Subgroups — D₇×#367 #11727098

Order 896.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₇×C₁₆	C₂	1	1	yes	maximal, characteristic
2	D₇×#107	C₂	1	2	yes	maximal
3	C₇×#367	C₂	1	1	yes	maximal, characteristic, complemented
4	#89300	C₂	1	1	yes	maximal, characteristic, complemented
5	#89285	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×D₇	C₂²	1	1	yes	characteristic
7	D₇×C₄⋊C₄	—	2	2	no	—
8	C₂×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₇×#107	C₂²	1	2	yes	—
10	#6392	C₂²	1	2	yes	—
11	C₂×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#6381	—	2	2	no	—
13	D₇×C₁₆	C₄	1	4	yes	complemented
14	C₂×#367	—	7	1	no	maximal, Sylow
15	C₂×C₄×D₇	—	4	2	no	—
16	C₂×C₄×D₇	D₄	1	1	yes	characteristic
17	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
18	C₇×C₄⋊C₄	—	2	2	no	—
19	C₄⋊Dic₇	—	2	2	no	—
20	Dic₇⋊C₄	—	4	2	no	—
21	C₂×C₇⋊C₈	D₄	1	1	yes	characteristic
22	C₈×D₇	C₂×C₄	1	2	yes	—
23	C₈×D₇	Q₈	1	2	yes	—
24	C₇×C₁₆	C₂×C₄	1	2	yes	complemented
25	C₇⋊C₁₆	C₂×C₄	1	2	yes	complemented
26	C₂²×C₁₆	—	7	1	no	—
27	C₂×#107	—	7	2	no	—
28	#367	—	7	1	no	—
29	#367	—	7	2	no	—
30	#367	D₇	1	1	yes	characteristic, direct product factor
31	C₂²×D₇	D₈	1	1	yes	characteristic
32	C₂×C₄×C₇	—	4	2	no	—
33	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₇	—	4	2	no	—
35	C₂×Dic₇	D₈	1	1	yes	characteristic
36	C₄×D₇	—	8	4	no	—
37	C₄×D₇	C₄⋊C₄	1	2	yes	—
38	C₄×D₇	Q₁₆	1	2	yes	—
39	C₇×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₇×C₈	C₂×Q₈	1	1	yes	characteristic
41	C₇⋊C₈	C₄⋊C₄	1	2	yes	—
42	C₂²×C₈	—	7	1	no	—
43	C₂×C₄⋊C₄	—	14	2	no	—
44	C₂×C₁₆	—	7	1	no	—
45	C₂×C₁₆	—	7	4	no	—
46	C₂×C₁₆	C₂×D₇	1	1	yes	characteristic
47	#107	—	7	2	no	—
48	#107	—	14	2	no	—
49	#107	C₂×D₇	1	2	yes	—
50	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
51	C₂×D₇	#107	1	2	yes	—
52	C₂×D₇	SD₃₂	1	4	yes	—
53	C₄×C₇	—	8	2	no	—
54	C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
56	Dic₇	—	8	2	no	—
57	Dic₇	#107	1	2	yes	—
58	C₂²×C₄	—	7	1	no	—
59	C₂²×C₄	—	28	2	no	—
60	C₂×C₈	—	7	1	no	—
61	C₂×C₈	—	7	2	no	—
62	C₂×C₈	—	7	2	no	—
63	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
64	C₄⋊C₄	—	2	2	no	—
65	C₄⋊C₄	—	14	2	no	—
66	C₄⋊C₄	—	28	2	no	—
67	C₁₆	—	7	2	no	—
68	C₁₆	C₄×D₇	1	2	yes	complemented
69	C₂×C₇	C₂×#107	1	1	yes	characteristic
70	C₂×C₇	C₂×SD₃₂	1	2	yes	—
71	D₇	#367	1	4	yes	direct product factor
72	C₂³	—	7	1	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	7	1	no	—
75	C₂×C₄	—	7	2	no	—
76	C₂×C₄	—	7	2	no	—
77	C₂×C₄	—	28	2	no	—
78	C₂×C₄	—	56	4	no	—
79	C₂×C₄	D₄×D₇	1	1	yes	characteristic
80	C₈	—	7	2	no	—
81	C₈	C₂×C₄×D₇	1	1	yes	characteristic
82	C₈	Q₈×D₇	1	1	yes	characteristic
83	C₇	C₂×#367	1	1	yes	characteristic, complemented, Sylow
84	C₂²	—	7	2	no	—
85	C₂²	—	7	4	no	—
86	C₂²	D₇×D₈	1	1	yes	characteristic
87	C₄	—	7	2	no	—
88	C₄	—	8	2	no	—
89	C₄	—	56	2	no	—
90	C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
91	C₄	D₇×Q₁₆	1	1	yes	characteristic
92	C₂	—	7	4	no	—
93	C₂	D₇×#107	1	1	yes	characteristic
94	C₂	D₇×SD₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²	D₇×D₈
commutator	C₇×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇
Fitting	C₇×#367	C₂
socle	C₂²×C₇	C₂×D₈
prime radical	C₂²×D₇	D₈
cosocle	C₂×C₇×C₈	C₂³
hypercenter	#367	D₇
hypocenter	C₇	C₂×#367
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
3. C₇×#367	C₂	semidirect
4. #89300	C₂	semidirect
5. #89285	C₂	semidirect
13. D₇×C₁₆	C₄	semidirect
24. C₇×C₁₆	C₂×C₄	semidirect
25. C₇⋊C₁₆	C₂×C₄	semidirect
30. #367	D₇	direct (N × H)
68. C₁₆	C₄×D₇	semidirect
71. D₇	#367	direct (N × H)
83. C₇	C₂×#367	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₇×C₁₆	2
D₇×#107	2
C₇×#367	2
#89300	2
#89285	2
C₂×#367	7

Subgroups — D7×#367 #11727098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#367 #11727098