Finite Group Databasebeta

← #27142182

Subgroups — #27142182

Order 12416.

358 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 89 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#20261246C211yesmaximal, characteristic
2C2×#15690769C211yesmaximal, characteristic, complemented
3C2×#15690736C212yesmaximal, complemented
4#20260496C212yesmaximal, complemented
5#407×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C2211yescharacteristic
7#15690793—21no—
8#15690789C412yescomplemented
9#15690784C412yescomplemented
10#15690779—21no—
11C4×C22⋊D97—22no—
12#15690769—24no—
13#15690769C2211yescharacteristic
14C2×C4⋊Dic97—21no—
15C2×C22⋊Dic97—21no—
16C2×C4×Dic97C2212yes—
17#15690736—24no—
18C23×C4×C97C2211yescharacteristic
19C2×M16×C97C2212yes—
20C22⋊C8×C97—22no—
21#13095511—22no—
22#13095511—22no—
23C2×Dic194C2×C411yescharacteristic
24C2×D388C2×C411yescharacteristic
25C2×C22⋊D97—22no—
26C2×C22⋊D97—21no—
27C22×Dic97—22no—
28C2×C4×D97—21no—
29C4⋊Dic97—24no—
30C4⋊Dic97C2×C412yes—
31#13095490—22no—
32C22⋊Dic97—24no—
33C22⋊Dic97—22no—
34C22⋊Dic97—21no—
35#13095488—22no—
36C4×Dic97—22no—
37C4×Dic97—21no—
38C4×Dic97D414yescomplemented
39C24×C97D411yescharacteristic
40C22×C4×C97—22no—
41C22×C4×C97—22no—
42C22×C4×C97C2×C412yes—
43C22×C4×C97D411yescharacteristic
44C22×C4×C97D412yes—
45C22×C4×C97C2311yescharacteristic
46M16×C97—22no—
47M16×C97—22no—
48C2×C8×C97—22no—
49C23×C97—21no—
50C23×C97—21no—
51C23×C97—42no—
52C23×C97C22⋊C412yes—
53C23×C97C2×D411yescharacteristic
54C22×D97—21no—
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—24no—
58C2×C4×C97—24no—
59C2×C4×C97C22⋊C412yes—
60C2×C4×C97C22×C411yescharacteristic
61C2×C4×C97C2×D412yes—
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97#4112yes—
64C22⋊D97—44no—
65C22⋊D97—41no—
66C22⋊D97—41no—
67C2×Dic97—22no—
68C2×Dic97—21no—
69C2×Dic97—24no—
70C2×Dic97—44no—
71D388—22no—
72C4×D97—42no—
73Dic194—22no—
74C8×C97—42no—
75C22×C97—22no—
76C22×C97—22no—
77C22×C97—22no—
78C22×C97—22no—
79C22×C97—42no—
80C22×C97—42no—
81C22×C97C2×C22⋊C411yescharacteristic
82C22×C97#12011yescharacteristic
83C22×C97#12311yescharacteristic
84C2×D97—41no—
85C2×D97—42no—
86C4×C97—22no—
87C4×C97C4×D412yes—
88C4×C97#12112yes—
89Dic97—44no—
90Dic97—41no—
91Dic97—42no—
92C2×C97—21no—
93C2×C97—21no—
94C2×C97—42no—
95C2×C97#38611yescharacteristic
96C2×C97#42112yes—
97D97—81no—
98#1897—971nomaximal, Sylow
99C97#189711yescharacteristic, complemented, Sylow
100#525—971no—
101C2×#117—971no—
102C2×#104—972no—
103#407D9711yescharacteristic, complemented
104#339—972no—
105C2×#41—971no—
106C23×C4C2×D9711yescharacteristic
107C2×M16C2×D9712yes—
108#123—1941no—
109C22⋊Q8—972no—
110#121—972no—
111#120—1941no—
112C4×D4—1942no—
113#117—971no—
114#117—1944no—
115C2×C4⋊C4—1941no—
116C2×C22⋊C4—1941no—
117C2×C42—972no—
118#104—1944no—
119C22⋊C8—22no—
120C24D38811yescharacteristic
121#41—1942no—
122#41—1942no—
123C2×Q8—971no—
124C2×D4—971no—
125C2×D4—1942no—
126C2×D4—1941no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—1942no—
130C22×C4—1941no—
131C22×C4C4×D9712yes—
132C22×C4D38811yescharacteristic
133C22×C4C22⋊D9712yes—
134C22×C4C22×D9711yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—972no—
139C4⋊C4—1944no—
140C4⋊C4—1942no—
141C22⋊C4—1944no—
142C22⋊C4—1942no—
143C22⋊C4—1941no—
144C22⋊C4—1942no—
145C42—974no—
146C42—1942no—
147C42—1941no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—1941no—
152C23#1309548812yes—
153C23C2×D38811yescharacteristic
154Q8—1942no—
155D4—3884no—
156D4—3881no—
157D4—3881no—
158D4—1942no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—3884no—
164C2×C4—3882no—
165C2×C4—1942no—
166C2×C4—1941no—
167C2×C4—1944no—
168C2×C4#1309548812yes—
169C2×C4C2×C4×D9711yescharacteristic
170C2×C4D4×D9712yes—
171C2×C4C2×C22⋊D9711yescharacteristic
172C2×C4#1309551012yes—
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—3881no—
181C22—3882no—
182C22C2×#1309548811yescharacteristic
183C22#1569078011yescharacteristic
184C22#1569079411yescharacteristic
185C4—22no—
186C4—3884no—
187C4—3881no—
188C4—3882no—
189C4#1569077012yes—
190C4#1569078212yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—7761no—
195C2#2026061211yescharacteristic
196C2#2026075112yes—
subgroupG/H
centerC2×C4C2×C22⋊D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#407×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261246C2
cosocleC22×C4×C97C23
hypercenter#407D97
hypocenterC97#1897
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#15690769C2semidirect
3. C2×#15690736C2semidirect
4. #20260496C2semidirect
5. #407×C97C2semidirect
8. #15690789C4semidirect
9. #15690784C4semidirect
38. C4×Dic97D4semidirect
99. C97#1897semidirect
103. #407D97semidirect

Maximal subgroups

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

subgroupindex
#202612462
C2×#156907692
C2×#156907362
#202604962
#407×C972
#189797