Subgroups — #89984

Order 448.

234 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₇	C₂	1	2	yes	maximal
3	C₂×C₄⋊Dic₇	C₂	1	2	yes	maximal, complemented
4	C₇⋊#95	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₇	—	2	4	no	—
7	C₂²×C₄×C₇	C₄	1	4	yes	complemented
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₇	C₂²	1	4	yes	—
11	C₂²⋊Dic₇	—	2	4	no	—
12	C₄⋊Dic₇	—	2	4	no	—
13	#380	—	7	1	no	maximal, Sylow
14	C₂³×C₇	—	2	2	no	—
15	C₂³×C₇	—	2	2	no	—
16	C₂³×C₇	C₂×C₄	1	2	yes	—
17	C₂³×C₇	D₄	1	2	yes	—
18	C₂³×C₇	Q₈	1	2	yes	—
19	C₂³×C₇	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₇	—	2	2	no	—
21	C₂×C₄×C₇	—	2	8	no	—
22	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₇	D₄	1	4	yes	—
25	C₂×Dic₇	—	2	4	no	—
26	C₂×Dic₇	—	2	4	no	—
27	C₂×Dic₇	—	2	4	no	—
28	C₂³×C₄	D₇	1	1	yes	characteristic
29	C₂×C₄⋊C₄	—	7	2	no	—
30	C₂×C₂²⋊C₄	—	7	2	no	—
31	#95	—	7	2	no	—
32	C₂²×C₇	—	2	2	no	—
33	C₂²×C₇	—	2	2	no	—
34	C₂²×C₇	—	2	4	no	—
35	C₂²×C₇	—	2	4	no	—
36	C₂²×C₇	C₄⋊C₄	1	4	yes	complemented
37	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₇	C₂×D₄	1	2	yes	—
40	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₇	#41	1	2	yes	—
42	C₄×C₇	—	2	2	no	—
43	C₄×C₇	C₂²⋊C₄	1	4	yes	complemented
44	Dic₇	—	4	4	no	—
45	C₂⁴	C₂×D₇	1	1	yes	characteristic
46	C₂²×C₄	—	2	4	no	—
47	C₂²×C₄	—	7	4	no	—
48	C₂²×C₄	Dic₇	1	4	yes	complemented
49	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
50	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
51	C₄⋊C₄	—	14	4	no	—
52	C₂²⋊C₄	—	14	4	no	—
53	C₂×C₇	—	2	4	no	—
54	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₇	#117	1	1	yes	characteristic
57	C₂×C₇	#121	1	2	yes	—
58	C₂×C₇	C₂²⋊Q₈	1	2	yes	—
59	C₂³	—	2	2	no	—
60	C₂³	—	2	2	no	—
61	C₂³	Dic₁₄	1	2	yes	—
62	C₂³	D₂₈	1	2	yes	—
63	C₂³	C₂×Dic₇	1	2	yes	—
64	C₂³	C₂²×D₇	1	1	yes	characteristic
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	8	no	—
67	C₂×C₄	—	14	4	no	—
68	C₂×C₄	—	14	4	no	—
69	C₂×C₄	—	14	4	no	—
70	C₂×C₄	C₂×Dic₇	1	2	yes	—
71	C₂×C₄	C₂×Dic₇	1	2	yes	—
72	C₂×C₄	C₂²⋊D₇	1	4	yes	—
73	C₇	#380	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	2	4	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	4	no	—
77	C₂²	—	2	2	no	—
78	C₂²	C₄⋊Dic₇	1	4	yes	complemented
79	C₂²	C₂×Dic₁₄	1	1	yes	characteristic
80	C₂²	C₂×D₂₈	1	1	yes	characteristic
81	C₂²	#1157	1	2	yes	—
82	C₂²	C₂²×Dic₇	1	1	yes	characteristic
83	C₂²	C₂×C₂²⋊D₇	1	2	yes	—
84	C₄	—	2	2	no	—
85	C₄	—	28	4	no	—
86	C₄	C₂²⋊Dic₇	1	4	yes	complemented
87	C₂	—	2	4	no	—
88	C₂	#6487	1	2	yes	—
89	C₂	C₂×C₄⋊Dic₇	1	1	yes	characteristic
90	C₂	#6489	1	1	yes	characteristic
91	C₂	#6493	1	2	yes	—
92	C₂	C₂×C₂²⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₄
Fitting	C₂³×C₄×C₇	C₂
socle, cosocle	C₂³×C₇	C₂³
prime radical	C₂⁴×C₇	C₂²
hypercenter	C₂³×C₄	D₇
hypocenter	C₇	#380
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊Dic₇	C₂	semidirect
4. C₇⋊#95	C₂	semidirect
7. C₂²×C₄×C₇	C₄	semidirect
36. C₂²×C₇	C₄⋊C₄	semidirect
43. C₄×C₇	C₂²⋊C₄	semidirect
48. C₂²×C₄	Dic₇	semidirect
73. C₇	#380	semidirect
78. C₂²	C₄⋊Dic₇	semidirect
86. C₄	C₂²⋊Dic₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₇	2
C₂×C₂²⋊Dic₇	2
C₂×C₄⋊Dic₇	2
C₇⋊#95	2
#380	7