Subgroups — C₂²×#15690810 #27155012

Order 12416.

1624 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₁₉₄	C₂	1	2	yes	maximal
2	C₂×#15690810	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊Dic₉₇	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₉₄	—	2	16	no	—
6	C₂²×Dic₁₉₄	C₂²	1	24	yes	—
7	C₂³×Dic₉₇	C₂²	1	4	yes	—
8	#15690810	C₂²	1	64	yes	direct product factor
9	C₂×C₄⋊Dic₉₇	C₂²	1	48	yes	—
10	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₉₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₉₇	C₂²	1	12	yes	—
13	C₂×Dic₁₉₄	—	2	96	no	—
14	C₂×Dic₁₉₄	C₂³	1	32	yes	—
15	C₂²×Dic₉₇	—	2	16	no	—
16	C₂²×Dic₉₇	C₂³	1	24	yes	—
17	C₄⋊Dic₉₇	C₂³	1	64	yes	—
18	C₂⁴×C₉₇	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₉₇	D₄	1	8	yes	—
20	C₂²×C₄×C₉₇	Q₈	1	16	yes	complemented
21	C₂²×C₄×C₉₇	C₂³	1	6	yes	—
22	C₂²×C₄×C₉₇	C₂³	1	12	yes	—
23	C₄²×C₉₇	C₂³	1	16	yes	—
24	C₂³×C₉₇	C₂×D₄	1	4	yes	—
25	C₂³×C₉₇	C₂×Q₈	1	8	yes	—
26	C₂³×C₉₇	C₂⁴	1	3	yes	—
27	C₂×C₄×C₉₇	C₂×D₄	1	24	yes	—
28	C₂×C₄×C₉₇	C₂×Q₈	1	48	yes	complemented
29	C₂×C₄×C₉₇	C₂⁴	1	4	yes	—
30	C₂×C₄×C₉₇	C₂⁴	1	8	yes	—
31	C₂×Dic₉₇	—	2	48	no	—
32	C₂×Dic₉₇	C₂⁴	1	16	yes	—
33	Dic₁₉₄	—	2	64	no	—
34	C₂²×C₉₇	C₄⋊Q₈	1	16	yes	complemented
35	C₂²×C₉₇	C₂²×D₄	1	6	yes	—
36	C₂²×C₉₇	C₂²×Q₈	1	12	yes	—
37	C₂²×C₉₇	C₂⁵	1	1	yes	characteristic
38	C₄×C₉₇	C₂²×D₄	1	8	yes	—
39	C₄×C₉₇	C₂²×Q₈	1	16	yes	complemented
40	Dic₉₇	—	2	16	no	—
41	C₂×C₉₇	C₂×C₄⋊Q₈	1	12	yes	complemented
42	C₂×C₉₇	C₂³×D₄	1	1	yes	characteristic
43	C₂×C₉₇	C₂³×Q₈	1	2	yes	—
44	C₂²×C₄⋊Q₈	—	97	1	no	maximal, Sylow
45	C₉₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
46	C₂³×Q₈	—	97	2	no	—
47	C₂×C₄⋊Q₈	—	97	24	no	—
48	C₂²×C₄⋊C₄	—	97	4	no	—
49	C₂²×C₄²	D₉₇	1	1	yes	characteristic
50	C₂²×Q₈	—	97	24	no	—
51	C₂²×Q₈	—	194	16	no	—
52	C₂³×C₄	—	97	4	no	—
53	C₂³×C₄	C₂×D₉₇	1	2	yes	—
54	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
55	C₄⋊Q₈	—	97	64	no	—
56	C₂×C₄⋊C₄	—	97	48	no	—
57	C₂×C₄²	C₂×D₉₇	1	12	yes	—
58	C₂⁴	C₂²×D₉₇	1	1	yes	characteristic
59	C₂×Q₈	—	97	32	no	—
60	C₂×Q₈	—	194	96	no	—
61	C₂²×C₄	—	97	24	no	—
62	C₂²×C₄	—	194	16	no	—
63	C₂²×C₄	Dic₁₉₄	1	16	yes	complemented
64	C₂²×C₄	D₃₈₈	1	8	yes	—
65	C₂²×C₄	C₂²×D₉₇	1	6	yes	—
66	C₂²×C₄	C₂²×D₉₇	1	12	yes	—
67	C₄⋊C₄	—	97	64	no	—
68	C₄²	C₂²×D₉₇	1	16	yes	—
69	C₂³	C₂×D₃₈₈	1	4	yes	—
70	C₂³	C₂×Dic₁₉₄	1	8	yes	—
71	C₂³	C₂³×D₉₇	1	3	yes	—
72	Q₈	—	194	64	no	—
73	C₂×C₄	—	97	16	no	—
74	C₂×C₄	—	194	48	no	—
75	C₂×C₄	C₂×D₃₈₈	1	24	yes	—
76	C₂×C₄	C₂×Dic₁₉₄	1	48	yes	complemented
77	C₂×C₄	C₂³×D₉₇	1	4	yes	—
78	C₂×C₄	C₂³×D₉₇	1	8	yes	—
79	C₂²	#15690810	1	16	yes	direct product factor
80	C₂²	C₂²×D₃₈₈	1	6	yes	—
81	C₂²	C₂²×Dic₁₉₄	1	12	yes	—
82	C₂²	C₂⁴×D₉₇	1	1	yes	characteristic
83	C₄	—	194	16	no	—
84	C₄	C₂²×D₃₈₈	1	8	yes	—
85	C₄	C₂²×Dic₁₉₄	1	16	yes	complemented
86	C₂	C₂×#15690810	1	12	yes	direct product factor
87	C₂	C₂³×D₃₈₈	1	1	yes	characteristic
88	C₂	C₂³×Dic₁₉₄	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₉₇
commutator, cosocle	C₂²×C₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₉₇
Fitting	C₂²×C₄²×C₉₇	C₂
socle, prime radical	C₂⁴×C₉₇	C₂³
hypercenter	C₂²×C₄²	D₉₇
hypocenter	C₉₇	C₂²×C₄⋊Q₈
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#15690810	C₂	direct (N × H)
8. #15690810	C₂²	direct (N × H)
20. C₂²×C₄×C₉₇	Q₈	semidirect
28. C₂×C₄×C₉₇	C₂×Q₈	semidirect
34. C₂²×C₉₇	C₄⋊Q₈	semidirect
39. C₄×C₉₇	C₂²×Q₈	semidirect
41. C₂×C₉₇	C₂×C₄⋊Q₈	semidirect
45. C₉₇	C₂²×C₄⋊Q₈	semidirect
63. C₂²×C₄	Dic₁₉₄	semidirect
76. C₂×C₄	C₂×Dic₁₉₄	semidirect
79. C₂²	#15690810	direct (N × H)
85. C₄	C₂²×Dic₁₉₄	semidirect
86. C₂	C₂×#15690810	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₁₉₄	2
C₂×#15690810	2
C₂²×C₄⋊Dic₉₇	2
C₂²×C₄²×C₉₇	2
C₂²×C₄⋊Q₈	97

Subgroups — C22×#15690810 #27155012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15690810 #27155012