Finite Group Databasebeta

Subgroups — #27141024

Order 12416.

262 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 45 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	#20261477	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261122	C₂	1	2	yes	maximal, complemented
3	#20260475	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260474	C₂	1	2	yes	maximal, complemented
5	#496×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13095499	—	2	2	no	—
7	C₂×#13095496	—	2	2	no	—
8	#15690799	—	2	1	no	—
9	#15690798	C₂²	1	2	yes	—
10	#15690796	—	2	2	no	—
11	#15690780	—	2	2	no	—
12	#15690778	—	2	1	no	—
13	#15690721	C₂²	1	2	yes	complemented
14	C₂×SD₁₆×C₉₇	—	2	1	no	—
15	C₂×D₈×C₉₇	—	2	1	no	—
16	#127×C₉₇	C₂²	1	1	yes	characteristic, complemented
17	#124×C₉₇	C₂²	1	1	yes	characteristic
18	#97×C₉₇	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₉₇	—	2	1	no	—
20	C₂×Dic₁₉₄	D₄	1	2	yes	—
21	C₂×D₃₈₈	—	2	1	no	—
22	C₂×D₃₈₈	D₄	1	2	yes	—
23	C₂×C₂²⋊D₉₇	—	2	2	no	—
24	#13095499	—	4	4	no	—
25	#13095496	—	4	4	no	—
26	C₂×#11715895	—	2	2	no	—
27	C₂²⋊Dic₉₇	—	2	1	no	—
28	#13095488	—	2	2	no	—
29	#13095488	—	2	2	no	—
30	#13095488	—	2	2	no	—
31	C₄×Dic₉₇	—	2	1	no	—
32	C₂×Q₈×C₉₇	D₄	1	1	yes	characteristic
33	C₂×D₄×C₉₇	—	2	2	no	—
34	C₂×D₄×C₉₇	—	2	1	no	—
35	C₂×D₄×C₉₇	D₄	1	1	yes	characteristic
36	SD₁₆×C₉₇	—	4	2	no	—
37	D₈×C₉₇	—	4	2	no	—
38	C₂×C₈×C₉₇	—	2	1	no	—
39	C₂²⋊C₄×C₉₇	—	2	2	no	—
40	C₄²×C₉₇	C₂³	1	1	yes	characteristic
41	C₂³×C₉₇	—	2	1	no	—
42	C₂³×C₉₇	—	4	1	no	—
43	C₂²×D₉₇	—	2	2	no	—
44	C₂²×D₉₇	—	8	2	no	—
45	C₂²×D₉₇	—	8	1	no	—
46	Q₈×C₉₇	—	4	1	no	—
47	D₄×C₉₇	—	4	1	no	—
48	D₄×C₉₇	—	4	1	no	—
49	D₄×C₉₇	—	4	1	no	—
50	D₄×C₉₇	—	4	2	no	—
51	D₄×C₉₇	—	4	2	no	—
52	C₂×C₄×C₉₇	—	2	1	no	—
53	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
55	C₂²⋊D₉₇	—	8	2	no	—
56	C₂×Dic₉₇	—	2	2	no	—
57	D₃₈₈	—	4	2	no	—
58	D₃₈₈	—	8	1	no	—
59	Dic₁₉₄	—	4	2	no	—
60	C₈×C₉₇	—	2	2	no	—
61	#11715895	—	2	4	no	—
62	C₂²×C₉₇	—	4	2	no	—
63	C₂²×C₉₇	—	4	1	no	—
64	C₂²×C₉₇	—	8	2	no	—
65	C₂²×C₉₇	—	8	1	no	—
66	C₂²×C₉₇	#120	1	1	yes	characteristic
67	C₂×D₉₇	—	4	2	no	—
68	C₂×D₉₇	—	4	2	no	—
69	C₂×D₉₇	—	4	2	no	—
70	C₂×D₉₇	—	8	4	no	—
71	C₄×C₉₇	—	2	2	no	—
72	C₄×C₉₇	—	2	1	no	—
73	C₄×C₉₇	—	4	1	no	—
74	Dic₉₇	—	4	2	no	—
75	C₂×C₉₇	—	8	1	no	—
76	C₂×C₉₇	—	16	1	no	—
77	C₂×C₉₇	#453	1	2	yes	—
78	C₂×C₉₇	#453	1	1	yes	characteristic
79	D₉₇	—	8	2	no	—
80	#1655	—	97	1	no	maximal, Sylow
81	C₉₇	#1655	1	1	yes	characteristic, complemented, Sylow
82	H(3,𝔽₄)	—	97	1	no	—
83	#496	—	97	2	no	—
84	#496	D₉₇	1	1	yes	characteristic, complemented
85	#329	—	97	1	no	—
86	#329	—	97	2	no	—
87	C₂×SD₁₆	—	2	1	no	—
88	C₂×SD₁₆	—	194	2	no	—
89	C₂×D₈	—	2	1	no	—
90	C₂×D₈	—	194	2	no	—
91	#127	C₂×D₉₇	1	1	yes	characteristic, complemented
92	#124	—	97	2	no	—
93	#124	—	194	2	no	—
94	#124	—	194	1	no	—
95	#124	C₂×D₉₇	1	1	yes	characteristic
96	#120	—	194	2	no	—
97	#120	—	194	1	no	—
98	#97	—	97	2	no	—
99	#97	C₂×D₉₇	1	1	yes	characteristic, complemented
100	C₂⁴	—	194	1	no	—
101	C₂×Q₈	—	97	2	no	—
102	C₂×Q₈	C₂²⋊D₉₇	1	1	yes	characteristic
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	97	2	no	—
106	C₂×D₄	—	194	1	no	—
107	C₂×D₄	—	194	2	no	—
108	C₂×D₄	C₂²⋊D₉₇	1	1	yes	characteristic
109	SD₁₆	—	4	2	no	—
110	SD₁₆	—	388	4	no	—
111	D₈	—	4	2	no	—
112	D₈	—	388	4	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	194	2	no	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	194	2	no	—
117	C₂²⋊C₄	—	194	1	no	—
118	C₂²⋊C₄	—	194	2	no	—
119	C₂²⋊C₄	—	194	2	no	—
120	C₄²	—	194	1	no	—
121	C₄²	C₂²×D₉₇	1	1	yes	characteristic
122	C₂³	—	2	1	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	776	2	no	—
125	C₂³	—	776	1	no	—
126	C₂³	—	194	2	no	—
127	Q₈	—	4	1	no	—
128	Q₈	—	388	2	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	2	no	—
134	D₄	—	776	1	no	—
135	D₄	—	776	2	no	—
136	D₄	—	388	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	194	2	no	—
139	C₂×C₄	D₄×D₉₇	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	194	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	8	2	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	776	4	no	—
148	C₂²	—	388	2	no	—
149	C₂²	—	388	2	no	—
150	C₂²	—	388	2	no	—
151	C₂²	#15690778	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	388	2	no	—
156	C₂	—	8	1	no	—
157	C₂	—	776	2	no	—
158	C₂	—	16	1	no	—
159	C₂	#20260901	1	2	yes	—
160	C₂	#20260902	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690778
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#496×C₉₇	C₂
socle	C₂²×C₉₇	#120
hypercenter	#496	D₉₇
hypocenter	C₉₇	#1655
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #496

Complements

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

N	H ≅ G/N	type
1. #20261477	C₂	semidirect
2. #20261122	C₂	semidirect
3. #20260475	C₂	semidirect
4. #20260474	C₂	semidirect
5. #496×C₉₇	C₂	semidirect
13. #15690721	C₂²	semidirect
16. #127×C₉₇	C₂²	semidirect
18. #97×C₉₇	C₂²	semidirect
81. C₉₇	#1655	semidirect
84. #496	D₉₇	semidirect
91. #127	C₂×D₉₇	semidirect
99. #97	C₂×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261477	2
#20261122	2
#20260475	2
#20260474	2
#496×C₉₇	2
#1655	97