Subgroups — #27144912

Order 12416.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690807	C₂	1	1	yes	maximal, characteristic
2	#20260613	C₂	1	12	yes	maximal, complemented
3	#20260603	C₂	1	1	yes	maximal, characteristic, complemented
4	#383×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₈₈	C₂²	1	6	yes	—
6	C₂²×C₄×D₉₇	C₂²	1	4	yes	—
7	#15690807	C₄	1	8	yes	complemented
8	C₂×#13095488	C₂²	1	12	yes	—
9	#15690717	C₂²	1	6	yes	—
10	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
11	#95×C₉₇	C₂²	1	6	yes	—
12	C₂³×D₉₇	C₂³	1	4	yes	—
13	C₂×D₃₈₈	—	4	6	no	—
14	C₂×D₃₈₈	—	4	6	no	—
15	C₂×D₃₈₈	C₂×C₄	1	24	yes	complemented
16	C₂²×Dic₉₇	C₂³	1	4	yes	—
17	C₂×C₄×D₉₇	—	4	12	no	—
18	#13095488	—	4	12	no	—
19	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
20	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
21	C₄²×C₉₇	C₂×C₄	1	4	yes	complemented
22	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
23	C₂²×D₉₇	—	2	12	no	—
24	C₂²×D₉₇	—	2	12	no	—
25	C₂²×D₉₇	C₂²×C₄	1	8	yes	—
26	C₂×C₄×C₉₇	—	2	6	no	—
27	C₂×C₄×C₉₇	—	2	12	no	—
28	C₂×C₄×C₉₇	C₂²×C₄	1	6	yes	—
29	C₂×Dic₉₇	—	2	12	no	—
30	D₃₈₈	—	4	24	no	—
31	C₄×D₉₇	—	4	16	no	—
32	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
33	C₂²×C₉₇	H(5,𝔽₂)	1	6	yes	—
34	C₂×D₉₇	—	2	24	no	—
35	C₂×D₉₇	—	4	4	no	—
36	C₂×D₉₇	—	4	12	no	—
37	C₄×C₉₇	—	2	6	no	—
38	C₄×C₉₇	—	4	4	no	—
39	Dic₉₇	—	4	4	no	—
40	C₂×C₉₇	#518	1	3	yes	—
41	C₂×C₉₇	#560	1	4	yes	—
42	D₉₇	—	4	8	no	—
43	#2380	—	97	1	no	maximal, Sylow
44	C₉₇	#2380	1	1	yes	characteristic, complemented, Sylow
45	C₂×#127	—	97	1	no	—
46	#386	—	97	12	no	—
47	#383	—	97	1	no	—
48	#383	D₉₇	1	1	yes	characteristic, complemented
49	C₂²×D₄	—	97	6	no	—
50	C₂³×C₄	—	97	4	no	—
51	#127	—	97	8	no	—
52	C₂×C₂²⋊C₄	—	97	12	no	—
53	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
54	#95	—	97	6	no	—
55	#95	C₂×D₉₇	1	6	yes	—
56	C₂⁴	—	97	4	no	—
57	C₂×D₄	—	97	24	no	—
58	C₂×D₄	—	388	6	no	—
59	C₂×D₄	—	388	6	no	—
60	C₂²×C₄	—	97	4	no	—
61	C₂²×C₄	—	388	12	no	—
62	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
63	C₂²×C₄	C₂²×D₉₇	1	3	yes	—
64	C₂²⋊C₄	—	388	12	no	—
65	C₄²	C₄×D₉₇	1	4	yes	complemented
66	C₂³	—	97	8	no	—
67	C₂³	—	194	12	no	—
68	C₂³	—	194	12	no	—
69	C₂³	C₂³×D₉₇	1	1	yes	characteristic
70	D₄	—	388	24	no	—
71	C₂×C₄	—	2	6	no	—
72	C₂×C₄	—	2	12	no	—
73	C₂×C₄	—	388	16	no	—
74	C₂×C₄	—	194	12	no	—
75	C₂×C₄	C₂×C₄×D₉₇	1	6	yes	—
76	C₂²	—	388	4	no	—
77	C₂²	—	388	12	no	—
78	C₂²	—	194	24	no	—
79	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
80	C₂²	#15690856	1	6	yes	—
81	C₄	—	2	6	no	—
82	C₄	—	4	4	no	—
83	C₄	—	388	4	no	—
84	C₂	—	388	8	no	—
85	C₂	#20261209	1	3	yes	—
86	C₂	#20261476	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator	C₂²×C₉₇	C₂³×C₄
Fitting	#383×C₉₇	C₂
socle, cosocle	C₂³×C₉₇	C₂⁴
prime radical	C₂×#15690807	C₂
hypercenter	#383	D₉₇
hypocenter	C₉₇	#2380
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #383

Complements

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

N	H ≅ G/N	type
2. #20260613	C₂	semidirect
3. #20260603	C₂	semidirect
4. #383×C₉₇	C₂	semidirect
7. #15690807	C₄	semidirect
15. C₂×D₃₈₈	C₂×C₄	semidirect
21. C₄²×C₉₇	C₂×C₄	semidirect
44. C₉₇	#2380	semidirect
48. #383	D₉₇	semidirect
65. C₄²	C₄×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690807	2
#20260613	2
#20260603	2
#383×C₉₇	2
#2380	97