Finite Group Databasebeta

← C2×#20261404 #27155214

Subgroups — C2×#20261404 #27155214

Order 12416.

1736 conjugacy classes of subgroups; only the 573 normal subgroups are shown.

normal subgroups only

This group has 1736 conjugacy classes of subgroups — too many to draw. Only its 573 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
1C22×#13095512C211yesmaximal, characteristic, complemented
2#20261404C2116yesmaximal, direct product factor
3C2×#15690806C213yesmaximal, complemented
4C2×#15690784C213yesmaximal, complemented
5C2×#15690781C213yesmaximal, complemented
6C2×Q8×Dic97C211yesmaximal, characteristic, complemented
7C2×C4×C22⋊D97C213yesmaximal, complemented
8C22×#41×C97C211yesmaximal, characteristic, complemented
9C2×#13095512—28no—
10C2×#13095512C2218yescomplemented
11C22×D388C2213yes—
12C22×C22⋊D97C2213yes—
13C22×C22⋊D97C2216yes—
14C22×C4×D97C2213yes—
15#15690806C22124yescomplemented
16#15690784C22124yescomplemented
17#15690781C22124yescomplemented
18Q8×Dic97C2218yescomplemented
19C4×C22⋊D97C22124yescomplemented
20C2×C4⋊Dic97C2213yes—
21C2×#13095490C2211yescharacteristic
22C2×C22⋊Dic97C2213yes—
23C2×#13095488C2213yes—
24C2×C4×Dic97C2213yes—
25C2×#41×C97—28no—
26C2×#41×C97C2218yescomplemented
27C22×Q8×C97C2211yescharacteristic
28C22×D4×C97C2213yes—
29C23×C4×C97C2213yes—
30C23×D97C2313yes—
31#13095512—232no—
32C2×D388—212no—
33C2×D388C23112yes—
34C2×C22⋊D97—212no—
35C2×C22⋊D97—412no—
36C2×C22⋊D97C23112yes—
37C2×C22⋊D97C23124yes—
38C22×Dic97C2313yes—
39C22×Dic97C2311yescharacteristic
40C22×Dic97C2311yescharacteristic
41C2×C4×D97—212no—
42C2×C4×D97C23112yes—
43C4⋊Dic97C23112yes—
44#13095490C2314yes—
45C22⋊Dic97C23112yes—
46#13095488C23112yes—
47C4×Dic97C23112yes—
48C24×C97C2313yes—
49#41×C97—232no—
50C2×Q8×C97D418yescomplemented
51C2×Q8×C97C2314yes—
52C2×D4×C97—212no—
53C2×D4×C97C23112yes—
54C22×C4×C97—212no—
55C22×C4×C97C2311yescharacteristic
56C22×C4×C97C23112yes—
57C22×C4×C97C2313yes—
58C23×C97—26no—
59C23×C97—26no—
60C23×C97—26no—
61C23×C97C2416yes—
62C23×C97C2411yescharacteristic
63C22×D97—26no—
64C22×D97—26no—
65C22×D97—26no—
66C22×D97C2416yes—
67Q8×C97C2×D4116yescomplemented
68D4×C97—224no—
69C2×C4×C97—22no—
70C2×C4×C97—224no—
71C2×C4×C97C2×D4112yes—
72C2×C4×C97C2412yes—
73C2×C4×C97C2416yes—
74C22⋊D97—224no—
75C22⋊D97—424no—
76C2×Dic97—22no—
77C2×Dic97—26no—
78C2×Dic97#4114yescomplemented
79C2×Dic97C2416yes—
80C2×Dic97C2412yes—
81C2×Dic97C2412yes—
82D388—224no—
83C4×D97—224no—
84C22×C97—26no—
85C22×C97—26no—
86C22×C97—26no—
87C22×C97—412no—
88C22×C97C22×D412yes—
89C22×C97C2×#4112yes—
90C22×C97H(5,𝔽2)12yes—
91C22×C97C2511yescharacteristic
92C2×D97—26no—
93C2×D97—26no—
94C2×D97—26no—
95C2×D97—412no—
96C4×C97—22no—
97C4×C97C22×D4112yes—
98Dic97—22no—
99Dic97—26no—
100Dic97C2×#4114yescomplemented
101C2×C97—46no—
102C2×C97#55014yescomplemented
103C2×C97C23×D411yescharacteristic
104C2×C97C22×#4111yescharacteristic
105C2×C97C2×H(5,𝔽2)11yescharacteristic
106D97—46no—
107C2×#550—971nomaximal, Sylow
108C97C2×#55011yescharacteristic, complemented, Sylow
109C22×#41—971no—
110C22×#41D9711yescharacteristic, complemented
111#550—9716no—
112C2×#127—973no—
113C2×#121—973no—
114C2×#121—973no—
115C2×C4×Q8—971no—
116C2×C4×D4—973no—
117C2×#41—28no—
118C2×#41—978no—
119C2×#41—1948no—
120C2×#41C2×D9718yescomplemented
121C22×Q8C2×D9711yescharacteristic
122C22×D4—976no—
123C22×D4—973no—
124C22×D4—973no—
125C22×D4C2×D9713yes—
126C23×C4—973no—
127C23×C4C2×D9713yes—
128#127—9724no—
129#121—9724no—
130#121—9724no—
131C4×Q8—978no—
132C4×D4—9724no—
133C2×C4⋊C4—971no—
134C2×C4⋊C4—973no—
135C2×C22⋊C4—973no—
136C2×C22⋊C4—973no—
137C2×C42—973no—
138C24—973no—
139C24C22×D9713yes—
140#41—232no—
141#41—19432no—
142C2×Q8C22⋊D9718yescomplemented
143C2×Q8C22×D9714yes—
144C2×D4—212no—
145C2×D4—9724no—
146C2×D4—9712no—
147C2×D4—9712no—
148C2×D4—38812no—
149C2×D4—19412no—
150C2×D4—19412no—
151C2×D4C22×D97112yes—
152C22×C4—212no—
153C22×C4—971no—
154C22×C4—973no—
155C22×C4—971no—
156C22×C4—9712no—
157C22×C4—19412no—
158C22×C4C22×D97112yes—
159C22×C4C22×D9711yescharacteristic
160C22×C4C22×D9713yes—
161C4⋊C4—974no—
162C4⋊C4—9712no—
163C22⋊C4—9712no—
164C22⋊C4—9712no—
165C42—9712no—
166C23—26no—
167C23—26no—
168C23—26no—
169C23—976no—
170C23—1946no—
171C23—1946no—
172C23—1946no—
173C23C23×D9716yes—
174C23C23×D9711yescharacteristic
175Q8C2×C22⋊D97116yescomplemented
176D4—224no—
177D4—38824no—
178D4—19424no—
179D4—19424no—
180C2×C4—22no—
181C2×C4—224no—
182C2×C4—974no—
183C2×C4—972no—
184C2×C4—976no—
185C2×C4—972no—
186C2×C4—1942no—
187C2×C4—1946no—
188C2×C4—19424no—
189C2×C4C2×C22⋊D97112yes—
190C2×C4C23×D9712yes—
191C2×C4C23×D9716yes—
192C22—26no—
193C22—26no—
194C22—26no—
195C22—412no—
196C22—38812no—
197C22—1946no—
198C22—1946no—
199C22—1946no—
200C22C22×C22⋊D9712yes—
201C22#41×D9712yes—
202C22#1569085612yes—
203C22C24×D9711yescharacteristic
204C4—22no—
205C4—974no—
206C4—1942no—
207C4—1946no—
208C4C22×C22⋊D97112yes—
209C2—46no—
210C2—3886no—
211C2#2026140414yesdirect product factor
212C2C23×C22⋊D9711yescharacteristic
213C2C2×#41×D9711yescharacteristic
214C2C2×#1569085611yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC22×#41×C97C2
socleC23×C97C24
hypercenterC22×#41D97
hypocenterC97C2×#550
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×#13095512C2semidirect
2. #20261404C2direct (N × H)
3. C2×#15690806C2semidirect
4. C2×#15690784C2semidirect
5. C2×#15690781C2semidirect
6. C2×Q8×Dic97C2semidirect
7. C2×C4×C22⋊D97C2semidirect
8. C22×#41×C97C2semidirect
10. C2×#13095512C22semidirect
15. #15690806C22semidirect
16. #15690784C22semidirect
17. #15690781C22semidirect
18. Q8×Dic97C22semidirect
19. C4×C22⋊D97C22semidirect
26. C2×#41×C97C22semidirect
50. C2×Q8×C97D4semidirect
67. Q8×C97C2×D4semidirect
78. C2×Dic97#41semidirect
100. Dic97C2×#41semidirect
102. C2×C97#550semidirect
108. C97C2×#550semidirect
110. C22×#41D97semidirect
120. C2×#41C2×D97semidirect
142. C2×Q8C22⋊D97semidirect
175. Q8C2×C22⋊D97semidirect
211. C2#20261404direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955122
#202614042
C2×#156908062
C2×#156907842
C2×#156907812
C2×Q8×Dic972
C2×C4×C22⋊D972
C22×#41×C972
C2×#55097