Finite Group Databasebeta

← #11734239

Subgroups — #11734239

Order 896.

420 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 129 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×#393C211yesmaximal, characteristic
2#90002C212yesmaximal, complemented
3#89725C212yesmaximal
4#89725C211yesmaximal, characteristic
5#89718C211yesmaximal, characteristic
6C22⋊C4×Dic7C212yesmaximal
7#89427C212yesmaximal, complemented
8#89426C212yesmaximal, complemented
9#89423C212yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C23×Dic7C2211yescharacteristic
12C2×C7×C22⋊C4C2212yes—
13C2×C7×C22⋊C4C2211yescharacteristic
14C2×C22⋊Dic7C2212yes—
15C2×C22⋊Dic7C2211yescharacteristic
16C2×C4⋊Dic7C2212yes—
17C2×C4⋊Dic7C2211yescharacteristic
18C2×Dic7⋊C4C2212yes—
19C2×C4×Dic7C2212yes—
20C2×C4×Dic7C2212yes—
21C7×#95C2211yescharacteristic
22C7×#95C2212yes—
23C7⋊#95C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2211yescharacteristic
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31#2523—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C2×C7×Q8—24no—
34C2×C7×Q8—41no—
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2312yes—
39C22×Dic7—24no—
40C22×Dic7—41no—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C7×C22⋊C4—24no—
46C7×C22⋊C4—41no—
47C22⋊Dic7—24no—
48C22⋊Dic7—22no—
49C4⋊Dic7—24no—
50C4⋊Dic7—41no—
51Dic7⋊C4—24no—
52C4×Dic7—24no—
53C4×Dic7—42no—
54#397—72no—
55#397—71no—
56#393D711yescharacteristic
57#391—72no—
58#387—72no—
59#387—72no—
60#385—71no—
61#383—72no—
62C4×C22⋊C4—72no—
63C23×C7—22no—
64C23×C7—22no—
65C23×C7—21no—
66C23×C7—21no—
67C23×C7#4112yes—
68C23×C7C2411yescharacteristic
69C7×Q8—44no—
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—21no—
77C2×C4×C7—21no—
78C2×C4×C7—22no—
79C2×C4×C7#4114yes—
80C2×Dic7—22no—
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—44no—
92C2×Dic7#4114yes—
93C22×Q8C2×D711yescharacteristic
94C23×C4—71no—
95C2×C4⋊C4—72no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4—71no—
98C2×C22⋊C4—72no—
99C2×C22⋊C4—71no—
100C2×C22⋊C4C2×D712yes—
101C2×C22⋊C4C2×D711yescharacteristic
102C2×C42—72no—
103C2×C42—72no—
104#95—72no—
105#95—72no—
106#95—72no—
107#95—72no—
108#95—72no—
109#95—72no—
110#95—71no—
111#95—72no—
112#95C2×D711yescharacteristic
113#95C2×D712yes—
114C22×C7—24no—
115C22×C7—22no—
116C22×C7—41no—
117C22×C7—41no—
118C22×C7—42no—
119C22×C7C2×#4112yes—
120C22×C7C2×#4111yescharacteristic
121C22×C7C2×#4112yes—
122C22×C7H(5,𝔽2)11yescharacteristic
123C22×C72−1+411yescharacteristic
124C4×C7—24no—
125C4×C7—41no—
126C4×C7—41no—
127C4×C7—42no—
128Dic7—24no—
129Dic7—42no—
130Dic7—42no—
131Dic7—42no—
132C24C22×D711yescharacteristic
133C2×Q8—24no—
134C2×Q8—41no—
135C22×C4—72no—
136C22×C4—72no—
137C22×C4—72no—
138C22×C4—72no—
139C22×C4—144no—
140C22×C4—281no—
141C22×C4C22×D711yescharacteristic
142C22×C4C22×D712yes—
143C22×C4C22×D711yescharacteristic
144C22×C4C22×D712yes—
145C4⋊C4—144no—
146C4⋊C4—144no—
147C4⋊C4—281no—
148C22⋊C4—24no—
149C22⋊C4—41no—
150C22⋊C4—144no—
151C22⋊C4—142no—
152C42—144no—
153C42—282no—
154C2×C7—42no—
155C2×C7#52912yes—
156C2×C7#53911yescharacteristic
157C2×C7#55111yescharacteristic
158C2×C7#55111yescharacteristic
159C2×C7#55612yes—
160C23—22no—
161C23—21no—
162C23—22no—
163C23—21no—
164C23#116112yes—
165C23C23×D711yescharacteristic
166Q8—44no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—74no—
177C2×C4—142no—
178C2×C4—142no—
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—284no—
189C2×C4#115914yes—
190C7#252311yescharacteristic, complemented, Sylow
191C22—24no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—42no—
196C22C2×#115912yes—
197C22#654811yescharacteristic
198C22C2×#116111yescharacteristic
199C22D7×#4112yes—
200C22#655411yescharacteristic
201C4—24no—
202C4—41no—
203C4—41no—
204C4—42no—
205C4—144no—
206C4—282no—
207C4—282no—
208C4—282no—
209C2—42no—
210C2#9027811yescharacteristic
211C2#9031512yes—
212C2#9033611yescharacteristic
213C2#9036212yes—
214C2#9037011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#393C2
prime radicalC24×C7C23
hypercenter#393D7
hypocenterC7#2523
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #90002C2semidirect
7. #89427C2semidirect
8. #89426C2semidirect
9. #89423C2semidirect
190. C7#2523semidirect

Maximal subgroups

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

subgroupindex
C7×#3932
#900022
#897252
#897182
C22⋊C4×Dic72
#894272
#894262
#894232
#25237