Finite Group Databasebeta

← #11739733

Subgroups — #11739733

Order 896.

548 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 149 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×C7×#117C211yesmaximal, characteristic, complemented
2C2×#6493C211yesmaximal, characteristic
3#89984C211yesmaximal, characteristic, complemented
4#89765C212yesmaximal, complemented
5#89764C212yesmaximal, complemented
6#89731C214yesmaximal, complemented
7#89711C212yesmaximal, complemented
8#89700C212yesmaximal, complemented
9C23×C4×C7C2211yescharacteristic
10C22×C22⋊D7C2212yes—
11C22×D28C2211yescharacteristic
12C7×#117—24no—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C22⋊C4C2212yes—
16C2×C42×C7C2212yes—
17C2×C22⋊Dic7C2212yes—
18#6493—24no—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2214yes—
21C2×(C2×D7)⋊C4C2214yes—
22C2×C4⋊Dic7C2211yescharacteristic
23C2×C4⋊Dic7C2212yes—
24C2×Dic7⋊C4C2214yes—
25C7⋊#95C2214yes—
26C7⋊#95C2212yes—
27#2624—71nomaximal, Sylow
28C24×C7C2311yescharacteristic
29C23×D7C2312yes—
30C22×C4×C7—24no—
31C22×C4×C7D414yes—
32C22×C4×C7C2312yes—
33C22×C4×C7C2312yes—
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2311yescharacteristic
36C2×C22⋊D7—24no—
37C2×C22⋊D7—44no—
38C22×Dic7C2312yes—
39C22×Dic7C2314yes—
40C2×D28—22no—
41C2×D28—24no—
42C7×C4⋊C4—22no—
43C7×C4⋊C4—22no—
44C7×C22⋊C4—24no—
45C42×C7—24no—
46C22⋊Dic7—42no—
47(C2×D7)⋊C4—24no—
48(C2×D7)⋊C4—44no—
49(C2×D7)⋊C4—44no—
50C4⋊Dic7—22no—
51C4⋊Dic7—24no—
52Dic7⋊C4—44no—
53C2×#121—71no—
54C2×#117D711yescharacteristic, complemented
55#399—72no—
56#397—72no—
57#394—74no—
58#390—72no—
59#389—72no—
60#380—71no—
61C23×C7—22no—
62C23×C7—21no—
63C23×C7—22no—
64C23×C7—21no—
65C23×C7C2×D412yes—
66C23×C7C2411yescharacteristic
67C22×D7—22no—
68C22×D7—24no—
69C22×D7—22no—
70C22×D7—24no—
71C22×D7—22no—
72C2×C4×C7—22no—
73C2×C4×C7—28no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7C2×D412yes—
81C2×C4×C7C2×D412yes—
82C2×C4×C7#4114yes—
83C22⋊D7—48no—
84C2×Dic7—22no—
85C2×Dic7—24no—
86C2×Dic7—24no—
87C2×Dic7—24no—
88C2×Dic7—24no—
89D28—44no—
90C22×D4—72no—
91C22×D4—71no—
92C23×C4C2×D711yescharacteristic
93#121—144no—
94#117—24no—
95C2×C4⋊C4—71no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4—74no—
98C2×C4⋊C4C2×D711yescharacteristic
99C2×C4⋊C4C2×D711yescharacteristic
100C2×C22⋊C4—72no—
101C2×C22⋊C4—72no—
102C2×C22⋊C4—74no—
103C2×C22⋊C4—74no—
104C2×C22⋊C4C2×D712yes—
105C2×C42C2×D712yes—
106#95—72no—
107#95—74no—
108C22×C7—22no—
109C22×C7—22no—
110C22×C7—22no—
111C22×C7—42no—
112C22×C7—42no—
113C22×C7C22×D411yescharacteristic
114C22×C7C2×#4112yes—
115C22×C7H(5,𝔽2)12yes—
116C22×C7H(5,𝔽2)11yescharacteristic
117C22×C72−1+411yescharacteristic
118C2×D7—42no—
119C2×D7—42no—
120C2×D7—42no—
121C2×D7—44no—
122C2×D7—44no—
123C4×C7—22no—
124C4×C7—42no—
125C4×C7—42no—
126C4×C7#12314yes—
127Dic7—42no—
128Dic7—44no—
129C24—72no—
130C24C22×D711yescharacteristic
131C2×D4—144no—
132C2×D4—144no—
133C2×D4—142no—
134C2×D4—284no—
135C22×C4—24no—
136C22×C4—74no—
137C22×C4—72no—
138C22×C4C22⋊D714yes—
139C22×C4C22×D712yes—
140C22×C4C22×D712yes—
141C22×C4C22×D711yescharacteristic
142C22×C4C22×D711yescharacteristic
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—144no—
146C4⋊C4—142no—
147C4⋊C4—284no—
148C22⋊C4—24no—
149C22⋊C4—144no—
150C22⋊C4—284no—
151C22⋊C4—284no—
152C22⋊C4—282no—
153C42—24no—
154C2×C7—42no—
155C2×C7C2×#12311yescharacteristic
156C2×C7#53511yescharacteristic
157C2×C7#53711yescharacteristic
158C2×C7#54012yes—
159C2×C7#54212yes—
160D7—82no—
161C23—22no—
162C23—21no—
163C23—22no—
164C23—21no—
165C23—144no—
166C23—142no—
167C23—144no—
168C23—142no—
169C23—142no—
170C23C2×C22⋊D712yes—
171C23C23×D711yescharacteristic
172D4—288no—
173D4—284no—
174C2×C4—22no—
175C2×C4—28no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—144no—
183C2×C4—144no—
184C2×C4—144no—
185C2×C4—144no—
186C2×C4—142no—
187C2×C4#115714yes—
188C2×C4C2×C22⋊D712yes—
189C2×C4C2×C22⋊D712yes—
190C7#262411yescharacteristic, complemented, Sylow
191C22—22no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22—282no—
197C22—284no—
198C22—284no—
199C22—282no—
200C22—282no—
201C22C2×#115712yes—
202C22#655312yes—
203C22#655311yescharacteristic
204C22#655411yescharacteristic
205C22C22×C22⋊D711yescharacteristic
206C4—22no—
207C4—42no—
208C4—42no—
209C4—284no—
210C4—282no—
211C4#649214yes—
212C2—42no—
213C2—562no—
214C2#9022512yes—
215C2#9022612yes—
216C2C2×#649211yescharacteristic
217C2#9052111yescharacteristic
218C2#9052411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#117C2
prime radicalC2×#6493C2
hypercenterC2×#117D7
hypocenterC7#2624
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C7×#117C2semidirect
3. #89984C2semidirect
4. #89765C2semidirect
5. #89764C2semidirect
6. #89731C2semidirect
7. #89711C2semidirect
8. #89700C2semidirect
54. C2×#117D7semidirect
190. C7#2624semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1172
C2×#64932
#899842
#897652
#897642
#897312
#897112
#897002
#26247