Subgroups — #11725852

Order 896.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#89471	C₂	1	1	yes	maximal, characteristic, complemented
2	#89466	C₂	1	2	yes	maximal
3	#89465	C₂	1	1	yes	maximal, characteristic
4	#89454	C₂	1	2	yes	maximal, complemented
5	C₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₈	—	2	2	no	—
7	C₂×#1132	—	2	2	no	—
8	#6437	C₂²	1	1	yes	characteristic
9	#6432	C₂²	1	1	yes	characteristic
10	#6432	C₂²	1	2	yes	—
11	C₄×C₇×C₈	C₂²	1	2	yes	—
12	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	#6395	—	2	4	no	—
14	#6392	—	2	4	no	—
15	#1722	—	7	1	no	maximal, Sylow
16	C₂×D₂₈	—	2	1	no	—
17	C₂×D₂₈	—	2	2	no	—
18	C₂×Dic₁₄	—	2	1	no	—
19	C₂×Dic₁₄	—	2	2	no	—
20	C₂×C₇×C₈	D₄	1	4	yes	—
21	C₂×C₇×C₈	D₄	1	2	yes	—
22	C₄²×C₇	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₇	—	2	2	no	—
24	C₄⋊Dic₇	—	2	2	no	—
25	C₄⋊Dic₇	—	2	2	no	—
26	Dic₂₈	—	4	4	no	—
27	#1132	—	4	4	no	—
28	#500	—	7	2	no	—
29	#494	—	7	1	no	—
30	#492	—	7	1	no	—
31	#486	—	7	2	no	—
32	#322	D₇	1	1	yes	characteristic, complemented
33	C₂²×D₇	—	4	1	no	—
34	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
36	C₂×Dic₇	—	4	1	no	—
37	C₂×Dic₇	—	4	2	no	—
38	D₂₈	—	4	2	no	—
39	D₂₈	—	4	4	no	—
40	Dic₁₄	—	4	2	no	—
41	Dic₁₄	—	4	4	no	—
42	C₇×C₈	—	2	4	no	—
43	C₇×C₈	D₈	1	4	yes	—
44	C₂×Q₁₆	—	14	2	no	—
45	C₂×SD₁₆	—	14	2	no	—
46	C₄⋊Q₈	—	7	2	no	—
47	C₄⋊Q₈	—	7	1	no	—
48	#127	—	7	1	no	—
49	#107	—	14	4	no	—
50	#102	—	14	4	no	—
51	C₄×C₈	C₂×D₇	1	2	yes	—
52	C₄×C₈	C₂×D₇	1	1	yes	characteristic
53	C₂²×C₇	#127	1	1	yes	characteristic
54	C₂×D₇	—	8	1	no	—
55	C₂×D₇	—	8	2	no	—
56	C₄×C₇	C₂×D₈	1	2	yes	—
57	C₄×C₇	#137	1	4	yes	—
58	Dic₇	—	8	1	no	—
59	Dic₇	—	8	2	no	—
60	C₂×Q₈	—	14	1	no	—
61	C₂×Q₈	—	14	2	no	—
62	C₂×D₄	—	14	2	no	—
63	C₂×D₄	—	14	1	no	—
64	Q₁₆	—	28	4	no	—
65	SD₁₆	—	28	4	no	—
66	C₂×C₈	D₂₈	1	4	yes	—
67	C₂×C₈	D₂₈	1	2	yes	—
68	C₄⋊C₄	—	14	2	no	—
69	C₄⋊C₄	—	14	2	no	—
70	C₄⋊C₄	—	14	2	no	—
71	C₄²	C₂²×D₇	1	1	yes	characteristic
72	C₂×C₇	#493	1	1	yes	characteristic
73	C₂×C₇	#497	1	2	yes	—
74	D₇	—	16	1	no	—
75	C₂³	—	28	1	no	—
76	Q₈	—	28	2	no	—
77	Q₈	—	28	4	no	—
78	D₄	—	28	2	no	—
79	D₄	—	28	4	no	—
80	C₂×C₄	—	28	2	no	—
81	C₂×C₄	—	28	1	no	—
82	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
83	C₂×C₄	C₂×D₂₈	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	D₅₆	1	4	yes	—
86	C₇	#1722	1	1	yes	characteristic, complemented, Sylow
87	C₂²	—	56	1	no	—
88	C₂²	—	56	2	no	—
89	C₂²	#6437	1	1	yes	characteristic
90	C₄	—	56	1	no	—
91	C₄	—	56	2	no	—
92	C₄	C₂×D₅₆	1	2	yes	—
93	C₄	#6472	1	4	yes	—
94	C₂	—	112	1	no	—
95	C₂	#89467	1	1	yes	characteristic
96	C₂	#89487	1	2	yes	—

	subgroup	G/H
center	C₂²	#6437
commutator, cosocle	C₄²×C₇	C₂³
Frattini	C₄²	C₂²×D₇
Fitting	C₇×#322	C₂
socle	C₂²×C₇	#127
prime radical	#6437	C₂²
hypercenter	#322	D₇
hypocenter	C₇	#1722
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
1. #89471	C₂	semidirect
4. #89454	C₂	semidirect
5. C₇×#322	C₂	semidirect
32. #322	D₇	semidirect
86. C₇	#1722	semidirect

Maximal subgroups

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

subgroup	index
#89471	2
#89466	2
#89465	2
#89454	2
C₇×#322	2
#1722	7