Finite Group Databasebeta

← #27150041

Subgroups — #27150041

Order 12416.

556 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 169 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×C42×D97C211yesmaximal, characteristic
2#20260631C212yesmaximal, complemented
3#20260616C214yesmaximal, complemented
4#20260606C214yesmaximal
5#20260601C211yesmaximal, characteristic, complemented
6#20260592C212yesmaximal
7C2×#125×C97C211yesmaximal, characteristic, complemented
8C22×C4×D97C2212yes—
9C22×C4×D97C2211yescharacteristic
10C2×C4⋊Dic97C2212yes—
11C2×#13095490C2214yes—
12C2×#13095488C2214yes—
13C2×C4×Dic97C2212yes—
14C2×C4×Dic97C2211yescharacteristic
15C42×D97—24no—
16#15690717C2214yes—
17#15690717C2214yes—
18#15690717C2214yes—
19#125×C97—24no—
20C2×C4⋊C4×C97C2214yes—
21C2×C4⋊C4×C97C2212yes—
22C2×C42×C97C2211yescharacteristic
23C23×D97C2311yescharacteristic
24C22×Dic97C2314yes—
25C22×Dic97C2311yescharacteristic
26C22×Dic97C2312yes—
27C2×C4×D97—24no—
28C2×C4×D97—24no—
29C2×C4×D97—22no—
30C2×C4×D97—22no—
31C2×C4×D97—24no—
32C2×C4×D97Q814yes—
33C4⋊Dic97—24no—
34#13095490—28no—
35#13095488—28no—
36C4×Dic97—24no—
37C4×Dic97—22no—
38C22×C4×C97C2314yes—
39C22×C4×C97C2312yes—
40C22×C4×C97C2311yescharacteristic
41C4⋊C4×C97—28no—
42C4⋊C4×C97—24no—
43C42×C97D414yes—
44C23×C97C2411yescharacteristic
45C22×D97—21no—
46C22×D97—22no—
47C22×D97—21no—
48C22×D97C2×Q812yes—
49C22×D97#4114yes—
50C2×C4×C97—22no—
51C2×C4×C97—22no—
52C2×C4×C97—24no—
53C2×C4×C97—24no—
54C2×C4×C97—24no—
55C2×C4×C97C2×D414yes—
56C2×C4×C97C2×D412yes—
57C2×C4×C97C2×Q812yes—
58C2×C4×C97#4112yes—
59C2×Dic97—21no—
60C2×Dic97—22no—
61C2×Dic97—22no—
62C2×Dic97—21no—
63C2×Dic97—24no—
64C2×Dic97—24no—
65C2×Dic97—24no—
66C2×Dic97C2×Q812yes—
67C2×Dic97#4114yes—
68C4×D97—216no—
69C4×D97—28no—
70C22×C97C22×D411yescharacteristic
71C22×C97C22×Q811yescharacteristic
72C22×C97C2×#4112yes—
73C22×C97C2×#4112yes—
74C22×C97C2×#4111yescharacteristic
75C2×D97—22no—
76C2×D97—24no—
77C2×D97—24no—
78C2×D97—22no—
79C2×D97#12514yescomplemented
80C4×C97—24no—
81C4×C97—44no—
82C4×C97C22⋊Q814yes—
83Dic97—22no—
84Dic97—24no—
85Dic97—44no—
86C2×C97C2×C22⋊Q811yescharacteristic
87C2×C97#52512yes—
88C2×C97C2×#12511yescharacteristic
89C2×C97#52912yes—
90C2×C97#53311yescharacteristic
91D97—24no—
92#2851—971nomaximal, Sylow
93C97#285111yescharacteristic, complemented, Sylow
94C2×#125D9711yescharacteristic, complemented
95C22×C42—971no—
96#389—972no—
97#387—974no—
98#385—974no—
99#382—971no—
100#380—972no—
101C23×C4—971no—
102C23×C4—972no—
103#125—24no—
104C2×C4⋊C4—974no—
105C2×C4⋊C4—972no—
106C2×C4⋊C4C2×D9712yes—
107C2×C4⋊C4C2×D9714yes—
108C2×C22⋊C4—974no—
109C2×C42—972no—
110C2×C42—971no—
111C2×C42—1944no—
112C2×C42C2×D9711yescharacteristic
113#95—974no—
114#95—974no—
115#95—974no—
116C24—971no—
117C22×C4—971no—
118C22×C4—972no—
119C22×C4—974no—
120C22×C4—974no—
121C22×C4—1944no—
122C22×C4—1944no—
123C22×C4—1944no—
124C22×C4—1942no—
125C22×C4—1942no—
126C22×C4C22×D9714yes—
127C22×C4C22×D9712yes—
128C22×C4C22×D9711yescharacteristic
129C4⋊C4—28no—
130C4⋊C4—24no—
131C4⋊C4—1948no—
132C4⋊C4—1944no—
133C22⋊C4—1948no—
134C42—1944no—
135C42—1942no—
136C42C22⋊D9714yes—
137C23—972no—
138C23—974no—
139C23—1941no—
140C23—1942no—
141C23—1941no—
142C23C23×D9711yescharacteristic
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—972no—
149C2×C4—974no—
150C2×C4—19416no—
151C2×C4—1944no—
152C2×C4—1944no—
153C2×C4—1944no—
154C2×C4—1941no—
155C2×C4—1942no—
156C2×C4—1942no—
157C2×C4—1941no—
158C2×C4—1948no—
159C2×C4C2×C22⋊D9714yes—
160C2×C4C2×C22⋊D9712yes—
161C2×C4Q8×D9712yes—
162C2×C4#1309551212yes—
163C22—974no—
164C22—1942no—
165C22—1944no—
166C22—1944no—
167C22—1942no—
168C22C22×C22⋊D9711yescharacteristic
169C22C2×Q8×D9711yescharacteristic
170C22#41×D9712yes—
171C22#41×D9712yes—
172C22C2×#1309551211yescharacteristic
173C4—24no—
174C4—44no—
175C4—3884no—
176C4—1942no—
177C4—1944no—
178C4#1569078614yes—
179C2—1944no—
180C2C2×#1569078611yescharacteristic
181C2#2026124212yes—
182C2#125×D9711yescharacteristic
183C2#2026127012yes—
184C2#2026129111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC2×#125×C97C2
prime radicalC23×D97C23
hypercenterC2×#125D97
hypocenterC97#2851
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #20260631C2semidirect
3. #20260616C2semidirect
5. #20260601C2semidirect
7. C2×#125×C97C2semidirect
79. C2×D97#125semidirect
93. C97#2851semidirect
94. C2×#125D97semidirect

Maximal subgroups

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

subgroupindex
C2×C42×D972
#202606312
#202606162
#202606062
#202606012
#202605922
C2×#125×C972
#285197