Finite Group Databasebeta

← #27142162

Subgroups — #27142162

Order 12416.

432 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C23×#11715895C211yesmaximal, characteristic, complemented
2C2×#15690784C211yesmaximal, characteristic
3C2×#15690730C212yesmaximal, complemented
4#20260497C212yesmaximal, complemented
5#380×C97C211yesmaximal, characteristic, complemented
6C22×D388C2211yescharacteristic
7C22×C22⋊D97—21no—
8C22×#11715895—24no—
9C22×#11715895—22no—
10C22×#11715895C2212yes—
11#15690784—22no—
12#15690784C414yescomplemented
13C2×C4⋊Dic97C2211yescharacteristic
14C2×#13095488—21no—
15#15690730—24no—
16C23×C4×C97C2211yescharacteristic
17C2×C4⋊C4×C97C2212yes—
18C2×C22⋊C4×C97—21no—
19#95×C97—21no—
20C23×D97—21no—
21C2×D388—22no—
22C2×D388—22no—
23C2×D388—21no—
24C2×D388C2×C412yes—
25C2×C22⋊D97—24no—
26C2×C22⋊D97—42no—
27C2×C22⋊D97—42no—
28C22×Dic97—21no—
29C2×#11715895—28no—
30C2×#11715895—22no—
31C2×#11715895—22no—
32C2×#11715895D414yescomplemented
33C4⋊Dic97—21no—
34C4⋊Dic97C2×C412yes—
35#13095488—24no—
36C24×C97D411yescharacteristic
37C22×C4×C97—22no—
38C22×C4×C97—22no—
39C22×C4×C97—22no—
40C22×C4×C97C2×C412yes—
41C22×C4×C97D412yes—
42C22×C4×C97D411yescharacteristic
43C22×C4×C97C2311yescharacteristic
44C4⋊C4×C97—22no—
45C4⋊C4×C97—22no—
46C22⋊C4×C97—42no—
47C23×C97—22no—
48C23×C97—22no—
49C23×C97C22⋊C412yes—
50C23×C97D812yes—
51C23×C97SD1612yes—
52C23×C97C2×D411yescharacteristic
53C22×D97—22no—
54C22×D97—42no—
55C22×D97—41no—
56C22×D97—41no—
57C22×D97—42no—
58C2×C4×C97—22no—
59C2×C4×C97—24no—
60C2×C4×C97—22no—
61C2×C4×C97—22no—
62C2×C4×C97—42no—
63C2×C4×C97—42no—
64C2×C4×C97—42no—
65C2×C4×C97C22⋊C412yes—
66C2×C4×C97C22×C411yescharacteristic
67C2×C4×C97C2×D412yes—
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97#4112yes—
70C22⋊D97—48no—
71C2×Dic97—22no—
72C2×Dic97—41no—
73C2×Dic97—41no—
74D388—24no—
75D388—41no—
76D388—41no—
77#11715895—24no—
78C22×C97—22no—
79C22×C97—22no—
80C22×C97—24no—
81C22×C97—24no—
82C22×C97#10214yescomplemented
83C22×C97C2×C22⋊C411yescharacteristic
84C22×C97#12112yes—
85C22×C97C2×D811yescharacteristic
86C22×C97C2×SD1611yescharacteristic
87C22×C97#13512yes—
88C2×D97—42no—
89C2×D97—42no—
90C2×D97—42no—
91C2×D97—82no—
92C2×D97—82no—
93C4×C97—22no—
94C4×C97—82no—
95C4×C97C4×D412yes—
96C4×C97#12011yescharacteristic
97C4×C97#12311yescharacteristic
98Dic97—42no—
99C2×C97—24no—
100C2×C97#38611yescharacteristic
101C2×C97C2×#10211yescharacteristic
102C2×C97#41611yescharacteristic
103C2×C97#46512yes—
104C2×C97#46612yes—
105D97—82no—
106#1893—971nomaximal, Sylow
107C97#189311yescharacteristic, complemented, Sylow
108C23×C8—971no—
109C2×#121—971no—
110C2×#102—972no—
111#380D9711yescharacteristic, complemented
112#340—972no—
113C22×D4—971no—
114C22×D4—1941no—
115C23×C4C2×D9711yescharacteristic
116C22×C8—972no—
117C22×C8—1944no—
118C22×C8—1942no—
119#121—974no—
120#121—1942no—
121C2×C4⋊C4—971no—
122C2×C4⋊C4C2×D9712yes—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—1941no—
125#102—1944no—
126#95—21no—
127C24—1941no—
128C24C22⋊D9711yescharacteristic
129C2×D4—972no—
130C2×D4—3882no—
131C2×D4—3882no—
132C2×D4—1944no—
133C2×D4—1942no—
134C2×D4—1941no—
135C2×D4—1942no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—1941no—
140C22×C4C4×D9712yes—
141C22×C4D38812yes—
142C22×C4C22⋊D9711yescharacteristic
143C22×C4C22×D9711yescharacteristic
144C2×C8—974no—
145C2×C8—1948no—
146C2×C8—1942no—
147C2×C8—1942no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—972no—
151C4⋊C4—1941no—
152C22⋊C4—42no—
153C22⋊C4—1944no—
154C23—22no—
155C23—22no—
156C23—3882no—
157C23—3881no—
158C23—3881no—
159C23—3882no—
160C23—1942no—
161C23#1309548812yes—
162C23#1309549612yes—
163C23#1309549812yes—
164C23C2×C22⋊D9711yescharacteristic
165D4—3888no—
166D4—3881no—
167D4—3881no—
168D4—1944no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—3881no—
177C2×C4—3881no—
178C2×C4—1942no—
179C2×C4#1309548812yes—
180C2×C4C2×C4×D9711yescharacteristic
181C2×C4D4×D9712yes—
182C2×C4C2×D38811yescharacteristic
183C2×C4#1309551012yes—
184C8—1944no—
185C22—22no—
186C22—24no—
187C22—24no—
188C22—22no—
189C22—7762no—
190C22—7762no—
191C22—3882no—
192C22—3882no—
193C22—3882no—
194C22#1569073014yescomplemented
195C22C2×#1309548811yescharacteristic
196C22#1569078212yes—
197C22C2×#1309549611yescharacteristic
198C22C2×#1309549811yescharacteristic
199C22#1569083012yes—
200C4—22no—
201C4—82no—
202C4—3882no—
203C4#1569077012yes—
204C4#1569078011yescharacteristic
205C4#1569079411yescharacteristic
206C2—24no—
207C2—7762no—
208C2#2026061211yescharacteristic
209C2C2×#1569073011yescharacteristic
210C2#2026072911yescharacteristic
211C2#2026096612yes—
212C2#2026096912yes—
subgroupG/H
centerC23C2×C22⋊D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#380×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690784C2
cosocleC22×C4×C97C23
hypercenter#380D97
hypocenterC97#1893
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×#11715895C2semidirect
3. C2×#15690730C2semidirect
4. #20260497C2semidirect
5. #380×C97C2semidirect
12. #15690784C4semidirect
32. C2×#11715895D4semidirect
82. C22×C97#102semidirect
107. C97#1893semidirect
111. #380D97semidirect
194. C22#15690730semidirect

Maximal subgroups

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

subgroupindex
C23×#117158952
C2×#156907842
C2×#156907302
#202604972
#380×C972
#189397