Subgroups — #27139729

Order 12416.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260993	C₂	1	1	yes	maximal, characteristic
2	#20260517	C₂	1	1	yes	maximal, characteristic
3	#338×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₈₈	C₄	1	1	yes	characteristic
5	#15690789	—	2	1	no	—
6	#15690750	—	2	1	no	—
7	#15690742	C₄	1	1	yes	characteristic
8	#15690735	—	2	1	no	—
9	C₂×M₁₆×C₉₇	—	2	1	no	—
10	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₄	—	2	1	no	—
12	Dic₃₈₈	—	4	1	no	—
13	C₄⋊Dic₉₇	—	2	1	no	—
14	#13095490	—	4	1	no	—
15	C₂²⋊Dic₉₇	—	4	1	no	—
16	#13095486	—	4	1	no	—
17	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
18	M₁₆×C₉₇	—	4	1	no	—
19	M₁₆×C₉₇	—	4	1	no	—
20	M₁₆×C₉₇	—	4	1	no	—
21	C₂×C₈×C₉₇	—	2	2	no	—
22	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
24	C₂³×C₉₇	SD₁₆	1	1	yes	characteristic
25	C₂×C₄×C₉₇	—	2	1	no	—
26	C₂×C₄×C₉₇	—	2	1	no	—
27	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
29	C₂×Dic₉₇	—	4	1	no	—
30	C₂×Dic₉₇	—	4	1	no	—
31	Dic₁₉₄	—	4	1	no	—
32	Dic₁₉₄	—	4	1	no	—
33	C₈×C₉₇	—	2	1	no	—
34	C₈×C₉₇	—	2	1	no	—
35	C₈×C₉₇	—	4	2	no	—
36	C₂²×C₉₇	—	2	1	no	—
37	C₂²×C₉₇	—	4	1	no	—
38	C₂²×C₉₇	#102	1	1	yes	characteristic
39	C₄×C₉₇	—	2	1	no	—
40	C₄×C₉₇	#99	1	1	yes	characteristic
41	C₄×C₉₇	#104	1	1	yes	characteristic
42	Dic₉₇	—	8	1	no	—
43	Dic₉₇	—	8	1	no	—
44	C₂×C₉₇	—	2	1	no	—
45	C₂×C₉₇	—	4	1	no	—
46	C₂×C₉₇	#327	1	1	yes	characteristic
47	#1342	—	97	1	no	maximal, Sylow
48	C₉₇	#1342	1	1	yes	characteristic, complemented, Sylow
49	#470	—	97	1	no	—
50	#349	—	97	1	no	—
51	#338	D₉₇	1	1	yes	characteristic
52	C₂×Q₁₆	—	97	1	no	—
53	C₂×M₁₆	—	2	1	no	—
54	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
55	C₂²⋊Q₈	—	194	1	no	—
56	M₃₂	—	194	1	no	—
57	#106	—	97	1	no	—
58	#103	—	194	1	no	—
59	C₂×Q₈	—	194	1	no	—
60	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
61	Q₁₆	—	388	1	no	—
62	M₁₆	—	4	1	no	—
63	M₁₆	—	4	1	no	—
64	M₁₆	—	4	1	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
67	C₂×C₈	D₃₈₈	1	1	yes	characteristic
68	C₄⋊C₄	—	388	1	no	—
69	C₄⋊C₄	—	194	1	no	—
70	C₂²⋊C₄	—	388	1	no	—
71	C₁₆	—	388	1	no	—
72	C₂³	#13095498	1	1	yes	characteristic
73	Q₈	—	388	1	no	—
74	Q₈	—	388	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	388	1	no	—
78	C₂×C₄	—	388	1	no	—
79	C₂×C₄	#13095488	1	1	yes	characteristic
80	C₂×C₄	#13095496	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	2	1	no	—
83	C₈	—	4	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#15690730	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	776	1	no	—
89	C₄	—	776	1	no	—
90	C₄	#15690724	1	1	yes	characteristic
91	C₄	#15690738	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	4	1	no	—
94	C₂	#20260468	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260468
commutator	C₂×C₈×C₉₇	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₉₇
Fitting	#338×C₉₇	C₂
socle	C₂×C₉₇	#327
prime radical	C₂³×C₉₇	SD₁₆
cosocle	C₂×M₁₆×C₉₇	C₂²
hypercenter	#338	D₉₇
hypocenter	C₉₇	#1342
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #338

Complements

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

N	H ≅ G/N	type
48. C₉₇	#1342	semidirect

Maximal subgroups

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

subgroup	index
#20260993	2
#20260517	2
#338×C₉₇	2
#1342	97