Subgroups — #27156659

Order 12416.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#13095514	C₂	1	2	yes	maximal, complemented
2	#20260703	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095514	—	2	2	no	—
4	C₂×#13095511	C₄	1	1	yes	characteristic
5	C₂²×#11715895	C₂²	1	1	yes	characteristic
6	#15690722	—	2	2	no	—
7	C₂×M₁₆×C₉₇	C₄	1	1	yes	characteristic
8	#13095514	—	4	2	no	—
9	#13095511	—	4	1	no	—
10	#13095511	—	4	1	no	—
11	C₂×Dic₁₉₄	C₈	1	1	yes	characteristic
12	C₂×D₃₈₈	C₈	1	1	yes	characteristic
13	C₂×C₂²⋊D₉₇	—	2	1	no	—
14	C₂×C₄×D₉₇	—	2	1	no	—
15	C₂×#11715895	—	4	1	no	—
16	C₂×#11715895	D₄	1	2	yes	—
17	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₉₇	—	4	1	no	—
19	C₂×C₈×C₉₇	—	2	1	no	—
20	C₂³×C₉₇	M₁₆	1	1	yes	characteristic
21	C₂²×D₉₇	—	2	1	no	—
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	—	2	1	no	—
24	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
26	C₂²⋊D₉₇	—	4	2	no	—
27	C₂×Dic₉₇	—	2	1	no	—
28	D₃₈₈	—	4	1	no	—
29	C₄×D₉₇	—	4	1	no	—
30	C₄×D₉₇	—	4	1	no	—
31	Dic₁₉₄	—	4	1	no	—
32	C₈×C₉₇	—	4	1	no	—
33	#11715895	—	2	2	no	—
34	C₂²×C₉₇	—	2	1	no	—
35	C₂²×C₉₇	—	2	2	no	—
36	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×D₉₇	—	4	1	no	—
38	C₂×D₉₇	—	4	2	no	—
39	C₄×C₉₇	—	2	1	no	—
40	C₄×C₉₇	#99	1	1	yes	characteristic
41	C₄×C₉₇	#100	1	1	yes	characteristic
42	Dic₉₇	—	4	1	no	—
43	C₂×C₉₇	—	4	1	no	—
44	C₂×C₉₇	#323	1	1	yes	characteristic
45	C₂×C₉₇	#350	1	2	yes	—
46	D₉₇	—	8	1	no	—
47	#1315	—	97	1	no	maximal, Sylow
48	C₉₇	#1315	1	1	yes	characteristic, complemented, Sylow
49	#408	—	97	1	no	—
50	C₂²⋊C₁₆	—	97	2	no	—
51	C₂×#41	—	97	1	no	—
52	C₂×M₁₆	#87604	1	1	yes	characteristic
53	C₂²×C₈	—	97	1	no	—
54	C₂×C₁₆	—	194	2	no	—
55	C₂²⋊C₈	—	194	2	no	—
56	#41	—	388	1	no	—
57	#41	—	388	1	no	—
58	C₂×Q₈	—	97	1	no	—
59	C₂×D₄	—	97	1	no	—
60	C₂×D₄	—	194	1	no	—
61	C₂²×C₄	—	194	1	no	—
62	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
63	M₁₆	—	4	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	97	2	no	—
66	C₂×C₈	—	388	1	no	—
67	C₁₆	—	388	2	no	—
68	C₂³	—	194	1	no	—
69	C₂³	#13095521	1	1	yes	characteristic
70	Q₈	—	388	1	no	—
71	D₄	—	388	1	no	—
72	D₄	—	388	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	388	1	no	—
76	C₂×C₄	—	388	1	no	—
77	C₂×C₄	—	194	1	no	—
78	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
79	C₂×C₄	#13095519	1	1	yes	characteristic
80	C₈	—	4	1	no	—
81	C₈	—	194	2	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	2	1	no	—
84	C₂²	—	388	2	no	—
85	C₂²	—	388	1	no	—
86	C₂²	#15690867	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	388	1	no	—
89	C₄	#15690869	1	1	yes	characteristic
90	C₄	#15690870	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	—	776	1	no	—
93	C₂	#20261579	1	1	yes	characteristic
94	C₂	#20261634	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#87604
commutator	C₂×C₄×C₉₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#87604
Fitting	C₂×M₁₆×C₉₇	C₄
socle	C₂²×C₉₇	C₂²⋊C₈
prime radical	C₂×#13095511	C₄
cosocle	C₂²×#11715895	C₂²
hypercenter	C₂×M₁₆	#87604
hypocenter	C₉₇	#1315
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²⋊#13095514	C₂	semidirect
48. C₉₇	#1315	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#13095514	2
#20260703	2
#1315	97