Subgroups — #89802

Order 448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#1144	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic
4	#6409	C₂	1	1	yes	maximal, characteristic
5	#6384	C₂	1	1	yes	maximal, characteristic
6	#6382	C₂	1	1	yes	maximal, characteristic
7	#6377	C₂	1	1	yes	maximal, characteristic
8	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊C₄	—	2	1	no	—
11	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₄²×C₇	—	2	1	no	—
13	C₄²×C₇	C₂²	1	1	yes	characteristic
14	#1144	C₄	1	4	yes	complemented
15	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	Dic₇⋊C₄	—	2	1	no	—
17	C₄×Dic₇	—	2	1	no	—
18	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
20	#441	—	7	1	no	maximal, Sylow
21	C₇×Q₈	—	2	1	no	—
22	C₇×Q₈	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₇	—	2	1	no	—
24	C₂×C₄×C₇	—	2	1	no	—
25	C₂×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₇	—	2	1	no	—
29	C₂×Dic₇	—	2	1	no	—
30	Dic₁₄	—	2	1	no	—
31	Dic₁₄	C₂×C₄	1	2	yes	—
32	C₇⋊C₈	—	2	1	no	—
33	C₇⋊C₈	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆	—	7	1	no	—
35	C₄×Q₈	—	7	1	no	—
36	C₄×Q₈	D₇	1	1	yes	characteristic
37	#106	—	7	1	no	—
38	#103	—	7	1	no	—
39	#103	—	7	1	no	—
40	#97	—	7	1	no	—
41	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
42	C₄×C₇	—	2	2	no	—
43	C₄×C₇	—	2	1	no	—
44	C₄×C₇	—	2	1	no	—
45	C₄×C₇	—	4	1	no	—
46	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₄×C₇	#41	1	1	yes	characteristic
48	Dic₇	—	2	2	no	—
49	Dic₇	—	4	1	no	—
50	C₂×Q₈	—	7	1	no	—
51	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
52	Q₁₆	—	7	4	no	—
53	C₂×C₈	—	7	1	no	—
54	C₂×C₈	—	7	1	no	—
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	—	7	1	no	—
57	C₄⋊C₄	—	14	1	no	—
58	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
59	C₄²	—	2	1	no	—
60	C₄²	—	14	1	no	—
61	C₄²	C₂×D₇	1	1	yes	characteristic
62	C₂×C₇	C₄×D₄	1	1	yes	characteristic
63	C₂×C₇	#137	1	1	yes	characteristic
64	C₂×C₇	#137	1	1	yes	characteristic
65	Q₈	—	2	1	no	—
66	Q₈	—	7	2	no	—
67	Q₈	—	14	1	no	—
68	Q₈	C₄×D₇	1	2	yes	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	14	1	no	—
72	C₂×C₄	—	14	1	no	—
73	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
76	C₈	—	7	2	no	—
77	C₈	—	14	1	no	—
78	C₇	#441	1	1	yes	characteristic, complemented, Sylow
79	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	14	2	no	—
85	C₄	—	28	1	no	—
86	C₄	C₂×C₄×D₇	1	1	yes	characteristic
87	C₄	#1157	1	1	yes	characteristic
88	C₂	C₄×C₂²⋊D₇	1	1	yes	characteristic
89	C₂	#6505	1	1	yes	characteristic
90	C₂	#6514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₄×C₇×Q₈	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₄×Q₈	D₇
hypocenter	C₇	#441
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
14. #1144	C₄	semidirect
78. C₇	#441	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×Q₈	2
C₂×#1144	2
C₄×Dic₁₄	2
#6409	2
#6384	2
#6382	2
#6377	2
#441	7