Finite Group Databasebeta

← #11734086

Subgroups — #11734086

Order 896.

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

normal subgroups only

This group has 528 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×#6509C211yesmaximal, characteristic
2C2×#6507C211yesmaximal, characteristic, complemented
3C2×#6460C211yesmaximal, characteristic
4C2×#6435C211yesmaximal, characteristic
5C7×#391C211yesmaximal, characteristic, complemented
6#89749C211yesmaximal, characteristic, complemented
7#89701C211yesmaximal, characteristic, complemented
8#89447C214yesmaximal
9#89429C214yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C4×D7C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C42×C7C2211yescharacteristic
15#6509—24no—
16#6507—24no—
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×C4⋊Dic7C2212yes—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2214yes—
25C2×C4×Dic7C2211yescharacteristic
26#6460—24no—
27#6435—24no—
28C7×#95C2214yes—
29C7⋊#95C2214yes—
30C7⋊#95C2214yes—
31#2621—71nomaximal, Sylow
32C23×D7C2311yescharacteristic
33C2×C7×Q8—22no—
34C2×C7×Q8—22no—
35C2×C7×Q8—22no—
36C22×C4×C7C2312yes—
37C22×C4×C7C2312yes—
38C22×C4×C7C2312yes—
39C22×C4×C7C2311yescharacteristic
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2311yescharacteristic
44C2×C4×D7—22no—
45C2×C4×D7—22no—
46C2×C4×D7D414yes—
47C2×Dic14—22no—
48C2×Dic14—22no—
49C2×Dic14—22no—
50C7×C4⋊C4—22no—
51C42×C7—22no—
52(C2×D7)⋊C4—24no—
53(C2×D7)⋊C4—24no—
54(C2×D7)⋊C4—24no—
55C4⋊Dic7—22no—
56C4⋊Dic7—24no—
57Dic7⋊C4—24no—
58Dic7⋊C4—28no—
59Dic7⋊C4—24no—
60C4×Dic7D414yes—
61C2×C4⋊Q8—71no—
62C2×C22⋊Q8—71no—
63C2×C22⋊Q8—71no—
64C2×#117—71no—
65#398—74no—
66#397—74no—
67#391—71no—
68#391D711yescharacteristic, complemented
69#382—71no—
70C23×C7C2411yescharacteristic
71C22×D7—21no—
72C22×D7—21no—
73C22×D7—22no—
74C22×D7—22no—
75C22×D7C2×D412yes—
76C7×Q8—44no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—24no—
85C2×C4×C7C2×D412yes—
86C2×C4×C7C2×D412yes—
87C2×C4×C7#4112yes—
88C2×Dic7—21no—
89C2×Dic7—21no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—24no—
97C2×Dic7C2×D414yes—
98C2×Dic7C2×D412yes—
99C4×D7—28no—
100Dic14—44no—
101C22×Q8—71no—
102C22×Q8C2×D711yescharacteristic
103C23×C4—71no—
104C4⋊Q8—144no—
105C22⋊Q8—144no—
106C22⋊Q8—144no—
107#117—144no—
108C2×C4⋊C4—72no—
109C2×C4⋊C4—72no—
110C2×C4⋊C4—71no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—74no—
113C2×C4⋊C4C2×D711yescharacteristic
114C2×C22⋊C4—72no—
115C2×C22⋊C4—72no—
116C2×C22⋊C4—72no—
117C2×C42—71no—
118C2×C42C2×D711yescharacteristic
119#95—74no—
120#95—74no—
121#95C2×D714yes—
122C22×C7C22×D411yescharacteristic
123C22×C7C22×D411yescharacteristic
124C22×C7C2×#4111yescharacteristic
125C22×C72−1+412yes—
126C22×C72−1+412yes—
127C2×D7—22no—
128C2×D7—22no—
129C2×D7—22no—
130C2×D7—42no—
131C2×D7—42no—
132C4×C7—42no—
133C4×C7—42no—
134C4×C7—42no—
135C4×C7#12114yes—
136Dic7—22no—
137Dic7—24no—
138Dic7—42no—
139Dic7—42no—
140C24—71no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—142no—
145C2×Q8—142no—
146C2×Q8—142no—
147C22×C4—72no—
148C22×C4—71no—
149C22×C4—74no—
150C22×C4—72no—
151C22×C4—72no—
152C22×C4—142no—
153C22×C4—142no—
154C22×C4C22×D712yes—
155C22×C4C22×D712yes—
156C22×C4C22×D712yes—
157C22×C4C22×D711yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—144no—
160C4⋊C4—144no—
161C4⋊C4—148no—
162C4⋊C4—144no—
163C4⋊C4—142no—
164C22⋊C4—144no—
165C22⋊C4—144no—
166C22⋊C4—144no—
167C42—22no—
168C42—74no—
169C2×C7C2×#12111yescharacteristic
170C2×C7#53611yescharacteristic
171C2×C7#53612yes—
172C2×C7#53611yescharacteristic
173C2×C7#54112yes—
174D7—42no—
175C23—72no—
176C23—141no—
177C23—141no—
178C23—142no—
179C23—142no—
180C23C23×D711yescharacteristic
181Q8—44no—
182Q8—284no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—74no—
192C2×C4—72no—
193C2×C4—141no—
194C2×C4—141no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—148no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—144no—
203C2×C4#115712yes—
204C2×C4D4×D712yes—
205C2×C4D4×D712yes—
206C7#262111yescharacteristic, complemented, Sylow
207C22—142no—
208C22—142no—
209C22—142no—
210C22—282no—
211C22—282no—
212C22C2×#115711yescharacteristic
213C22C2×D4×D711yescharacteristic
214C22C2×D4×D711yescharacteristic
215C22#655112yes—
216C22#655412yes—
217C4—42no—
218C4—42no—
219C4—42no—
220C4—142no—
221C4—144no—
222C4—282no—
223C4—282no—
224C4#644714yes—
225C2—282no—
226C2C2×#644711yescharacteristic
227C2#9027412yes—
228C2#9031912yes—
229C2#9036611yescharacteristic
230C2#9041511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#391C2
prime radicalC23×D7C23
hypercenter#391D7
hypocenterC7#2621
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#6507C2semidirect
5. C7×#391C2semidirect
6. #89749C2semidirect
7. #89701C2semidirect
9. #89429C2semidirect
68. #391D7semidirect
206. C7#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#65092
C2×#65072
C2×#64602
C2×#64352
C7×#3912
#897492
#897012
#894472
#894292
#26217