Subgroups — D₇×#108 #89664

Order 448.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₇×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₈×D₇	C₂	1	1	yes	maximal, characteristic
3	C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	#6396	C₂	1	2	yes	maximal, complemented
5	#6393	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
7	C₇×M₁₆	C₂²	1	2	yes	—
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	#1136	C₂²	1	2	yes	—
10	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
11	#1131	—	2	2	no	—
12	C₈×D₇	—	2	2	no	—
13	C₈×D₇	C₄	1	2	yes	—
14	C₈×D₇	C₄	1	2	yes	—
15	C₂×#108	—	7	1	no	maximal, Sylow
16	C₂²×D₇	Q₈	1	1	yes	characteristic
17	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂×Dic₇	Q₈	1	1	yes	characteristic
19	C₄×D₇	C₂×C₄	1	1	yes	characteristic
20	C₄×D₇	C₂×C₄	1	1	yes	characteristic
21	C₄×D₇	D₄	1	1	yes	characteristic
22	C₄×D₇	D₄	1	1	yes	characteristic
23	C₇×C₈	—	2	2	no	—
24	C₇×C₈	C₂×C₄	1	2	yes	—
25	C₇⋊C₈	—	2	2	no	—
26	C₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×M₁₆	—	7	2	no	—
28	C₂²×C₈	—	7	1	no	—
29	#108	—	7	2	no	—
30	#108	—	7	1	no	—
31	#108	D₇	1	1	yes	characteristic, direct product factor
32	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×D₇	—	2	2	no	—
34	C₂×D₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×D₇	C₄⋊C₄	1	1	yes	characteristic
36	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₇	C₂×D₄	1	1	yes	characteristic
38	Dic₇	C₄⋊C₄	1	1	yes	characteristic
39	Dic₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄	—	7	1	no	—
41	M₁₆	—	7	2	no	—
42	M₁₆	—	14	2	no	—
43	M₁₆	C₂×D₇	1	2	yes	—
44	C₂×C₈	—	7	1	no	—
45	C₂×C₈	—	7	2	no	—
46	C₂×C₈	—	7	2	no	—
47	C₂×C₈	—	14	2	no	—
48	C₂×C₈	C₂×D₇	1	1	yes	characteristic
49	C₂×C₇	—	2	1	no	—
50	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	D₇	—	2	1	no	—
52	D₇	#108	1	2	yes	direct product factor
53	C₂³	—	7	1	no	—
54	C₂×C₄	—	7	1	no	—
55	C₂×C₄	—	7	1	no	—
56	C₂×C₄	—	7	1	no	—
57	C₂×C₄	—	7	1	no	—
58	C₂×C₄	—	7	1	no	—
59	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	—	7	2	no	—
62	C₈	—	14	2	no	—
63	C₈	C₄×D₇	1	2	yes	—
64	C₇	C₂×#108	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	7	1	no	—
66	C₂²	—	7	1	no	—
67	C₂²	—	14	2	no	—
68	C₂²	Q₈×D₇	1	1	yes	characteristic
69	C₄	—	7	1	no	—
70	C₄	—	7	1	no	—
71	C₄	C₂×C₄×D₇	1	1	yes	characteristic
72	C₄	D₄×D₇	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	7	2	no	—
75	C₂	—	14	1	no	—
76	C₂	D₇×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#108	C₂
socle	C₂×C₇	C₂×C₄⋊C₄
prime radical	C₂²×D₇	Q₈
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#108	D₇
hypocenter	C₇	C₂×#108
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
3. C₇×#108	C₂	semidirect
4. #6396	C₂	semidirect
5. #6393	C₂	semidirect
31. #108	D₇	direct (N × H)
52. D₇	#108	direct (N × H)
64. C₇	C₂×#108	semidirect

Maximal subgroups

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

subgroup	index
D₇×M₁₆	2
C₂×C₈×D₇	2
C₇×#108	2
#6396	2
#6393	2
C₂×#108	7

Subgroups — D7×#108 #89664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#108 #89664