Subgroups — #27139892

Order 12416.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261125	C₂	1	1	yes	maximal, characteristic
2	#20260532	C₂	1	1	yes	maximal, characteristic, complemented
3	#354×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13095498	—	2	1	no	—
5	C₂×#13095496	—	2	1	no	—
6	#15690807	C₄	1	1	yes	characteristic, complemented
7	#15690729	—	2	1	no	—
8	#15690721	C₄	1	1	yes	characteristic, complemented
9	#124×C₉₇	C₂²	1	1	yes	characteristic
10	#99×C₉₇	—	2	1	no	—
11	C₂×D₃₈₈	—	2	1	no	—
12	C₂×D₃₈₈	—	4	1	no	—
13	#13095498	—	4	2	no	—
14	#13095496	—	4	2	no	—
15	#13095495	—	4	1	no	—
16	C₂×#11715895	—	2	1	no	—
17	C₂×Q₈×C₉₇	D₄	1	1	yes	characteristic
18	C₂×D₄×C₉₇	D₄	1	1	yes	characteristic
19	C₂²⋊C₄×C₉₇	—	4	1	no	—
20	C₂²⋊C₄×C₉₇	—	4	1	no	—
21	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₉₇	—	2	1	no	—
23	C₂²×D₉₇	—	4	1	no	—
24	Q₈×C₉₇	—	4	1	no	—
25	D₄×C₉₇	—	4	1	no	—
26	C₂×C₄×C₉₇	—	2	1	no	—
27	C₂×C₄×C₉₇	—	2	1	no	—
28	C₂×C₄×C₉₇	—	8	1	no	—
29	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
30	D₃₈₈	—	4	1	no	—
31	D₃₈₈	—	4	1	no	—
32	D₃₈₈	—	8	2	no	—
33	#11715895	—	2	2	no	—
34	#11715895	—	8	1	no	—
35	C₂²×C₉₇	—	4	1	no	—
36	C₂²×C₉₇	—	8	1	no	—
37	C₂²×C₉₇	#99	1	1	yes	characteristic
38	C₂×D₉₇	—	8	2	no	—
39	C₂×D₉₇	—	8	1	no	—
40	C₄×C₉₇	—	2	1	no	—
41	C₄×C₉₇	—	4	1	no	—
42	C₄×C₉₇	—	4	1	no	—
43	C₄×C₉₇	—	16	1	no	—
44	C₂×C₉₇	—	2	1	no	—
45	C₂×C₉₇	—	8	1	no	—
46	C₂×C₉₇	#353	1	1	yes	characteristic
47	D₉₇	—	16	1	no	—
48	#1406	—	97	1	no	maximal, Sylow
49	C₉₇	#1406	1	1	yes	characteristic, complemented, Sylow
50	#496	—	97	1	no	—
51	#355	—	97	1	no	—
52	#354	D₉₇	1	1	yes	characteristic, complemented
53	C₂×SD₁₆	—	194	1	no	—
54	C₂×D₈	—	194	1	no	—
55	#127	—	97	1	no	—
56	#124	C₂×D₉₇	1	1	yes	characteristic
57	#101	—	194	1	no	—
58	#99	—	2	1	no	—
59	#97	—	97	1	no	—
60	C₂×Q₈	D₃₈₈	1	1	yes	characteristic
61	C₂×D₄	—	388	1	no	—
62	C₂×D₄	—	194	1	no	—
63	C₂×D₄	C₂²⋊D₉₇	1	1	yes	characteristic
64	SD₁₆	—	388	2	no	—
65	D₈	—	388	2	no	—
66	M₁₆	—	388	1	no	—
67	C₂×C₈	—	194	1	no	—
68	C₂²⋊C₄	—	4	1	no	—
69	C₂²⋊C₄	—	4	1	no	—
70	C₄²	C₄×D₉₇	1	1	yes	characteristic
71	C₂³	—	2	1	no	—
72	C₂³	—	388	1	no	—
73	Q₈	—	4	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	776	2	no	—
76	D₄	—	388	1	no	—
77	D₄	—	388	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	8	1	no	—
81	C₂×C₄	#13095488	1	1	yes	characteristic
82	C₈	—	776	1	no	—
83	C₈	—	194	2	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	8	1	no	—
86	C₂²	—	776	2	no	—
87	C₂²	—	776	1	no	—
88	C₂²	#15690724	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	4	1	no	—
92	C₄	—	16	1	no	—
93	C₂	—	2	1	no	—
94	C₂	—	8	1	no	—
95	C₂	—	1552	1	no	—
96	C₂	#20260527	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260527
commutator	C₄²×C₉₇	C₂×C₄
Frattini	#124	C₂×D₉₇
Fitting	#354×C₉₇	C₂
socle	C₂×C₉₇	#353
prime radical	#20261125	C₂
cosocle	#124×C₉₇	C₂²
hypercenter	#354	D₉₇
hypocenter	C₉₇	#1406
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #354

Complements

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

N	H ≅ G/N	type
2. #20260532	C₂	semidirect
3. #354×C₉₇	C₂	semidirect
6. #15690807	C₄	semidirect
8. #15690721	C₄	semidirect
49. C₉₇	#1406	semidirect
52. #354	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261125	2
#20260532	2
#354×C₉₇	2
#1406	97