Finite Group Databasebeta

← #11731537

Subgroups — #11731537

Order 896.

424 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C7×#381C211yesmaximal, characteristic
2#89984C211yesmaximal, characteristic
3C4×C22⋊Dic7C211yesmaximal, characteristic
4#89720C212yesmaximal
5#89719C212yesmaximal
6#89432C212yesmaximal, complemented
7#89428C212yesmaximal, complemented
8#89426C212yesmaximal, complemented
9#89418C212yesmaximal, complemented
10C23×C4×C7C2211yescharacteristic
11C22×Dic14C2212yes—
12C2×C7×C22⋊C4C2212yes—
13C2×C22⋊Dic7C2212yes—
14C2×C22⋊Dic7C2211yescharacteristic
15C2×C22⋊Dic7C2211yescharacteristic
16C2×C4⋊Dic7C2212yes—
17C2×C4⋊Dic7C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×C4×Dic7C2212yes—
21C7×#95C2211yescharacteristic
22C7×#95C2211yescharacteristic
23C7×#95C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2211yescharacteristic
31#2665—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C22×C4×C7—42no—
34C22×C4×C7Q814yescomplemented
35C22×C4×C7C2312yes—
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C2×Dic14—24no—
44C2×Dic14—42no—
45C2×Dic14—42no—
46C7×C22⋊C4—42no—
47C22⋊Dic7—24no—
48C22⋊Dic7—41no—
49C22⋊Dic7—41no—
50C4⋊Dic7—24no—
51C4⋊Dic7—42no—
52Dic7⋊C4—42no—
53Dic7⋊C4—42no—
54C4×Dic7—42no—
55#400—72no—
56#396—72no—
57#395—72no—
58#393—72no—
59#391—72no—
60#387—72no—
61#381D711yescharacteristic
62#380—71no—
63C4×C22⋊C4—71no—
64C23×C7—22no—
65C23×C7—21no—
66C23×C7—22no—
67C23×C7—21no—
68C23×C7C2×Q812yes—
69C23×C7C2411yescharacteristic
70C2×C4×C7—22no—
71C2×C4×C7—22no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—44no—
79C2×C4×C7C2×Q812yes—
80C2×C4×C7C2×Q812yes—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7#4114yes—
93Dic14—48no—
94C22×Q8—72no—
95C23×C4C2×D711yescharacteristic
96C2×C4⋊C4—72no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—72no—
99C2×C4⋊C4—72no—
100C2×C22⋊C4—72no—
101C2×C22⋊C4—71no—
102C2×C22⋊C4—71no—
103C2×C22⋊C4C2×D712yes—
104C2×C42—72no—
105#95—71no—
106#95—71no—
107#95—72no—
108#95—72no—
109#95—72no—
110#95—72no—
111#95—72no—
112#95C2×D711yescharacteristic
113#95C2×D711yescharacteristic
114#95C2×D712yes—
115C22×C7—22no—
116C22×C7—22no—
117C22×C7—22no—
118C22×C7—42no—
119C22×C7—42no—
120C22×C7C22×Q811yescharacteristic
121C22×C7C2×#4112yes—
122C22×C7H(5,𝔽2)12yes—
123C22×C7H(5,𝔽2)11yescharacteristic
124C22×C72−1+411yescharacteristic
125C4×C7—22no—
126C4×C7—22no—
127C4×C7—42no—
128C4×C7—42no—
129Dic7—24no—
130Dic7—42no—
131Dic7—42no—
132Dic7—42no—
133C24C22×D711yescharacteristic
134C2×Q8—144no—
135C2×Q8—282no—
136C2×Q8—282no—
137C22×C4—42no—
138C22×C4—72no—
139C22×C4—72no—
140C22×C4—72no—
141C22×C4—72no—
142C22×C4Dic1414yescomplemented
143C22×C4C22×D712yes—
144C22×C4C22×D712yes—
145C22×C4C22×D711yescharacteristic
146C22×C4C22×D711yescharacteristic
147C4⋊C4—144no—
148C4⋊C4—282no—
149C4⋊C4—282no—
150C4⋊C4—282no—
151C22⋊C4—42no—
152C22⋊C4—144no—
153C22⋊C4—281no—
154C22⋊C4—281no—
155C42—282no—
156C2×C7—42no—
157C2×C7#53311yescharacteristic
158C2×C7#53812yes—
159C2×C7#54212yes—
160C2×C7#54311yescharacteristic
161C2×C7#54411yescharacteristic
162C23—22no—
163C23—22no—
164C23—21no—
165C23—21no—
166C23C2×Dic1412yes—
167C23C23×D711yescharacteristic
168Q8—288no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—44no—
178C2×C4—74no—
179C2×C4—142no—
180C2×C4—142no—
181C2×C4—142no—
182C2×C4—142no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—142no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—142no—
190C2×C4C2×Dic1412yes—
191C2×C4C2×Dic1412yes—
192C7#266511yescharacteristic, complemented, Sylow
193C22—22no—
194C22—22no—
195C22—22no—
196C22—42no—
197C22—42no—
198C22C22×Dic1411yescharacteristic
199C22#654812yes—
200C22D7×#4112yes—
201C22#655311yescharacteristic
202C22#655411yescharacteristic
203C4—22no—
204C4—22no—
205C4—42no—
206C4—42no—
207C4—144no—
208C4—282no—
209C4—282no—
210C4—282no—
211C2—42no—
212C2#9017411yescharacteristic
213C2#9020811yescharacteristic
214C2#9021011yescharacteristic
215C2#9034812yes—
216C2#9035312yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#381C2
prime radicalC24×C7C23
hypercenter#381D7
hypocenterC7#2665
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
6. #89432C2semidirect
7. #89428C2semidirect
8. #89426C2semidirect
9. #89418C2semidirect
34. C22×C4×C7Q8semidirect
142. C22×C4Dic14semidirect
192. C7#2665semidirect

Maximal subgroups

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

subgroupindex
C7×#3812
#899842
C4×C22⋊Dic72
#897202
#897192
#894322
#894282
#894262
#894182
#26657