Subgroups — #89954

Order 448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₇	C₂	1	1	yes	maximal, characteristic
3	#6507	C₂	1	1	yes	maximal, characteristic
4	C₂×#1144	C₂	1	1	yes	maximal, characteristic
5	#6409	C₂	1	1	yes	maximal, characteristic
6	#6390	C₂	1	1	yes	maximal, characteristic
7	Dic₇⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
11	C₇×Q₁₆	—	2	2	no	—
12	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
13	#1144	—	2	2	no	—
14	C₄⋊Dic₇	—	2	1	no	—
15	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	Dic₇⋊C₄	—	2	1	no	—
17	Dic₇⋊C₄	—	2	1	no	—
18	C₄×Dic₇	—	2	1	no	—
19	C₄×Dic₇	C₂²	1	1	yes	characteristic
20	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
21	#462	—	7	1	no	maximal, Sylow
22	C₇×Q₈	—	2	1	no	—
23	C₇×Q₈	—	2	1	no	—
24	C₇×Q₈	—	2	1	no	—
25	C₇×Q₈	D₄	1	2	yes	—
26	C₂×C₄×C₇	—	2	1	no	—
27	C₂×C₄×C₇	—	2	1	no	—
28	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₇	—	2	1	no	—
30	C₂×Dic₇	—	2	1	no	—
31	C₂×Dic₇	D₄	1	1	yes	characteristic
32	C₂×Dic₇	D₄	1	1	yes	characteristic
33	Dic₁₄	—	2	1	no	—
34	Dic₁₄	—	2	1	no	—
35	C₇×C₈	—	2	1	no	—
36	C₇⋊C₈	—	2	1	no	—
37	C₂×Q₁₆	—	7	1	no	—
38	C₂×Q₁₆	D₇	1	1	yes	characteristic
39	C₄⋊Q₈	—	7	1	no	—
40	C₄×Q₈	—	7	1	no	—
41	C₄⋊C₈	—	7	1	no	—
42	#103	—	7	1	no	—
43	#103	—	7	1	no	—
44	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
45	C₄×C₇	—	2	2	no	—
46	C₄×C₇	—	4	1	no	—
47	C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₄×C₇	#41	1	1	yes	characteristic
49	Dic₇	—	2	1	no	—
50	Dic₇	—	4	1	no	—
51	Dic₇	—	4	1	no	—
52	Dic₇	Q₁₆	1	2	yes	complemented
53	C₂×Q₈	—	7	1	no	—
54	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
55	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
56	Q₁₆	—	2	2	no	—
57	Q₁₆	—	14	2	no	—
58	C₂×C₈	—	7	1	no	—
59	C₂×C₈	C₂×D₇	1	1	yes	characteristic
60	C₄⋊C₄	—	7	1	no	—
61	C₄⋊C₄	—	14	1	no	—
62	C₄⋊C₄	—	14	1	no	—
63	C₄⋊C₄	—	14	1	no	—
64	C₄²	—	7	1	no	—
65	C₄²	—	14	1	no	—
66	C₂×C₇	#121	1	1	yes	characteristic
67	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₇	#137	1	1	yes	characteristic
69	Q₈	—	2	1	no	—
70	Q₈	—	2	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	—	14	1	no	—
73	Q₈	—	14	1	no	—
74	Q₈	C₂²⋊D₇	1	2	yes	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	7	1	no	—
78	C₂×C₄	—	7	1	no	—
79	C₂×C₄	—	14	1	no	—
80	C₂×C₄	—	14	1	no	—
81	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₈	—	14	1	no	—
84	C₇	#462	1	1	yes	characteristic, complemented, Sylow
85	C₂²	D₄×D₇	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	4	1	no	—
88	C₄	—	7	2	no	—
89	C₄	—	14	1	no	—
90	C₄	—	28	1	no	—
91	C₄	—	28	1	no	—
92	C₄	#1159	1	1	yes	characteristic
93	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
94	C₂	D₇×Q₁₆	1	1	yes	characteristic
95	C₂	#6481	1	1	yes	characteristic
96	C₂	#6502	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₇×Q₁₆	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	C₂×Q₁₆	D₇
hypocenter	C₇	#462
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
52. Dic₇	Q₁₆	semidirect
84. C₇	#462	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×Q₁₆	2
Q₈×Dic₇	2
#6507	2
C₂×#1144	2
#6409	2
#6390	2
Dic₇⋊C₈	2
#462	7