Subgroups — C₂²×#15690744 #27150311

Order 12416.

600 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₉₇	C₂	1	2	yes	maximal
2	C₂×#15690744	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₉₇	—	2	2	no	—
5	C₂×C₄⋊Dic₉₇	—	2	6	no	—
6	C₂×C₄⋊Dic₉₇	C₂²	1	12	yes	—
7	#15690744	C₂²	1	16	yes	direct product factor
8	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₉₇	C₄	1	8	yes	complemented
10	C₂²×C₈×C₉₇	C₂²	1	6	yes	—
11	C₂²×Dic₉₇	—	2	12	no	—
12	C₂²×Dic₉₇	—	4	8	no	—
13	C₄⋊Dic₉₇	—	2	12	no	—
14	C₄⋊Dic₉₇	C₂³	1	8	yes	—
15	C₂⁴×C₉₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₉₇	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₉₇	D₄	1	3	yes	—
18	C₂²×C₄×C₉₇	Q₈	1	4	yes	—
19	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
20	C₂×C₈×C₉₇	C₂×C₄	1	24	yes	complemented
21	C₂×C₈×C₉₇	C₂³	1	4	yes	—
22	C₂³×C₉₇	C₄⋊C₄	1	4	yes	—
23	C₂³×C₉₇	D₈	1	4	yes	—
24	C₂³×C₉₇	Q₁₆	1	4	yes	—
25	C₂³×C₉₇	C₂×D₄	1	3	yes	—
26	C₂×C₄×C₉₇	C₄⋊C₄	1	12	yes	—
27	C₂×C₄×C₉₇	C₂²×C₄	1	6	yes	—
28	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
29	C₂×C₄×C₉₇	C₂×Q₈	1	6	yes	—
30	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
31	C₂×Dic₉₇	—	2	8	no	—
32	C₂×Dic₉₇	—	4	24	no	—
33	C₈×C₉₇	C₂²×C₄	1	8	yes	complemented
34	C₂²×C₉₇	#107	1	16	yes	complemented
35	C₂²×C₉₇	C₂×C₄⋊C₄	1	6	yes	—
36	C₂²×C₉₇	C₂×D₈	1	6	yes	—
37	C₂²×C₉₇	C₂×Q₁₆	1	6	yes	—
38	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
39	C₄×C₉₇	C₂×C₄⋊C₄	1	6	yes	—
40	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
41	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
42	Dic₉₇	—	4	8	no	—
43	C₂×C₉₇	C₂×#107	1	12	yes	complemented
44	C₂×C₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₉₇	C₂²×D₈	1	1	yes	characteristic
46	C₂×C₉₇	C₂²×Q₁₆	1	1	yes	characteristic
47	C₂²×#107	—	97	1	no	maximal, Sylow
48	C₉₇	C₂²×#107	1	1	yes	characteristic, complemented, Sylow
49	C₂³×C₈	D₉₇	1	1	yes	characteristic
50	C₂²×C₄⋊C₄	—	97	2	no	—
51	C₂×#107	—	97	12	no	—
52	C₂³×C₄	—	194	2	no	—
53	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
54	C₂²×C₈	Dic₉₇	1	8	yes	complemented
55	C₂²×C₈	C₂×D₉₇	1	6	yes	—
56	C₂×C₄⋊C₄	—	97	12	no	—
57	C₂×C₄⋊C₄	—	194	6	no	—
58	#107	—	97	16	no	—
59	C₂⁴	D₃₈₈	1	1	yes	characteristic
60	C₂²×C₄	—	388	8	no	—
61	C₂²×C₄	—	194	12	no	—
62	C₂²×C₄	Dic₁₉₄	1	4	yes	—
63	C₂²×C₄	D₃₈₈	1	3	yes	—
64	C₂²×C₄	C₂×Dic₉₇	1	4	yes	—
65	C₂²×C₄	C₂²×D₉₇	1	3	yes	—
66	C₂×C₈	C₂×Dic₉₇	1	24	yes	complemented
67	C₂×C₈	C₂²×D₉₇	1	4	yes	—
68	C₄⋊C₄	—	97	8	no	—
69	C₄⋊C₄	—	194	12	no	—
70	C₂³	C₄⋊Dic₉₇	1	4	yes	—
71	C₂³	D₇₇₆	1	4	yes	—
72	C₂³	Dic₃₈₈	1	4	yes	—
73	C₂³	C₂×D₃₈₈	1	3	yes	—
74	C₂×C₄	—	388	24	no	—
75	C₂×C₄	—	194	8	no	—
76	C₂×C₄	C₄⋊Dic₉₇	1	12	yes	—
77	C₂×C₄	C₂²×Dic₉₇	1	6	yes	—
78	C₂×C₄	C₂×D₃₈₈	1	3	yes	—
79	C₂×C₄	C₂×Dic₁₉₄	1	6	yes	—
80	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
81	C₈	C₂²×Dic₉₇	1	8	yes	complemented
82	C₂²	#15690744	1	16	yes	direct product factor
83	C₂²	C₂×C₄⋊Dic₉₇	1	6	yes	—
84	C₂²	C₂×D₇₇₆	1	6	yes	—
85	C₂²	C₂×Dic₃₈₈	1	6	yes	—
86	C₂²	C₂²×D₃₈₈	1	1	yes	characteristic
87	C₄	—	388	8	no	—
88	C₄	C₂×C₄⋊Dic₉₇	1	6	yes	—
89	C₄	C₂³×Dic₉₇	1	1	yes	characteristic
90	C₄	C₂²×Dic₁₉₄	1	1	yes	characteristic
91	C₂	C₂×#15690744	1	12	yes	direct product factor
92	C₂	C₂²×C₄⋊Dic₉₇	1	1	yes	characteristic
93	C₂	C₂²×D₇₇₆	1	1	yes	characteristic
94	C₂	C₂²×Dic₃₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₃₈₈
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂³×C₈×C₉₇	C₂
socle, prime radical	C₂⁴×C₉₇	D₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂³×C₈	D₉₇
hypocenter	C₉₇	C₂²×#107
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#15690744	C₂	direct (N × H)
7. #15690744	C₂²	direct (N × H)
9. C₂²×C₈×C₉₇	C₄	semidirect
20. C₂×C₈×C₉₇	C₂×C₄	semidirect
33. C₈×C₉₇	C₂²×C₄	semidirect
34. C₂²×C₉₇	#107	semidirect
43. C₂×C₉₇	C₂×#107	semidirect
48. C₉₇	C₂²×#107	semidirect
54. C₂²×C₈	Dic₉₇	semidirect
66. C₂×C₈	C₂×Dic₉₇	semidirect
81. C₈	C₂²×Dic₉₇	semidirect
82. C₂²	#15690744	direct (N × H)
91. C₂	C₂×#15690744	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×C₄⋊Dic₉₇	2
C₂×#15690744	2
C₂³×C₈×C₉₇	2
C₂²×#107	97

Subgroups — C22×#15690744 #27150311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15690744 #27150311