Subgroups — #89987

Order 448.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	#6489	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1136	C₂	1	1	yes	maximal, characteristic, complemented
4	#6406	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₇	—	2	1	no	—
6	C₂×C₇×D₄	—	2	1	no	—
7	C₂×C₇×D₄	—	2	2	no	—
8	C₂×C₇×D₄	C₄	1	4	yes	complemented
9	C₂×C₇×D₄	C₂²	1	2	yes	—
10	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₇	—	2	1	no	—
12	C₄⋊Dic₇	C₂²	1	2	yes	complemented
13	C₄×Dic₇	—	2	1	no	—
14	#1136	—	2	2	no	—
15	C₂×C₇⋊C₈	C₂²	1	2	yes	complemented
16	#418	—	7	1	no	maximal, Sylow
17	C₂³×C₇	—	2	4	no	—
18	C₂³×C₇	—	2	2	no	—
19	C₂³×C₇	—	4	4	no	—
20	C₂³×C₇	D₄	1	1	yes	characteristic
21	C₇×D₄	—	2	4	no	—
22	C₇×D₄	—	2	2	no	—
23	C₇×D₄	C₂×C₄	1	4	yes	—
24	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₇	D₄	1	2	yes	—
27	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₇	—	2	2	no	—
29	C₇⋊C₈	—	2	2	no	—
30	C₂²×D₄	D₇	1	1	yes	characteristic
31	C₂×M₁₆	—	7	1	no	—
32	#117	—	7	1	no	—
33	#102	—	7	4	no	—
34	C₂²×C₇	—	2	2	no	—
35	C₂²×C₇	—	2	4	no	—
36	C₂²×C₇	—	4	8	no	—
37	C₂²×C₇	—	4	4	no	—
38	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₇	C₂²⋊C₄	1	2	yes	—
41	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
42	C₄×C₇	C₂×D₄	1	1	yes	characteristic
43	Dic₇	—	4	2	no	—
44	C₂⁴	—	2	1	no	—
45	C₂×D₄	—	2	1	no	—
46	C₂×D₄	—	2	2	no	—
47	C₂×D₄	Dic₇	1	4	yes	complemented
48	C₂×D₄	C₂×D₇	1	2	yes	—
49	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
50	M₁₆	—	14	2	no	—
51	C₂×C₈	—	7	2	no	—
52	C₄⋊C₄	—	7	2	no	—
53	C₂²⋊C₄	—	14	1	no	—
54	C₄²	—	14	1	no	—
55	C₂×C₇	—	2	2	no	—
56	C₂×C₇	—	4	4	no	—
57	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₇	AGL(1,ℤ/8)	1	2	yes	—
59	C₂³	—	2	4	no	—
60	C₂³	—	2	2	no	—
61	C₂³	—	4	4	no	—
62	C₂³	C₂²⋊D₇	1	1	yes	characteristic
63	D₄	—	2	4	no	—
64	D₄	—	2	2	no	—
65	D₄	C₂×Dic₇	1	4	yes	—
66	C₂×C₄	—	14	2	no	—
67	C₂×C₄	C₂×Dic₇	1	2	yes	—
68	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₇	1	2	yes	—
70	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
71	C₈	—	14	2	no	—
72	C₇	#418	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	2	2	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	4	8	no	—
76	C₂²	—	4	4	no	—
77	C₂²	C₂²⋊Dic₇	1	2	yes	—
78	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
79	C₄	—	28	2	no	—
80	C₄	C₂²⋊Dic₇	1	2	yes	—
81	C₄	C₂²×Dic₇	1	1	yes	characteristic
82	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	4	4	no	—
85	C₂	#6495	1	2	yes	—
86	C₂	C₂×C₂²⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting, prime radical	C₂²×C₇×D₄	C₂
socle	C₂²×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂²×D₄	D₇
hypocenter	C₇	#418
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
2. #6489	C₂	semidirect
3. C₂×#1136	C₂	semidirect
4. #6406	C₂	semidirect
8. C₂×C₇×D₄	C₄	semidirect
12. C₄⋊Dic₇	C₂²	semidirect
15. C₂×C₇⋊C₈	C₂²	semidirect
47. C₂×D₄	Dic₇	semidirect
72. C₇	#418	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×D₄	2
#6489	2
C₂×#1136	2
#6406	2
#418	7