Finite Group Databasebeta

← #27142177

Subgroups — #27142177

Order 12416.

442 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 97 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
2C22×C4×Dic97C211yesmaximal, characteristic, complemented
3C2×#15690736C212yesmaximal, complemented
4#20260496C212yesmaximal, complemented
5#407×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C2211yescharacteristic
7C23×Dic97—21no—
8#15690793—21no—
9#15690789C412yescomplemented
10#15690784C412yescomplemented
11#15690779—21no—
12C4×C22⋊D97—22no—
13#15690769C2211yescharacteristic
14C2×C4×Dic97—24no—
15C2×C4×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, complemented
24C2×D388C2×C411yescharacteristic, complemented
25C2×C22⋊D97—22no—
26C2×C22⋊D97—21no—
27C22×Dic97—24no—
28C22×Dic97—28no—
29C22×Dic97—22no—
30C2×C4×D97—21no—
31C4⋊Dic97C2×C412yescomplemented
32#13095490—22no—
33C22⋊Dic97—22no—
34C22⋊Dic97—21no—
35#13095488—22no—
36C4×Dic97—24no—
37C4×Dic97—22no—
38C4×Dic97—21no—
39C4×Dic97D414yescomplemented
40C24×C97D411yescharacteristic
41C22×C4×C97—22no—
42C22×C4×C97—22no—
43C22×C4×C97C2×C412yes—
44C22×C4×C97D411yescharacteristic
45C22×C4×C97D412yes—
46C22×C4×C97C2311yescharacteristic
47M16×C97—22no—
48M16×C97—22no—
49C2×C8×C97—22no—
50C23×C97—24no—
51C23×C97—21no—
52C23×C97—21no—
53C23×C97C22⋊C412yes—
54C23×C97C2×D411yescharacteristic
55C22×D97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97—24no—
59C2×C4×C97—24no—
60C2×C4×C97C22⋊C412yes—
61C2×C4×C97C22×C411yescharacteristic
62C2×C4×C97C2×D412yes—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97#4112yes—
65C22⋊D97—44no—
66C22⋊D97—41no—
67C22⋊D97—41no—
68C2×Dic97—216no—
69C2×Dic97—28no—
70C2×Dic97—24no—
71C2×Dic97—22no—
72C2×Dic97—21no—
73D388—22no—
74C4×D97—42no—
75Dic194—22no—
76C8×C97—42no—
77C22×C97—24no—
78C22×C97—24no—
79C22×C97—22no—
80C22×C97—22no—
81C22×C97—22no—
82C22×C97#10414yescomplemented
83C22×C97C2×C22⋊C411yescharacteristic
84C22×C97#12011yescharacteristic
85C22×C97#12311yescharacteristic
86C2×D97—41no—
87C2×D97—42no—
88C4×C97—22no—
89C4×C97C4×D412yes—
90C4×C97#12112yes—
91Dic97—28no—
92Dic97—41no—
93Dic97—42no—
94C2×C97—21no—
95C2×C97—24no—
96C2×C97—21no—
97C2×C97#38611yescharacteristic
98C2×C97C2×#10412yes—
99D97—81no—
100#1896—971nomaximal, Sylow
101C97#189611yescharacteristic, complemented, Sylow
102#525—971no—
103C22×C42—971no—
104C2×#104—972no—
105#407D9711yescharacteristic, complemented
106#339—972no—
107C2×#41—971no—
108C23×C4—1941no—
109C23×C4C2×D9711yescharacteristic
110C2×M16C2×D9712yes—
111#123—1941no—
112C22⋊Q8—972no—
113#121—972no—
114#120—1941no—
115C4×D4—1942no—
116#117—971no—
117C2×C42—972no—
118C2×C42—1944no—
119C2×C42—1941no—
120#104—1944no—
121C22⋊C8—22no—
122C24D38811yescharacteristic
123#41—1942no—
124#41—1942no—
125C2×Q8—971no—
126C2×D4—971no—
127C2×D4—1942no—
128C2×D4—1941no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—1941no—
132C22×C4—1944no—
133C22×C4—1948no—
134C22×C4—1942no—
135C22×C4C4×D9712yes—
136C22×C4D38811yescharacteristic
137C22×C4C22⋊D9712yes—
138C22×C4C22×D9711yescharacteristic
139M16—22no—
140M16—22no—
141C2×C8—22no—
142C4⋊C4—972no—
143C4⋊C4—1942no—
144C22⋊C4—1942no—
145C22⋊C4—1941no—
146C22⋊C4—1942no—
147C42—974no—
148C42—1944no—
149C42—1942no—
150C42—1941no—
151C23—21no—
152C23—21no—
153C23—24no—
154C23—1941no—
155C23#1309548812yes—
156C23C2×D38811yescharacteristic
157Q8—1942no—
158D4—3881no—
159D4—3881no—
160D4—3884no—
161D4—1942no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—3882no—
167C2×C4—1942no—
168C2×C4—1941no—
169C2×C4—19416no—
170C2×C4—1948no—
171C2×C4—1944no—
172C2×C4#1309548812yes—
173C2×C4C2×C4×D9711yescharacteristic
174C2×C4D4×D9712yes—
175C2×C4C2×C22⋊D9711yescharacteristic
176C2×C4#1309551012yes—
177C8—42no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—3881no—
184C22—3882no—
185C22#1569073614yescomplemented
186C22C2×#1309548811yescharacteristic
187C22#1569078011yescharacteristic
188C22#1569079411yescharacteristic
189C4—22no—
190C4—3881no—
191C4—3882no—
192C4—1948no—
193C4#1569077012yes—
194C4#1569078212yes—
195C2—21no—
196C2—21no—
197C2—24no—
198C2—7761no—
199C2#2026061211yescharacteristic
200C2C2×#1569073612yes—
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#1896
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. C22×C4×Dic97C2semidirect
3. C2×#15690736C2semidirect
4. #20260496C2semidirect
5. #407×C97C2semidirect
9. #15690789C4semidirect
10. #15690784C4semidirect
23. C2×Dic194C2×C4semidirect
24. C2×D388C2×C4semidirect
31. C4⋊Dic97C2×C4semidirect
39. C4×Dic97D4semidirect
82. C22×C97#104semidirect
101. C97#1896semidirect
105. #407D97semidirect
185. C22#15690736semidirect

Maximal subgroups

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

subgroupindex
#202612462
C22×C4×Dic972
C2×#156907362
#202604962
#407×C972
#189697