Finite Group Databasebeta

← #11734048

Subgroups — #11734048

Order 896.

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

normal subgroups only

This group has 368 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×#391C211yesmaximal, characteristic
2#90002C211yesmaximal, characteristic
3#90002C211yesmaximal, characteristic
4#89756C212yesmaximal
5#89705C212yesmaximal
6#89432C212yesmaximal
7#89429C212yesmaximal
8#89426C212yesmaximal
9#89419C212yesmaximal
10C22×C7×Q8C2211yescharacteristic
11C2×C7×C4⋊C4C2211yescharacteristic
12C2×C42×C7C2211yescharacteristic
13C2×C4⋊Dic7C2212yes—
14C2×C4⋊Dic7C2211yescharacteristic
15C2×C4⋊Dic7C2211yescharacteristic
16C2×Dic7⋊C4C2212yes—
17C2×Dic7⋊C4C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×C4×Dic7C2211yescharacteristic
21C2×C4×Dic7C2211yescharacteristic
22C7×#95C2212yes—
23C7×#95C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31#2809—71nomaximal, Sylow
32C2×C7×Q8—41no—
33C2×C7×Q8—41no—
34C2×C7×Q8Q814yes—
35C22×C4×C7C2312yes—
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C7×C4⋊C4—41no—
45C42×C7—41no—
46C4⋊Dic7—24no—
47C4⋊Dic7—41no—
48C4⋊Dic7—41no—
49Dic7⋊C4—24no—
50Dic7⋊C4—24no—
51Dic7⋊C4—42no—
52Dic7⋊C4—42no—
53C4×Dic7—41no—
54C4×Dic7—41no—
55#400—72no—
56#398—72no—
57#398—72no—
58#391—71no—
59#391—71no—
60#391D711yescharacteristic
61#389—72no—
62#387—72no—
63#387—72no—
64C23×C7C2411yescharacteristic
65C7×Q8—44no—
66C2×C4×C7—21no—
67C2×C4×C7—21no—
68C2×C4×C7—21no—
69C2×C4×C7—21no—
70C2×C4×C7—21no—
71C2×C4×C7—21no—
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×C7C2×Q812yes—
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—24no—
91C2×Dic7#4114yes—
92C22×Q8C2×D711yescharacteristic
93C2×C4⋊C4—72no—
94C2×C4⋊C4—72no—
95C2×C4⋊C4—72no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—71no—
99C2×C4⋊C4—71no—
100C2×C4⋊C4C2×D711yescharacteristic
101C2×C42—71no—
102C2×C42—71no—
103C2×C42C2×D711yescharacteristic
104#95—72no—
105#95—72no—
106#95—72no—
107#95—72no—
108#95—72no—
109#95—72no—
110#95—72no—
111#95C2×D712yes—
112#95C2×D712yes—
113C22×C7C22×Q811yescharacteristic
114C22×C7C2×#4112yes—
115C22×C7H(5,𝔽2)11yescharacteristic
116C22×C7H(5,𝔽2)11yescharacteristic
117C22×C72−1+412yes—
118C4×C7—22no—
119C4×C7—22no—
120C4×C7—22no—
121C4×C7—42no—
122C4×C7—42no—
123Dic7—24no—
124Dic7—42no—
125Dic7—42no—
126Dic7—42no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8Dic1414yes—
130C22×C4—72no—
131C22×C4—72no—
132C22×C4—72no—
133C22×C4—72no—
134C22×C4C22×D712yes—
135C22×C4C22×D712yes—
136C22×C4C22×D711yescharacteristic
137C22×C4C22×D711yescharacteristic
138C22×C4C22×D711yescharacteristic
139C4⋊C4—41no—
140C4⋊C4—144no—
141C4⋊C4—144no—
142C4⋊C4—144no—
143C4⋊C4—282no—
144C4⋊C4—282no—
145C4⋊C4—281no—
146C4⋊C4—281no—
147C42—41no—
148C42—281no—
149C42—281no—
150C2×C7#53912yes—
151C2×C7#54311yescharacteristic
152C2×C7#55511yescharacteristic
153C2×C7#55712yes—
154C2×C7#56311yescharacteristic
155C23C23×D711yescharacteristic
156Q8—44no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—74no—
170C2×C4—142no—
171C2×C4—142no—
172C2×C4—142no—
173C2×C4—142no—
174C2×C4—142no—
175C2×C4—144no—
176C2×C4—142no—
177C2×C4—142no—
178C2×C4—142no—
179C2×C4—142no—
180C2×C4C2×Dic1412yes—
181C2×C4C2×Dic1412yes—
182C2×C4C2×Dic1412yes—
183C7#280911yescharacteristic, complemented, Sylow
184C22C22×Dic1411yescharacteristic
185C22#654811yescharacteristic
186C22#654811yescharacteristic
187C22#655112yes—
188C22D7×#4112yes—
189C4—22no—
190C4—22no—
191C4—22no—
192C4—42no—
193C4—42no—
194C4—144no—
195C4—282no—
196C4—282no—
197C4—282no—
198C2#9017411yescharacteristic
199C2#9026012yes—
200C2#9032912yes—
201C2#9036111yescharacteristic
202C2#9042611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#391C2
hypercenter#391D7
hypocenterC7#2809
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
183. C7#2809semidirect

Maximal subgroups

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

subgroupindex
C7×#3912
#900022
#897562
#897052
#894322
#894292
#894262
#894192
#28097