Finite Group Databasebeta

← #11733981

Subgroups — #11733981

Order 896.

604 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 157 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×#6510C211yesmaximal, characteristic
2C2×#6507C211yesmaximal, characteristic, complemented
3C2×#6456C211yesmaximal, characteristic
4C2×#6438C211yesmaximal, characteristic
5C7×#391C211yesmaximal, characteristic, complemented
6#89746C211yesmaximal, characteristic, complemented
7C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
8#89447C214yesmaximal, complemented
9#89439C214yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×D28C2211yescharacteristic
12C22×C4×D7C2212yes—
13C22×Dic14C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C42×C7C2211yescharacteristic
16#6510—24no—
17#6507—24no—
18C2×(C2×D7)⋊C4C2214yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2214yes—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×C4×Dic7C2211yescharacteristic
25C2×C4×Dic7C2211yescharacteristic
26#6456—24no—
27#6438—24no—
28C7×#95C2214yes—
29C7⋊#95C2214yes—
30C7⋊#95C2212yes—
31#2547—71nomaximal, Sylow
32C23×D7C2312yes—
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×Dic7C2311yescharacteristic
43C22×Dic7C2311yescharacteristic
44C2×D28—22no—
45C2×D28—22no—
46C2×D28D414yes—
47C2×C4×D7—24no—
48C2×C4×D7—28no—
49C2×Dic14—22no—
50C2×Dic14—22no—
51C2×Dic14—22no—
52C7×C4⋊C4—22no—
53C42×C7—22no—
54(C2×D7)⋊C4—28no—
55(C2×D7)⋊C4—28no—
56(C2×D7)⋊C4—24no—
57Dic7⋊C4—24no—
58Dic7⋊C4—24no—
59Dic7⋊C4—24no—
60C4×Dic7—22no—
61C4×Dic7D414yes—
62C2×C4⋊Q8—71no—
63C2×#124—71no—
64C2×#124—71no—
65C2×C4×D4—71no—
66#397—74no—
67#391—71no—
68#391D711yescharacteristic, complemented
69#380—74no—
70C4×C4⋊C4—71no—
71C23×C7C2411yescharacteristic
72C22×D7—22no—
73C22×D7—24no—
74C22×D7—22no—
75C22×D7—24no—
76C22×D7C2×D414yes—
77C7×Q8—44no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—24no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7C2×D412yes—
87C2×C4×C7C2×D412yes—
88C2×C4×C7#4112yes—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7C2×D412yes—
97C2×Dic7C2×D412yes—
98C2×Dic7#4114yes—
99D28—28no—
100C4×D7—48no—
101Dic14—44no—
102C22×Q8—71no—
103C22×Q8C2×D711yescharacteristic
104C22×D4—71no—
105C23×C4—72no—
106C4⋊Q8—144no—
107#124—144no—
108#124—144no—
109C4×D4—144no—
110C2×C4⋊C4—72no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4C2×D711yescharacteristic
114C2×C22⋊C4—72no—
115C2×C22⋊C4—74no—
116C2×C22⋊C4—74no—
117C2×C42—71no—
118C2×C42—71no—
119C2×C42C2×D711yescharacteristic
120#95—72no—
121#95—74no—
122#95C2×D714yes—
123C22×C7C22×D411yescharacteristic
124C22×C7C22×D411yescharacteristic
125C22×C7C2×#4111yescharacteristic
126C22×C7C2×#4112yes—
127C22×C72−1+412yes—
128C2×D7—24no—
129C2×D7—28no—
130C2×D7—42no—
131C2×D7—42no—
132C2×D7—44no—
133C4×C7—42no—
134C4×C7—42no—
135C4×C7—42no—
136C4×C7#12114yes—
137Dic7—22no—
138Dic7—42no—
139Dic7—42no—
140Dic7#12414yescomplemented
141C24—72no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—142no—
146C2×Q8—142no—
147C2×Q8—142no—
148C2×D4—74no—
149C2×D4—142no—
150C2×D4—142no—
151C22×C4—71no—
152C22×C4—71no—
153C22×C4—72no—
154C22×C4—72no—
155C22×C4—144no—
156C22×C4—148no—
157C22×C4C22×D712yes—
158C22×C4C22×D712yes—
159C22×C4C22×D712yes—
160C22×C4C22×D711yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—144no—
163C4⋊C4—144no—
164C4⋊C4—144no—
165C22⋊C4—144no—
166C22⋊C4—148no—
167C22⋊C4—148no—
168C42—22no—
169C42—74no—
170C42—142no—
171C2×C7C2×#12111yescharacteristic
172C2×C7C2×#12411yescharacteristic
173C2×C7#53611yescharacteristic
174C2×C7#54712yes—
175C2×C7#55612yes—
176D7—44no—
177C23—74no—
178C23—142no—
179C23—142no—
180C23—144no—
181C23—144no—
182C23C23×D711yescharacteristic
183Q8—44no—
184Q8—284no—
185D4—148no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—72no—
195C2×C4—74no—
196C2×C4—72no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4—288no—
205C2×C4#115712yes—
206C2×C4D4×D712yes—
207C2×C4D4×D712yes—
208C7#254711yescharacteristic, complemented, Sylow
209C22—144no—
210C22—148no—
211C22—282no—
212C22—284no—
213C22—282no—
214C22C2×#115711yescharacteristic
215C22C2×D4×D711yescharacteristic
216C22C2×D4×D711yescharacteristic
217C22#655112yes—
218C22D7×#4112yes—
219C4—42no—
220C4—42no—
221C4—42no—
222C4—74no—
223C4—142no—
224C4—282no—
225C4—282no—
226C4#644714yes—
227C2—284no—
228C2C2×#644711yescharacteristic
229C2#9025912yes—
230C2#9032212yes—
231C2D7×#12411yescharacteristic
232C2#9041511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#391C2
prime radicalC22×D28C22
hypercenter#391D7
hypocenterC7#2547
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. #89746C2semidirect
7. C4×Dic7⋊C4C2semidirect
8. #89447C2semidirect
9. #89439C2semidirect
68. #391D7semidirect
140. Dic7#124semidirect
208. C7#2547semidirect

Maximal subgroups

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

subgroupindex
C2×#65102
C2×#65072
C2×#64562
C2×#64382
C7×#3912
#897462
C4×Dic7⋊C42
#894472
#894392
#25477