Subgroups — #89781

Order 448.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	#6433	C₂	1	1	yes	maximal, characteristic, complemented
3	#6406	C₂	1	1	yes	maximal, characteristic
4	#6406	C₂	1	1	yes	maximal, characteristic
5	#6382	C₂	1	1	yes	maximal, characteristic, complemented
6	#6381	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₇	—	2	1	no	—
10	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₇×C₂²⋊C₄	—	2	1	no	—
12	C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
15	Dic₇⋊C₄	—	2	1	no	—
16	Dic₇⋊C₄	—	2	1	no	—
17	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#478	—	7	1	no	maximal, Sylow
20	C₂³×C₇	—	2	1	no	—
21	C₇×D₄	—	2	1	no	—
22	C₇×D₄	Q₈	1	2	yes	—
23	C₂×C₄×C₇	—	2	1	no	—
24	C₂×C₄×C₇	—	4	2	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	C₇⋊C₈	—	2	1	no	—
31	C₇⋊C₈	—	2	1	no	—
32	#125	—	7	1	no	—
33	C₄×D₄	D₇	1	1	yes	characteristic
34	#107	—	7	1	no	—
35	#106	—	7	1	no	—
36	C₄⋊C₈	—	7	1	no	—
37	#102	—	7	1	no	—
38	#102	—	7	1	no	—
39	C₂²×C₇	—	2	2	no	—
40	C₂²×C₇	—	4	1	no	—
41	C₂²×C₇	—	4	1	no	—
42	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
43	C₄×C₇	—	2	1	no	—
44	C₄×C₇	—	2	1	no	—
45	C₄×C₇	—	4	1	no	—
46	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
47	C₄×C₇	#41	1	1	yes	characteristic
48	Dic₇	—	4	1	no	—
49	Dic₇	—	4	1	no	—
50	C₂×D₄	C₂×D₇	1	1	yes	characteristic
51	C₂²×C₄	—	2	1	no	—
52	C₂×C₈	—	7	1	no	—
53	C₂×C₈	—	7	1	no	—
54	C₄⋊C₄	—	7	1	no	—
55	C₄⋊C₄	—	7	1	no	—
56	C₄⋊C₄	—	14	1	no	—
57	C₄⋊C₄	—	14	1	no	—
58	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
59	C₂²⋊C₄	—	2	1	no	—
60	C₄²	C₂×D₇	1	1	yes	characteristic
61	C₂×C₇	—	4	2	no	—
62	C₂×C₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂×C₇	#135	1	1	yes	characteristic
64	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂³	—	2	1	no	—
66	D₄	—	2	1	no	—
67	D₄	Dic₁₄	1	2	yes	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	4	2	no	—
70	C₂×C₄	—	14	1	no	—
71	C₂×C₄	—	14	1	no	—
72	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
74	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
75	C₈	—	14	1	no	—
76	C₈	—	14	1	no	—
77	C₇	#478	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	2	2	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	28	1	no	—
86	C₄	—	28	1	no	—
87	C₄	C₂×Dic₁₄	1	1	yes	characteristic
88	C₄	#1157	1	1	yes	characteristic
89	C₂	—	4	2	no	—
90	C₂	#6487	1	1	yes	characteristic
91	C₂	#6495	1	1	yes	characteristic
92	C₂	#6513	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₄×C₇×D₄	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×C₇×D₄	C₂²
hypercenter	C₄×D₄	D₇
hypocenter	C₇	#478
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #6433	C₂	semidirect
5. #6382	C₂	semidirect
6. #6381	C₂	semidirect
7. C₄⋊(C₇⋊C₈)	C₂	semidirect
77. C₇	#478	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×D₄	2
#6433	2
#6406	2
#6382	2
#6381	2
C₄⋊(C₇⋊C₈)	2
#478	7