	—	2	1	no	—
31	C₂×C₇	—	4	2	no	—
32	C₂×C₇	#99	1	1	yes	characteristic
33	C₂³	C₇×D₄	1	2	yes	—
34	Q₈	—	4	1	no	—
35	D₄	—	4	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	4	2	no	—
39	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
40	C₇	#354	1	1	yes	characteristic, direct product factor, Sylow
41	C₂²	—	2	2	no	—
42	C₂²	—	4	2	no	—
43	C₂²	C₇×C₂²⋊C₄	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	4	1	no	—
46	C₄	—	4	1	no	—
47	C₄	—	8	2	no	—
48	C₂	—	2	1	no	—
49	C₂	—	4	2	no	—
50	C₂	C₇×#99	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#99
commutator	C₂×C₄	C₂×C₄×C₇
Frattini, cosocle	C₂×D₄	C₂²×C₇
prime radical	C₂×C₇×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₇×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₇×Q₈	C₄	semidirect
7. C₄²×C₇	C₄	semidirect
8. #354	C₇	direct (N × H)
25. C₂×Q₈	C₄×C₇	semidirect
29. C₄²	C₄×C₇	semidirect
40. C₇	#354	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#124	2
C₇×#99	2
#354	7

Subgroups — C7×#354 #89396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#354 #89396