Finite Group Databasebeta

← C2×#90522 #11744338

Subgroups — C2×#90522 #11744338

Order 896.

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×C7×#41C211yesmaximal, characteristic, complemented
2C22×Q8×D7C211yesmaximal, characteristic, complemented
3#90522C2116yesmaximal, direct product factor
4C2×Q8×Dic7C211yesmaximal, characteristic, complemented
5C2×#6501C213yesmaximal, complemented
6C2×#6499C213yesmaximal, complemented
7C2×C4×C22⋊D7C213yesmaximal, complemented
8C2×#6487C213yesmaximal, complemented
9C2×C7×#41—28no—
10C2×C7×#41C2218yescomplemented
11C22×C7×Q8C2211yescharacteristic
12C22×C7×D4C2213yes—
13C23×C4×C7C2213yes—
14C22×C22⋊D7C2213yes—
15C2×Q8×D7—28no—
16C2×Q8×D7C2218yescomplemented
17C22×C4×D7C2213yes—
18C22×Dic14C2213yes—
19C2×C22⋊Dic7C2216yes—
20C2×C22⋊Dic7C2213yes—
21Q8×Dic7C2218yescomplemented
22#6501C22124yescomplemented
23#6499C22124yescomplemented
24C4×C22⋊D7C22124yescomplemented
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×C4⋊Dic7C2213yes—
27#6487C22124yescomplemented
28C2×Dic7⋊C4C2213yes—
29C2×C4×Dic7C2213yes—
30C2×#548—71nomaximal, Sylow
31C24×C7C2313yes—
32C23×D7C2311yescharacteristic
33C7×#41—232no—
34C2×C7×Q8D418yescomplemented
35C2×C7×Q8C2314yes—
36C2×C7×D4—212no—
37C2×C7×D4C23112yes—
38C22×C4×C7—212no—
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C23112yes—
41C22×C4×C7C2313yes—
42C2×C22⋊D7—212no—
43C2×C22⋊D7C23112yes—
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2313yes—
46C22×Dic7C2313yes—
47Q8×D7—232no—
48C2×C4×D7—212no—
49C2×C4×D7C23112yes—
50C2×Dic14—212no—
51C2×Dic14C23112yes—
52C22⋊Dic7C23124yes—
53C22⋊Dic7C23112yes—
54(C2×D7)⋊C4C2314yes—
55C4⋊Dic7C23112yes—
56Dic7⋊C4C23112yes—
57C4×Dic7C23112yes—
58C22×#41D711yescharacteristic, complemented
59C23×Q8—71no—
60#548—716no—
61C2×#124—73no—
62C2×C22⋊Q8—73no—
63C2×#121—73no—
64C2×C4×Q8—71no—
65C2×C4×D4—73no—
66C23×C7—26no—
67C23×C7—26no—
68C23×C7—26no—
69C23×C7C2416yes—
70C23×C7C2411yescharacteristic
71C22×D7—22no—
72C22×D7—22no—
73C22×D7#4114yescomplemented
74C22×D7C2412yes—
75C7×Q8C2×D4116yescomplemented
76C7×D4—224no—
77C2×C4×C7—22no—
78C2×C4×C7—224no—
79C2×C4×C7C2×D4112yes—
80C2×C4×C7C2412yes—
81C2×C4×C7C2416yes—
82C22⋊D7—224no—
83C2×Dic7—26no—
84C2×Dic7—26no—
85C2×Dic7—22no—
86C2×Dic7C2412yes—
87C2×Dic7C2416yes—
88C2×Dic7C2416yes—
89C4×D7—224no—
90Dic14—224no—
91C2×#41—28no—
92C2×#41C2×D718yescomplemented
93C22×Q8—78no—
94C22×Q8—73no—
95C22×Q8—148no—
96C22×Q8C2×D711yescharacteristic
97C22×D4—73no—
98C22×D4C2×D713yes—
99C23×C4—73no—
100C23×C4C2×D713yes—
101#124—724no—
102C22⋊Q8—724no—
103#121—724no—
104C4×Q8—78no—
105C4×D4—724no—
106C2×C4⋊C4—73no—
107C2×C4⋊C4—73no—
108C2×C22⋊C4—73no—
109C2×C22⋊C4—76no—
110C2×C22⋊C4—71no—
111C2×C42—73no—
112C22×C7—26no—
113C22×C7—26no—
114C22×C7—26no—
115C22×C7—412no—
116C22×C7C22×D412yes—
117C22×C7C2×#4112yes—
118C22×C72−1+412yes—
119C22×C7C2511yescharacteristic
120C2×D7—28no—
121C2×D7—22no—
122C2×D7—22no—
123C2×D7C2×#4114yescomplemented
124C4×C7—22no—
125C4×C7C22×D4112yes—
126Dic7—26no—
127Dic7—26no—
128Dic7—22no—
129C24—71no—
130C24C22×D713yes—
131#41—232no—
132C2×Q8—712no—
133C2×Q8—1432no—
134C2×Q8—1412no—
135C2×Q8C22⋊D718yescomplemented
136C2×Q8C22×D714yes—
137C2×D4—212no—
138C2×D4—712no—
139C2×D4—1412no—
140C2×D4C22×D7112yes—
141C22×C4—212no—
142C22×C4—71no—
143C22×C4—73no—
144C22×C4—73no—
145C22×C4—712no—
146C22×C4—1412no—
147C22×C4C22×D7112yes—
148C22×C4C22×D711yescharacteristic
149C22×C4C22×D713yes—
150C4⋊C4—712no—
151C4⋊C4—712no—
152C22⋊C4—712no—
153C22⋊C4—724no—
154C22⋊C4—74no—
155C42—712no—
156C2×C7—46no—
157C2×C7#54814yescomplemented
158C2×C7C23×D411yescharacteristic
159C2×C7C22×#4111yescharacteristic
160C2×C7C2×2−1+411yescharacteristic
161D7—24no—
162C23—26no—
163C23—26no—
164C23—26no—
165C23—74no—
166C23—72no—
167C23—142no—
168C23—142no—
169C23C23×D716yes—
170C23C23×D711yescharacteristic
171Q8—1424no—
172Q8C2×C22⋊D7116yescomplemented
173D4—224no—
174D4—1424no—
175C2×C4—22no—
176C2×C4—224no—
177C2×C4—72no—
178C2×C4—76no—
179C2×C4—76no—
180C2×C4—146no—
181C2×C4—142no—
182C2×C4—146no—
183C2×C4—1424no—
184C2×C4C2×C22⋊D7112yes—
185C2×C4C23×D712yes—
186C2×C4C23×D716yes—
187C7C2×#54811yescharacteristic, complemented, Sylow
188C22—26no—
189C22—26no—
190C22—26no—
191C22—412no—
192C22—74no—
193C22—148no—
194C22—142no—
195C22—142no—
196C22D7×#4112yes—
197C22#655412yes—
198C22C22×C22⋊D712yes—
199C22C24×D711yescharacteristic
200C4—22no—
201C4—146no—
202C4—142no—
203C4—146no—
204C4C22×C22⋊D7112yes—
205C2—46no—
206C2—144no—
207C2#9052214yesdirect product factor
208C2C2×D7×#4111yescharacteristic
209C2C2×#655411yescharacteristic
210C2C23×C22⋊D711yescharacteristic
subgroupG/H
centerC23C23×D7
commutator, cosocleC22×C7C25
FrattiniC22C24×D7
FittingC22×C7×#41C2
socleC23×C7C24
hypercenterC22×#41D7
hypocenterC7C2×#548
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×C7×#41C2semidirect
2. C22×Q8×D7C2semidirect
3. #90522C2direct (N × H)
4. C2×Q8×Dic7C2semidirect
5. C2×#6501C2semidirect
6. C2×#6499C2semidirect
7. C2×C4×C22⋊D7C2semidirect
8. C2×#6487C2semidirect
10. C2×C7×#41C22semidirect
16. C2×Q8×D7C22semidirect
21. Q8×Dic7C22semidirect
22. #6501C22semidirect
23. #6499C22semidirect
24. C4×C22⋊D7C22semidirect
27. #6487C22semidirect
34. C2×C7×Q8D4semidirect
58. C22×#41D7semidirect
73. C22×D7#41semidirect
75. C7×Q8C2×D4semidirect
92. C2×#41C2×D7semidirect
123. C2×D7C2×#41semidirect
135. C2×Q8C22⋊D7semidirect
157. C2×C7#548semidirect
172. Q8C2×C22⋊D7semidirect
187. C7C2×#548semidirect
207. C2#90522direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#412
C22×Q8×D72
#905222
C2×Q8×Dic72
C2×#65012
C2×#64992
C2×C4×C22⋊D72
C2×#64872
C2×#5487