Finite Group Databasebeta

← #27142155

Subgroups — #27142155

Order 12416.

342 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 93 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#20261502C211yesmaximal, characteristic, complemented
2#20260719C211yesmaximal, characteristic, complemented
3#20260699C212yesmaximal, complemented
4#20260505C212yesmaximal, complemented
5#525×C97C211yesmaximal, characteristic
6#15690817—24no—
7C2×#13095495—22no—
8C2×#13095495C2212yes—
9C2×#13095495C2212yes—
10#15690729—21no—
11#15690727—21no—
12C22⋊#11715895—24no—
13C2×#41×C97C2211yescharacteristic
14C23×C4×C97C2211yescharacteristic
15#123×C97C412yes—
16C22⋊Q8×C97—21no—
17#121×C97—21no—
18#120×C97C412yes—
19C4×D4×C97—22no—
20#117×C97C2211yescharacteristic
21#13095495—24no—
22#13095495—22no—
23#13095495—22no—
24#13095495—22no—
25#13095495—22no—
26C2×#11715895—24no—
27C2×#11715895—22no—
28C2×#11715895D414yescomplemented
29C24×C97C2×C411yescharacteristic
30#41×C97—22no—
31#41×C97—22no—
32C2×Q8×C97D411yescharacteristic
33C2×D4×C97—22no—
34C2×D4×C97C2×C412yes—
35C2×D4×C97D411yescharacteristic
36C22×C4×C97—21no—
37C22×C4×C97—21no—
38C22×C4×C97—22no—
39C22×C4×C97—22no—
40C22×C4×C97C2×C411yescharacteristic
41C22×C4×C97D412yes—
42C22×C4×C97C2311yescharacteristic
43C4⋊C4×C97—22no—
44C4⋊C4×C97—21no—
45C22⋊C4×C97—22no—
46C22⋊C4×C97—22no—
47C22⋊C4×C97C2×C412yes—
48C42×C97—21no—
49C23×C97—21no—
50C23×C97—22no—
51C23×C97—42no—
52C23×C97C22⋊C412yes—
53C23×C97C22×C411yescharacteristic
54Q8×C97—22no—
55D4×C97—24no—
56D4×C97—22no—
57D4×C97—44no—
58C2×C4×C97—22no—
59C2×C4×C97—22no—
60C2×C4×C97—22no—
61C2×C4×C97—21no—
62C2×C4×C97—22no—
63C2×C4×C97—41no—
64C2×C4×C97—42no—
65C2×C4×C97—41no—
66C2×C4×C97C22⋊C412yes—
67C2×C4×C97C2×D412yes—
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97#4112yes—
71#11715895—24no—
72#11715895—42no—
73C22×C97—22no—
74C22×C97—24no—
75C22×C97—22no—
76C22×C97—41no—
77C22×C97—42no—
78C22×C97—41no—
79C22×C97—42no—
80C22×C97—41no—
81C22×C97C2×C22⋊C411yescharacteristic
82C22×C97C4×D412yes—
83C4×C97—22no—
84C4×C97—42no—
85C4×C97—41no—
86C4×C97#12011yescharacteristic
87C4×C97#12112yes—
88C4×C97#12311yescharacteristic
89C2×C97—21no—
90C2×C97—22no—
91C2×C97—42no—
92C2×C97—42no—
93C2×C97#38611yescharacteristic
94#1891—971nomaximal, Sylow
95C97#189111yescharacteristic, complemented, Sylow
96#568—971no—
97#525D9711yescharacteristic
98#413—971no—
99#407—972no—
100#344—972no—
101C2×#41C2×D9711yescharacteristic
102C23×C4C2×D9711yescharacteristic
103#131—1944no—
104C2×M16—972no—
105C2×M16—972no—
106C2×M16—1942no—
107#123Dic9712yes—
108C22⋊Q8—21no—
109#121—21no—
110#120Dic9712yes—
111C4×D4—22no—
112#117C2×D9711yescharacteristic
113#101—1941no—
114#100—1941no—
115C22⋊C8—1944no—
116C24C2×Dic9711yescharacteristic
117#41—22no—
118#41—22no—
119C2×Q8C22⋊D9711yescharacteristic
120C2×D4—22no—
121C2×D4C2×Dic9712yes—
122C2×D4C22⋊D9711yescharacteristic
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4C2×Dic9711yescharacteristic
128C22×C4C22⋊D9712yes—
129C22×C4C22×D9711yescharacteristic
130M16—1942no—
131M16—1942no—
132M16—1944no—
133M16—1942no—
134M16—1942no—
135C2×C8—974no—
136C2×C8—1942no—
137C2×C8—1944no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4C2×Dic9712yes—
143C42—21no—
144C23—21no—
145C23—22no—
146C23—42no—
147C23C22⋊Dic9712yes—
148C23C22×Dic9711yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—44no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4C22⋊Dic9712yes—
162C2×C4D4×D9712yes—
163C2×C4C2×C22⋊D9711yescharacteristic
164C2×C4C2×C22⋊D9711yescharacteristic
165C2×C4#1309551012yes—
166C8—3882no—
167C8—1944no—
168C22—22no—
169C22—24no—
170C22—22no—
171C22—41no—
172C22—42no—
173C22—41no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊Dic9711yescharacteristic
177C22D4×Dic9712yes—
178C4—22no—
179C4—42no—
180C4—41no—
181C4#1569077811yescharacteristic
182C4#1569078212yes—
183C4#1569079111yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—42no—
188C2#2026061511yescharacteristic
subgroupG/H
centerC4#15690778
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radical#525×C97C2
socleC2×C97#386
cosocleC22×C4×C97C23
hypercenter#525D97
hypocenterC97#1891
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #20261502C2semidirect
2. #20260719C2semidirect
3. #20260699C2semidirect
4. #20260505C2semidirect
28. C2×#11715895D4semidirect
95. C97#1891semidirect

Maximal subgroups

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

subgroupindex
#202615022
#202607192
#202606992
#202605052
#525×C972
#189197