Subgroups — #89867

Order 448.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6551	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#6505	C₂	1	2	yes	maximal, complemented
4	#6410	C₂	1	1	yes	maximal, characteristic, complemented
5	#6379	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
8	#1161	—	2	2	no	—
9	Q₈×D₇	—	2	2	no	—
10	#1157	—	2	2	no	—
11	#1157	C₂²	1	2	yes	—
12	C₇×C₄⋊C₄	—	2	2	no	—
13	C₄²×C₇	C₂²	1	1	yes	characteristic
14	#1144	—	2	2	no	—
15	#1143	—	2	2	no	—
16	#1136	C₂²	1	2	yes	—
17	#456	—	7	1	no	maximal, Sylow
18	C₇×Q₈	—	2	2	no	—
19	C₇×Q₈	—	2	2	no	—
20	C₂×C₄×C₇	—	2	1	no	—
21	C₂×C₄×C₇	—	2	1	no	—
22	C₂×C₄×C₇	D₄	1	2	yes	—
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂²⋊D₇	—	2	2	no	—
25	D₂₈	—	2	2	no	—
26	D₂₈	D₄	1	2	yes	—
27	C₄×D₇	—	2	4	no	—
28	C₄×D₇	—	2	2	no	—
29	Dic₁₄	—	2	2	no	—
30	Dic₁₄	D₄	1	2	yes	—
31	C₇⋊C₈	—	2	2	no	—
32	2₋¹⁺⁴	—	7	1	no	—
33	#137	—	7	2	no	—
34	C₄⋊Q₈	D₇	1	1	yes	characteristic, complemented
35	#104	—	7	2	no	—
36	#101	—	7	1	no	—
37	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
38	C₂×D₇	—	2	2	no	—
39	C₄×C₇	—	2	2	no	—
40	C₄×C₇	—	2	2	no	—
41	C₄×C₇	—	4	1	no	—
42	C₄×C₇	C₂×D₄	1	2	yes	—
43	Dic₇	—	2	2	no	—
44	#41	—	7	2	no	—
45	#41	—	14	2	no	—
46	#41	—	14	2	no	—
47	C₂×Q₈	—	14	2	no	—
48	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
49	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
50	Q₁₆	—	14	2	no	—
51	SD₁₆	—	14	2	no	—
52	M₁₆	—	7	2	no	—
53	C₄⋊C₄	—	2	2	no	—
54	C₄²	C₂×D₇	1	1	yes	characteristic
55	C₂×C₇	—	2	1	no	—
56	C₂×C₇	#120	1	1	yes	characteristic
57	D₇	—	4	2	no	—
58	Q₈	—	2	2	no	—
59	Q₈	—	2	2	no	—
60	Q₈	—	7	2	no	—
61	Q₈	—	14	2	no	—
62	D₄	—	7	2	no	—
63	D₄	—	14	2	no	—
64	D₄	—	14	2	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	14	4	no	—
68	C₂×C₄	—	14	2	no	—
69	C₂×C₄	C₂²⋊D₇	1	2	yes	—
70	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
71	C₈	—	14	2	no	—
72	C₇	#456	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	14	2	no	—
74	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	14	2	no	—
79	C₄	D₄×D₇	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	28	2	no	—
82	C₂	#6500	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6500
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×C₄⋊Q₈	C₂
socle	C₂×C₇	#120
prime radical	#6551	C₂
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	#456
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₇×C₄⋊Q₈	C₂	semidirect
3. #6505	C₂	semidirect
4. #6410	C₂	semidirect
5. #6379	C₂	semidirect
34. C₄⋊Q₈	D₇	semidirect
72. C₇	#456	semidirect

Maximal subgroups

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

subgroup	index
#6551	2
C₇×C₄⋊Q₈	2
#6505	2
#6410	2
#6379	2
#456	7