Subgroups — #27156601

Order 12416.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261756	C₂	1	2	yes	maximal
2	C₂×#15690739	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11715897	—	2	2	no	—
4	C₂²⋊#11715897	—	2	2	no	—
5	#15690865	C₄	1	4	yes	—
6	C₂²×#11715895	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
8	#15690739	—	2	2	no	—
9	C₂²×C₈×C₉₇	C₄	1	1	yes	characteristic
10	C₂×#11715897	—	2	4	no	—
11	C₂²×Dic₉₇	D₄	1	1	yes	characteristic
12	C₂²×Dic₉₇	Q₈	1	1	yes	characteristic
13	C₂×#11715895	—	2	1	no	—
14	C₂×#11715895	—	4	1	no	—
15	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
16	C₄×Dic₉₇	D₄	1	2	yes	—
17	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₉₇	—	2	1	no	—
19	C₂×C₈×C₉₇	—	4	1	no	—
20	C₂³×C₉₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₉₇	—	2	1	no	—
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×Dic₉₇	—	2	1	no	—
26	C₂×Dic₉₇	—	2	1	no	—
27	C₂×Dic₉₇	—	4	1	no	—
28	C₂×Dic₉₇	C₂²⋊C₄	1	2	yes	—
29	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
30	#11715897	—	4	4	no	—
31	C₈×C₉₇	—	4	1	no	—
32	#11715895	—	4	1	no	—
33	C₂²×C₉₇	—	2	1	no	—
34	C₂²×C₉₇	—	2	1	no	—
35	C₂²×C₉₇	#95	1	1	yes	characteristic
36	C₂²×C₉₇	#108	1	1	yes	characteristic
37	C₂²×C₉₇	#108	1	1	yes	characteristic
38	C₄×C₉₇	—	2	1	no	—
39	C₄×C₉₇	#100	1	1	yes	characteristic
40	C₄×C₉₇	#101	1	1	yes	characteristic
41	Dic₉₇	—	2	2	no	—
42	Dic₉₇	—	2	2	no	—
43	C₂×C₉₇	—	2	1	no	—
44	C₂×C₉₇	—	2	1	no	—
45	C₂×C₉₇	H(3,ℤ/4)	1	1	yes	characteristic
46	C₂×C₉₇	#341	1	1	yes	characteristic
47	C₂×C₉₇	#343	1	1	yes	characteristic
48	#1290	—	97	1	no	maximal, Sylow
49	C₉₇	#1290	1	1	yes	characteristic, complemented, Sylow
50	#432	—	97	2	no	—
51	C₂×C₄⋊C₈	—	97	1	no	—
52	C₂²×C₈	—	97	1	no	—
53	C₂²×C₈	#87604	1	1	yes	characteristic
54	C₂×C₄²	—	97	1	no	—
55	C₄⋊C₈	—	194	2	no	—
56	C₄⋊C₈	—	194	2	no	—
57	C₂²⋊C₈	—	194	2	no	—
58	#97	—	97	4	no	—
59	C₂²×C₄	—	97	1	no	—
60	C₂²×C₄	—	97	1	no	—
61	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	—	388	1	no	—
65	C₂×C₈	—	194	4	no	—
66	C₂×C₈	—	194	1	no	—
67	C₄²	—	97	2	no	—
68	C₄²	—	97	2	no	—
69	C₂³	C₄⋊#87604	1	1	yes	characteristic
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	97	2	no	—
73	C₂×C₄	—	97	2	no	—
74	C₂×C₄	—	388	1	no	—
75	C₂×C₄	—	194	1	no	—
76	C₂×C₄	—	194	1	no	—
77	C₂×C₄	C₄×#87604	1	1	yes	characteristic
78	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
79	C₈	—	4	1	no	—
80	C₈	—	388	4	no	—
81	C₈	—	388	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	1	no	—
84	C₂²	(C₂×#87604)⋊C₄	1	1	yes	characteristic
85	C₂²	#15690880	1	1	yes	characteristic
86	C₂²	#15690881	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	194	2	no	—
89	C₄	—	194	2	no	—
90	C₄	#15690870	1	1	yes	characteristic
91	C₄	#15690871	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	2	1	no	—
94	C₂	#20261603	1	1	yes	characteristic
95	C₂	#20261611	1	1	yes	characteristic
96	C₂	#20261615	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×#87604)⋊C₄
commutator	C₂×C₄×C₉₇	C₄²
Frattini	C₂²×C₄	C₂×#87604
Fitting	C₂²×C₈×C₉₇	C₄
socle	C₂²×C₉₇	#95
prime radical	C₂³×C₉₇	C₄⋊C₄
cosocle	C₂×C₄×Dic₉₇	C₂²
hypercenter	C₂²×C₈	#87604
hypocenter	C₉₇	#1290
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
49. C₉₇	#1290	semidirect

Maximal subgroups

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

subgroup	index
#20261756	2
C₂×#15690739	2
#1290	97