Subgroups — #27150133

Order 12416.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690809	C₂	1	6	yes	maximal
2	#20260605	C₂	1	1	yes	maximal, characteristic
3	#20260604	C₂	1	6	yes	maximal
4	C₄²×Dic₉₇	C₂	1	1	yes	maximal, characteristic
5	#384×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₄	C₂²	1	4	yes	—
7	#15690809	—	2	24	no	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	6	yes	—
9	C₂×#13095490	C₂²	1	12	yes	—
10	C₂×C₄×Dic₉₇	C₂²	1	3	yes	—
11	C₂×C₄×Dic₉₇	C₂²	1	3	yes	—
12	C₂×C₄⋊C₄×C₉₇	C₂²	1	6	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₉₄	—	2	24	no	—
15	C₂²×Dic₉₇	C₂³	1	4	yes	—
16	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
17	C₂²×Dic₉₇	C₂³	1	3	yes	—
18	C₄⋊Dic₉₇	—	2	12	no	—
19	#13095490	—	2	24	no	—
20	C₄×Dic₉₇	D₄	1	12	yes	—
21	C₄×Dic₉₇	Q₈	1	12	yes	complemented
22	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
23	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
24	C₄⋊C₄×C₉₇	—	2	12	no	—
25	C₄²×C₉₇	Q₈	1	4	yes	complemented
26	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
27	C₂×C₄×C₉₇	—	2	12	no	—
28	C₂×C₄×C₉₇	C₂×D₄	1	6	yes	—
29	C₂×C₄×C₉₇	C₂×Q₈	1	6	yes	—
30	C₂×C₄×C₉₇	C₂×Q₈	1	6	yes	—
31	C₂×Dic₉₇	—	2	12	no	—
32	C₂×Dic₉₇	C₂×D₄	1	6	yes	—
33	C₂×Dic₉₇	C₂×D₄	1	6	yes	—
34	C₂×Dic₉₇	C₂×Q₈	1	12	yes	—
35	Dic₁₉₄	—	4	16	no	—
36	C₂²×C₉₇	C₂²×D₄	1	3	yes	—
37	C₂²×C₉₇	C₂²×Q₈	1	3	yes	—
38	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
39	C₄×C₉₇	—	4	4	no	—
40	C₄×C₉₇	C₄⋊Q₈	1	12	yes	complemented
41	Dic₉₇	—	4	4	no	—
42	Dic₉₇	#127	1	4	yes	—
43	Dic₉₇	C₄⋊Q₈	1	12	yes	complemented
44	C₂×C₉₇	C₂×#127	1	1	yes	characteristic
45	C₂×C₉₇	C₂×C₄⋊Q₈	1	3	yes	—
46	C₂×C₉₇	C₂×C₄⋊Q₈	1	3	yes	—
47	#2868	—	97	1	no	maximal, Sylow
48	C₉₇	#2868	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₄⋊Q₈	—	97	6	no	—
50	#384	—	97	6	no	—
51	#384	—	97	1	no	—
52	#384	D₉₇	1	1	yes	characteristic
53	C₄³	—	97	1	no	—
54	C₂²×Q₈	—	97	4	no	—
55	C₄⋊Q₈	—	194	24	no	—
56	C₂×C₄⋊C₄	—	97	12	no	—
57	C₂×C₄⋊C₄	—	97	6	no	—
58	C₂×C₄⋊C₄	C₂×D₉₇	1	6	yes	—
59	C₂×C₄²	—	97	3	no	—
60	C₂×C₄²	—	97	3	no	—
61	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
62	C₂×Q₈	—	194	24	no	—
63	C₂²×C₄	—	97	3	no	—
64	C₂²×C₄	—	97	1	no	—
65	C₂²×C₄	—	97	4	no	—
66	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
67	C₂²×C₄	C₂²×D₉₇	1	3	yes	—
68	C₄⋊C₄	—	2	12	no	—
69	C₄⋊C₄	—	194	24	no	—
70	C₄⋊C₄	—	194	12	no	—
71	C₄²	—	97	12	no	—
72	C₄²	—	97	12	no	—
73	C₄²	Dic₁₉₄	1	4	yes	complemented
74	C₂³	C₂³×D₉₇	1	1	yes	characteristic
75	Q₈	—	388	16	no	—
76	C₂×C₄	—	2	12	no	—
77	C₂×C₄	—	97	6	no	—
78	C₂×C₄	—	97	6	no	—
79	C₂×C₄	—	97	12	no	—
80	C₂×C₄	—	194	12	no	—
81	C₂×C₄	D₄×D₉₇	1	6	yes	—
82	C₂×C₄	Q₈×D₉₇	1	6	yes	—
83	C₂×C₄	C₂×Dic₁₉₄	1	6	yes	—
84	C₂²	C₂×D₄×D₉₇	1	3	yes	—
85	C₂²	C₂×Q₈×D₉₇	1	3	yes	—
86	C₂²	C₂²×Dic₁₉₄	1	1	yes	characteristic
87	C₄	—	4	4	no	—
88	C₄	—	97	4	no	—
89	C₄	—	97	12	no	—
90	C₄	—	388	4	no	—
91	C₄	#15690809	1	12	yes	complemented
92	C₂	#127×D₉₇	1	1	yes	characteristic
93	C₂	C₄⋊Q₈×D₉₇	1	3	yes	—
94	C₂	C₂×#15690809	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, prime radical, cosocle	C₂³×C₉₇	C₂⁴
Fitting	#384×C₉₇	C₂
hypercenter	#384	D₉₇
hypocenter	C₉₇	#2868
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
21. C₄×Dic₉₇	Q₈	semidirect
25. C₄²×C₉₇	Q₈	semidirect
40. C₄×C₉₇	C₄⋊Q₈	semidirect
43. Dic₉₇	C₄⋊Q₈	semidirect
48. C₉₇	#2868	semidirect
73. C₄²	Dic₁₉₄	semidirect
91. C₄	#15690809	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690809	2
#20260605	2
#20260604	2
C₄²×Dic₉₇	2
#384×C₉₇	2
#2868	97