Finite Group Databasebeta

← C2×#20261392 #27155200

Subgroups — C2×#20261392 #27155200

Order 12416.

1620 conjugacy classes of subgroups; only the 573 normal subgroups are shown.

normal subgroups only

This group has 1620 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×Q8×D97C211yesmaximal, characteristic, complemented
2#20261392C2116yesmaximal, direct product factor
3C2×#15690797C213yesmaximal, complemented
4C2×#15690789C213yesmaximal, complemented
5C2×#15690781C213yesmaximal, complemented
6C2×Q8×Dic97C211yesmaximal, characteristic, complemented
7C2×C4×C22⋊D97C213yesmaximal, complemented
8C22×#41×C97C211yesmaximal, characteristic, complemented
9C22×Dic194C2213yes—
10C2×Q8×D97—28no—
11C2×Q8×D97C2218yescomplemented
12C22×C22⋊D97C2213yes—
13C22×C4×D97C2213yes—
14#15690797C22124yescomplemented
15#15690789C22124yescomplemented
16#15690781C22124yescomplemented
17Q8×Dic97C2218yescomplemented
18C4×C22⋊D97C22124yescomplemented
19C2×C4⋊Dic97C2213yes—
20C2×#13095490C2213yes—
21C2×C22⋊Dic97C2213yes—
22C2×C22⋊Dic97C2216yes—
23C2×#13095488C2211yescharacteristic
24C2×C4×Dic97C2213yes—
25C2×#41×C97—28no—
26C2×#41×C97C2218yescomplemented
27C22×Q8×C97C2211yescharacteristic
28C22×D4×C97C2213yes—
29C23×C4×C97C2213yes—
30C23×D97C2311yescharacteristic
31C2×Dic194—212no—
32C2×Dic194C23112yes—
33Q8×D97—232no—
34C2×C22⋊D97—212no—
35C2×C22⋊D97C23112yes—
36C22×Dic97C2311yescharacteristic
37C22×Dic97C2313yes—
38C22×Dic97C2313yes—
39C2×C4×D97—212no—
40C2×C4×D97C23112yes—
41C4⋊Dic97C23112yes—
42#13095490C23112yes—
43C22⋊Dic97C23112yes—
44C22⋊Dic97C23124yes—
45#13095488C2314yes—
46C4×Dic97C23112yes—
47C24×C97C2313yes—
48#41×C97—232no—
49C2×Q8×C97D418yescomplemented
50C2×Q8×C97C2314yes—
51C2×D4×C97—212no—
52C2×D4×C97C23112yes—
53C22×C4×C97—212no—
54C22×C4×C97C2311yescharacteristic
55C22×C4×C97C23112yes—
56C22×C4×C97C2313yes—
57C23×C97—26no—
58C23×C97—26no—
59C23×C97—26no—
60C23×C97C2416yes—
61C23×C97C2411yescharacteristic
62C22×D97—22no—
63C22×D97—22no—
64C22×D97#4114yescomplemented
65C22×D97C2412yes—
66Q8×C97C2×D4116yescomplemented
67D4×C97—224no—
68C2×C4×C97—22no—
69C2×C4×C97—224no—
70C2×C4×C97C2×D4112yes—
71C2×C4×C97C2412yes—
72C2×C4×C97C2416yes—
73C22⋊D97—224no—
74C2×Dic97—26no—
75C2×Dic97—26no—
76C2×Dic97—22no—
77C2×Dic97C2412yes—
78C2×Dic97C2416yes—
79C2×Dic97C2416yes—
80C4×D97—224no—
81Dic194—224no—
82C22×C97—26no—
83C22×C97—26no—
84C22×C97—26no—
85C22×C97—412no—
86C22×C97C22×D412yes—
87C22×C97C2×#4112yes—
88C22×C972−1+412yes—
89C22×C97C2511yescharacteristic
90C2×D97—28no—
91C2×D97—22no—
92C2×D97—22no—
93C2×D97C2×#4114yescomplemented
94C4×C97—22no—
95C4×C97C22×D4112yes—
96Dic97—26no—
97Dic97—26no—
98Dic97—22no—
99C2×C97—46no—
100C2×C97#54814yescomplemented
101C2×C97C23×D411yescharacteristic
102C2×C97C22×#4111yescharacteristic
103C2×C97C2×2−1+411yescharacteristic
104D97—24no—
105C2×#548—971nomaximal, Sylow
106C97C2×#54811yescharacteristic, complemented, Sylow
107C22×#41D9711yescharacteristic, complemented
108C23×Q8—971no—
109#548—9716no—
110C2×#124—973no—
111C2×C22⋊Q8—973no—
112C2×#121—973no—
113C2×C4×Q8—971no—
114C2×C4×D4—973no—
115C2×#41—28no—
116C2×#41C2×D9718yescomplemented
117C22×Q8—978no—
118C22×Q8—973no—
119C22×Q8—1948no—
120C22×Q8C2×D9711yescharacteristic
121C22×D4—973no—
122C22×D4C2×D9713yes—
123C23×C4—973no—
124C23×C4C2×D9713yes—
125#124—9724no—
126C22⋊Q8—9724no—
127#121—9724no—
128C4×Q8—978no—
129C4×D4—9724no—
130C2×C4⋊C4—973no—
131C2×C4⋊C4—973no—
132C2×C22⋊C4—973no—
133C2×C22⋊C4—976no—
134C2×C22⋊C4—971no—
135C2×C42—973no—
136C24—971no—
137C24C22×D9713yes—
138#41—232no—
139C2×Q8—9712no—
140C2×Q8—19432no—
141C2×Q8—19412no—
142C2×Q8C22⋊D9718yescomplemented
143C2×Q8C22×D9714yes—
144C2×D4—212no—
145C2×D4—9712no—
146C2×D4—19412no—
147C2×D4C22×D97112yes—
148C22×C4—212no—
149C22×C4—971no—
150C22×C4—973no—
151C22×C4—973no—
152C22×C4—9712no—
153C22×C4—19412no—
154C22×C4C22×D97112yes—
155C22×C4C22×D9711yescharacteristic
156C22×C4C22×D9713yes—
157C4⋊C4—9712no—
158C4⋊C4—9712no—
159C22⋊C4—9712no—
160C22⋊C4—9724no—
161C22⋊C4—974no—
162C42—9712no—
163C23—26no—
164C23—26no—
165C23—26no—
166C23—974no—
167C23—972no—
168C23—1942no—
169C23—1942no—
170C23C23×D9716yes—
171C23C23×D9711yescharacteristic
172Q8—19424no—
173Q8C2×C22⋊D97116yescomplemented
174D4—224no—
175D4—19424no—
176C2×C4—22no—
177C2×C4—224no—
178C2×C4—972no—
179C2×C4—976no—
180C2×C4—976no—
181C2×C4—1946no—
182C2×C4—1946no—
183C2×C4—1942no—
184C2×C4—19424no—
185C2×C4C2×C22⋊D97112yes—
186C2×C4C23×D9712yes—
187C2×C4C23×D9716yes—
188C22—26no—
189C22—26no—
190C22—26no—
191C22—412no—
192C22—974no—
193C22—1948no—
194C22—1942no—
195C22—1942no—
196C22C22×C22⋊D9712yes—
197C22#41×D9712yes—
198C22#1569085712yes—
199C22C24×D9711yescharacteristic
200C4—22no—
201C4—1946no—
202C4—1946no—
203C4—1942no—
204C4C22×C22⋊D97112yes—
205C2—46no—
206C2—1944no—
207C2#2026139214yesdirect product factor
208C2C23×C22⋊D9711yescharacteristic
209C2C2×#41×D9711yescharacteristic
210C2C2×#1569085711yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC22×#41×C97C2
socleC23×C97C24
hypercenterC22×#41D97
hypocenterC97C2×#548
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D97C2semidirect
2. #20261392C2direct (N × H)
3. C2×#15690797C2semidirect
4. C2×#15690789C2semidirect
5. C2×#15690781C2semidirect
6. C2×Q8×Dic97C2semidirect
7. C2×C4×C22⋊D97C2semidirect
8. C22×#41×C97C2semidirect
11. C2×Q8×D97C22semidirect
14. #15690797C22semidirect
15. #15690789C22semidirect
16. #15690781C22semidirect
17. Q8×Dic97C22semidirect
18. C4×C22⋊D97C22semidirect
26. C2×#41×C97C22semidirect
49. C2×Q8×C97D4semidirect
64. C22×D97#41semidirect
66. Q8×C97C2×D4semidirect
93. C2×D97C2×#41semidirect
100. C2×C97#548semidirect
106. C97C2×#548semidirect
107. C22×#41D97semidirect
116. C2×#41C2×D97semidirect
142. C2×Q8C22⋊D97semidirect
173. Q8C2×C22⋊D97semidirect
207. C2#20261392direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D972
#202613922
C2×#156907972
C2×#156907892
C2×#156907812
C2×Q8×Dic972
C2×C4×C22⋊D972
C22×#41×C972
C2×#54897