Finite Group Databasebeta

← #27150970

Subgroups — #27150970

Order 12416.

738 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 173 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×#15690841C212yesmaximal, complemented
2#20261246C211yesmaximal, characteristic
3#20260891C214yesmaximal, complemented
4#20260874C214yesmaximal, complemented
5#20260744C212yesmaximal, complemented
6#20260699C211yesmaximal, characteristic, complemented
7C23×Q8×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9#15690841—28no—
10C2×#13095501C2214yescomplemented
11C2×#13095498C2214yescomplemented
12C2×#13095495C2212yes—
13#15690793—21no—
14#15690789C2212yes—
15#15690784C2212yes—
16#15690779—21no—
17C4×C22⋊D97—22no—
18#15690769C2211yescharacteristic
19#15690734C2218yescomplemented
20C22⋊#11715895C2214yes—
21C22×Q8×C97—28no—
22C22×Q8×C97—22no—
23C22×Q8×C97—21no—
24C22×Q8×C97C2214yes—
25C22×Q8×C97C2212yes—
26C23×C4×C97—21no—
27C23×C4×C97C2211yescharacteristic
28#13095511—24no—
29C2×Dic194C2311yescharacteristic
30C2×D388C2311yescharacteristic
31C2×C22⋊D97—22no—
32C2×C22⋊D97—21no—
33C2×C4×D97—21no—
34#13095501—28no—
35#13095498—28no—
36#13095495—24no—
37C2×#11715895C2314yes—
38C4⋊Dic97C2312yes—
39#13095490—22no—
40C22⋊Dic97—21no—
41C22⋊Dic97—22no—
42#13095488—22no—
43C4×Dic97—21no—
44C24×C97D411yescharacteristic
45C2×Q8×C97—216no—
46C2×Q8×C97—28no—
47C2×Q8×C97—216no—
48C2×Q8×C97—24no—
49C2×Q8×C97—24no—
50C2×Q8×C97—22no—
51C2×Q8×C97D418yescomplemented
52C2×Q8×C97C2314yes—
53C22×C4×C97—28no—
54C22×C4×C97—24no—
55C22×C4×C97—22no—
56C22×C4×C97—24no—
57C22×C4×C97D412yes—
58C22×C4×C97D411yescharacteristic
59C22×C4×C97C2312yes—
60C22×C4×C97C2311yescharacteristic
61C23×C97—22no—
62C23×C97—24no—
63C23×C97C2×D411yescharacteristic
64C23×C97C2×D412yes—
65C22×D97—21no—
66Q8×C97—216no—
67Q8×C97—28no—
68Q8×C97—24no—
69Q8×C97C2×D418yes—
70C2×C4×C97—216no—
71C2×C4×C97—28no—
72C2×C4×C97—22no—
73C2×C4×C97—24no—
74C2×C4×C97—24no—
75C2×C4×C97—24no—
76C2×C4×C97C2×D412yes—
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D414yes—
79C2×C4×C97C2411yescharacteristic
80C22⋊D97—42no—
81C22⋊D97—44no—
82C2×Dic97—22no—
83C2×Dic97—21no—
84D388—22no—
85C4×D97—42no—
86Dic194—22no—
87#11715895—24no—
88C22×C97—28no—
89C22×C97—22no—
90C22×C97—22no—
91C22×C97—22no—
92C22×C97#12012yes—
93C22×C97#13714yescomplemented
94C22×C97C22×D411yescharacteristic
95C2×D97—41no—
96C2×D97—42no—
97C4×C97—28no—
98C4×C97—22no—
99C4×C97#12012yes—
100C4×C97C22×D412yes—
101Dic97—41no—
102Dic97—42no—
103C2×C97—22no—
104C2×C97—24no—
105C2×C97C2×#12011yescharacteristic
106C2×C97C2×#13712yes—
107D97—81no—
108#3004—971nomaximal, Sylow
109C97#300411yescharacteristic, complemented, Sylow
110C23×Q8D9711yescharacteristic, complemented
111C2×#137—972no—
112#525—971no—
113#451—974no—
114#448—974no—
115#419—972no—
116#407—971no—
117C2×#41—971no—
118C22×Q8—28no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8C2×D9714yes—
122C22×Q8C2×D9712yes—
123C23×C4—21no—
124C23×C4C2×D9711yescharacteristic
125#137—1948no—
126C2×Q16—974no—
127C2×SD16—974no—
128C2×M16—972no—
129#123—1941no—
130C22⋊Q8—972no—
131#121—972no—
132#120—1941no—
133C4×D4—1942no—
134#117—971no—
135#103—978no—
136C22⋊C8—974no—
137C24C22⋊D9711yescharacteristic
138#41—1944no—
139C2×Q8—216no—
140C2×Q8—28no—
141C2×Q8—216no—
142C2×Q8—24no—
143C2×Q8—24no—
144C2×Q8—22no—
145C2×Q8—971no—
146C2×Q8C22⋊D9718yescomplemented
147C2×Q8C22×D9714yes—
148C2×D4—971no—
149C2×D4—1942no—
150C2×D4—1941no—
151C22×C4—28no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—1941no—
156C22×C4C22⋊D9712yes—
157C22×C4C22⋊D9711yescharacteristic
158C22×C4C22×D9712yes—
159C22×C4C22×D9711yescharacteristic
160Q16—1948no—
161SD16—1948no—
162M16—1944no—
163C2×C8—974no—
164C4⋊C4—972no—
165C4⋊C4—1942no—
166C22⋊C4—1941no—
167C22⋊C4—1942no—
168C22⋊C4—1942no—
169C42—1941no—
170C23—22no—
171C23—24no—
172C23—1941no—
173C23C2×C22⋊D9711yescharacteristic
174C23C2×C22⋊D9712yes—
175Q8—216no—
176Q8—28no—
177Q8—24no—
178Q8—1942no—
179Q8C2×C22⋊D9718yes—
180D4—3882no—
181D4—3884no—
182D4—1942no—
183C2×C4—216no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—3882no—
190C2×C4—1942no—
191C2×C4—1941no—
192C2×C4C2×C22⋊D9712yes—
193C2×C4C2×C22⋊D9711yescharacteristic
194C2×C4C2×C22⋊D9714yes—
195C2×C4C23×D9711yescharacteristic
196C8—1944no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—3881no—
202C22—3882no—
203C22#1569077912yes—
204C22#1569084114yescomplemented
205C22C22×C22⋊D9711yescharacteristic
206C4—28no—
207C4—22no—
208C4—3881no—
209C4—3882no—
210C4#1569077912yes—
211C4C22×C22⋊D9712yes—
212C2—22no—
213C2—24no—
214C2—7761no—
215C2C2×#1569077911yescharacteristic
216C2C2×#1569084112yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC23×Q8×C97C2
socleC22×C97C22×D4
prime radical#20261246C2
hypercenterC23×Q8D97
hypocenterC97#3004
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15690841C2semidirect
3. #20260891C2semidirect
4. #20260874C2semidirect
5. #20260744C2semidirect
6. #20260699C2semidirect
7. C23×Q8×C97C2semidirect
10. C2×#13095501C22semidirect
11. C2×#13095498C22semidirect
19. #15690734C22semidirect
51. C2×Q8×C97D4semidirect
93. C22×C97#137semidirect
109. C97#3004semidirect
110. C23×Q8D97semidirect
146. C2×Q8C22⋊D97semidirect
204. C22#15690841semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
#202612462
#202608912
#202608742
#202607442
#202606992
C23×Q8×C972
#300497