Finite Group Databasebeta

← #11738451

Subgroups — #11738451

Order 896.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#90412C211yesmaximal, characteristic
2#90261C211yesmaximal, characteristic
3C7×#475C211yesmaximal, characteristic
4#89850C211yesmaximal, characteristic
5C4×#1144C211yesmaximal, characteristic
6#89630C212yesmaximal
7#89629C212yesmaximal
8#89623C211yesmaximal, characteristic
9#89602C211yesmaximal, characteristic
10#89571C212yesmaximal
11#89562C212yesmaximal
12C7×C4⋊Q8C2211yescharacteristic
13C4×C7×Q8C2211yescharacteristic
14Q8×Dic7—21no—
15Q8×Dic7C2212yes—
16#6507—21no—
17C2×#1144C2211yescharacteristic
18#6453—22no—
19#6453C2212yes—
20#6453C2212yes—
21#6452—22no—
22#6450C2212yes—
23#6432—21no—
24#6432C2211yescharacteristic
25C4×Dic14—21no—
26C4×Dic14C2211yescharacteristic
27C4×Dic14C2211yescharacteristic
28C7×#106C2212yes—
29C7×C4⋊C8C2211yescharacteristic
30C7×#103C2212yes—
31#6409C2211yescharacteristic
32#6391C2212yes—
33#6390C2212yes—
34Dic7⋊C8C2212yes—
35C8×Dic7C2212yes—
36#6384C2211yescharacteristic
37#6381C2212yes—
38#6381C2211yescharacteristic
39#6381C2212yes—
40C4×C7⋊C8C2211yescharacteristic
41#3383—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8C2311yescharacteristic
44C2×Dic14—21no—
45C2×Dic14D411yescharacteristic
46C2×Dic14C2311yescharacteristic
47C2×C7×C8C2312yes—
48C7×C4⋊C4—21no—
49C7×C4⋊C4—21no—
50C7×C4⋊C4C2312yes—
51C7×C4⋊C4C2311yescharacteristic
52C42×C7—21no—
53C42×C7C2311yescharacteristic
54#1144Q814yescomplemented
55C4⋊Dic7—21no—
56C4⋊Dic7—22no—
57C4⋊Dic7—22no—
58C4⋊Dic7—22no—
59C4⋊Dic7D411yescharacteristic
60C4⋊Dic7C2311yescharacteristic
61C4⋊Dic7C2312yes—
62Dic7⋊C4—21no—
63Dic7⋊C4—22no—
64Dic7⋊C4—22no—
65Dic7⋊C4—22no—
66Dic7⋊C4D412yes—
67C4×Dic7—21no—
68C4×Dic7—22no—
69C4×Dic7—22no—
70C4×Dic7C2312yes—
71C2×C7⋊C8C2311yescharacteristic
72C2×C7⋊C8C2311yescharacteristic
73#557—71no—
74#557—71no—
75#500—71no—
76#499—72no—
77#479—72no—
78#479—72no—
79#475—71no—
80#475D711yescharacteristic
81C8×Q8—71no—
82C4×Q16—71no—
83C4×Q16—72no—
84C7×Q8—21no—
85C7×Q8—42no—
86C7×Q8C2×Q812yes—
87C2×C4×C7—21no—
88C2×C4×C7—22no—
89C2×C4×C7—21no—
90C2×C4×C7C2×D411yescharacteristic
91C2×C4×C7C2×D411yescharacteristic
92C2×C4×C7C2411yescharacteristic
93C2×Dic7—21no—
94C2×Dic7—21no—
95C2×Dic7—22no—
96C2×Dic7C2×D412yes—
97C2×Dic7C2×D412yes—
98Dic14—22no—
99Dic14—21no—
100Dic14—42no—
101Dic14C2×Q812yes—
102C7×C8—22no—
103C7⋊C8—21no—
104C7⋊C8C2×Q812yes—
105C2×Q16—71no—
106C4⋊Q8—71no—
107C4⋊Q8—141no—
108C4⋊Q8—141no—
109C4⋊Q8C2×D711yescharacteristic
110#125—72no—
111#125—72no—
112#125—142no—
113#125—142no—
114C4×Q8—71no—
115C4×Q8—72no—
116C4×Q8—72no—
117C4×Q8—71no—
118C4×Q8—141no—
119C4×Q8—141no—
120C4×Q8C2×D711yescharacteristic
121#107—71no—
122#107—72no—
123#107—72no—
124#106—72no—
125#106C2×D712yes—
126C4⋊C8—72no—
127C4⋊C8C2×D711yescharacteristic
128#103—71no—
129#103—72no—
130#103—71no—
131#103C2×D712yes—
132C4×C8—72no—
133C4×C8—71no—
134C22×C7C22×D411yescharacteristic
135C4×C7—22no—
136C4×C7—21no—
137C4×C7—42no—
138C4×C7—41no—
139C4×C7#13512yes—
140C4×C7C22×Q811yescharacteristic
141C4×C72−1+411yescharacteristic
142Dic7—22no—
143Dic7—22no—
144Dic7—22no—
145Dic7—42no—
146Dic7—41no—
147C2×Q8—21no—
148C2×Q8—71no—
149C2×Q8—71no—
150C2×Q8—141no—
151C2×Q8C22×D711yescharacteristic
152Q16—74no—
153C2×C8—71no—
154C2×C8—71no—
155C2×C8C22×D712yes—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—72no—
159C4⋊C4—71no—
160C4⋊C4—71no—
161C4⋊C4—72no—
162C4⋊C4—142no—
163C4⋊C4—142no—
164C4⋊C4—142no—
165C4⋊C4—142no—
166C4⋊C4—142no—
167C4⋊C4—142no—
168C4⋊C4—141no—
169C4⋊C4—141no—
170C4⋊C4C22×D711yescharacteristic
171C4⋊C4C22×D712yes—
172C42—21no—
173C42—72no—
174C42—142no—
175C42—142no—
176C42—141no—
177C42C22×D711yescharacteristic
178C2×C7D4×Q811yescharacteristic
179C2×C7C2×#13511yescharacteristic
180C2×C7#57811yescharacteristic
181Q8—21no—
182Q8—42no—
183Q8—72no—
184Q8—142no—
185Q8—141no—
186Q8—282no—
187Q8Q8×D712yes—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—72no—
192C2×C4—72no—
193C2×C4—141no—
194C2×C4—141no—
195C2×C4—142no—
196C2×C4D4×D711yescharacteristic
197C2×C4D4×D711yescharacteristic
198C2×C4C23×D711yescharacteristic
199C8—22no—
200C8—72no—
201C8—141no—
202C7#338311yescharacteristic, complemented, Sylow
203C22C2×D4×D711yescharacteristic
204C4—21no—
205C4—22no—
206C4—42no—
207C4—41no—
208C4—142no—
209C4—142no—
210C4—142no—
211C4—282no—
212C4—281no—
213C4#647912yes—
214C4C2×Q8×D711yescharacteristic
215C4#655411yescharacteristic
216C2#9032311yescharacteristic
217C2C2×#647911yescharacteristic
218C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#475C2
socle, prime radicalC22×C7C22×D4
hypercenter#475D7
hypocenterC7#3383
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
54. #1144Q8semidirect
202. C7#3383semidirect

Maximal subgroups

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

subgroupindex
#904122
#902612
C7×#4752
#898502
C4×#11442
#896302
#896292
#896232
#896022
#895712
#895622
#33837