Finite Group Databasebeta

← #11739641

Subgroups — #11739641

Order 896.

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

normal subgroups only

This group has 624 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
1C22×C7×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#6492C212yesmaximal
3C2×C4×C22⋊D7C211yesmaximal, characteristic
4#89984C211yesmaximal, characteristic, complemented
5#89983C212yesmaximal, complemented
6#89765C212yesmaximal, complemented
7#89759C212yesmaximal, complemented
8#89752C212yesmaximal, complemented
9#89745C212yesmaximal, complemented
10C23×C4×C7C2212yes—
11C23×C4×C7C2211yescharacteristic
12C22×C22⋊D7C2211yescharacteristic
13C22×C4×D7C2211yescharacteristic
14C2×C7×C4⋊C4—24no—
15C2×C7×C4⋊C4C2212yes—
16C2×C7×C4⋊C4C2212yes—
17C2×C22⋊Dic7C2211yescharacteristic
18C2×C22⋊Dic7C2212yes—
19#6492—28no—
20C4×C22⋊D7—24no—
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2214yes—
23C2×C4⋊Dic7C2212yes—
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2214yes—
26C2×C4×Dic7C2211yescharacteristic
27C7⋊#95C2214yes—
28C7⋊#95C2214yes—
29C7⋊#95C2212yes—
30#2486—71nomaximal, Sylow
31C24×C7C2311yescharacteristic
32C23×D7C2311yescharacteristic
33C22×C4×C7—24no—
34C22×C4×C7—24no—
35C22×C4×C7—24no—
36C22×C4×C7—24no—
37C22×C4×C7D414yes—
38C22×C4×C7C2314yes—
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2311yescharacteristic
41C2×C22⋊D7—22no—
42C2×C22⋊D7—22no—
43C2×C22⋊D7D414yes—
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2314yes—
47C22×Dic7C2311yescharacteristic
48C2×C4×D7—22no—
49C2×C4×D7—24no—
50C7×C4⋊C4—24no—
51C7×C4⋊C4—24no—
52C22⋊Dic7—22no—
53C22⋊Dic7—24no—
54(C2×D7)⋊C4—28no—
55(C2×D7)⋊C4—22no—
56C4⋊Dic7—24no—
57Dic7⋊C4—22no—
58Dic7⋊C4—28no—
59C4×Dic7—22no—
60C2×#123—72no—
61C2×C4×D4—71no—
62C22×C4⋊C4D711yescharacteristic, complemented
63#398—72no—
64#397—72no—
65#389—72no—
66#381—72no—
67#380—71no—
68#380—72no—
69C23×C7—22no—
70C23×C7—22no—
71C23×C7C2×D412yes—
72C23×C7C2×D412yes—
73C23×C7#4112yes—
74C23×C7C2411yescharacteristic
75C22×D7—21no—
76C22×D7—22no—
77C22×D7—21no—
78C22×D7—22no—
79C22×D7C2×D412yes—
80C2×C4×C7—22no—
81C2×C4×C7—28no—
82C2×C4×C7—24no—
83C2×C4×C7—24no—
84C2×C4×C7—24no—
85C2×C4×C7—48no—
86C2×C4×C7C2×D412yes—
87C2×C4×C7C2×D412yes—
88C2×C4×C7#4114yes—
89C22⋊D7—28no—
90C2×Dic7—21no—
91C2×Dic7—21no—
92C2×Dic7—21no—
93C2×Dic7—24no—
94C2×Dic7—21no—
95C2×Dic7—24no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—24no—
99C2×Dic7C2×D412yes—
100C4×D7—44no—
101C22×D4—71no—
102C23×C4—71no—
103C23×C4C2×D712yes—
104C23×C4C2×D711yescharacteristic
105#123—148no—
106C4×D4—144no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—71no—
109C2×C4⋊C4—74no—
110C2×C4⋊C4—72no—
111C2×C4⋊C4C2×D712yes—
112C2×C4⋊C4C2×D712yes—
113C2×C22⋊C4—71no—
114C2×C22⋊C4—72no—
115C2×C22⋊C4—71no—
116C2×C22⋊C4—74no—
117C2×C42—71no—
118#95—74no—
119#95—72no—
120#95—74no—
121C22×C7—22no—
122C22×C7—22no—
123C22×C7—24no—
124C22×C7—24no—
125C22×C7#12114yes—
126C22×C7C22×D411yescharacteristic
127C22×C7C22×D411yescharacteristic
128C22×C7C2×#4112yes—
129C22×C7C2×#4111yescharacteristic
130C22×C72−1+412yes—
131C2×D7—22no—
132C2×D7—24no—
133C2×D7—41no—
134C2×D7—41no—
135C2×D7—42no—
136C4×C7—22no—
137C4×C7—44no—
138C4×C7#12314yescomplemented
139Dic7—22no—
140Dic7—41no—
141Dic7—41no—
142Dic7—44no—
143C24—71no—
144C24C22×D711yescharacteristic
145C2×D4—74no—
146C2×D4—142no—
147C2×D4—142no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—71no—
153C22×C4—71no—
154C22×C4—71no—
155C22×C4—74no—
156C22×C4—144no—
157C22×C4—142no—
158C22×C4C22⋊D714yes—
159C22×C4C22×D714yes—
160C22×C4C22×D711yescharacteristic
161C22×C4C22×D711yescharacteristic
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—148no—
165C4⋊C4—144no—
166C4⋊C4—142no—
167C22⋊C4—148no—
168C22⋊C4—144no—
169C22⋊C4—142no—
170C22⋊C4—142no—
171C42—142no—
172C2×C7—24no—
173C2×C7C2×#12111yescharacteristic
174C2×C7C2×#12311yescharacteristic
175C2×C7#53611yescharacteristic
176C2×C7#54712yes—
177C2×C7#55212yes—
178D7—42no—
179C23—22no—
180C23—22no—
181C23—72no—
182C23—142no—
183C23—141no—
184C23—142no—
185C23—141no—
186C23D4×D712yes—
187C23#115912yes—
188C23C2×C22⋊D712yes—
189C23C23×D711yescharacteristic
190D4—148no—
191C2×C4—22no—
192C2×C4—28no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—48no—
197C2×C4—72no—
198C2×C4—141no—
199C2×C4—141no—
200C2×C4—144no—
201C2×C4—142no—
202C2×C4—144no—
203C2×C4—144no—
204C2×C4—142no—
205C2×C4—141no—
206C2×C4—141no—
207C2×C4—284no—
208C2×C4#115714yes—
209C2×C4C2×C22⋊D712yes—
210C2×C4C2×C22⋊D712yes—
211C7#248611yescharacteristic, complemented, Sylow
212C22—24no—
213C22—24no—
214C22—22no—
215C22—22no—
216C22—144no—
217C22—142no—
218C22—282no—
219C22—281no—
220C22—281no—
221C22#650114yes—
222C22C2×#115712yes—
223C22C2×D4×D711yescharacteristic
224C22C2×#115911yescharacteristic
225C22#655112yes—
226C22C22×C22⋊D711yescharacteristic
227C4—22no—
228C4—44no—
229C4—142no—
230C4—284no—
231C4—281no—
232C4—281no—
233C4#649214yescomplemented
234C2—24no—
235C2—282no—
236C2#9019812yes—
237C2#9020412yes—
238C2C2×#649211yescharacteristic
239C2C2×#650111yescharacteristic
240C2#9050711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC22×C7×C4⋊C4C2
prime radicalC22×C22⋊D7C22
hypercenterC22×C4⋊C4D7
hypocenterC7#2486
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C7×C4⋊C4C2semidirect
4. #89984C2semidirect
5. #89983C2semidirect
6. #89765C2semidirect
7. #89759C2semidirect
8. #89752C2semidirect
9. #89745C2semidirect
62. C22×C4⋊C4D7semidirect
138. C4×C7#123semidirect
211. C7#2486semidirect
233. C4#6492semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C4⋊C42
C2×#64922
C2×C4×C22⋊D72
#899842
#899832
#897652
#897592
#897522
#897452
#24867