Subgroups — C₂×#384×C₉₇ #27138242

Order 12416.

888 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₄×C₉₇	C₂	1	6	yes	maximal
2	C₂²×C₄²×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#384×C₉₇	C₂	1	8	yes	maximal, direct product factor
4	C₂³×C₄×C₉₇	C₂²	1	4	yes	—
5	C₂³×C₄×C₉₇	C₂²	1	3	yes	—
6	C₂×C₄⋊C₄×C₉₇	—	2	24	no	—
7	C₂×C₄⋊C₄×C₉₇	C₂²	1	24	yes	—
8	C₂×C₄²×C₉₇	C₄	1	8	yes	complemented
9	C₂×C₄²×C₉₇	C₂²	1	4	yes	—
10	C₂⁴×C₉₇	C₂³	1	1	yes	characteristic
11	C₂²×C₄×C₉₇	—	2	24	no	—
12	C₂²×C₄×C₉₇	C₂×C₄	1	12	yes	—
13	C₂²×C₄×C₉₇	D₄	1	12	yes	—
14	C₂²×C₄×C₉₇	Q₈	1	12	yes	—
15	C₂²×C₄×C₉₇	C₂³	1	8	yes	—
16	C₂²×C₄×C₉₇	C₂³	1	6	yes	—
17	C₄⋊C₄×C₉₇	—	2	48	no	—
18	C₄²×C₉₇	C₂×C₄	1	16	yes	complemented
19	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
20	C₂³×C₉₇	C₂×D₄	1	6	yes	—
21	C₂³×C₉₇	C₂×Q₈	1	6	yes	—
22	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
23	C₂×C₄×C₉₇	—	2	24	no	—
24	C₂×C₄×C₉₇	—	4	16	no	—
25	C₂×C₄×C₉₇	C₄⋊C₄	1	48	yes	complemented
26	C₂×C₄×C₉₇	C₂²×C₄	1	12	yes	—
27	C₂×C₄×C₉₇	C₂×D₄	1	12	yes	—
28	C₂×C₄×C₉₇	C₂×Q₈	1	12	yes	—
29	C₂²×C₉₇	C₂×C₄⋊C₄	1	12	yes	—
30	C₂²×C₉₇	#127	1	4	yes	—
31	C₂²×C₉₇	C₄⋊Q₈	1	12	yes	—
32	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
33	C₂²×C₉₇	C₂²×D₄	1	3	yes	—
34	C₂²×C₉₇	C₂²×Q₈	1	3	yes	—
35	C₄×C₉₇	—	4	8	no	—
36	C₄×C₉₇	C₂×C₄⋊C₄	1	24	yes	complemented
37	C₂×C₉₇	#384	1	8	yes	direct product factor
38	C₂×C₉₇	C₂²×C₄⋊C₄	1	3	yes	—
39	C₂×C₉₇	C₂×#127	1	1	yes	characteristic
40	C₂×C₉₇	C₂×C₄⋊Q₈	1	3	yes	—
41	C₂×#384	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₉₇	C₂×#384	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²×C₄⋊C₄	C₂×C₉₇	1	6	yes	—
44	C₂²×C₄²	C₂×C₉₇	1	1	yes	characteristic
45	#384	C₂×C₉₇	1	8	yes	direct product factor
46	C₂³×C₄	C₂²×C₉₇	1	4	yes	—
47	C₂³×C₄	C₂²×C₉₇	1	3	yes	—
48	C₂×C₄⋊C₄	—	2	24	no	—
49	C₂×C₄⋊C₄	C₂²×C₉₇	1	24	yes	—
50	C₂×C₄²	C₄×C₉₇	1	8	yes	complemented
51	C₂×C₄²	C₂²×C₉₇	1	4	yes	—
52	C₂⁴	C₂³×C₉₇	1	1	yes	characteristic
53	C₂²×C₄	—	2	24	no	—
54	C₂²×C₄	C₂×C₄×C₉₇	1	12	yes	—
55	C₂²×C₄	D₄×C₉₇	1	12	yes	—
56	C₂²×C₄	Q₈×C₉₇	1	12	yes	—
57	C₂²×C₄	C₂³×C₉₇	1	8	yes	—
58	C₂²×C₄	C₂³×C₉₇	1	6	yes	—
59	C₄⋊C₄	—	2	48	no	—
60	C₄²	C₂×C₄×C₉₇	1	16	yes	complemented
61	C₂³	C₂²×C₄×C₉₇	1	2	yes	—
62	C₂³	C₂×D₄×C₉₇	1	6	yes	—
63	C₂³	C₂×Q₈×C₉₇	1	6	yes	—
64	C₂³	C₂⁴×C₉₇	1	1	yes	characteristic
65	C₂×C₄	—	2	24	no	—
66	C₂×C₄	—	4	16	no	—
67	C₂×C₄	C₄⋊C₄×C₉₇	1	48	yes	complemented
68	C₂×C₄	C₂²×C₄×C₉₇	1	12	yes	—
69	C₂×C₄	C₂×D₄×C₉₇	1	12	yes	—
70	C₂×C₄	C₂×Q₈×C₉₇	1	12	yes	—
71	C₂²	C₂×C₄⋊C₄×C₉₇	1	12	yes	—
72	C₂²	#127×C₉₇	1	4	yes	—
73	C₂²	C₄⋊Q₈×C₉₇	1	12	yes	—
74	C₂²	C₂³×C₄×C₉₇	1	1	yes	characteristic
75	C₂²	C₂²×D₄×C₉₇	1	3	yes	—
76	C₂²	C₂²×Q₈×C₉₇	1	3	yes	—
77	C₄	—	4	8	no	—
78	C₄	C₂×C₄⋊C₄×C₉₇	1	24	yes	complemented
79	C₂	#384×C₉₇	1	8	yes	direct product factor
80	C₂	C₂²×C₄⋊C₄×C₉₇	1	3	yes	—
81	C₂	C₂×#127×C₉₇	1	1	yes	characteristic
82	C₂	C₂×C₄⋊Q₈×C₉₇	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₉₇	C₂³
commutator	C₂²	C₂³×C₄×C₉₇
Frattini, cosocle	C₂³	C₂⁴×C₉₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #384×C₉₇	C₂	direct (N × H)
8. C₂×C₄²×C₉₇	C₄	semidirect
18. C₄²×C₉₇	C₂×C₄	semidirect
25. C₂×C₄×C₉₇	C₄⋊C₄	semidirect
36. C₄×C₉₇	C₂×C₄⋊C₄	semidirect
37. C₂×C₉₇	#384	direct (N × H)
41. C₂×#384	C₉₇	direct (N × H)
42. C₉₇	C₂×#384	direct (N × H)
45. #384	C₂×C₉₇	direct (N × H)
50. C₂×C₄²	C₄×C₉₇	semidirect
60. C₄²	C₂×C₄×C₉₇	semidirect
67. C₂×C₄	C₄⋊C₄×C₉₇	semidirect
78. C₄	C₂×C₄⋊C₄×C₉₇	semidirect
79. C₂	#384×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×C₄⋊C₄×C₉₇	2
C₂²×C₄²×C₉₇	2
#384×C₉₇	2
C₂×#384	97

Subgroups — C2×#384×C97 #27138242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#384×C₉₇ #27138242