Finite Group Databasebeta

← #11740109

Subgroups — #11740109

Order 896.

708 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 181 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×#121C211yesmaximal, characteristic
2C2×#6498C214yesmaximal, complemented
3C22×C4⋊Dic7C211yesmaximal, characteristic, complemented
4#89993C212yesmaximal, complemented
5C4×C22⋊Dic7C211yesmaximal, characteristic, complemented
6#89748C212yesmaximal, complemented
7#89721C214yesmaximal, complemented
8C22×C7×D4C2212yes—
9C22×C7×D4C2211yescharacteristic
10C23×C4×C7C2211yescharacteristic
11C23×Dic7C2212yes—
12C7×#121—24no—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C22⋊C4C2212yes—
15C2×C22⋊Dic7C2218yes—
16C2×C22⋊Dic7C2212yes—
17#6498—216no—
18C2×C4⋊Dic7—24no—
19C2×C4⋊Dic7C2212yes—
20C2×C4⋊Dic7C2212yes—
21C2×Dic7⋊C4C2218yes—
22C2×C4×Dic7C2212yes—
23C7⋊#95C2212yes—
24#2501—71nomaximal, Sylow
25C24×C7C2312yes—
26C24×C7C2311yescharacteristic
27C2×C7×D4—28no—
28C2×C7×D4—24no—
29C2×C7×D4—22no—
30C2×C7×D4—24no—
31C22×C4×C7—24no—
32C22×C4×C7D414yes—
33C22×C4×C7C2312yes—
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2311yescharacteristic
36C22×Dic7—24no—
37C22×Dic7—28no—
38C22×Dic7C2314yes—
39C22×Dic7C2314yes—
40C7×C4⋊C4—22no—
41C7×C22⋊C4—24no—
42C22⋊Dic7—216no—
43C22⋊Dic7D418yescomplemented
44C4⋊Dic7—24no—
45C4⋊Dic7—24no—
46Dic7⋊C4—216no—
47C4×Dic7—24no—
48C2×#123—74no—
49C2×#121D711yescharacteristic
50C22×C4⋊C4—71no—
51#399—74no—
52#390—72no—
53#384—72no—
54C4×C22⋊C4—71no—
55C23×C7—22no—
56C23×C7—22no—
57C23×C7—24no—
58C23×C7—22no—
59C23×C7—24no—
60C23×C7—24no—
61C23×C7C2×D414yes—
62C23×C7C2×D412yes—
63C23×C7C2411yescharacteristic
64C7×D4—48no—
65C7×D4—44no—
66C2×C4×C7—22no—
67C2×C4×C7—28no—
68C2×C4×C7—24no—
69C2×C4×C7—22no—
70C2×C4×C7C2×D412yes—
71C2×C4×C7C2×D412yes—
72C2×C4×C7#4114yes—
73C2×Dic7—24no—
74C2×Dic7—24no—
75C2×Dic7—28no—
76C2×Dic7—24no—
77C2×Dic7—48no—
78C2×Dic7C2×D418yes—
79C22×D4C2×D712yes—
80C22×D4C2×D711yescharacteristic
81C23×C4—72no—
82C23×C4C2×D711yescharacteristic
83#123—1416no—
84#121—24no—
85C2×C4⋊C4—72no—
86C2×C4⋊C4—72no—
87C2×C4⋊C4—78no—
88C2×C4⋊C4—144no—
89C2×C4⋊C4C2×D711yescharacteristic
90C2×C22⋊C4—72no—
91C2×C22⋊C4—78no—
92C2×C22⋊C4C2×D712yes—
93C2×C42—72no—
94#95—72no—
95C22×C7—22no—
96C22×C7—22no—
97C22×C7—28no—
98C22×C7—42no—
99C22×C7—42no—
100C22×C7—42no—
101C22×C7—48no—
102C22×C7#12714yes—
103C22×C7C22×D412yes—
104C22×C7C22×D411yescharacteristic
105C22×C7C2×#4112yes—
106C22×C72−1+412yes—
107C4×C7—22no—
108C4×C7—42no—
109C4×C7#12314yescomplemented
110Dic7—28no—
111Dic7—44no—
112C24C22×D712yes—
113C24C22×D711yescharacteristic
114C2×D4—28no—
115C2×D4—24no—
116C2×D4—24no—
117C2×D4—22no—
118C22×C4—24no—
119C22×C4—74no—
120C22×C4—74no—
121C22×C4—148no—
122C22×C4—144no—
123C22×C4C22⋊D714yes—
124C22×C4C22×D712yes—
125C22×C4C22×D711yescharacteristic
126C22×C4C22×D711yescharacteristic
127C4⋊C4—22no—
128C4⋊C4—1416no—
129C4⋊C4—144no—
130C4⋊C4—144no—
131C22⋊C4—24no—
132C22⋊C4—78no—
133C22⋊C4—1416no—
134C42—144no—
135C2×C7—24no—
136C2×C7—82no—
137C2×C7C2×#12311yescharacteristic
138C2×C7C2×#12711yescharacteristic
139C2×C7#53611yescharacteristic
140C2×C7#54714yes—
141C23—22no—
142C23—22no—
143C23—24no—
144C23—24no—
145C23—24no—
146C23—22no—
147C23D4×D714yes—
148C23C2×C22⋊D712yes—
149C23C23×D711yescharacteristic
150D4—48no—
151D4—44no—
152C2×C4—22no—
153C2×C4—28no—
154C2×C4—24no—
155C2×C4—22no—
156C2×C4—78no—
157C2×C4—144no—
158C2×C4—144no—
159C2×C4—148no—
160C2×C4—144no—
161C2×C4—288no—
162C2×C4#115914yes—
163C2×C4C2×C22⋊D712yes—
164C2×C4C2×C22⋊D712yes—
165C7#250111yescharacteristic, complemented, Sylow
166C22—22no—
167C22—22no—
168C22—28no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—48no—
173C22#650314yes—
174C22C2×D4×D712yes—
175C22C2×#115912yes—
176C22#655412yes—
177C22C22×C22⋊D711yescharacteristic
178C4—22no—
179C4—42no—
180C4—148no—
181C4—284no—
182C4#649814yescomplemented
183C2—24no—
184C2—82no—
185C2#9030214yes—
186C2C2×#649811yescharacteristic
187C2C2×#650311yescharacteristic
188C2#9051611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC2×C7×#121C2
hypercenterC2×#121D7
hypocenterC7#2501
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#6498C2semidirect
3. C22×C4⋊Dic7C2semidirect
4. #89993C2semidirect
5. C4×C22⋊Dic7C2semidirect
6. #89748C2semidirect
7. #89721C2semidirect
43. C22⋊Dic7D4semidirect
109. C4×C7#123semidirect
165. C7#2501semidirect
182. C4#6498semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1212
C2×#64982
C22×C4⋊Dic72
#899932
C4×C22⋊Dic72
#897482
#897212
#25017