Subgroups — #89901

Order 448.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1157	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇)⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₂²⋊D₇	C₄	1	2	yes	—
7	#1157	—	2	2	no	—
8	#1157	—	2	2	no	—
9	C₂×D₂₈	C₄	1	1	yes	characteristic
10	C₂×C₄×D₇	C₂²	1	2	yes	—
11	C₂×Dic₁₄	C₄	1	1	yes	characteristic
12	C₇×M₁₆	—	2	2	no	—
13	C₂×C₇×C₈	C₂²	1	2	yes	—
14	C₂×C₇⋊C₈	—	2	2	no	—
15	C₂×C₇⋊C₈	C₂²	1	2	yes	—
16	#408	—	7	1	no	maximal, Sylow
17	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×D₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₇	D₄	1	2	yes	—
21	C₂×C₄×C₇	D₄	1	2	yes	—
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂²⋊D₇	—	2	4	no	—
24	C₂×Dic₇	C₂×C₄	1	2	yes	—
25	D₂₈	—	2	2	no	—
26	C₄×D₇	—	2	2	no	—
27	C₄×D₇	—	2	2	no	—
28	Dic₁₄	—	2	2	no	—
29	C₇×C₈	—	2	2	no	—
30	C₇⋊C₈	—	2	2	no	—
31	C₂×#41	—	7	1	no	—
32	C₂×M₁₆	D₇	1	1	yes	characteristic, complemented
33	C₂²×C₈	—	7	1	no	—
34	C₂²⋊C₈	—	7	4	no	—
35	C₂²×C₇	—	2	2	no	—
36	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×D₇	—	2	2	no	—
39	C₂×D₇	—	2	4	no	—
40	C₄×C₇	C₂²⋊C₄	1	2	yes	—
41	C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	C₄×C₇	C₂×D₄	1	1	yes	characteristic
43	Dic₇	—	2	2	no	—
44	#41	—	14	2	no	—
45	#41	—	14	2	no	—
46	C₂×Q₈	—	7	1	no	—
47	C₂×D₄	—	7	2	no	—
48	C₂×D₄	—	7	1	no	—
49	C₂²×C₄	—	7	2	no	—
50	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
51	M₁₆	—	2	2	no	—
52	C₂×C₈	—	7	2	no	—
53	C₂×C₈	—	14	2	no	—
54	C₂×C₈	C₂×D₇	1	2	yes	—
55	C₂×C₇	—	2	2	no	—
56	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₇	#131	1	2	yes	—
58	D₇	—	4	2	no	—
59	C₂³	—	7	2	no	—
60	C₂³	C₄×D₇	1	1	yes	characteristic
61	Q₈	—	14	2	no	—
62	D₄	—	14	4	no	—
63	D₄	—	14	2	no	—
64	C₂×C₄	—	7	2	no	—
65	C₂×C₄	—	14	2	no	—
66	C₂×C₄	—	14	2	no	—
67	C₂×C₄	C₄×D₇	1	1	yes	characteristic
68	C₂×C₄	D₂₈	1	2	yes	—
69	C₂×C₄	C₂²⋊D₇	1	2	yes	—
70	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₈	—	14	2	no	—
73	C₇	#408	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	2	2	no	—
75	C₂²	—	14	4	no	—
76	C₂²	—	14	2	no	—
77	C₂²	(C₂×D₇)⋊C₄	1	2	yes	—
78	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
79	C₄	—	14	2	no	—
80	C₄	(C₂×D₇)⋊C₄	1	2	yes	—
81	C₄	C₂×D₂₈	1	1	yes	characteristic
82	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	28	2	no	—
85	C₂	#6470	1	2	yes	—
86	C₂	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₄
Fitting	C₂×C₇×M₁₆	C₂
prime radical	C₂×#1157	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×M₁₆	D₇
hypocenter	C₇	#408
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×M₁₆	C₂	semidirect
3. C₂²×C₇⋊C₈	C₂	semidirect
4. (C₂×D₇)⋊C₈	C₂	semidirect
32. C₂×M₁₆	D₇	semidirect
73. C₇	#408	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1157	2
C₂×C₇×M₁₆	2
C₂²×C₇⋊C₈	2
(C₂×D₇)⋊C₈	2
#408	7