Finite Group Databasebeta

← #11738484

Subgroups — #11738484

Order 896.

370 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 133 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#90418C211yesmaximal, characteristic
2Q8×Dic14C211yesmaximal, characteristic, complemented
3C7×#475C211yesmaximal, characteristic, complemented
4#89851C211yesmaximal, characteristic, complemented
5C4×#1143C211yesmaximal, characteristic
6#89642C212yesmaximal
7#89627C212yesmaximal, complemented
8#89614C211yesmaximal, characteristic
9#89602C211yesmaximal, characteristic, complemented
10#89563C212yesmaximal, complemented
11#89560C212yesmaximal
12C2×Q8×D7—21no—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×Q8C2211yescharacteristic
15#6509—21no—
16Q8×Dic7C2212yes—
17C2×#1143C2211yescharacteristic
18#6459—22no—
19#6456C2212yes—
20#6451—22no—
21#6451C2212yes—
22#6451C2212yes—
23C4×D28C2211yescharacteristic
24#6432—21no—
25#6432C2211yescharacteristic
26C4×Dic14—21no—
27C4×Dic14C2211yescharacteristic
28C7×#106C2212yes—
29C7×C4⋊C8C2211yescharacteristic
30C7×#103C2212yes—
31#6409C2211yescharacteristic
32#6395C2212yes—
33#6391C2212yes—
34Dic7⋊C8C2212yes—
35C8×Dic7C2212yes—
36#6383C2211yescharacteristic
37#6382C2211yescharacteristic
38#6382C2212yes—
39#6382C2212yes—
40C4×C7⋊C8C2211yescharacteristic
41Q8×SD16—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8C2311yescharacteristic
44Q8×D7—44no—
45C2×D28C2311yescharacteristic
46C2×C4×D7—21no—
47C2×C4×D7—22no—
48C2×Dic14—22no—
49C2×Dic14—21no—
50C2×Dic14—22no—
51C2×Dic14D411yescharacteristic
52C2×C7×C8C2312yes—
53C7×C4⋊C4—21no—
54C7×C4⋊C4—21no—
55C7×C4⋊C4C2312yes—
56C7×C4⋊C4C2311yescharacteristic
57C42×C7—21no—
58C42×C7C2311yescharacteristic
59#1143Q814yescomplemented
60(C2×D7)⋊C4—21no—
61(C2×D7)⋊C4—22no—
62C4⋊Dic7—21no—
63C4⋊Dic7—22no—
64C4⋊Dic7D411yescharacteristic
65C4⋊Dic7C2311yescharacteristic
66C4⋊Dic7C2312yes—
67Dic7⋊C4—22no—
68Dic7⋊C4—22no—
69Dic7⋊C4D412yes—
70C4×Dic7—22no—
71C4×Dic7C2312yes—
72C2×C7⋊C8C2311yescharacteristic
73C2×C7⋊C8C2311yescharacteristic
74Q82—71no—
75D4×Q8—71no—
76#498—71no—
77#498—72no—
78#476—72no—
79#476—71no—
80#475—72no—
81#475D711yescharacteristic, complemented
82C8×Q8—71no—
83C4×SD16—71no—
84C4×SD16—72no—
85C22×D7—21no—
86C7×Q8—21no—
87C7×Q8—42no—
88C7×Q8C2×Q812yescomplemented
89C2×C4×C7—21no—
90C2×C4×C7—21no—
91C2×C4×C7—22no—
92C2×C4×C7C2×D411yescharacteristic
93C2×C4×C7C2×D411yescharacteristic
94C2×C4×C7C2411yescharacteristic
95C2×Dic7—22no—
96C2×Dic7—21no—
97C2×Dic7C2×D412yes—
98C2×Dic7C2×D412yes—
99D28—21no—
100D28C2×Q812yes—
101C4×D7—42no—
102C4×D7—44no—
103Dic14—44no—
104Dic14—82no—
105Dic14—81no—
106Dic14SD1614yescomplemented
107C7×C8—22no—
108C7⋊C8—21no—
109C7⋊C8C2×Q812yescomplemented
110C22×Q8—141no—
111C2×SD16—71no—
112C4⋊Q8—72no—
113C4⋊Q8—72no—
114C4⋊Q8—71no—
115C4⋊Q8—141no—
116C4⋊Q8—142no—
117C4⋊Q8C2×D711yescharacteristic
118C22⋊Q8—142no—
119C22⋊Q8—141no—
120C4×Q8—71no—
121C4×Q8—72no—
122C4×Q8—141no—
123C4×Q8C2×D711yescharacteristic
124C4×D4—72no—
125C4×D4—71no—
126#106—72no—
127#106—72no—
128#106—71no—
129#106—72no—
130#106C2×D712yes—
131C4⋊C8—72no—
132C4⋊C8C2×D711yescharacteristic
133#103—71no—
134#103C2×D712yes—
135#102—72no—
136#102—71no—
137C4×C8—72no—
138C4×C8—71no—
139C22×C7C22×D411yescharacteristic
140C2×D7—22no—
141C2×D7—41no—
142C2×D7—41no—
143C4×C7—22no—
144C4×C7—21no—
145C4×C7—42no—
146C4×C7—41no—
147C4×C7C2×SD1612yes—
148C4×C7C22×Q811yescharacteristic
149C4×C72−1+411yescharacteristic
150Dic7—22no—
151Dic7—41no—
152Dic7—42no—
153Dic7C2×SD1614yes—
154C2×Q8—21no—
155C2×Q8—71no—
156C2×Q8—142no—
157C2×Q8—142no—
158C2×Q8—141no—
159C2×Q8—284no—
160C2×Q8C22×D711yescharacteristic
161C2×D4—71no—
162C22×C4—142no—
163C22×C4—141no—
164SD16—74no—
165C2×C8—71no—
166C2×C8—71no—
167C2×C8C22×D712yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—72no—
171C4⋊C4—71no—
172C4⋊C4—72no—
173C4⋊C4—71no—
174C4⋊C4—142no—
175C4⋊C4—142no—
176C4⋊C4—142no—
177C4⋊C4—141no—
178C4⋊C4C22×D711yescharacteristic
179C4⋊C4C22×D712yes—
180C22⋊C4—142no—
181C22⋊C4—141no—
182C42—21no—
183C42—72no—
184C42—142no—
185C42C22×D711yescharacteristic
186C2×C7D4×Q811yescharacteristic
187C2×C7C22×SD1611yescharacteristic
188C2×C7#57711yescharacteristic
189D7—42no—
190C23—141no—
191Q8—21no—
192Q8—42no—
193Q8—74no—
194Q8—284no—
195Q8—561no—
196Q8—562no—
197Q8Q8×D712yescomplemented
198D4—72no—
199D4—141no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—72no—
204C2×C4—72no—
205C2×C4—142no—
206C2×C4—141no—
207C2×C4—284no—
208C2×C4—282no—
209C2×C4D4×D711yescharacteristic
210C2×C4D4×D711yescharacteristic
211C2×C4C23×D711yescharacteristic
212C8—22no—
213C8—72no—
214C8—141no—
215C7Q8×SD1611yescharacteristic, complemented, Sylow
216C22—142no—
217C22—281no—
218C22—281no—
219C22C2×D4×D711yescharacteristic
220C4—21no—
221C4—22no—
222C4—42no—
223C4—41no—
224C4—74no—
225C4—142no—
226C4—281no—
227C4—282no—
228C4D7×SD1612yes—
229C4C2×Q8×D711yescharacteristic
230C4#655411yescharacteristic
231C2—282no—
232C2#9032311yescharacteristic
233C2C2×D7×SD1611yescharacteristic
234C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#475C2
socleC22×C7C22×D4
prime radicalC2×D28C23
hypercenter#475D7
hypocenterC7Q8×SD16
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
2. Q8×Dic14C2semidirect
3. C7×#475C2semidirect
4. #89851C2semidirect
7. #89627C2semidirect
9. #89602C2semidirect
10. #89563C2semidirect
59. #1143Q8semidirect
81. #475D7semidirect
88. C7×Q8C2×Q8semidirect
106. Dic14SD16semidirect
109. C7⋊C8C2×Q8semidirect
197. Q8Q8×D7semidirect
215. C7Q8×SD16semidirect

Maximal subgroups

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

subgroupindex
#904182
Q8×Dic142
C7×#4752
#898512
C4×#11432
#896422
#896272
#896142
#896022
#895632
#895602
Q8×SD167