Finite Group Databasebeta

Subgroups — #27141401

Order 12416.

242 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 61 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	#20261111	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261111	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261107	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261082	C₂	1	1	yes	maximal, characteristic, complemented
5	#20261081	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260457	C₂	1	1	yes	maximal, characteristic, complemented
7	#493×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13095499	—	2	2	no	—
9	C₂×#13095496	—	2	2	no	—
10	C₂×#13095496	—	2	2	no	—
11	#15690810	C₂²	1	1	yes	characteristic
12	#15690807	C₂²	1	1	yes	characteristic
13	#15690732	—	2	2	no	—
14	#15690731	—	2	2	no	—
15	C₄×#11715895	C₂²	1	1	yes	characteristic
16	C₄×#11715895	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₉₇	—	2	2	no	—
18	#127×C₉₇	C₂²	1	1	yes	characteristic
19	#127×C₉₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₉₄	—	2	1	no	—
22	C₂×D₃₈₈	—	2	1	no	—
23	C₂×D₃₈₈	—	2	1	no	—
24	C₂×D₃₈₈	—	2	1	no	—
25	#13095499	—	4	4	no	—
26	#13095496	—	4	4	no	—
27	#13095496	—	4	4	no	—
28	C₂×#11715895	D₄	1	2	yes	—
29	C₂×#11715895	D₄	1	2	yes	—
30	C₄⋊Dic₉₇	—	2	1	no	—
31	C₄⋊Dic₉₇	—	2	1	no	—
32	C₂×D₄×C₉₇	—	2	1	no	—
33	C₂×D₄×C₉₇	—	2	1	no	—
34	C₂×D₄×C₉₇	—	2	1	no	—
35	C₂×D₄×C₉₇	—	2	1	no	—
36	C₂×D₄×C₉₇	—	2	1	no	—
37	C₂×D₄×C₉₇	—	2	1	no	—
38	D₈×C₉₇	—	4	4	no	—
39	C₂×C₈×C₉₇	D₄	1	2	yes	—
40	C₄²×C₉₇	C₂³	1	1	yes	characteristic
41	C₂³×C₉₇	—	4	1	no	—
42	C₂³×C₉₇	—	4	1	no	—
43	C₂²×D₉₇	—	4	1	no	—
44	D₄×C₉₇	—	4	2	no	—
45	D₄×C₉₇	—	4	2	no	—
46	D₄×C₉₇	—	4	2	no	—
47	D₄×C₉₇	—	4	2	no	—
48	D₄×C₉₇	—	4	2	no	—
49	D₄×C₉₇	—	4	2	no	—
50	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₉₇	—	4	1	no	—
54	D₃₈₈	—	4	2	no	—
55	D₃₈₈	—	4	2	no	—
56	D₃₈₈	—	4	2	no	—
57	Dic₁₉₄	—	4	2	no	—
58	C₈×C₉₇	—	2	2	no	—
59	#11715895	—	2	2	no	—
60	#11715895	D₈	1	4	yes	complemented
61	C₂²×C₉₇	—	8	1	no	—
62	C₂²×C₉₇	—	8	1	no	—
63	C₂²×C₉₇	—	8	1	no	—
64	C₂²×C₉₇	—	8	1	no	—
65	C₂²×C₉₇	—	8	1	no	—
66	C₂²×C₉₇	—	8	1	no	—
67	C₂²×C₉₇	#127	1	1	yes	characteristic
68	C₂×D₉₇	—	8	1	no	—
69	C₂×D₉₇	—	8	1	no	—
70	C₂×D₉₇	—	8	1	no	—
71	C₄×C₉₇	C₂×D₈	1	2	yes	—
72	C₄×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
73	C₄×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
74	Dic₉₇	—	8	1	no	—
75	C₂×C₉₇	—	16	1	no	—
76	C₂×C₉₇	—	16	1	no	—
77	C₂×C₉₇	#493	1	1	yes	characteristic
78	C₂×C₉₇	#496	1	1	yes	characteristic
79	C₂×C₉₇	#496	1	1	yes	characteristic
80	D₉₇	—	16	1	no	—
81	#1721	—	97	1	no	maximal, Sylow
82	C₉₇	#1721	1	1	yes	characteristic, complemented, Sylow
83	#493	—	97	1	no	—
84	#493	—	97	1	no	—
85	#493	D₉₇	1	1	yes	characteristic, complemented
86	#492	—	97	1	no	—
87	#486	—	97	1	no	—
88	#486	—	97	1	no	—
89	#322	—	97	1	no	—
90	C₂×SD₁₆	—	194	2	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	—	194	2	no	—
93	C₂×D₈	—	194	2	no	—
94	C₄⋊Q₈	—	97	1	no	—
95	#127	—	97	1	no	—
96	#127	C₂×D₉₇	1	1	yes	characteristic
97	#127	C₂×D₉₇	1	1	yes	characteristic
98	#102	—	194	2	no	—
99	#102	—	194	2	no	—
100	C₄×C₈	—	97	1	no	—
101	C₄×C₈	—	97	1	no	—
102	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
103	C₂×Q₈	—	194	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	194	1	no	—
111	C₂×D₄	—	194	1	no	—
112	C₂×D₄	—	194	1	no	—
113	SD₁₆	—	388	4	no	—
114	D₈	—	4	4	no	—
115	D₈	—	388	4	no	—
116	D₈	—	388	4	no	—
117	C₂×C₈	—	97	2	no	—
118	C₂×C₈	—	97	2	no	—
119	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
120	C₄⋊C₄	—	194	1	no	—
121	C₄⋊C₄	—	194	1	no	—
122	C₄²	C₂²×D₉₇	1	1	yes	characteristic
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	—	388	1	no	—
126	Q₈	—	388	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	2	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	388	2	no	—
134	D₄	—	388	2	no	—
135	D₄	—	388	2	no	—
136	C₂×C₄	—	388	1	no	—
137	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
138	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	97	4	no	—
142	C₈	—	194	2	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	776	1	no	—
150	C₂²	—	776	1	no	—
151	C₂²	—	776	1	no	—
152	C₂²	#15690806	1	1	yes	characteristic
153	C₄	—	776	1	no	—
154	C₄	D₈×D₉₇	1	2	yes	—
155	C₄	#15690834	1	2	yes	—
156	C₄	#15690836	1	2	yes	—
157	C₂	—	16	1	no	—
158	C₂	—	16	1	no	—
159	C₂	—	1552	1	no	—
160	C₂	#20261110	1	1	yes	characteristic
161	C₂	#20261122	1	1	yes	characteristic
162	C₂	#20261123	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690806
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#493×C₉₇	C₂
socle	C₂²×C₉₇	#127
hypercenter	#493	D₉₇
hypocenter	C₉₇	#1721
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #20261111	C₂	semidirect
2. #20261111	C₂	semidirect
3. #20261107	C₂	semidirect
4. #20261082	C₂	semidirect
5. #20261081	C₂	semidirect
6. #20260457	C₂	semidirect
7. #493×C₉₇	C₂	semidirect
60. #11715895	D₈	semidirect
82. C₉₇	#1721	semidirect
85. #493	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261111	2
#20261107	2
#20261082	2
#20261081	2
#20260457	2
#493×C₉₇	2
#1721	97