Finite Group Databasebeta

← #27142175

Subgroups — #27142175

Order 12416.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C22×#13095495C211yesmaximal, characteristic, complemented
2C2×#15690731C212yesmaximal, complemented
3#20260595C211yesmaximal, characteristic, complemented
4#20260497C212yesmaximal, complemented
5C2×#121×C97C211yesmaximal, characteristic
6C2×#13095495—24no—
7C2×#13095495—22no—
8C22×#11715895C2212yes—
9C2×C4⋊Dic97C2212yes—
10C2×C22⋊Dic97—21no—
11#15690731—24no—
12#15690717—21no—
13C22×D4×C97—21no—
14C22×D4×C97C2211yescharacteristic
15C23×C4×C97C2211yescharacteristic
16#121×C97—22no—
17#121×C97C414yescomplemented
18C2×C4⋊C4×C97C2211yescharacteristic
19C2×C22⋊C4×C97—21no—
20C22×Dic97—22no—
21#13095495—28no—
22C2×#11715895—22no—
23C2×#11715895—22no—
24C2×#11715895D414yescomplemented
25C4⋊Dic97—22no—
26C4⋊Dic97—22no—
27C22⋊Dic97—42no—
28C24×C97—21no—
29C24×C97D411yescharacteristic
30C2×D4×C97—24no—
31C2×D4×C97—22no—
32C2×D4×C97—21no—
33C2×D4×C97—22no—
34C2×D4×C97—44no—
35C2×D4×C97C2×C412yes—
36C22×C4×C97—22no—
37C22×C4×C97—22no—
38C22×C4×C97—21no—
39C22×C4×C97C2×C412yes—
40C22×C4×C97D411yescharacteristic
41C22×C4×C97D412yes—
42C22×C4×C97C2311yescharacteristic
43C4⋊C4×C97—21no—
44C4⋊C4×C97C2×C412yes—
45C22⋊C4×C97—24no—
46C23×C97—22no—
47C23×C97—22no—
48C23×C97—21no—
49C23×C97—21no—
50C23×C97—22no—
51C23×C97—42no—
52C23×C97—42no—
53C23×C97—41no—
54C23×C97—41no—
55C23×C97C22⋊C412yes—
56C23×C97C2×D411yescharacteristic
57D4×C97—24no—
58D4×C97—48no—
59D4×C97—41no—
60D4×C97—41no—
61C2×C4×C97—22no—
62C2×C4×C97—24no—
63C2×C4×C97—22no—
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—42no—
67C2×C4×C97C22⋊C412yes—
68C2×C4×C97C22×C411yescharacteristic
69C2×C4×C97C2×D412yes—
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97#4112yes—
72C2×Dic97—42no—
73C2×Dic97—42no—
74C2×Dic97—42no—
75#11715895—24no—
76C22×C97—22no—
77C22×C97—22no—
78C22×C97—22no—
79C22×C97—42no—
80C22×C97—42no—
81C22×C97—42no—
82C22×C97—42no—
83C22×C97—42no—
84C22×C97—82no—
85C22×C97—82no—
86C22×C97C2×C22⋊C411yescharacteristic
87C22×C97#12112yes—
88C22×C97AGL(1,ℤ/8)11yescharacteristic
89C22×C97AGL(1,ℤ/8)12yes—
90C22×C97#13711yescharacteristic
91C4×C97—22no—
92C4×C97—42no—
93C4×C97C4×D412yes—
94C4×C97#12011yescharacteristic
95C4×C97#12311yescharacteristic
96Dic97—82no—
97C2×C97—42no—
98C2×C97—82no—
99C2×C97#38611yescharacteristic
100C2×C97#41711yescharacteristic
101C2×C97#41811yescharacteristic
102C2×C97#46812yes—
103C2×C97#46912yes—
104#1895—971nomaximal, Sylow
105C97#189511yescharacteristic, complemented, Sylow
106C22×M16—971no—
107C2×#121D9711yescharacteristic
108C2×#102—972no—
109#380—971no—
110#340—972no—
111C22×D4—21no—
112C22×D4C2×D9711yescharacteristic
113C23×C4C2×D9711yescharacteristic
114C2×M16—1944no—
115C2×M16—1942no—
116C22×C8—972no—
117#121—22no—
118#121Dic9714yescomplemented
119C2×C4⋊C4—972no—
120C2×C4⋊C4C2×D9711yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—1941no—
123#102—1944no—
124#95—1941no—
125C24—21no—
126C24C22⋊D9711yescharacteristic
127C2×D4—24no—
128C2×D4—22no—
129C2×D4—21no—
130C2×D4—22no—
131C2×D4—44no—
132C2×D4C2×Dic9712yes—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—1942no—
137C22×C4C2×Dic9712yes—
138C22×C4C22⋊D9711yescharacteristic
139C22×C4C22⋊D9712yes—
140C22×C4C22×D9711yescharacteristic
141M16—1948no—
142C2×C8—974no—
143C2×C8—1942no—
144C2×C8—1942no—
145C4⋊C4—21no—
146C4⋊C4—1942no—
147C4⋊C4—1942no—
148C4⋊C4C2×Dic9712yes—
149C22⋊C4—24no—
150C22⋊C4—3882no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—22no—
156C23—42no—
157C23—41no—
158C23—42no—
159C23—41no—
160C23C22⋊Dic9712yes—
161C23C2×C22⋊D9711yescharacteristic
162D4—24no—
163D4—48no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—3882no—
173C2×C4—3882no—
174C2×C4—3882no—
175C2×C4C22⋊Dic9712yes—
176C2×C4C22×Dic9711yescharacteristic
177C2×C4D4×D9712yes—
178C2×C4C2×C22⋊D9711yescharacteristic
179C2×C4#1309551012yes—
180C8—1944no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—42no—
189C22—82no—
190C22—82no—
191C22C2×C22⋊Dic9711yescharacteristic
192C22#1569078212yes—
193C22#1569083511yescharacteristic
194C22#1569083612yes—
195C22#1569084011yescharacteristic
196C4—22no—
197C4—42no—
198C4—7762no—
199C4D4×Dic9712yes—
200C4#1569077811yescharacteristic
201C4#1569079111yescharacteristic
202C2—42no—
203C2—82no—
204C2#2026061511yescharacteristic
205C2#2026073611yescharacteristic
206C2#2026074011yescharacteristic
207C2#2026097912yes—
208C2#2026098612yes—
subgroupG/H
centerC23C2×C22⋊D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radicalC2×#121×C97C2
socleC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenterC2×#121D97
hypocenterC97#1895
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#13095495C2semidirect
2. C2×#15690731C2semidirect
3. #20260595C2semidirect
4. #20260497C2semidirect
17. #121×C97C4semidirect
24. C2×#11715895D4semidirect
105. C97#1895semidirect
118. #121Dic97semidirect

Maximal subgroups

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

subgroupindex
C22×#130954952
C2×#156907312
#202605952
#202604972
C2×#121×C972
#189597