Finite Group Databasebeta

← #11732029

Subgroups — #11732029

Order 896.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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×#385C211yesmaximal, characteristic
2#89982C212yesmaximal
3#89756C212yesmaximal, complemented
4#89752C212yesmaximal, complemented
5#89725C211yesmaximal, characteristic
6#89724C212yesmaximal
7#89721C211yesmaximal, characteristic
8C7⋊#401C212yesmaximal, complemented
9#89429C212yesmaximal, complemented
10C23×C4×C7C2211yescharacteristic
11C2×C7×C4⋊C4C2212yes—
12C2×C7×C22⋊C4C2211yescharacteristic
13C2×C7×C22⋊C4C2211yescharacteristic
14C2×C22⋊Dic7C2212yes—
15C2×C22⋊Dic7C2212yes—
16C2×C4⋊Dic7C2211yescharacteristic
17C2×C4⋊Dic7C2211yescharacteristic
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C7×#95C2212yes—
23C7⋊#95C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2211yescharacteristic
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31#2834—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C22×C4×C7—41no—
34C22×C4×C7—42no—
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2312yes—
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C7×C4⋊C4—42no—
44C7×C22⋊C4—22no—
45C7×C22⋊C4—41no—
46C22⋊Dic7—24no—
47C22⋊Dic7—42no—
48C4⋊Dic7—41no—
49C4⋊Dic7—41no—
50Dic7⋊C4—42no—
51Dic7⋊C4—42no—
52Dic7⋊C4—42no—
53Dic7⋊C4—42no—
54#401—72no—
55#399—71no—
56#398—72no—
57#398—72no—
58#398—72no—
59#397—72no—
60#397—71no—
61#385—72no—
62#385D711yescharacteristic
63C23×C7—22no—
64C23×C7—21no—
65C23×C7—21no—
66C23×C7—22no—
67C23×C7#4112yes—
68C23×C7C2411yescharacteristic
69C2×C4×C7—22no—
70C2×C4×C7—22no—
71C2×C4×C7—22no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—21no—
75C2×C4×C7—21no—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7—44no—
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—22no—
92C23×C4C2×D711yescharacteristic
93C2×C4⋊C4—72no—
94C2×C4⋊C4—72no—
95C2×C4⋊C4—72no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4—71no—
98C2×C4⋊C4—71no—
99C2×C4⋊C4C2×D712yes—
100C2×C22⋊C4—72no—
101C2×C22⋊C4—72no—
102C2×C22⋊C4C2×D711yescharacteristic
103C2×C22⋊C4C2×D711yescharacteristic
104#95—72no—
105#95—72no—
106#95—71no—
107#95—72no—
108#95—72no—
109#95—72no—
110#95—72no—
111#95—71no—
112#95C2×D712yes—
113C22×C7—22no—
114C22×C7—24no—
115C22×C7—41no—
116C22×C7—42no—
117C22×C7—41no—
118C22×C7C2×#4111yescharacteristic
119C22×C7H(5,𝔽2)11yescharacteristic
120C22×C7H(5,𝔽2)12yes—
121C22×C72−1+411yescharacteristic
122C22×C72−1+412yes—
123C4×C7—42no—
124C4×C7—41no—
125C4×C7—42no—
126C4×C7—41no—
127Dic7—42no—
128Dic7—42no—
129Dic7—42no—
130Dic7—42no—
131C24C22×D711yescharacteristic
132C22×C4—42no—
133C22×C4—41no—
134C22×C4—72no—
135C22×C4—72no—
136C22×C4—72no—
137C22×C4—72no—
138C22×C4C22×D711yescharacteristic
139C22×C4C22×D712yes—
140C22×C4C22×D712yes—
141C22×C4C22×D711yescharacteristic
142C4⋊C4—42no—
143C4⋊C4—282no—
144C4⋊C4—282no—
145C4⋊C4—282no—
146C4⋊C4—282no—
147C4⋊C4—281no—
148C4⋊C4—281no—
149C22⋊C4—22no—
150C22⋊C4—41no—
151C22⋊C4—144no—
152C22⋊C4—282no—
153C2×C7—42no—
154C2×C7#53911yescharacteristic
155C2×C7#53912yes—
156C2×C7#56011yescharacteristic
157C2×C7#56312yes—
158C2×C7#56311yescharacteristic
159C23—21no—
160C23—22no—
161C23—21no—
162C23—22no—
163C23#115712yes—
164C23C23×D711yescharacteristic
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
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—142no—
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—
188C7#283411yescharacteristic, complemented, Sylow
189C22—22no—
190C22—24no—
191C22—41no—
192C22—42no—
193C22—41no—
194C22C2×#115711yescharacteristic
195C22#654811yescharacteristic
196C22#654812yes—
197C22#655111yescharacteristic
198C22#655412yes—
199C4—42no—
200C4—42no—
201C4—41no—
202C4—41no—
203C4—282no—
204C4—282no—
205C4—282no—
206C4—282no—
207C2—42no—
208C2#9020611yescharacteristic
209C2#9025512yes—
210C2#9028611yescharacteristic
211C2#9033712yes—
212C2#9034211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC24×C7C23
hypercenter#385D7
hypocenterC7#2834
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #89756C2semidirect
4. #89752C2semidirect
8. C7⋊#401C2semidirect
9. #89429C2semidirect
188. C7#2834semidirect

Maximal subgroups

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

subgroupindex
C7×#3852
#899822
#897562
#897522
#897252
#897242
#897212
C7⋊#4012
#894292
#28347