Subgroups — #27158105

Order 12416.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261658	C₂	1	2	yes	maximal, complemented
2	#20261631	C₂	1	2	yes	maximal
3	#20261627	C₂	1	2	yes	maximal, complemented
4	C₄×#13095495	C₂	1	1	yes	maximal, characteristic
5	#15690884	C₂²	1	4	yes	—
6	C₂×#13095495	C₄	1	1	yes	characteristic
7	C₂×#13095495	C₂²	1	1	yes	characteristic
8	#15690721	C₄	1	1	yes	characteristic
9	#15690721	C₂²	1	1	yes	characteristic
10	C₄×#11715895	C₄	1	1	yes	characteristic
11	C₄×#11715895	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₉₇	C₄	1	1	yes	characteristic
13	#13095514	—	2	4	no	—
14	#13095495	—	2	2	no	—
15	#13095495	—	2	2	no	—
16	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
17	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
18	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
19	C₂×#11715895	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₉₇	—	2	1	no	—
21	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₉₇	—	2	1	no	—
23	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₉₇	M₁₆	1	1	yes	characteristic
26	C₂×C₄×C₉₇	—	2	1	no	—
27	C₂×C₄×C₉₇	—	2	1	no	—
28	C₂×C₄×C₉₇	—	2	1	no	—
29	C₂×C₄×C₉₇	—	2	1	no	—
30	C₂×C₄×C₉₇	—	4	2	no	—
31	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	#11715895	—	2	1	no	—
34	#11715895	—	2	1	no	—
35	#11715895	M₁₆	1	2	yes	—
36	#11715895	#41	1	2	yes	—
37	C₂²×C₉₇	—	2	1	no	—
38	C₂²×C₉₇	—	4	1	no	—
39	C₂²×C₉₇	C₂×M₁₆	1	1	yes	characteristic
40	C₄×C₉₇	—	2	2	no	—
41	C₄×C₉₇	—	2	1	no	—
42	C₄×C₉₇	—	4	1	no	—
43	C₄×C₉₇	#117	1	1	yes	characteristic
44	C₄×C₉₇	C₂×M₁₆	1	1	yes	characteristic
45	C₂×C₉₇	—	2	1	no	—
46	C₂×C₉₇	—	4	1	no	—
47	C₂×C₉₇	#432	1	1	yes	characteristic
48	#2165	—	97	1	no	maximal, Sylow
49	C₉₇	#2165	1	1	yes	characteristic, complemented, Sylow
50	C₄×M₁₆	—	97	1	no	—
51	#364	—	97	2	no	—
52	#349	—	97	2	no	—
53	#347	—	97	2	no	—
54	C₂×M₁₆	—	97	1	no	—
55	C₂×M₁₆	—	97	1	no	—
56	C₂×C₄²	#87604	1	1	yes	characteristic
57	M₃₂	—	97	4	no	—
58	#97	—	97	1	no	—
59	#97	—	97	1	no	—
60	C₄×C₈	—	97	1	no	—
61	C₄×C₈	—	97	1	no	—
62	C₂²×C₄	—	2	1	no	—
63	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
64	M₁₆	—	194	2	no	—
65	M₁₆	—	194	2	no	—
66	C₂×C₈	—	97	1	no	—
67	C₂×C₈	—	97	1	no	—
68	C₂×C₈	—	97	1	no	—
69	C₂×C₈	—	97	1	no	—
70	C₄²	—	2	1	no	—
71	C₄²	C₂×#87604	1	1	yes	characteristic
72	C₄²	C₂×#87604	1	1	yes	characteristic
73	C₁₆	—	194	4	no	—
74	C₂³	#13095520	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	#13095520	1	1	yes	characteristic
81	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
82	C₈	—	97	2	no	—
83	C₈	—	97	2	no	—
84	C₈	—	194	1	no	—
85	C₈	—	194	1	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	4	1	no	—
88	C₂²	C₂×#13095520	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	—	2	1	no	—
91	C₄	—	4	1	no	—
92	C₄	#15690889	1	1	yes	characteristic
93	C₄	D₉₇⋊M₁₆	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#20261755	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690889
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87604
Fitting	C₂×C₄²×C₉₇	C₄
socle	C₂×C₉₇	#432
prime radical	C₂³×C₉₇	M₁₆
cosocle	C₂×#11715895	C₂³
hypercenter	C₂×C₄²	#87604
hypocenter	C₉₇	#2165
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #20261658	C₂	semidirect
3. #20261627	C₂	semidirect
49. C₉₇	#2165	semidirect

Maximal subgroups

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

subgroup	index
#20261658	2
#20261631	2
#20261627	2
C₄×#13095495	2
#2165	97