Finite Group Databasebeta

← #11739621

Subgroups — #11739621

Order 896.

512 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C22×C7×C4⋊C4C211yesmaximal, characteristic
2C2×#6489C211yesmaximal, characteristic
3#89983C211yesmaximal, characteristic
4#89983C212yesmaximal
5#89982C212yesmaximal
6#89758C212yesmaximal, complemented
7#89755C212yesmaximal, complemented
8#89750C214yesmaximal, complemented
9C23×C4×C7C2211yescharacteristic
10C23×C4×C7C2211yescharacteristic
11C23×C4×C7C2211yescharacteristic
12C2×C7×C4⋊C4—24no—
13C2×C7×C4⋊C4C2212yes—
14C2×C7×C4⋊C4C2212yes—
15C2×C22⋊Dic7C2212yes—
16C2×C22⋊Dic7C2212yes—
17#6489—24no—
18C2×C4⋊Dic7C2211yescharacteristic
19C2×C4⋊Dic7C2211yescharacteristic
20C2×Dic7⋊C4C2214yes—
21C2×C4×Dic7C2212yes—
22C7⋊#95C2214yes—
23C7⋊#95C2212yes—
24C7⋊#95C2214yes—
25C7⋊#95C2214yes—
26C7⋊#95C2212yes—
27#2484—71nomaximal, Sylow
28C24×C7C2311yescharacteristic
29C22×C4×C7—22no—
30C22×C4×C7—24no—
31C22×C4×C7—24no—
32C22×C4×C7—22no—
33C22×C4×C7—24no—
34C22×C4×C7D414yes—
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2312yes—
39C22×Dic7C2314yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C7×C4⋊C4—24no—
43C7×C4⋊C4—24no—
44C22⋊Dic7—24no—
45C22⋊Dic7—24no—
46C4⋊Dic7—22no—
47C4⋊Dic7—22no—
48Dic7⋊C4—28no—
49C4×Dic7—24no—
50C2×#117—71no—
51C22×C4⋊C4D711yescharacteristic
52#400—72no—
53#398—72no—
54#387—74no—
55#385—72no—
56#381—71no—
57#381—72no—
58C23×C7—22no—
59C23×C7—22no—
60C23×C7C2×D412yes—
61C23×C7#4114yes—
62C23×C7C2411yescharacteristic
63C2×C4×C7—28no—
64C2×C4×C7—28no—
65C2×C4×C7—22no—
66C2×C4×C7—22no—
67C2×C4×C7—22no—
68C2×C4×C7—22no—
69C2×C4×C7—22no—
70C2×C4×C7—24no—
71C2×C4×C7—44no—
72C2×C4×C7C2×D412yes—
73C2×C4×C7C2×D412yes—
74C2×C4×C7#4114yes—
75C2×Dic7—22no—
76C2×Dic7—22no—
77C2×Dic7—24no—
78C2×Dic7—24no—
79C2×Dic7—24no—
80C2×Dic7—24no—
81C2×Dic7—24no—
82C23×C4C2×D711yescharacteristic
83C23×C4C2×D711yescharacteristic
84C23×C4C2×D711yescharacteristic
85#117—144no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—71no—
88C2×C4⋊C4—71no—
89C2×C4⋊C4—74no—
90C2×C4⋊C4C2×D712yes—
91C2×C4⋊C4C2×D712yes—
92C2×C22⋊C4—72no—
93C2×C22⋊C4—72no—
94C2×C42—72no—
95#95—72no—
96#95—74no—
97#95—74no—
98#95—72no—
99#95—74no—
100C22×C7—22no—
101C22×C7—22no—
102C22×C7—28no—
103C22×C7#12314yes—
104C22×C7C22×D411yescharacteristic
105C22×C7C2×#4112yes—
106C22×C7C2×#4112yes—
107C22×C72−1+412yes—
108C4×C7—22no—
109C4×C7—22no—
110C4×C7—24no—
111C4×C7—42no—
112Dic7—42no—
113Dic7—42no—
114Dic7—44no—
115C24C22×D711yescharacteristic
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—74no—
122C22×C4—72no—
123C22×C4—72no—
124C22×C4C22⋊D714yes—
125C22×C4C22×D712yes—
126C22×C4C22×D712yes—
127C22×C4C22×D711yescharacteristic
128C22×C4C22×D711yescharacteristic
129C4⋊C4—24no—
130C4⋊C4—24no—
131C4⋊C4—148no—
132C4⋊C4—142no—
133C4⋊C4—142no—
134C22⋊C4—144no—
135C22⋊C4—144no—
136C42—144no—
137C2×C7—24no—
138C2×C7C2×#12311yescharacteristic
139C2×C7#52511yescharacteristic
140C2×C7#53611yescharacteristic
141C2×C7#55214yes—
142C23—22no—
143C23—22no—
144C23#115914yes—
145C23C2×C22⋊D712yes—
146C23C23×D711yescharacteristic
147C2×C4—28no—
148C2×C4—28no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—22no—
155C2×C4—44no—
156C2×C4—144no—
157C2×C4—144no—
158C2×C4—144no—
159C2×C4—144no—
160C2×C4—144no—
161C2×C4—142no—
162C2×C4—142no—
163C2×C4#115714yes—
164C2×C4C2×C22⋊D712yes—
165C2×C4C2×C22⋊D712yes—
166C7#248411yescharacteristic, complemented, Sylow
167C22—28no—
168C22—22no—
169C22—22no—
170C22#649814yes—
171C22C2×#115712yes—
172C22C2×#115912yes—
173C22#655112yes—
174C22C22×C22⋊D711yescharacteristic
175C4—24no—
176C4—22no—
177C4—22no—
178C4—42no—
179C4—284no—
180C4—282no—
181C4—282no—
182C2—24no—
183C2#9020414yes—
184C2#9048211yescharacteristic
185C2C2×#649811yescharacteristic
186C2#9050711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC22×C7×C4⋊C4C2
prime radicalC24×C7C23
hypercenterC22×C4⋊C4D7
hypocenterC7#2484
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #89758C2semidirect
7. #89755C2semidirect
8. #89750C2semidirect
166. C7#2484semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C4⋊C42
C2×#64892
#899832
#899822
#897582
#897552
#897502
#24847