Finite Group Databasebeta

← #27150046

Subgroups — #27150046

Order 12416.

556 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690802C211yesmaximal, characteristic, complemented
2#20260633C212yesmaximal, complemented
3#20260619C214yesmaximal, complemented
4#20260608C214yesmaximal
5#20260601C211yesmaximal, characteristic, complemented
6#20260595C212yesmaximal
7C22×C42×C97C211yesmaximal, characteristic
8#15690802—24no—
9C2×C4⋊Dic97C2212yes—
10C2×C4⋊Dic97C2212yes—
11C2×#13095490C2214yes—
12C2×#13095490C2214yes—
13C2×C22⋊Dic97C2214yes—
14#15690717C2214yes—
15#15690717C2214yes—
16#15690717C2214yes—
17C23×C4×C97C2212yes—
18C23×C4×C97C2211yescharacteristic
19C2×C42×C97—24no—
20C2×C42×C97C2212yes—
21C2×C42×C97C2211yescharacteristic
22C2×C42×C97C2211yescharacteristic
23C22×Dic97C2314yes—
24C22×Dic97C2314yes—
25C4⋊Dic97—24no—
26C4⋊Dic97—24no—
27#13095490—28no—
28#13095490—28no—
29C22⋊Dic97—28no—
30C24×C97C2311yescharacteristic
31C22×C4×C97—24no—
32C22×C4×C97—24no—
33C22×C4×C97—22no—
34C22×C4×C97—24no—
35C22×C4×C97—22no—
36C22×C4×C97Q814yes—
37C22×C4×C97C2312yes—
38C22×C4×C97C2311yescharacteristic
39C22×C4×C97C2312yes—
40C22×C4×C97C2311yescharacteristic
41C42×C97—24no—
42C42×C97—22no—
43C42×C97D414yes—
44C23×C97—21no—
45C23×C97—21no—
46C23×C97—22no—
47C23×C97C2×Q812yes—
48C23×C97#4114yes—
49C23×C97C2411yescharacteristic
50C2×C4×C97—216no—
51C2×C4×C97—21no—
52C2×C4×C97—22no—
53C2×C4×C97—21no—
54C2×C4×C97—22no—
55C2×C4×C97—22no—
56C2×C4×C97—22no—
57C2×C4×C97—28no—
58C2×C4×C97C2×D414yes—
59C2×C4×C97C2×D412yes—
60C2×C4×C97C2×Q812yes—
61C2×C4×C97C2×Q812yes—
62C2×C4×C97#4114yes—
63C2×C4×C97#4112yes—
64C2×Dic97—24no—
65C2×Dic97—24no—
66C2×Dic97—24no—
67C2×Dic97—24no—
68C2×Dic97—24no—
69C2×Dic97—24no—
70C22×C97—22no—
71C22×C97—24no—
72C22×C97—24no—
73C22×C97—22no—
74C22×C97#12514yescomplemented
75C22×C97C22×D411yescharacteristic
76C22×C97C22×Q811yescharacteristic
77C22×C97C2×#4111yescharacteristic
78C22×C97C2×#4112yes—
79C22×C97C2×#4112yes—
80C4×C97—22no—
81C4×C97—24no—
82C4×C97—24no—
83C4×C97C22⋊Q814yes—
84Dic97—44no—
85Dic97—44no—
86C2×C97—24no—
87C2×C97C2×C22⋊Q811yescharacteristic
88C2×C97#52512yes—
89C2×C97C2×#12511yescharacteristic
90C2×C97#52912yes—
91C2×C97#53311yescharacteristic
92#2851—971nomaximal, Sylow
93C97#285111yescharacteristic, complemented, Sylow
94C2×#125—971no—
95C22×C42D9711yescharacteristic
96#389—972no—
97#387—974no—
98#385—974no—
99#382—971no—
100#380—972no—
101C23×C4C2×D9712yes—
102C23×C4C2×D9711yescharacteristic
103#125—1944no—
104C2×C4⋊C4—974no—
105C2×C4⋊C4—974no—
106C2×C4⋊C4—972no—
107C2×C4⋊C4—972no—
108C2×C22⋊C4—974no—
109C2×C42—24no—
110C2×C42C2×D9712yes—
111C2×C42C2×D9711yescharacteristic
112C2×C42C2×D9711yescharacteristic
113#95—974no—
114#95—974no—
115#95—974no—
116C24C22×D9711yescharacteristic
117C22×C4—24no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—974no—
123C22×C4—974no—
124C22×C4Dic19414yes—
125C22×C4C22×D9712yes—
126C22×C4C22×D9711yescharacteristic
127C22×C4C22×D9712yes—
128C22×C4C22×D9711yescharacteristic
129C4⋊C4—1948no—
130C4⋊C4—1944no—
131C4⋊C4—1948no—
132C4⋊C4—1944no—
133C22⋊C4—1948no—
134C42—24no—
135C42—22no—
136C42C22⋊D9714yes—
137C23—21no—
138C23—21no—
139C23—22no—
140C23C2×Dic19412yes—
141C23#1309551114yes—
142C23C23×D9711yescharacteristic
143C2×C4—216no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—28no—
151C2×C4—1944no—
152C2×C4—1944no—
153C2×C4—1944no—
154C2×C4—1944no—
155C2×C4—1944no—
156C2×C4—1944no—
157C2×C4C2×C22⋊D9714yes—
158C2×C4C2×C22⋊D9712yes—
159C2×C4C2×Dic19412yes—
160C2×C4C2×Dic19412yes—
161C2×C4#1309551114yes—
162C2×C4#1309551112yes—
163C22—22no—
164C22—24no—
165C22—24no—
166C22—22no—
167C22#1569080214yescomplemented
168C22C22×C22⋊D9711yescharacteristic
169C22C22×Dic19411yescharacteristic
170C22C2×#1309551111yescharacteristic
171C22C2×#1309551112yes—
172C22C2×#1309551112yes—
173C4—22no—
174C4—24no—
175C4—24no—
176C4—3884no—
177C4—3884no—
178C4#1569078914yes—
179C2—24no—
180C2C2×#1569078911yescharacteristic
181C2#2026124612yes—
182C2C2×#1569080211yescharacteristic
183C2#2026126912yes—
184C2#2026129211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC22×C42×C97C2
prime radicalC24×C97C23
hypercenterC22×C42D97
hypocenterC97#2851
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
1. C2×#15690802C2semidirect
2. #20260633C2semidirect
3. #20260619C2semidirect
5. #20260601C2semidirect
74. C22×C97#125semidirect
93. C97#2851semidirect
167. C22#15690802semidirect

Maximal subgroups

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

subgroupindex
C2×#156908022
#202606332
#202606192
#202606082
#202606012
#202605952
C22×C42×C972
#285197