Finite Group Databasebeta

Subgroups — #27157333

Order 12416.

192 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 64 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	#20261763	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#11715897	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261597	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261583	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊#11715897	C₂	1	1	yes	maximal, characteristic, complemented
6	#20261576	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260815	C₂	1	1	yes	maximal, characteristic
8	C₂×#13095519	—	2	1	no	—
9	C₂²×#11715897	—	2	1	no	—
10	C₄⋊#11715897	C₂²	1	1	yes	characteristic
11	C₄⋊#11715897	C₂²	1	1	yes	characteristic
12	#15690867	—	2	1	no	—
13	C₂²⋊#11715897	—	2	1	no	—
14	C₄×#11715897	C₂²	1	1	yes	characteristic
15	C₄×#11715897	C₂²	1	1	yes	characteristic
16	C₂×#13095496	C₄	1	1	yes	characteristic
17	D₄×Dic₉₇	C₂²	1	1	yes	characteristic
18	#15690772	C₂²	1	1	yes	characteristic
19	#15690743	C₄	1	1	yes	characteristic
20	#15690732	C₄	1	1	yes	characteristic
21	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
22	#102×C₉₇	C₄	1	1	yes	characteristic
23	#13095519	—	4	2	no	—
24	C₂×#11715897	—	2	1	no	—
25	C₂×#11715897	—	2	1	no	—
26	C₂×#11715897	—	2	1	no	—
27	C₂×#11715897	—	4	2	no	—
28	C₂×#11715897	D₄	1	1	yes	characteristic
29	C₂×#11715897	D₄	1	1	yes	characteristic
30	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₉₇	—	2	1	no	—
32	C₂×C₄×D₉₇	—	2	1	no	—
33	#13095496	C₈	1	4	yes	complemented
34	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
35	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
36	C₂²⋊Dic₉₇	—	2	1	no	—
37	#13095488	—	2	1	no	—
38	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
39	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₉₇	—	2	1	no	—
43	C₂²×D₉₇	—	2	1	no	—
44	D₄×C₉₇	—	2	1	no	—
45	D₄×C₉₇	C₂×C₈	1	2	yes	complemented
46	C₂×C₄×C₉₇	—	2	1	no	—
47	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₉₇	—	2	1	no	—
49	C₂×Dic₉₇	—	4	1	no	—
50	C₂×Dic₉₇	—	4	1	no	—
51	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₉₇	#41	1	1	yes	characteristic
53	D₃₈₈	—	2	1	no	—
54	D₃₈₈	C₂×C₈	1	2	yes	complemented
55	C₄×D₉₇	—	4	1	no	—
56	C₄×D₉₇	—	4	1	no	—
57	#11715897	—	2	1	no	—
58	#11715897	—	2	1	no	—
59	#11715897	—	4	1	no	—
60	#11715897	—	4	1	no	—
61	#11715897	—	4	1	no	—
62	C₈×C₉₇	—	2	1	no	—
63	#11715895	C₂×C₈	1	2	yes	complemented
64	C₂²×C₉₇	—	2	2	no	—
65	C₂²×C₉₇	—	4	1	no	—
66	C₂²×C₉₇	—	4	1	no	—
67	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
68	C₂×D₉₇	—	2	2	no	—
69	C₂×D₉₇	—	4	1	no	—
70	C₂×D₉₇	—	4	1	no	—
71	C₄×C₉₇	—	4	1	no	—
72	C₄×C₉₇	C₂²×C₈	1	1	yes	characteristic
73	C₄×C₉₇	#131	1	1	yes	characteristic
74	Dic₉₇	—	2	1	no	—
75	Dic₉₇	—	4	1	no	—
76	Dic₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Dic₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₉₇	—	4	2	no	—
79	C₂×C₉₇	C₈×D₄	1	1	yes	characteristic
80	C₂×C₉₇	#442	1	1	yes	characteristic
81	C₂×C₉₇	#444	1	1	yes	characteristic
82	D₉₇	—	4	2	no	—
83	#1580	—	97	1	no	maximal, Sylow
84	C₉₇	#1580	1	1	yes	characteristic, complemented, Sylow
85	C₄×D₈	—	97	1	no	—
86	C₈×D₄	—	97	1	no	—
87	C₈×D₄	—	97	1	no	—
88	#334	—	97	1	no	—
89	D₄⋊C₈	—	97	1	no	—
90	D₄⋊C₈	—	97	1	no	—
91	#322	—	97	1	no	—
92	C₂×D₈	—	97	1	no	—
93	C₂²×C₈	—	194	1	no	—
94	C₂²×C₈	—	194	1	no	—
95	C₄×D₄	—	97	1	no	—
96	C₄×D₄	—	97	1	no	—
97	#107	—	97	1	no	—
98	C₄⋊C₈	—	97	1	no	—
99	C₄⋊C₈	—	97	1	no	—
100	#102	—	97	1	no	—
101	#102	#87604	1	1	yes	characteristic
102	C₂²⋊C₈	—	194	1	no	—
103	C₂²⋊C₈	—	194	1	no	—
104	C₄×C₈	—	97	1	no	—
105	C₄×C₈	—	97	1	no	—
106	C₄×C₈	—	97	1	no	—
107	C₂×D₄	—	97	1	no	—
108	C₂×D₄	C₂×#87604	1	1	yes	characteristic
109	C₂²×C₄	—	194	1	no	—
110	C₂²×C₄	—	194	1	no	—
111	D₈	—	97	4	no	—
112	C₂×C₈	—	97	1	no	—
113	C₂×C₈	—	97	1	no	—
114	C₂×C₈	—	97	1	no	—
115	C₂×C₈	—	388	2	no	—
116	C₂×C₈	—	388	2	no	—
117	C₂×C₈	—	194	1	no	—
118	C₂×C₈	—	194	1	no	—
119	C₂×C₈	—	194	1	no	—
120	C₂×C₈	C₂×#87604	1	1	yes	characteristic
121	C₄⋊C₄	—	97	1	no	—
122	C₄⋊C₄	C₂×#87604	1	1	yes	characteristic
123	C₂²⋊C₄	—	194	1	no	—
124	C₂²⋊C₄	—	194	1	no	—
125	C₄²	—	97	1	no	—
126	C₂³	—	2	1	no	—
127	C₂³	—	194	1	no	—
128	D₄	—	2	1	no	—
129	D₄	—	97	2	no	—
130	D₄	—	194	1	no	—
131	D₄	#13095519	1	2	yes	complemented
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	97	1	no	—
134	C₂×C₄	—	97	1	no	—
135	C₂×C₄	—	388	1	no	—
136	C₂×C₄	—	388	1	no	—
137	C₂×C₄	—	388	1	no	—
138	C₂×C₄	—	388	1	no	—
139	C₂×C₄	—	194	1	no	—
140	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	97	2	no	—
143	C₈	—	388	1	no	—
144	C₈	—	388	1	no	—
145	C₈	—	388	1	no	—
146	C₈	—	194	1	no	—
147	C₈	—	194	1	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	388	1	no	—
152	C₂²	—	388	1	no	—
153	C₂²	—	194	2	no	—
154	C₂²	D₄×#87604	1	1	yes	characteristic
155	C₄	—	4	1	no	—
156	C₄	—	97	1	no	—
157	C₄	—	97	1	no	—
158	C₄	—	388	1	no	—
159	C₄	—	194	1	no	—
160	C₄	C₂×#13095519	1	1	yes	characteristic
161	C₄	#15690897	1	1	yes	characteristic
162	C₂	—	4	2	no	—
163	C₂	—	388	2	no	—
164	C₂	#20261762	1	1	yes	characteristic
165	C₂	#20261776	1	1	yes	characteristic
166	C₂	#20261780	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×#87604
commutator	C₄×C₉₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#87604
Fitting	#102×C₉₇	C₄
socle	C₂²×C₉₇	C₄×D₄
prime radical	C₂×#13095496	C₄
cosocle	C₄×Dic₉₇	C₂³
hypercenter	#102	#87604
hypocenter	C₉₇	#1580
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. #20261763	C₂	semidirect
2. D₄×#11715897	C₂	semidirect
3. #20261597	C₂	semidirect
4. #20261583	C₂	semidirect
5. D₄⋊#11715897	C₂	semidirect
6. #20261576	C₂	semidirect
33. #13095496	C₈	semidirect
45. D₄×C₉₇	C₂×C₈	semidirect
54. D₃₈₈	C₂×C₈	semidirect
63. #11715895	C₂×C₈	semidirect
84. C₉₇	#1580	semidirect
131. D₄	#13095519	semidirect

Maximal subgroups

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

subgroup	index
#20261763	2
D₄×#11715897	2
#20261597	2
#20261583	2
D₄⋊#11715897	2
#20261576	2
#20260815	2
#1580	97