Finite Group Databasebeta

← C2×#20261067 #27151615

Subgroups — C2×#20261067 #27151615

Order 12416.

460 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15690790C211yesmaximal, characteristic
2C2×C4⋊C4×D97C211yesmaximal, characteristic
3#20261067C218yesmaximal, direct product factor
4C2×#15690741C211yesmaximal, characteristic, complemented
5C2×#15690735C211yesmaximal, characteristic, complemented
6C2×#15690733C211yesmaximal, characteristic, complemented
7C2×#15690722C211yesmaximal, characteristic
8C2×#106×C97C211yesmaximal, characteristic, complemented
9C22×Dic194C2211yescharacteristic
10C22×C4×D97—21no—
11C22×C4×D97C2211yescharacteristic
12C22×#11715895C2211yescharacteristic
13#15690790—22no—
14#15690790C2214yes—
15C2×C4⋊Dic97—21no—
16C2×C4⋊Dic97C2211yescharacteristic
17C2×#13095490—21no—
18C4⋊C4×D97—22no—
19C4⋊C4×D97C2214yes—
20C2×#13095488—21no—
21#15690741C2214yescomplemented
22#15690735C2214yescomplemented
23#15690733C2214yescomplemented
24#15690722C2214yes—
25C22×C8×C97C2211yescharacteristic
26C2×C4⋊C4×C97C2211yescharacteristic
27C2×C4⋊C4×C97C2211yescharacteristic
28#106×C97C2214yescomplemented
29C23×D97D411yescharacteristic
30C2×Dic194—22no—
31C2×Dic194—22no—
32C2×Dic194—21no—
33C2×Dic194C2312yes—
34C22×Dic97—21no—
35C22×Dic97—21no—
36C22×Dic97D411yescharacteristic
37C2×C4×D97—22no—
38C2×C4×D97—22no—
39C2×C4×D97—24no—
40C2×C4×D97—44no—
41C2×C4×D97D412yes—
42C2×C4×D97C2312yes—
43C2×#11715895—22no—
44C2×#11715895C2312yes—
45C4⋊Dic97—21no—
46C4⋊Dic97—24no—
47C4⋊Dic97C2312yes—
48#13095490—24no—
49#13095488—24no—
50C22×C4×C97—21no—
51C22×C4×C97—21no—
52C22×C4×C97C2311yescharacteristic
53C2×C8×C97—22no—
54C2×C8×C97C2312yes—
55C4⋊C4×C97—21no—
56C4⋊C4×C97—21no—
57C4⋊C4×C97C2312yes—
58C4⋊C4×C97C2312yes—
59C23×C97C2×D411yescharacteristic
60C22×D97—22no—
61C22×D97—22no—
62C22×D97SD1614yes—
63C22×D97C2×D412yes—
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—42no—
67C2×C4×C97—42no—
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97#4112yes—
70C2×C4×C97#4112yes—
71C2×C4×C97C2411yescharacteristic
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—42no—
76C2×Dic97—42no—
77C2×Dic97C2×D412yes—
78C4×D97—24no—
79C4×D97—22no—
80C4×D97—22no—
81C4×D97—48no—
82Dic194—22no—
83Dic194—22no—
84Dic194—42no—
85C8×C97—22no—
86#11715895—22no—
87C22×C97#12312yes—
88C22×C97C2×SD1612yes—
89C22×C97#13712yes—
90C22×C97C22×D411yescharacteristic
91C2×D97—22no—
92C2×D97—22no—
93C2×D97—28no—
94C2×D97C2×SD1614yes—
95C4×C97—42no—
96C4×C97—42no—
97C4×C97#12312yes—
98C4×C97C2×#4111yescharacteristic
99C4×C97C2×#4111yescharacteristic
100Dic97—22no—
101Dic97—42no—
102Dic97—42no—
103C2×C97#48314yescomplemented
104C2×C97C2×#12311yescharacteristic
105C2×C97C22×SD1611yescharacteristic
106C2×C97C2×#13711yescharacteristic
107D97—24no—
108C2×#483—971nomaximal, Sylow
109C97C2×#48311yescharacteristic, complemented, Sylow
110C2×C22⋊Q8—971no—
111C22×C4⋊C4—971no—
112#483—978no—
113C2×#106—971no—
114C2×#106D9711yescharacteristic, complemented
115C2×#103—971no—
116C2×#103—971no—
117C2×C22⋊C8—971no—
118C22×Q8—971no—
119C23×C4—971no—
120C23×C4—1941no—
121C22×C8—971no—
122C22×C8C2×D9711yescharacteristic
123C22⋊Q8—974no—
124C22⋊Q8—1942no—
125C2×C4⋊C4—971no—
126C2×C4⋊C4—974no—
127C2×C4⋊C4—1942no—
128C2×C4⋊C4—1941no—
129C2×C4⋊C4—1941no—
130C2×C4⋊C4C2×D9711yescharacteristic
131C2×C4⋊C4C2×D9711yescharacteristic
132C2×C22⋊C4—1941no—
133#106—974no—
134#106C2×D9714yescomplemented
135#103—974no—
136#103—974no—
137C22⋊C8—974no—
138C24—971no—
139C2×Q8—972no—
140C2×Q8—1942no—
141C2×Q8—1942no—
142C2×Q8—1941no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—971no—
146C22×C4—972no—
147C22×C4—972no—
148C22×C4—3884no—
149C22×C4—1944no—
150C22×C4—1941no—
151C22×C4—1941no—
152C22×C4—1942no—
153C22×C4—1942no—
154C22×C4C22×D9711yescharacteristic
155C2×C8—22no—
156C2×C8—972no—
157C2×C8—1942no—
158C2×C8C22×D9712yes—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—972no—
162C4⋊C4—1944no—
163C4⋊C4—1944no—
164C4⋊C4—1941no—
165C4⋊C4C22×D9712yes—
166C4⋊C4C22×D9712yes—
167C22⋊C4—1944no—
168C23—974no—
169C23—972no—
170C23—1942no—
171C23—1942no—
172C23D4×D9711yescharacteristic
173Q8—3882no—
174Q8—1942no—
175Q8—1942no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—972no—
181C2×C4—3888no—
182C2×C4—3882no—
183C2×C4—3882no—
184C2×C4—1942no—
185C2×C4—1942no—
186C2×C4—1942no—
187C2×C4—1944no—
188C2×C4—1942no—
189C2×C4—1942no—
190C2×C4D4×D9711yescharacteristic
191C2×C4#1309551112yes—
192C2×C4#1309551212yes—
193C2×C4C23×D9711yescharacteristic
194C8—22no—
195C8—1942no—
196C22—974no—
197C22—1942no—
198C22—1942no—
199C22—1948no—
200C22#1569079512yes—
201C22SD16×D9712yes—
202C22#1569083912yes—
203C22C2×D4×D9711yescharacteristic
204C4—42no—
205C4—42no—
206C4—3882no—
207C4—3882no—
208C4—1942no—
209C4#1569079512yes—
210C4C2×#1309551111yescharacteristic
211C4C2×#1309551211yescharacteristic
212C2—1944no—
213C2#2026106714yesdirect product factor
214C2C2×#1569079511yescharacteristic
215C2C2×SD16×D9711yescharacteristic
216C2C2×#1569083911yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#106×C97C2
socleC23×C97C2×D4
prime radicalC23×D97D4
hypercenterC2×#106D97
hypocenterC97C2×#483
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
3. #20261067C2direct (N × H)
4. C2×#15690741C2semidirect
5. C2×#15690735C2semidirect
6. C2×#15690733C2semidirect
8. C2×#106×C97C2semidirect
21. #15690741C22semidirect
22. #15690735C22semidirect
23. #15690733C22semidirect
28. #106×C97C22semidirect
103. C2×C97#483semidirect
109. C97C2×#483semidirect
114. C2×#106D97semidirect
134. #106C2×D97semidirect
213. C2#20261067direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156907902
C2×C4⋊C4×D972
#202610672
C2×#156907412
C2×#156907352
C2×#156907332
C2×#156907222
C2×#106×C972
C2×#48397