Finite Group Databasebeta

← #11733980

Subgroups — #11733980

Order 896.

540 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 177 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×#6507C211yesmaximal, characteristic
2C2×#6507C211yesmaximal, characteristic
3C2×#6450C211yesmaximal, characteristic
4C2×#6432C211yesmaximal, characteristic
5C7×#391C211yesmaximal, characteristic
6#89746C211yesmaximal, characteristic
7C4×Dic7⋊C4C211yesmaximal, characteristic
8#89429C214yesmaximal
9#89419C214yesmaximal
10C22×C7×Q8C2211yescharacteristic
11C22×Dic14C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C42×C7C2211yescharacteristic
15#6507—24no—
16#6507—24no—
17C2×C4⋊Dic7C2214yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2214yes—
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×C4×Dic7C2212yes—
24C2×C4×Dic7C2211yescharacteristic
25C2×C4×Dic7C2211yescharacteristic
26#6450—24no—
27#6432—24no—
28C7×#95C2214yes—
29C7⋊#95C2214yes—
30C7⋊#95C2212yes—
31#2546—71nomaximal, Sylow
32C2×C7×Q8—22no—
33C2×C7×Q8—22no—
34C2×C7×Q8—22no—
35C22×C4×C7C2312yes—
36C22×C4×C7C2312yes—
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×Dic7C2312yes—
40C22×Dic7C2311yescharacteristic
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2311yescharacteristic
44C2×Dic14—22no—
45C2×Dic14—22no—
46C2×Dic14—22no—
47C2×Dic14—22no—
48C2×Dic14—22no—
49C2×Dic14D414yes—
50C7×C4⋊C4—22no—
51C42×C7—22no—
52C4⋊Dic7—28no—
53Dic7⋊C4—24no—
54Dic7⋊C4—28no—
55Dic7⋊C4—24no—
56Dic7⋊C4—24no—
57Dic7⋊C4Q818yescomplemented
58C4×Dic7—22no—
59C4×Dic7—24no—
60C4×Dic7D414yes—
61C2×C4⋊Q8—71no—
62C2×C4⋊Q8—71no—
63C2×C4⋊Q8—71no—
64C2×C4×Q8—71no—
65#398—74no—
66#391—71no—
67#391D711yescharacteristic
68#389—74no—
69C4×C4⋊C4—71no—
70C23×C7C2411yescharacteristic
71C7×Q8—44no—
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×C7—24no—
79C2×C4×C7C2×D412yes—
80C2×C4×C7C2×D412yes—
81C2×C4×C7C2×Q814yes—
82C2×C4×C7#4112yes—
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×Dic7C2×D412yes—
92C2×Dic7C2×D414yes—
93C2×Dic7C2×D412yes—
94C2×Dic7C2×Q814yes—
95C2×Dic7C2×Q814yes—
96Dic14—28no—
97Dic14—44no—
98C22×Q8—71no—
99C22×Q8—71no—
100C22×Q8C2×D711yescharacteristic
101C4⋊Q8—144no—
102C4⋊Q8—144no—
103C4⋊Q8—144no—
104C4×Q8—144no—
105C2×C4⋊C4—72no—
106C2×C4⋊C4—72no—
107C2×C4⋊C4—72no—
108C2×C4⋊C4—74no—
109C2×C4⋊C4—72no—
110C2×C4⋊C4—74no—
111C2×C4⋊C4C2×D711yescharacteristic
112C2×C42—71no—
113C2×C42—72no—
114C2×C42—71no—
115C2×C42C2×D711yescharacteristic
116#95—72no—
117#95—74no—
118#95C2×D714yes—
119C22×C7C22×D411yescharacteristic
120C22×C7C22×D411yescharacteristic
121C22×C7C22×Q812yes—
122C22×C7C2×#4111yescharacteristic
123C22×C72−1+412yes—
124C4×C7—24no—
125C4×C7—42no—
126C4×C7—42no—
127C4×C7#12114yes—
128Dic7—24no—
129Dic7—22no—
130Dic7—24no—
131Dic7—42no—
132Dic7C4⋊Q814yescomplemented
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—74no—
137C2×Q8—142no—
138C2×Q8—142no—
139C2×Q8—142no—
140C2×Q8—142no—
141C2×Q8—142no—
142C22×C4—71no—
143C22×C4—72no—
144C22×C4—71no—
145C22×C4—72no—
146C22×C4—72no—
147C22×C4C22×D712yes—
148C22×C4C22×D712yes—
149C22×C4C22×D712yes—
150C22×C4C22×D711yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—78no—
153C4⋊C4—144no—
154C4⋊C4—144no—
155C4⋊C4—144no—
156C4⋊C4—148no—
157C4⋊C4—148no—
158C42—22no—
159C42—74no—
160C42—144no—
161C42—142no—
162C2×C7C2×#12111yescharacteristic
163C2×C7C2×C4⋊Q811yescharacteristic
164C2×C7#53611yescharacteristic
165C2×C7D4×Q812yes—
166C2×C7#55712yes—
167C23C23×D711yescharacteristic
168Q8—44no—
169Q8—148no—
170Q8—284no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—72no—
179C2×C4—74no—
180C2×C4—72no—
181C2×C4—74no—
182C2×C4—74no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—142no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—142no—
190C2×C4—142no—
191C2×C4#115712yes—
192C2×C4D4×D712yes—
193C2×C4D4×D712yes—
194C2×C4Q8×D714yes—
195C7#254611yescharacteristic, complemented, Sylow
196C22C2×#115711yescharacteristic
197C22C2×D4×D711yescharacteristic
198C22C2×D4×D711yescharacteristic
199C22C2×Q8×D712yes—
200C22#655412yes—
201C4—24no—
202C4—42no—
203C4—42no—
204C4—74no—
205C4—144no—
206C4—144no—
207C4—142no—
208C4—282no—
209C4#644714yes—
210C2C2×#644711yescharacteristic
211C2#9025812yes—
212C2#9032312yes—
213C2#9036611yescharacteristic
214C2D7×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#391C2
hypercenter#391D7
hypocenterC7#2546
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
57. Dic7⋊C4Q8semidirect
132. Dic7C4⋊Q8semidirect
195. C7#2546semidirect

Maximal subgroups

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

subgroupindex
C2×#65072
C2×#64502
C2×#64322
C7×#3912
#897462
C4×Dic7⋊C42
#894292
#894192
#25467