Subgroups — C₂×#367×C₉₇ #27138115

Order 12416.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#107×C₉₇	C₂	1	2	yes	maximal
3	#367×C₉₇	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
6	C₂×C₁₆×C₉₇	C₄	1	4	yes	complemented
7	C₂×C₁₆×C₉₇	C₂²	1	2	yes	—
8	#107×C₉₇	—	2	2	no	—
9	#107×C₉₇	C₂²	1	4	yes	—
10	C₂²×C₄×C₉₇	—	4	2	no	—
11	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
12	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
13	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₉₇	Q₈	1	2	yes	—
15	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
16	C₄⋊C₄×C₉₇	—	2	4	no	—
17	C₄⋊C₄×C₉₇	—	4	2	no	—
18	C₁₆×C₉₇	C₂×C₄	1	4	yes	complemented
19	C₂³×C₉₇	D₈	1	1	yes	characteristic
20	C₂×C₄×C₉₇	—	4	4	no	—
21	C₂×C₄×C₉₇	—	8	4	no	—
22	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
23	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
24	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
25	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
26	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
27	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
28	C₈×C₉₇	C₂×Q₈	1	1	yes	characteristic
29	C₂²×C₉₇	#107	1	2	yes	—
30	C₂²×C₉₇	SD₃₂	1	4	yes	—
31	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₉₇	—	8	4	no	—
33	C₄×C₉₇	#107	1	2	yes	—
34	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₄×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
36	C₂×C₉₇	#367	1	4	yes	direct product factor
37	C₂×C₉₇	C₂×#107	1	1	yes	characteristic
38	C₂×C₉₇	C₂×SD₃₂	1	2	yes	—
39	C₂×#367	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₉₇	C₂×#367	1	1	yes	characteristic, direct product factor, Sylow
41	C₂²×C₁₆	C₂×C₉₇	1	1	yes	characteristic
42	C₂×#107	C₂×C₉₇	1	2	yes	—
43	#367	C₂×C₉₇	1	4	yes	direct product factor
44	C₂²×C₈	C₂²×C₉₇	1	1	yes	characteristic
45	C₂×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₆	C₄×C₉₇	1	4	yes	complemented
47	C₂×C₁₆	C₂²×C₉₇	1	2	yes	—
48	#107	—	2	2	no	—
49	#107	C₂²×C₉₇	1	4	yes	—
50	C₂²×C₄	—	4	2	no	—
51	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
52	C₂×C₈	C₂×C₄×C₉₇	1	2	yes	—
53	C₂×C₈	D₄×C₉₇	1	1	yes	characteristic
54	C₂×C₈	Q₈×C₉₇	1	2	yes	—
55	C₂×C₈	C₂³×C₉₇	1	1	yes	characteristic
56	C₄⋊C₄	—	2	4	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₁₆	C₂×C₄×C₉₇	1	4	yes	complemented
59	C₂³	D₈×C₉₇	1	1	yes	characteristic
60	C₂×C₄	—	4	4	no	—
61	C₂×C₄	—	8	4	no	—
62	C₂×C₄	C₄⋊C₄×C₉₇	1	2	yes	—
63	C₂×C₄	D₈×C₉₇	1	1	yes	characteristic
64	C₂×C₄	Q₁₆×C₉₇	1	2	yes	—
65	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
66	C₈	C₄⋊C₄×C₉₇	1	2	yes	—
67	C₈	C₂²×C₄×C₉₇	1	1	yes	characteristic
68	C₈	C₂×Q₈×C₉₇	1	1	yes	characteristic
69	C₂²	#107×C₉₇	1	2	yes	—
70	C₂²	SD₃₂×C₉₇	1	4	yes	—
71	C₂²	C₂×D₈×C₉₇	1	1	yes	characteristic
72	C₄	—	8	4	no	—
73	C₄	#107×C₉₇	1	2	yes	—
74	C₄	C₂×C₄⋊C₄×C₉₇	1	1	yes	characteristic
75	C₄	C₂×Q₁₆×C₉₇	1	1	yes	characteristic
76	C₂	#367×C₉₇	1	4	yes	direct product factor
77	C₂	C₂×#107×C₉₇	1	1	yes	characteristic
78	C₂	C₂×SD₃₂×C₉₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₉₇	D₈
commutator	C₈	C₂²×C₄×C₉₇
Frattini, cosocle	C₂×C₈	C₂³×C₉₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ C₂²×C₄×C₉₇ ◁ C₂²×C₈×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #367×C₉₇	C₂	direct (N × H)
6. C₂×C₁₆×C₉₇	C₄	semidirect
18. C₁₆×C₉₇	C₂×C₄	semidirect
36. C₂×C₉₇	#367	direct (N × H)
39. C₂×#367	C₉₇	direct (N × H)
40. C₉₇	C₂×#367	direct (N × H)
43. #367	C₂×C₉₇	direct (N × H)
46. C₂×C₁₆	C₄×C₉₇	semidirect
58. C₁₆	C₂×C₄×C₉₇	semidirect
76. C₂	#367×C₉₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×C₉₇	2
C₂×#107×C₉₇	2
#367×C₉₇	2
C₂×#367	97

Subgroups — C2×#367×C97 #27138115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#367×C₉₇ #27138115