Finite Group Databasebeta

← #11738485

Subgroups — #11738485

Order 896.

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

normal subgroups only

This group has 360 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#90418C211yesmaximal, characteristic
2#90261C211yesmaximal, characteristic, complemented
3C7×#475C211yesmaximal, characteristic, complemented
4#89850C211yesmaximal, characteristic, complemented
5C4×#1143C211yesmaximal, characteristic
6#89643C212yesmaximal
7#89628C212yesmaximal, complemented
8#89618C211yesmaximal, characteristic
9#89603C211yesmaximal, characteristic, complemented
10#89589C212yesmaximal
11#89566C212yesmaximal, complemented
12C2×Q8×D7—21no—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×Q8C2211yescharacteristic
15#6509—21no—
16Q8×Dic7C2212yes—
17C2×#1143C2211yescharacteristic
18#6459—22no—
19#6456C2212yes—
20#6453—22no—
21#6453C2212yes—
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#6392C2212yes—
34#6389C2212yes—
35Dic7⋊C8C2212yes—
36#6383C2211yescharacteristic
37#6382C2211yescharacteristic
38#6381C2212yes—
39#6381C2212yes—
40C4×C7⋊C8C2211yescharacteristic
41#3386—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8C2311yescharacteristic
44Q8×D7—44no—
45C2×D28C2311yescharacteristic
46C2×C4×D7—21no—
47C2×C4×D7—22no—
48C2×Dic14—21no—
49C2×Dic14—22no—
50C2×Dic14D411yescharacteristic
51C2×C7×C8C2312yes—
52C7×C4⋊C4—21no—
53C7×C4⋊C4—21no—
54C7×C4⋊C4C2312yes—
55C7×C4⋊C4C2311yescharacteristic
56C42×C7—21no—
57C42×C7C2311yescharacteristic
58#1143Q814yescomplemented
59(C2×D7)⋊C4—21no—
60(C2×D7)⋊C4—22no—
61C4⋊Dic7—21no—
62C4⋊Dic7—22no—
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
74#557—71no—
75D4×Q8—71no—
76#501—72no—
77#500—71no—
78#478—72no—
79#477—72no—
80#475D711yescharacteristic, complemented
81#474—71no—
82#446—71no—
83#440—72no—
84C4×SD16—71no—
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—44no—
102C4×D7—42no—
103Dic14—22no—
104Dic14—44no—
105Dic14—42no—
106C7×C8—22no—
107C7⋊C8—21no—
108C7⋊C8C2×Q812yescomplemented
109C22×Q8—141no—
110C2×SD16—71no—
111C4⋊Q8—72no—
112C4⋊Q8—71no—
113C4⋊Q8—141no—
114C4⋊Q8C2×D711yescharacteristic
115#125—72no—
116#125—142no—
117C22⋊Q8—142no—
118C22⋊Q8—141no—
119C4×Q8—71no—
120C4×Q8—72no—
121C4×Q8—141no—
122C4×Q8C2×D711yescharacteristic
123C4×D4—72no—
124C4×D4—71no—
125#107—72no—
126#107—72no—
127#107—72no—
128#106—71no—
129#106C2×D712yes—
130C4⋊C8—72no—
131C4⋊C8C2×D711yescharacteristic
132#103—71no—
133#103C2×D712yes—
134#102—72no—
135#102—71no—
136#97—72no—
137C4×C8—71no—
138C22×C7C22×D411yescharacteristic
139C2×D7—22no—
140C2×D7—41no—
141C2×D7—41no—
142C4×C7—22no—
143C4×C7—21no—
144C4×C7—42no—
145C4×C7—41no—
146C4×C7AGL(1,ℤ/8)12yes—
147C4×C7C22×Q811yescharacteristic
148C4×C72−1+411yescharacteristic
149Dic7—22no—
150Dic7—22no—
151Dic7—41no—
152Dic7—42no—
153C2×Q8—21no—
154C2×Q8—71no—
155C2×Q8—142no—
156C2×Q8—141no—
157C2×Q8—284no—
158C2×Q8C22×D711yescharacteristic
159C2×D4—71no—
160C22×C4—142no—
161C22×C4—141no—
162SD16—74no—
163C2×C8—71no—
164C2×C8—71no—
165C2×C8C22×D712yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—72no—
169C4⋊C4—71no—
170C4⋊C4—72no—
171C4⋊C4—71no—
172C4⋊C4—142no—
173C4⋊C4—142no—
174C4⋊C4—142no—
175C4⋊C4—142no—
176C4⋊C4—141no—
177C4⋊C4C22×D711yescharacteristic
178C4⋊C4C22×D712yes—
179C22⋊C4—142no—
180C22⋊C4—141no—
181C42—21no—
182C42—72no—
183C42—142no—
184C42C22×D711yescharacteristic
185C2×C7D4×Q811yescharacteristic
186C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
187C2×C7#57811yescharacteristic
188D7—42no—
189C23—141no—
190Q8—21no—
191Q8—42no—
192Q8—142no—
193Q8—284no—
194Q8—282no—
195Q8Q8×D712yescomplemented
196D4—72no—
197D4—141no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—72no—
202C2×C4—72no—
203C2×C4—142no—
204C2×C4—141no—
205C2×C4—284no—
206C2×C4—282no—
207C2×C4D4×D711yescharacteristic
208C2×C4D4×D711yescharacteristic
209C2×C4C23×D711yescharacteristic
210C8—22no—
211C8—72no—
212C8—141no—
213C7#338611yescharacteristic, complemented, Sylow
214C22—142no—
215C22—281no—
216C22—281no—
217C22C2×D4×D711yescharacteristic
218C4—21no—
219C4—22no—
220C4—42no—
221C4—41no—
222C4—142no—
223C4—142no—
224C4—281no—
225C4—282no—
226C4#647712yes—
227C4C2×Q8×D711yescharacteristic
228C4#655411yescharacteristic
229C2—282no—
230C2#9032311yescharacteristic
231C2C2×#647711yescharacteristic
232C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#475C2
socleC22×C7C22×D4
prime radicalC2×D28C23
hypercenter#475D7
hypocenterC7#3386
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. #90261C2semidirect
3. C7×#475C2semidirect
4. #89850C2semidirect
7. #89628C2semidirect
9. #89603C2semidirect
11. #89566C2semidirect
58. #1143Q8semidirect
80. #475D7semidirect
88. C7×Q8C2×Q8semidirect
108. C7⋊C8C2×Q8semidirect
195. Q8Q8×D7semidirect
213. C7#3386semidirect

Maximal subgroups

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

subgroupindex
#904182
#902612
C7×#4752
#898502
C4×#11432
#896432
#896282
#896182
#896032
#895892
#895662
#33867