Finite Group Databasebeta

← #27151665

Subgroups — #27151665

Order 12416.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15690769C211yesmaximal, characteristic, complemented
2#20261073C214yesmaximal, complemented
3#20261056C214yesmaximal, complemented
4#20260778C212yesmaximal, complemented
5#20260736C212yesmaximal, complemented
6#20260699C211yesmaximal, characteristic, complemented
7#525×C97C211yesmaximal, characteristic
8C2×#13095495C2212yes—
9#15690769—22no—
10#15690769C2214yes—
11C2×C4⋊Dic97C2212yes—
12C2×C22⋊Dic97—21no—
13C2×C4×Dic97—21no—
14#15690743C2214yescomplemented
15#15690741C2214yescomplemented
16#15690734C2214yescomplemented
17#15690731C2214yescomplemented
18C22⋊#11715895C2214yes—
19C2×#41×C97C2211yescharacteristic
20C23×C4×C97C2211yescharacteristic
21#123×C97—21no—
22C22⋊Q8×C97C2212yes—
23#121×C97C2212yes—
24#120×C97—21no—
25C4×D4×C97—22no—
26#117×C97C2211yescharacteristic
27C22×Dic97—22no—
28#13095495—24no—
29C2×#11715895C2314yes—
30C4⋊Dic97—22no—
31C4⋊Dic97C2314yes—
32C22⋊Dic97—24no—
33C4×Dic97—24no—
34C24×C97D411yescharacteristic
35#41×C97—24no—
36C2×Q8×C97C2311yescharacteristic
37C2×D4×C97—22no—
38C2×D4×C97—21no—
39C2×D4×C97C2311yescharacteristic
40C22×C4×C97—24no—
41C22×C4×C97—21no—
42C22×C4×C97D411yescharacteristic
43C22×C4×C97D412yes—
44C22×C4×C97C2312yes—
45C22×C4×C97C2311yescharacteristic
46C4⋊C4×C97—22no—
47C4⋊C4×C97C2312yes—
48C22⋊C4×C97—21no—
49C22⋊C4×C97—22no—
50C22⋊C4×C97—22no—
51C42×C97—21no—
52C23×C97—22no—
53C23×C97—21no—
54C23×C97—42no—
55C23×C97C2×D412yes—
56C23×C97C2×D411yescharacteristic
57Q8×C97—22no—
58D4×C97—22no—
59D4×C97—44no—
60D4×C97—42no—
61C2×C4×C97—22no—
62C2×C4×C97—24no—
63C2×C4×C97—24no—
64C2×C4×C97—22no—
65C2×C4×C97—21no—
66C2×C4×C97—42no—
67C2×C4×C97C2×D412yes—
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97#4114yes—
70C2×C4×C97C2411yescharacteristic
71C2×Dic97—24no—
72C2×Dic97—44no—
73#11715895—24no—
74C22×C97—22no—
75C22×C97—22no—
76C22×C97—22no—
77C22×C97—22no—
78C22×C97—44no—
79C22×C97—41no—
80C22×C97—42no—
81C22×C97#12312yes—
82C22×C97C22×D411yescharacteristic
83C4×C97—22no—
84C4×C97—41no—
85C4×C97—42no—
86C4×C97#12312yes—
87C4×C97C2×#4112yes—
88Dic97—44no—
89C2×C97—22no—
90C2×C97—42no—
91C2×C97—81no—
92C2×C97C2×#12311yescharacteristic
93C2×C97#57512yes—
94#3093—971nomaximal, Sylow
95C97#309311yescharacteristic, complemented, Sylow
96#525D9711yescharacteristic
97C2×#117—971no—
98#485—974no—
99#482—974no—
100#428—972no—
101#417—972no—
102#407—971no—
103C2×#41C2×D9711yescharacteristic
104C23×C4C2×D9711yescharacteristic
105C2×M16—972no—
106#123—21no—
107C22⋊Q8C2×D9712yes—
108#121C2×D9712yes—
109#120—21no—
110C4×D4—22no—
111#117—974no—
112#117—1942no—
113#117C2×D9711yescharacteristic
114C2×C4⋊C4—972no—
115C2×C22⋊C4—1941no—
116C2×C42—1941no—
117#107—974no—
118#106—974no—
119#103—974no—
120#102—974no—
121C22⋊C8—974no—
122C24C22⋊D9711yescharacteristic
123#41—24no—
124C2×Q8C22×D9711yescharacteristic
125C2×D4—22no—
126C2×D4—21no—
127C2×D4C22×D9711yescharacteristic
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—1942no—
131C22×C4C22⋊D9711yescharacteristic
132C22×C4C22⋊D9712yes—
133C22×C4C22×D9712yes—
134C22×C4C22×D9711yescharacteristic
135M16—1944no—
136C2×C8—974no—
137C4⋊C4—22no—
138C4⋊C4—974no—
139C4⋊C4—1942no—
140C4⋊C4C22×D9712yes—
141C22⋊C4—21no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—1944no—
145C42—21no—
146C42—1944no—
147C23—22no—
148C23—21no—
149C23—42no—
150C23C2×C22⋊D9712yes—
151C23C2×C22⋊D9711yescharacteristic
152Q8—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—3884no—
163C2×C4—1944no—
164C2×C4C2×C22⋊D9712yes—
165C2×C4C2×C22⋊D9711yescharacteristic
166C2×C4#1309551014yes—
167C2×C4C23×D9711yescharacteristic
168C8—1944no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—41no—
175C22—42no—
176C22#1569079112yes—
177C22C22×C22⋊D9711yescharacteristic
178C4—22no—
179C4—41no—
180C4—42no—
181C4—3884no—
182C4#1569079112yes—
183C4C2×#1309551012yes—
184C2—22no—
185C2—42no—
186C2—81no—
187C2C2×#1569079111yescharacteristic
188C2#2026153312yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting, prime radical#525×C97C2
socleC22×C97C22×D4
hypercenter#525D97
hypocenterC97#3093
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#15690769C2semidirect
2. #20261073C2semidirect
3. #20261056C2semidirect
4. #20260778C2semidirect
5. #20260736C2semidirect
6. #20260699C2semidirect
14. #15690743C22semidirect
15. #15690741C22semidirect
16. #15690734C22semidirect
17. #15690731C22semidirect
95. C97#3093semidirect

Maximal subgroups

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

subgroupindex
C2×#156907692
#202610732
#202610562
#202607782
#202607362
#202606992
#525×C972
#309397