Finite Group Databasebeta

← #11731556

Subgroups — #11731556

Order 896.

400 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 121 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#89982C212yesmaximal
3#89725C211yesmaximal, characteristic
4#89719C211yesmaximal, characteristic
5#89717C212yesmaximal
6#89430C212yesmaximal, complemented
7#89429C212yesmaximal, complemented
8#89423C212yesmaximal, complemented
9#89418C212yesmaximal, complemented
10C23×C4×C7C2211yescharacteristic
11C22×Dic14C2211yescharacteristic
12C2×C7×C22⋊C4C2211yescharacteristic
13C2×C7×C22⋊C4C2211yescharacteristic
14C2×C22⋊Dic7C2212yes—
15C2×C22⋊Dic7C2212yes—
16C2×C4⋊Dic7C2211yescharacteristic
17C2×Dic7⋊C4C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×C4×Dic7C2212yes—
21C7×#95C2212yes—
22C7×#95C2212yes—
23C7⋊#95C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2211yescharacteristic
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31#2764—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C22×C4×C7—22no—
34C22×C4×C7—41no—
35C22×C4×C7—41no—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2312yes—
39C22×C4×C7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C2×Dic14—24no—
45C2×Dic14—41no—
46C7×C22⋊C4—41no—
47C7×C22⋊C4—41no—
48C22⋊Dic7—24no—
49C22⋊Dic7—24no—
50C4⋊Dic7—41no—
51Dic7⋊C4—42no—
52Dic7⋊C4—42no—
53Dic7⋊C4—42no—
54C4×Dic7—24no—
55#400—72no—
56#398—72no—
57#397—71no—
58#393—71no—
59#391—72no—
60#388—72no—
61#385—72no—
62#383—72no—
63#381D711yescharacteristic
64C23×C7—22no—
65C23×C7—21no—
66C23×C7—21no—
67C23×C7—22no—
68C23×C7#4112yes—
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—21no—
77C2×C4×C7—21no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7—44no—
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—44no—
94C22×Q8—71no—
95C23×C4C2×D711yescharacteristic
96C2×C4⋊C4—72no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—72no—
99C2×C4⋊C4—71no—
100C2×C22⋊C4—72no—
101C2×C22⋊C4—72no—
102C2×C22⋊C4C2×D711yescharacteristic
103C2×C22⋊C4C2×D711yescharacteristic
104C2×C42—72no—
105#95—71no—
106#95—72no—
107#95—72no—
108#95—72no—
109#95—72no—
110#95—72no—
111#95—72no—
112#95—71no—
113#95C2×D712yes—
114#95C2×D712yes—
115C22×C7—22no—
116C22×C7—24no—
117C22×C7—42no—
118C22×C7—41no—
119C22×C7—41no—
120C22×C7C2×#4111yescharacteristic
121C22×C7C2×#4112yes—
122C22×C7H(5,𝔽2)11yescharacteristic
123C22×C7H(5,𝔽2)11yescharacteristic
124C22×C72−1+412yes—
125C4×C7—42no—
126C4×C7—42no—
127C4×C7—41no—
128C4×C7—41no—
129Dic7—24no—
130Dic7—42no—
131Dic7—42no—
132Dic7—42no—
133C24C22×D711yescharacteristic
134C2×Q8—144no—
135C2×Q8—281no—
136C22×C4—22no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—72no—
140C22×C4—72no—
141C22×C4—72no—
142C22×C4—72no—
143C22×C4C22×D711yescharacteristic
144C22×C4C22×D711yescharacteristic
145C22×C4C22×D712yes—
146C22×C4C22×D712yes—
147C4⋊C4—282no—
148C4⋊C4—282no—
149C4⋊C4—282no—
150C4⋊C4—281no—
151C22⋊C4—41no—
152C22⋊C4—41no—
153C22⋊C4—144no—
154C22⋊C4—144no—
155C42—144no—
156C2×C7—42no—
157C2×C7#53811yescharacteristic
158C2×C7#54212yes—
159C2×C7#55111yescharacteristic
160C2×C7#55212yes—
161C2×C7#56311yescharacteristic
162C23—21no—
163C23—21no—
164C23—22no—
165C23—22no—
166C23#115712yes—
167C23C23×D711yescharacteristic
168Q8—284no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—74no—
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×C4—142no—
191C2×C4—142no—
192C7#276411yescharacteristic, complemented, Sylow
193C22—22no—
194C22—24no—
195C22—41no—
196C22—41no—
197C22—42no—
198C22C2×#115711yescharacteristic
199C22#654811yescharacteristic
200C22#654811yescharacteristic
201C22D7×#4112yes—
202C22#655412yes—
203C4—42no—
204C4—41no—
205C4—41no—
206C4—42no—
207C4—144no—
208C4—282no—
209C4—282no—
210C4—282no—
211C2—42no—
212C2#9018511yescharacteristic
213C2#9022212yes—
214C2#9034211yescharacteristic
215C2#9035011yescharacteristic
216C2#9035312yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#381C2
prime radicalC24×C7C23
hypercenter#381D7
hypocenterC7#2764
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
6. #89430C2semidirect
7. #89429C2semidirect
8. #89423C2semidirect
9. #89418C2semidirect
192. C7#2764semidirect

Maximal subgroups

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

subgroupindex
C7×#3812
#899822
#897252
#897192
#897172
#894302
#894292
#894232
#894182
#27647