Finite Group Databasebeta

← #27142174

Subgroups — #27142174

Order 12416.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#15690784C211yesmaximal, characteristic
2C2×#15690732C212yesmaximal, complemented
3#20260595C211yesmaximal, characteristic, complemented
4#20260498C212yesmaximal, complemented
5C22×M16×C97C211yesmaximal, characteristic, complemented
6C22×D388C2211yescharacteristic
7C22×C22⋊D97—21no—
8#15690784—22no—
9#15690784C414yescomplemented
10C2×C4⋊Dic97C2212yes—
11C2×C4⋊Dic97C2211yescharacteristic
12C2×C22⋊Dic97—21no—
13C2×#13095488—21no—
14#15690732—24no—
15#15690717—21no—
16C23×C4×C97C2211yescharacteristic
17C2×M16×C97—22no—
18C2×M16×C97—24no—
19C22×C8×C97C2212yes—
20C23×D97—21no—
21C2×D388—22no—
22C2×D388—22no—
23C2×D388—21no—
24C2×D388C2×C412yes—
25C2×C22⋊D97—24no—
26C2×C22⋊D97—44no—
27C22×Dic97—21no—
28C22×Dic97—22no—
29C4⋊Dic97—21no—
30C4⋊Dic97—22no—
31C4⋊Dic97—22no—
32C4⋊Dic97C2×C412yes—
33C22⋊Dic97—42no—
34#13095488—24no—
35C24×C97D411yescharacteristic
36C22×C4×C97—22no—
37C22×C4×C97—22no—
38C22×C4×C97C2×C412yes—
39C22×C4×C97D411yescharacteristic
40C22×C4×C97D412yes—
41C22×C4×C97C2311yescharacteristic
42M16×C97—28no—
43C2×C8×C97—22no—
44C2×C8×C97—22no—
45C2×C8×C97D414yescomplemented
46C23×C97—22no—
47C23×C97—21no—
48C23×C97—22no—
49C23×C97—21no—
50C23×C97C22⋊C412yes—
51C23×C97C2×D411yescharacteristic
52C22×D97—22no—
53C22×D97—42no—
54C22×D97—42no—
55C22×D97—41no—
56C22×D97—41no—
57C2×C4×C97—22no—
58C2×C4×C97—24no—
59C2×C4×C97—22no—
60C2×C4×C97—22no—
61C2×C4×C97C22⋊C412yes—
62C2×C4×C97C22×C411yescharacteristic
63C2×C4×C97C2×D412yes—
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97#4112yes—
66C22⋊D97—48no—
67C2×Dic97—22no—
68C2×Dic97—42no—
69C2×Dic97—42no—
70C2×Dic97—42no—
71C2×Dic97—42no—
72D388—24no—
73D388—41no—
74D388—41no—
75C8×C97—24no—
76C22×C97—22no—
77C22×C97—22no—
78C22×C97—22no—
79C22×C97—42no—
80C22×C97—42no—
81C22×C97C2×C22⋊C411yescharacteristic
82C22×C97#12112yes—
83C22×C97AGL(1,ℤ/8)11yescharacteristic
84C22×C97AGL(1,ℤ/8)12yes—
85C22×C97#13711yescharacteristic
86C2×D97—42no—
87C2×D97—42no—
88C2×D97—42no—
89C2×D97—82no—
90C2×D97—82no—
91C4×C97—22no—
92C4×C97C4×D412yes—
93C4×C97#12011yescharacteristic
94C4×C97#12311yescharacteristic
95Dic97—42no—
96Dic97—82no—
97C2×C97—42no—
98C2×C97#38611yescharacteristic
99C2×C97#41711yescharacteristic
100C2×C97#41811yescharacteristic
101C2×C97#46812yes—
102C2×C97#46912yes—
103D97—82no—
104#1895—971nomaximal, Sylow
105C97#189511yescharacteristic, complemented, Sylow
106C22×M16D9711yescharacteristic, complemented
107C2×#121—971no—
108C2×#102—972no—
109#380—971no—
110#340—972no—
111C22×D4—971no—
112C22×D4—1941no—
113C23×C4C2×D9711yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C2×D9712yes—
117#121—974no—
118#121—1942no—
119C2×C4⋊C4—971no—
120C2×C4⋊C4—972no—
121C2×C22⋊C4—1941no—
122C2×C22⋊C4—1941no—
123#102—1944no—
124#95—1941no—
125C24—1941no—
126C24D38811yescharacteristic
127C2×D4—972no—
128C2×D4—3884no—
129C2×D4—1944no—
130C2×D4—1941no—
131C2×D4—1942no—
132C2×D4—1942no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—1942no—
136C22×C4—1941no—
137C22×C4C4×D9712yes—
138C22×C4D38811yescharacteristic
139C22×C4C22⋊D9712yes—
140C22×C4C22×D9711yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D9714yescomplemented
145C4⋊C4—972no—
146C4⋊C4—1942no—
147C4⋊C4—1942no—
148C4⋊C4—1941no—
149C22⋊C4—3882no—
150C22⋊C4—1944no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—3882no—
156C23—3882no—
157C23—3881no—
158C23—3881no—
159C23—1942no—
160C23#1309548812yes—
161C23C2×D38811yescharacteristic
162D4—3888no—
163D4—3881no—
164D4—3881no—
165D4—1944no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—3882no—
171C2×C4—3882no—
172C2×C4—3882no—
173C2×C4—3882no—
174C2×C4—1942no—
175C2×C4#1309548812yes—
176C2×C4C2×C4×D9711yescharacteristic
177C2×C4C2×C22⋊D9712yes—
178C2×C4C2×C22⋊D9711yescharacteristic
179C2×C4#1309551112yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—7762no—
187C22—7762no—
188C22—3882no—
189C22—3882no—
190C22—3882no—
191C22C2×#1309548811yescharacteristic
192C22#1569078412yes—
193C22#1569083812yes—
194C22#1569083811yescharacteristic
195C22#1569084311yescharacteristic
196C4—22no—
197C4—7762no—
198C4—3882no—
199C4C4×C22⋊D9712yes—
200C4#1569077911yescharacteristic
201C4#1569079311yescharacteristic
202C2—42no—
203C2—7762no—
204C2#2026061411yescharacteristic
205C2#2026073711yescharacteristic
206C2#2026074111yescharacteristic
207C2#2026098112yes—
208C2#2026098812yes—
subgroupG/H
centerC23C2×D388
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC22×M16×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690784C2
cosocleC22×C4×C97C23
hypercenterC22×M16D97
hypocenterC97#1895
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#15690732C2semidirect
3. #20260595C2semidirect
4. #20260498C2semidirect
5. C22×M16×C97C2semidirect
9. #15690784C4semidirect
45. C2×C8×C97D4semidirect
105. C97#1895semidirect
106. C22×M16D97semidirect
144. C2×C8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907842
C2×#156907322
#202605952
#202604982
C22×M16×C972
#189597