Finite Group Databasebeta

← #11739002

Subgroups — #11739002

Order 896.

456 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 129 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#90416C211yesmaximal, characteristic, complemented
2#90407C211yesmaximal, characteristic, complemented
3C7×#486C211yesmaximal, characteristic, complemented
4#89850C211yesmaximal, characteristic, complemented
5#89845C211yesmaximal, characteristic, complemented
6#89547C214yesmaximal, complemented
7#89528C214yesmaximal, complemented
8#89466C211yesmaximal, characteristic, complemented
9C4×C8×D7C211yesmaximal, characteristic, complemented
10C2×#1161—21no—
11C2×#1159—21no—
12C7×C4⋊Q8C2211yescharacteristic
13C7×#127C2211yescharacteristic
14#6510—21no—
15#6501—21no—
16#6501C2212yes—
17#6499—21no—
18D4×Dic7—21no—
19D4×Dic7C2212yes—
20C2×#1141C2214yes—
21C2×C8×D7C2212yes—
22#6458—21no—
23#6458C2212yes—
24#6456—21no—
25#6456C2212yes—
26#6437C2211yescharacteristic
27C42×D7C2211yescharacteristic
28#6432C2211yescharacteristic
29C7×#102C2214yes—
30C4×C7×C8C2211yescharacteristic
31#6395C2214yes—
32C8×Dic7C2212yes—
33#6381C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3169—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×D4—21no—
38C2×C7×D4—21no—
39C2×C7×D4C2312yes—
40C2×C22⋊D7—22no—
41C22×Dic7—22no—
42#1161—44no—
43#1159—44no—
44C2×D28—22no—
45C2×D28—21no—
46C2×D28—21no—
47C2×D28C2312yes—
48C2×C4×D7—22no—
49C2×C4×D7D411yescharacteristic
50C2×C4×D7D411yescharacteristic
51C2×C4×D7C2311yescharacteristic
52C2×Dic14—21no—
53C2×C7×C8C2312yes—
54C7×C4⋊C4—21no—
55C7×C4⋊C4C2312yes—
56C42×C7C2311yescharacteristic
57C22⋊Dic7—22no—
58C22⋊Dic7—22no—
59#1141—28no—
60(C2×D7)⋊C4—22no—
61(C2×D7)⋊C4—22no—
62C4⋊Dic7—21no—
63C4⋊Dic7C2312yes—
64C4×Dic7D411yescharacteristic
65C4×Dic7D411yescharacteristic
66C4×Dic7C2311yescharacteristic
67C2×C7⋊C8C2312yes—
68C8×D7—24no—
69#532—71no—
70#532—71no—
71#500—71no—
72#493—71no—
73#486—71no—
74#486D711yescharacteristic, complemented
75#466—74no—
76C4×D8—74no—
77C2×C4×C8—71no—
78C23×C7—22no—
79C22×D7—22no—
80C22×D7C2×D411yescharacteristic
81C7×Q8—42no—
82C7×D4—24no—
83C7×D4—42no—
84C7×D4—42no—
85C2×C4×C7—22no—
86C2×C4×C7C2×D411yescharacteristic
87C2×C4×C7C2×D411yescharacteristic
88C2×C4×C7C2411yescharacteristic
89C22⋊D7—44no—
90C2×Dic7—22no—
91C2×Dic7—44no—
92C2×Dic7C2×D411yescharacteristic
93C2×Dic7C2×D411yescharacteristic
94C2×Dic7C2×D411yescharacteristic
95D28—24no—
96D28—44no—
97D28—42no—
98D28—42no—
99C4×D7—22no—
100C4×D7—22no—
101C4×D7—44no—
102C4×D7D814yescomplemented
103Dic14—42no—
104C7×C8—22no—
105C7⋊C8#4114yescomplemented
106C2×#41—141no—
107C2×#41—141no—
108C2×D8—74no—
109C22×C8—72no—
110C4⋊Q8—71no—
111C4⋊Q8C2×D711yescharacteristic
112#127—71no—
113#127C2×D711yescharacteristic
114#124—141no—
115#124—141no—
116#121—72no—
117#121—72no—
118#121—141no—
119#121—141no—
120C4×D4—72no—
121C4×D4—72no—
122C4×D4—141no—
123C4×D4—141no—
124C2×C42—71no—
125#107—74no—
126#102—74no—
127#102C2×D714yes—
128C4×C8—72no—
129C4×C8—71no—
130C4×C8C2×D711yescharacteristic
131C22×C7—42no—
132C22×C7—42no—
133C22×C7—42no—
134C22×C7C22×D411yescharacteristic
135C2×D7—21no—
136C2×D7—21no—
137C2×D7—42no—
138C2×D7—42no—
139C2×D7—42no—
140C2×D7C2×D812yes—
141C4×C7—42no—
142C4×C7C2×D812yes—
143C4×C7#13512yes—
144C4×C7C2×#4112yes—
145Dic7—21no—
146Dic7—21no—
147Dic7—42no—
148Dic7C2×D812yes—
149#41—284no—
150#41—284no—
151C2×Q8—21no—
152C2×Q8—141no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—72no—
156C2×D4—142no—
157C2×D4—142no—
158C2×D4—141no—
159C2×D4—141no—
160C2×D4C22×D712yes—
161C22×C4—71no—
162C22×C4—71no—
163C22×C4—71no—
164C22×C4—142no—
165C22×C4—142no—
166D8—148no—
167C2×C8—72no—
168C2×C8—144no—
169C2×C8C22×D712yes—
170C4⋊C4—21no—
171C4⋊C4—72no—
172C4⋊C4—141no—
173C4⋊C4C22×D712yes—
174C22⋊C4—142no—
175C22⋊C4—142no—
176C22⋊C4—142no—
177C22⋊C4—142no—
178C42—71no—
179C42—71no—
180C42—71no—
181C42C22×D711yescharacteristic
182C2×C7—82no—
183C2×C7#53211yescharacteristic
184C2×C7C22×D811yescharacteristic
185C2×C7C2×#13511yescharacteristic
186D7—22no—
187D7—82no—
188C23—22no—
189C23—71no—
190C23—142no—
191Q8—42no—
192Q8—282no—
193D4—24no—
194D4—42no—
195D4—42no—
196D4—144no—
197D4—284no—
198D4—284no—
199D4—282no—
200D4—282no—
201C2×C4—22no—
202C2×C4—71no—
203C2×C4—74no—
204C2×C4—71no—
205C2×C4—71no—
206C2×C4—142no—
207C2×C4—142no—
208C2×C4—142no—
209C2×C4—284no—
210C2×C4—284no—
211C2×C4D4×D711yescharacteristic
212C2×C4D4×D711yescharacteristic
213C2×C4C23×D711yescharacteristic
214C8—22no—
215C8—74no—
216C7#316911yescharacteristic, complemented, Sylow
217C22—42no—
218C22—42no—
219C22—42no—
220C22—72no—
221C22—141no—
222C22—141no—
223C22—282no—
224C22—282no—
225C22—282no—
226C22C2×D4×D711yescharacteristic
227C4—42no—
228C4—72no—
229C4—141no—
230C4—141no—
231C4—282no—
232C4D7×D812yes—
233C4#647912yes—
234C4D7×#4112yes—
235C2—82no—
236C2—142no—
237C2—562no—
238C2#9035911yescharacteristic
239C2C2×D7×D811yescharacteristic
240C2C2×#647911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#486C2
socleC22×C7C22×D4
hypercenter#486D7
hypocenterC7#3169
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #90416C2semidirect
2. #90407C2semidirect
3. C7×#486C2semidirect
4. #89850C2semidirect
5. #89845C2semidirect
6. #89547C2semidirect
7. #89528C2semidirect
8. #89466C2semidirect
9. C4×C8×D7C2semidirect
74. #486D7semidirect
102. C4×D7D8semidirect
105. C7⋊C8#41semidirect
216. C7#3169semidirect

Maximal subgroups

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

subgroupindex
#904162
#904072
C7×#4862
#898502
#898452
#895472
#895282
#894662
C4×C8×D72
#31697