Finite Group Databasebeta

Subgroups — #27154884

Order 12416.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261552	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261540	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261178	C₂	1	3	yes	maximal, complemented
4	#20261173	C₂	1	3	yes	maximal, complemented
5	#20261167	C₂	1	3	yes	maximal, complemented
6	C₂×#15690755	C₂	1	3	yes	maximal, complemented
7	#504×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#15690857	—	2	1	no	—
9	#15690856	—	2	1	no	—
10	#15690843	—	2	3	no	—
11	#15690838	—	2	3	no	—
12	#15690832	C₂²	1	3	yes	—
13	#15690832	C₂²	1	3	yes	complemented
14	C₂×Dic₃₈₈	C₂²	1	3	yes	complemented
15	C₂×D₇₇₆	C₂²	1	3	yes	—
16	Dic₇₇₆	C₂²	1	3	yes	complemented
17	#15690755	C₂²	1	3	yes	complemented
18	#15690755	C₂²	1	6	yes	complemented
19	#15690755	C₂²	1	1	yes	characteristic, complemented
20	D₁₅₅₂	C₂²	1	3	yes	complemented
21	#131×C₉₇	C₂²	1	1	yes	characteristic
22	M₃₂×C₉₇	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₉₇	C₂²	1	3	yes	complemented
24	#13095512	—	4	1	no	—
25	#13095511	—	2	3	no	—
26	#13095511	—	2	3	no	—
27	#13095510	—	4	3	no	—
28	C₂×Dic₁₉₄	—	2	3	no	—
29	Q₈×D₉₇	—	4	1	no	—
30	C₂×D₃₈₈	—	2	3	no	—
31	D₄×D₉₇	—	4	3	no	—
32	Dic₃₈₈	—	2	3	no	—
33	Dic₃₈₈	C₂³	1	3	yes	—
34	Dic₃₈₈	C₂³	1	1	yes	characteristic
35	#13095500	—	2	3	no	—
36	#13095500	—	2	3	no	—
37	D₇₇₆	—	2	3	no	—
38	D₇₇₆	C₂³	1	1	yes	characteristic
39	D₇₇₆	C₂³	1	3	yes	—
40	#41×C₉₇	D₄	1	1	yes	characteristic
41	M₁₆×C₉₇	D₄	1	3	yes	—
42	C₂×C₈×C₉₇	C₂³	1	3	yes	—
43	C₁₆×C₉₇	C₂³	1	3	yes	—
44	C₁₆×C₉₇	C₂³	1	1	yes	characteristic
45	C₂²×D₉₇	—	4	3	no	—
46	Q₈×C₉₇	D₈	1	1	yes	characteristic
47	D₄×C₉₇	D₈	1	3	yes	—
48	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
49	C₂²⋊D₉₇	—	4	3	no	—
50	C₂²⋊D₉₇	—	4	3	no	—
51	C₂×Dic₉₇	—	4	3	no	—
52	D₃₈₈	—	2	3	no	—
53	D₃₈₈	—	2	1	no	—
54	D₃₈₈	—	4	3	no	—
55	C₄×D₉₇	—	4	3	no	—
56	C₄×D₉₇	—	4	3	no	—
57	Dic₁₉₄	—	2	1	no	—
58	Dic₁₉₄	—	2	3	no	—
59	Dic₁₉₄	—	4	3	no	—
60	C₈×C₉₇	C₂×D₄	1	3	yes	—
61	C₈×C₉₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₉₇	C₂×D₈	1	3	yes	—
63	C₂×D₉₇	—	4	1	no	—
64	C₂×D₉₇	—	4	3	no	—
65	C₂×D₉₇	—	16	3	no	—
66	C₄×C₉₇	C₂×D₈	1	3	yes	—
67	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
68	Dic₉₇	—	4	3	no	—
69	Dic₉₇	—	4	1	no	—
70	C₂×C₉₇	—	2	3	no	—
71	C₂×C₉₇	C₂²×D₈	1	1	yes	characteristic
72	D₉₇	—	8	1	no	—
73	D₉₇	—	8	3	no	—
74	#3416	—	97	1	no	maximal, Sylow
75	C₉₇	#3416	1	1	yes	characteristic, complemented, Sylow
76	#578	—	97	1	no	—
77	#576	—	97	1	no	—
78	#510	—	97	3	no	—
79	#509	—	97	3	no	—
80	#508	—	97	3	no	—
81	C₂×SD₃₂	—	97	3	no	—
82	#504	D₉₇	1	1	yes	characteristic, complemented
83	2₋¹⁺⁴	—	194	1	no	—
84	H(5,𝔽₂)	—	194	1	no	—
85	#137	—	194	3	no	—
86	AGL(1,ℤ/8)	—	194	3	no	—
87	#135	—	97	3	no	—
88	#135	—	97	3	no	—
89	C₂×Q₁₆	—	97	3	no	—
90	C₂×D₈	—	97	3	no	—
91	#131	C₂×D₉₇	1	1	yes	characteristic
92	Q₃₂	—	97	3	no	—
93	SD₃₂	—	97	3	no	—
94	SD₃₂	—	97	6	no	—
95	SD₃₂	—	97	1	no	—
96	D₁₆	—	97	3	no	—
97	M₃₂	C₂×D₉₇	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₉₇	1	3	yes	complemented
99	#41	—	388	1	no	—
100	#41	—	388	3	no	—
101	#41	—	194	3	no	—
102	#41	—	194	3	no	—
103	#41	D₃₈₈	1	1	yes	characteristic
104	C₂×Q₈	—	388	1	no	—
105	C₂×Q₈	—	194	3	no	—
106	C₂×D₄	—	388	3	no	—
107	C₂×D₄	—	194	3	no	—
108	Q₁₆	—	97	3	no	—
109	Q₁₆	—	97	1	no	—
110	Q₁₆	—	194	3	no	—
111	SD₁₆	—	194	3	no	—
112	SD₁₆	—	194	3	no	—
113	D₈	—	97	1	no	—
114	D₈	—	97	3	no	—
115	D₈	—	194	3	no	—
116	M₁₆	D₃₈₈	1	3	yes	—
117	C₂×C₈	C₂²×D₉₇	1	3	yes	—
118	C₁₆	C₂²×D₉₇	1	1	yes	characteristic
119	C₁₆	C₂²×D₉₇	1	3	yes	—
120	C₂³	—	388	3	no	—
121	Q₈	—	388	3	no	—
122	Q₈	—	194	1	no	—
123	Q₈	—	194	3	no	—
124	Q₈	D₇₇₆	1	1	yes	characteristic
125	D₄	—	388	3	no	—
126	D₄	—	388	3	no	—
127	D₄	—	388	3	no	—
128	D₄	—	194	3	no	—
129	D₄	—	194	1	no	—
130	D₄	D₇₇₆	1	3	yes	—
131	C₂×C₄	—	388	3	no	—
132	C₂×C₄	—	388	3	no	—
133	C₂×C₄	—	388	3	no	—
134	C₂×C₄	C₂×D₃₈₈	1	3	yes	—
135	C₈	C₂×D₃₈₈	1	3	yes	—
136	C₈	C₂³×D₉₇	1	1	yes	characteristic
137	C₂²	—	1552	3	no	—
138	C₂²	—	388	1	no	—
139	C₂²	—	388	3	no	—
140	C₂²	C₂×D₇₇₆	1	3	yes	—
141	C₄	—	388	1	no	—
142	C₄	—	388	3	no	—
143	C₄	C₂×D₇₇₆	1	3	yes	—
144	C₄	C₂²×D₃₈₈	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	776	3	no	—
147	C₂	—	776	1	no	—
148	C₂	C₂²×D₇₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₇₇₆
commutator, cosocle	C₈×C₉₇	C₂⁴
Frattini	C₈	C₂³×D₉₇
Fitting	#504×C₉₇	C₂
socle	C₂×C₉₇	C₂²×D₈
hypercenter	#504	D₉₇
hypocenter	C₉₇	#3416
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #20261552	C₂	semidirect
2. #20261540	C₂	semidirect
3. #20261178	C₂	semidirect
4. #20261173	C₂	semidirect
5. #20261167	C₂	semidirect
6. C₂×#15690755	C₂	semidirect
7. #504×C₉₇	C₂	semidirect
13. #15690832	C₂²	semidirect
14. C₂×Dic₃₈₈	C₂²	semidirect
16. Dic₇₇₆	C₂²	semidirect
17. #15690755	C₂²	semidirect
18. #15690755	C₂²	semidirect
19. #15690755	C₂²	semidirect
20. D₁₅₅₂	C₂²	semidirect
22. M₃₂×C₉₇	C₂²	semidirect
23. C₂×C₁₆×C₉₇	C₂²	semidirect
75. C₉₇	#3416	semidirect
82. #504	D₉₇	semidirect
97. M₃₂	C₂×D₉₇	semidirect
98. C₂×C₁₆	C₂×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261552	2
#20261540	2
#20261178	2
#20261173	2
#20261167	2
C₂×#15690755	2
#504×C₉₇	2
#3416	97