Subgroups — #89898

Order 448.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1136	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇)⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂³×D₇	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₇	—	2	2	no	—
9	C₂×C₄×D₇	—	2	2	no	—
10	C₂×C₄×D₇	C₄	1	2	yes	—
11	C₂×C₄×D₇	C₂²	1	2	yes	—
12	C₇×M₁₆	—	2	2	no	—
13	C₂×C₇×C₈	C₂²	1	2	yes	complemented
14	#1136	—	2	2	no	—
15	C₂×C₇⋊C₈	C₂²	1	2	yes	complemented
16	#407	—	7	1	no	maximal, Sylow
17	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×D₇	—	2	2	no	—
19	C₂²×D₇	—	2	4	no	—
20	C₂²×D₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₇	D₄	1	2	yes	—
23	C₂×C₄×C₇	D₄	1	2	yes	—
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₇	—	2	2	no	—
26	C₂×Dic₇	C₂×C₄	1	2	yes	—
27	C₄×D₇	—	2	4	no	—
28	C₄×D₇	—	2	2	no	—
29	C₄×D₇	—	2	2	no	—
30	C₇×C₈	—	2	2	no	—
31	C₇⋊C₈	—	2	2	no	—
32	C₂³×C₄	—	7	1	no	—
33	C₂×M₁₆	—	7	1	no	—
34	C₂×M₁₆	D₇	1	1	yes	characteristic, complemented
35	C₂²⋊C₈	—	7	4	no	—
36	C₂²×C₇	—	2	2	no	—
37	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₇	—	2	8	no	—
40	C₂×D₇	—	2	2	no	—
41	C₂×D₇	—	2	2	no	—
42	C₂×D₇	M₁₆	1	4	yes	complemented
43	C₄×C₇	C₂²⋊C₄	1	2	yes	—
44	C₄×C₇	C₂×D₄	1	1	yes	characteristic
45	C₄×C₇	C₂×D₄	1	1	yes	characteristic
46	Dic₇	—	2	2	no	—
47	C₂⁴	—	7	1	no	—
48	C₂²×C₄	—	7	1	no	—
49	C₂²×C₄	—	7	2	no	—
50	C₂²×C₄	—	7	2	no	—
51	C₂²×C₄	—	14	2	no	—
52	C₂²×C₄	—	14	2	no	—
53	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
54	M₁₆	—	2	2	no	—
55	M₁₆	—	14	2	no	—
56	C₂×C₈	—	7	2	no	—
57	C₂×C₈	C₂×D₇	1	2	yes	complemented
58	C₂×C₇	—	2	2	no	—
59	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₇	C₂×M₁₆	1	2	yes	—
61	D₇	—	2	4	no	—
62	C₂³	—	7	2	no	—
63	C₂³	—	14	4	no	—
64	C₂³	—	14	2	no	—
65	C₂³	C₄×D₇	1	1	yes	characteristic
66	C₂×C₄	—	7	2	no	—
67	C₂×C₄	—	14	2	no	—
68	C₂×C₄	—	14	4	no	—
69	C₂×C₄	—	14	2	no	—
70	C₂×C₄	—	14	2	no	—
71	C₂×C₄	C₄×D₇	1	1	yes	characteristic
72	C₂×C₄	D₂₈	1	2	yes	—
73	C₂×C₄	C₂²⋊D₇	1	2	yes	—
74	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	14	2	no	—
77	C₇	#407	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	2	2	no	—
79	C₂²	—	7	4	no	—
80	C₂²	—	14	8	no	—
81	C₂²	—	14	2	no	—
82	C₂²	—	14	2	no	—
83	C₂²	(C₂×D₇)⋊C₄	1	2	yes	—
84	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
85	C₄	—	14	2	no	—
86	C₄	(C₂×D₇)⋊C₄	1	2	yes	—
87	C₄	C₂×D₂₈	1	1	yes	characteristic
88	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
89	C₂	—	2	2	no	—
90	C₂	—	14	4	no	—
91	C₂	D₇×M₁₆	1	2	yes	—
92	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	C₂³×D₇	C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×M₁₆	D₇
hypocenter	C₇	#407
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. C₂×#1136	C₂	semidirect
4. (C₂×D₇)⋊C₈	C₂	semidirect
13. C₂×C₇×C₈	C₂²	semidirect
15. C₂×C₇⋊C₈	C₂²	semidirect
34. C₂×M₁₆	D₇	semidirect
42. C₂×D₇	M₁₆	semidirect
57. C₂×C₈	C₂×D₇	semidirect
77. C₇	#407	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₇	2
C₂×C₇×M₁₆	2
C₂×#1136	2
(C₂×D₇)⋊C₈	2
#407	7