Finite Group Databasebeta

← #15671935

Subgroups — #15671935

Order 2944.

222 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 77 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
1#13092474C211yesmaximal, characteristic, complemented
2#13092283C211yesmaximal, characteristic, complemented
3#13092154C212yesmaximal, complemented
4#13092151C212yesmaximal
5C23×#529C211yesmaximal, characteristic
6C23×#126—21no—
7C23×#125C411yescharacteristic, complemented
8C23×#124C411yescharacteristic, complemented
9C23×#123—21no—
10C23×C22⋊Q8C2211yescharacteristic
11C23×#121C2211yescharacteristic
12C4×Q8×C23C411yescharacteristic, complemented
13C4×D4×C23—21no—
14C4×D4×C23C411yescharacteristic, complemented
15C23×#117—21no—
16C2×C42×C23C2211yescharacteristic
17C2×C4⋊Dic23—21no—
18C23⋊#95—21no—
19C23⋊#95C2212yes—
20C22⋊(C23⋊C8)C2212yes—
21C4⋊(C23⋊C8)—21no—
22#10624985—21no—
23C2×Q8×C23C2×C411yescharacteristic
24C2×D4×C23—21no—
25C2×D4×C23C2×C411yescharacteristic
26C22×C4×C23—21no—
27C22×C4×C23D411yescharacteristic
28C22×C4×C23D411yescharacteristic
29C22×C4×C23C2311yescharacteristic
30C22×Dic23—22no—
31C4⋊C4×C23—21no—
32C4⋊C4×C23—21no—
33C4⋊C4×C23—21no—
34C4⋊C4×C23—21no—
35C4⋊C4×C23C2×C411yescharacteristic
36C4⋊C4×C23C2×C411yescharacteristic
37C22⋊C4×C23—21no—
38C22⋊C4×C23—21no—
39C22⋊C4×C23—21no—
40C22⋊C4×C23—21no—
41C22⋊C4×C23—21no—
42C42×C23—21no—
43C42×C23C2×C411yescharacteristic
44C42×C23C2×C411yescharacteristic
45C42×C23D411yescharacteristic
46C42×C23D411yescharacteristic
47C4⋊Dic23—42no—
48C2×C23⋊C8—22no—
49C23×C23—21no—
50C23×C23C2×D411yescharacteristic
51Q8×C23—41no—
52D4×C23—41no—
53D4×C23—41no—
54D4×C23—41no—
55C2×C4×C23—21no—
56C2×C4×C23—21no—
57C2×C4×C23—21no—
58C2×C4×C23—21no—
59C2×C4×C23—21no—
60C2×C4×C23—41no—
61C2×C4×C23—41no—
62C2×C4×C23—42no—
63C2×C4×C23C22⋊C411yescharacteristic
64C2×C4×C23C22⋊C411yescharacteristic
65C2×C4×C23C22⋊C411yescharacteristic
66C2×C4×C23C22⋊C411yescharacteristic
67C2×C4×C23C22×C411yescharacteristic
68C2×C4×C23C2×D411yescharacteristic
69C2×Dic23—42no—
70C2×Dic23—42no—
71C2×Dic23—42no—
72C23⋊C8—42no—
73#1512—231nomaximal, Sylow
74C22×C23—21no—
75C22×C23—21no—
76C22×C23—41no—
77C22×C23—41no—
78C22×C23—41no—
79C22×C23C2×C22⋊C411yescharacteristic
80C22×C23AGL(1,ℤ/8)11yescharacteristic
81C22×C23#13711yescharacteristic
82C4×C23—21no—
83C4×C23—21no—
84C4×C23—21no—
85C4×C23—21no—
86C4×C23—21no—
87C4×C23—21no—
88C4×C23—41no—
89C4×C23—41no—
90C4×C23—41no—
91Dic23—82no—
92#529D2311yescharacteristic
93#432—231no—
94#382—231no—
95#328—232no—
96#327—232no—
97C2×C23—21no—
98C2×C23—21no—
99C2×C23—81no—
100C2×C23#41011yescharacteristic
101C2×C23#41711yescharacteristic
102C2×C23#42111yescharacteristic
103#126—21no—
104#125Dic2311yescharacteristic, complemented
105#124Dic2311yescharacteristic, complemented
106#123—21no—
107C22⋊Q8C2×D2311yescharacteristic
108#121C2×D2311yescharacteristic
109C4×Q8Dic2311yescharacteristic, complemented
110C4×D4—21no—
111C4×D4Dic2311yescharacteristic, complemented
112#117—21no—
113C2×C4⋊C4—461no—
114C2×C42C2×D2311yescharacteristic
115C4⋊C8—461no—
116C22⋊C8—232no—
117#97—461no—
118#95—232no—
119#95—461no—
120C23#151211yescharacteristic, complemented, Sylow
121C2×Q8C2×Dic2311yescharacteristic
122C2×D4—21no—
123C2×D4C2×Dic2311yescharacteristic
124C22×C4—21no—
125C22×C4—462no—
126C22×C4C22⋊D2311yescharacteristic
127C22×C4C22⋊D2311yescharacteristic
128C22×C4C22×D2311yescharacteristic
129C2×C8—462no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—922no—
135C4⋊C4C2×Dic2311yescharacteristic
136C4⋊C4C2×Dic2311yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C2×Dic2311yescharacteristic
144C42C2×Dic2311yescharacteristic
145C42C22⋊D2311yescharacteristic
146C42C22⋊D2311yescharacteristic
147C23—21no—
148C23C2×C22⋊D2311yescharacteristic
149Q8—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—922no—
162C2×C4—922no—
163C2×C4—922no—
164C2×C4C22⋊Dic2311yescharacteristic
165C2×C4C22⋊Dic2311yescharacteristic
166C2×C4C22⋊Dic2311yescharacteristic
167C2×C4C22⋊Dic2311yescharacteristic
168C2×C4C22×Dic2311yescharacteristic
169C2×C4C2×C22⋊D2311yescharacteristic
170C8—922no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22#1062512011yescharacteristic
177C22#1062512211yescharacteristic
178C22C2×C22⋊Dic2311yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—1842no—
189C2—21no—
190C2—21no—
191C2—81no—
192C2#1309239111yescharacteristic
193C2#1309241811yescharacteristic
194C2#1309243411yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic23
commutatorC2×C4×C23C22×C4
FrattiniC22×C4C22×D23
FittingC23×#529C2
socleC22×C23C2×C22⋊C4
prime radicalC23×#121C22
cosocleC22×C4×C23C23
hypercenter#529D23
hypocenterC23#1512
LCSG ▷ C2×C4×C23 ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #13092474C2semidirect
2. #13092283C2semidirect
3. #13092154C2semidirect
7. C23×#125C4semidirect
8. C23×#124C4semidirect
12. C4×Q8×C23C4semidirect
14. C4×D4×C23C4semidirect
104. #125Dic23semidirect
105. #124Dic23semidirect
109. C4×Q8Dic23semidirect
111. C4×D4Dic23semidirect
120. C23#1512semidirect

Maximal subgroups

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

subgroupindex
#130924742
#130922832
#130921542
#130921512
C23×#5292
#151223