Finite Group Databasebeta

← #11731538

Subgroups — #11731538

Order 896.

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

normal subgroups only

This group has 408 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#89721C212yesmaximal
5#89720C212yesmaximal
6#89433C212yesmaximal, complemented
7#89429C212yesmaximal, complemented
8#89427C212yesmaximal, complemented
9#89419C212yesmaximal, complemented
10C23×C4×C7C2211yescharacteristic
11C2×C7×C22⋊C4C2212yes—
12C2×C22⋊Dic7C2212yes—
13C2×C22⋊Dic7C2211yescharacteristic
14C2×C22⋊Dic7C2211yescharacteristic
15C2×C4⋊Dic7C2212yes—
16C2×C4⋊Dic7C2212yes—
17C2×C4⋊Dic7C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×C4×Dic7C2212yes—
23C7×#95C2211yescharacteristic
24C7×#95C2211yescharacteristic
25C7×#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2211yescharacteristic
31#2667—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C22×C4×C7—42no—
34C22×C4×C7Q814yes—
35C22×C4×C7C2312yes—
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C7×C22⋊C4—42no—
44C22⋊Dic7—24no—
45C22⋊Dic7—41no—
46C22⋊Dic7—41no—
47C4⋊Dic7—42no—
48C4⋊Dic7—42no—
49C4⋊Dic7—42no—
50Dic7⋊C4—24no—
51Dic7⋊C4—24no—
52Dic7⋊C4—42no—
53Dic7⋊C4—42no—
54C4×Dic7—42no—
55#399—72no—
56#398—72no—
57#398—72no—
58#396—72no—
59#389—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—
93C23×C4C2×D711yescharacteristic
94C2×C4⋊C4—72no—
95C2×C4⋊C4—72no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—72no—
99C2×C4⋊C4—72no—
100C2×C4⋊C4—72no—
101C2×C22⋊C4—72no—
102C2×C22⋊C4—71no—
103C2×C22⋊C4—71no—
104C2×C22⋊C4C2×D712yes—
105C2×C42—72no—
106#95—71no—
107#95—71no—
108#95—72no—
109#95—72no—
110#95—72no—
111#95C2×D711yescharacteristic
112#95C2×D711yescharacteristic
113#95C2×D712yes—
114C22×C7—22no—
115C22×C7—22no—
116C22×C7—22no—
117C22×C7—42no—
118C22×C7—42no—
119C22×C7C22×Q811yescharacteristic
120C22×C7C2×#4112yes—
121C22×C7H(5,𝔽2)12yes—
122C22×C7H(5,𝔽2)11yescharacteristic
123C22×C72−1+411yescharacteristic
124C4×C7—22no—
125C4×C7—22no—
126C4×C7—42no—
127C4×C7—42no—
128Dic7—24no—
129Dic7—42no—
130Dic7—42no—
131Dic7—42no—
132C24C22×D711yescharacteristic
133C22×C4—42no—
134C22×C4—72no—
135C22×C4—72no—
136C22×C4—72no—
137C22×C4—72no—
138C22×C4Dic1414yes—
139C22×C4C22×D712yes—
140C22×C4C22×D712yes—
141C22×C4C22×D711yescharacteristic
142C22×C4C22×D711yescharacteristic
143C4⋊C4—144no—
144C4⋊C4—144no—
145C4⋊C4—282no—
146C4⋊C4—282no—
147C4⋊C4—282no—
148C4⋊C4—282no—
149C4⋊C4—282no—
150C22⋊C4—42no—
151C22⋊C4—144no—
152C22⋊C4—281no—
153C22⋊C4—281no—
154C42—282no—
155C2×C7—42no—
156C2×C7#53311yescharacteristic
157C2×C7#53912yes—
158C2×C7#54012yes—
159C2×C7#54311yescharacteristic
160C2×C7#54411yescharacteristic
161C23—22no—
162C23—22no—
163C23—21no—
164C23—21no—
165C23C2×Dic1412yes—
166C23C23×D711yescharacteristic
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—44no—
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×C4C2×Dic1412yes—
189C2×C4C2×Dic1412yes—
190C7#266711yescharacteristic, complemented, Sylow
191C22—22no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22C22×Dic1411yescharacteristic
197C22#654812yes—
198C22D7×#4112yes—
199C22#655311yescharacteristic
200C22#655411yescharacteristic
201C4—22no—
202C4—22no—
203C4—42no—
204C4—42no—
205C4—144no—
206C4—282no—
207C4—282no—
208C4—282no—
209C2—42no—
210C2#9017411yescharacteristic
211C2#9020811yescharacteristic
212C2#9021011yescharacteristic
213C2#9035112yes—
214C2#9035412yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#381C2
prime radicalC24×C7C23
hypercenter#381D7
hypocenterC7#2667
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
6. #89433C2semidirect
7. #89429C2semidirect
8. #89427C2semidirect
9. #89419C2semidirect
190. C7#2667semidirect

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
#897212
#897202
#894332
#894292
#894272
#894192
#26677