Finite Group Databasebeta

← #11739387

Subgroups — #11739387

Order 896.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#90419C211yesmaximal, characteristic, complemented
2#90416C211yesmaximal, characteristic
3C7×#500C211yesmaximal, characteristic, complemented
4#89860C211yesmaximal, characteristic, complemented
5#89851C211yesmaximal, characteristic, complemented
6#89663C214yesmaximal, complemented
7#89657C214yesmaximal, complemented
8#89466C211yesmaximal, characteristic, complemented
9#89457C211yesmaximal, characteristic, complemented
10C2×#1161—21no—
11C7×C4⋊Q8C2211yescharacteristic
12C7×C4⋊Q8C2211yescharacteristic
13#6510—21no—
14#6509—21no—
15Q8×Dic7—21no—
16#6460—21no—
17#6458—21no—
18#6458C2212yes—
19#6456—21no—
20#6456C2212yes—
21#6455C2212yes—
22#6453C2212yes—
23#6450—21no—
24#6437C2211yescharacteristic
25C42×D7C2211yescharacteristic
26#6432C2211yescharacteristic
27C7×#107C2214yes—
28C4×C7×C8C2211yescharacteristic
29#6395C2214yes—
30(C2×D7)⋊C8C2212yes—
31Dic7⋊C8C2212yes—
32#6383C2214yes—
33#6382C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3228—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×Q8—21no—
38#1161—44no—
39C2×D28—22no—
40C2×D28—21no—
41C2×D28—21no—
42C2×D28C2312yes—
43C2×C4×D7—22no—
44C2×C4×D7D411yescharacteristic
45C2×C4×D7D411yescharacteristic
46C2×C4×D7C2311yescharacteristic
47C2×Dic14—21no—
48C2×C7×C8C2312yes—
49C7×C4⋊C4—21no—
50C7×C4⋊C4—21no—
51C7×C4⋊C4C2312yes—
52C7×C4⋊C4C2312yes—
53C42×C7C2311yescharacteristic
54(C2×D7)⋊C4—22no—
55(C2×D7)⋊C4—22no—
56(C2×D7)⋊C4—22no—
57C4⋊Dic7—21no—
58C4⋊Dic7—22no—
59C4⋊Dic7C2312yes—
60Dic7⋊C4—22no—
61C4×Dic7—22no—
62C4×Dic7D411yescharacteristic
63C4×Dic7D411yescharacteristic
64C4×Dic7C2311yescharacteristic
65C2×C7⋊C8C2312yes—
66#533—71no—
67#532—71no—
68#500D711yescharacteristic, complemented
69#498—71no—
70#486—71no—
71#486—71no—
72#482—74no—
73#478—74no—
74#431—71no—
75C22×D7—22no—
76C22×D7C2×D411yescharacteristic
77C7×Q8—42no—
78C7×Q8—42no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7C2×D411yescharacteristic
82C2×C4×C7C2×D411yescharacteristic
83C2×C4×C7C2411yescharacteristic
84C2×Dic7—22no—
85C2×Dic7C2×D411yescharacteristic
86C2×Dic7C2×D411yescharacteristic
87C2×Dic7C2×D411yescharacteristic
88D28—24no—
89D28—44no—
90D28—42no—
91D28—42no—
92C4×D7—22no—
93C4×D7—22no—
94C4×D7—22no—
95C4×D7—44no—
96Dic14—42no—
97C7×C8—22no—
98C7⋊C8—22no—
99C2×#41—141no—
100C4⋊Q8—71no—
101C4⋊Q8C2×D711yescharacteristic
102C4⋊Q8C2×D711yescharacteristic
103#127—71no—
104#125—72no—
105#124—141no—
106C22⋊Q8—141no—
107C22⋊Q8—141no—
108#121—72no—
109#121—141no—
110C4×Q8—141no—
111C4×Q8—141no—
112C4×D4—72no—
113C4×D4—141no—
114#117—72no—
115C2×C42—71no—
116#107C2×D714yes—
117#106—74no—
118C4⋊C8—72no—
119#102—74no—
120#102—74no—
121C22⋊C8—72no—
122C4×C8—71no—
123C4×C8C2×D711yescharacteristic
124C22×C7C22×D411yescharacteristic
125C2×D7—21no—
126C2×D7—21no—
127C2×D7—21no—
128C2×D7—42no—
129C2×D7—42no—
130C2×D7—42no—
131C4×C7—42no—
132C4×C7—42no—
133C4×C7#13512yes—
134C4×C7AGL(1,ℤ/8)12yes—
135C4×C72−1+412yes—
136Dic7—21no—
137Dic7—21no—
138Dic7—21no—
139Dic7—42no—
140#41—284no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—141no—
144C2×D4—72no—
145C2×D4—142no—
146C2×D4—141no—
147C2×D4—141no—
148C22×C4—71no—
149C22×C4—71no—
150C22×C4—71no—
151C22×C4—142no—
152C2×C8—72no—
153C2×C8C22×D712yes—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—72no—
157C4⋊C4—142no—
158C4⋊C4—142no—
159C4⋊C4—141no—
160C4⋊C4C22×D712yes—
161C4⋊C4C22×D712yes—
162C22⋊C4—142no—
163C22⋊C4—142no—
164C22⋊C4—142no—
165C42—71no—
166C42—71no—
167C42—71no—
168C42—142no—
169C42C22×D711yescharacteristic
170C2×C7#53611yescharacteristic
171C2×C7C2×#13511yescharacteristic
172C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
173D7—41no—
174D7—82no—
175C23—71no—
176C23—142no—
177Q8—42no—
178Q8—42no—
179Q8—282no—
180D4—144no—
181D4—284no—
182D4—282no—
183D4—282no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—71no—
187C2×C4—71no—
188C2×C4—71no—
189C2×C4—142no—
190C2×C4—142no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—284no—
194C2×C4D4×D711yescharacteristic
195C2×C4D4×D711yescharacteristic
196C2×C4C23×D711yescharacteristic
197C8—22no—
198C8—142no—
199C7#322811yescharacteristic, complemented, Sylow
200C22—141no—
201C22—141no—
202C22—141no—
203C22—282no—
204C22—282no—
205C22—282no—
206C22C2×D4×D711yescharacteristic
207C4—42no—
208C4—42no—
209C4—141no—
210C4—141no—
211C4—141no—
212C4—282no—
213C4#647412yes—
214C4#648212yes—
215C4#655112yes—
216C2—281no—
217C2—562no—
218C2#9041511yescharacteristic
219C2C2×#647411yescharacteristic
220C2C2×#648211yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#500C2
socleC22×C7C22×D4
prime radical#90416C2
hypercenter#500D7
hypocenterC7#3228
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #90419C2semidirect
3. C7×#500C2semidirect
4. #89860C2semidirect
5. #89851C2semidirect
6. #89663C2semidirect
7. #89657C2semidirect
8. #89466C2semidirect
9. #89457C2semidirect
68. #500D7semidirect
199. C7#3228semidirect

Maximal subgroups

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

subgroupindex
#904192
#904162
C7×#5002
#898602
#898512
#896632
#896572
#894662
#894572
#32287