Finite Group Databasebeta

← #11738355

Subgroups — #11738355

Order 896.

418 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 109 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#90318C212yesmaximal, complemented
2C7×#470C211yesmaximal, characteristic, complemented
3#89959C212yesmaximal, complemented
4#89907C211yesmaximal, characteristic, complemented
5#89898C211yesmaximal, characteristic, complemented
6#89821C212yesmaximal, complemented
7#89638C212yesmaximal, complemented
8#89587C212yesmaximal, complemented
9#89583C212yesmaximal, complemented
10C2×#1161C2212yes—
11C22×C4×D7C2211yescharacteristic
12C2×C7×Q16C2211yescharacteristic, complemented
13C2×C7×M16C2211yescharacteristic
14C7×C22⋊Q8C2212yes—
15#6509C2212yes—
16C2×#1143C2212yescomplemented
17#6493C2212yes—
18C2×#1136C2211yescharacteristic
19C2×D56C2211yescharacteristic, complemented
20#6458C2212yes—
21#6457—22no—
22#6456—22no—
23#6455C2212yes—
24#6445—22no—
25(C22⋊D7)⋊C4—22no—
26C7×#106C2211yescharacteristic, complemented
27C7×#103C2212yescomplemented
28#6409C2212yescomplemented
29#6395C2212yescomplemented
30(C2×D7)⋊C8C2212yes—
31(C2×D7)⋊C8C2212yes—
32#6391C2211yescharacteristic, complemented
33#6383C2212yescomplemented
34#6381C2212yescomplemented
35#3197—71nomaximal, Sylow
36C23×D7D411yescharacteristic
37C2×C7×Q8C2312yes—
38C22×C4×C7C2311yescharacteristic
39C2×C22⋊D7—22no—
40C22×Dic7D411yescharacteristic
41#1161—42no—
42#1161—42no—
43C2×D28—22no—
44C2×D28—22no—
45C2×D28C2312yes—
46C2×C4×D7—22no—
47C2×C4×D7—41no—
48C2×C4×D7—41no—
49C2×C4×D7D412yes—
50C2×C4×D7C2312yes—
51C7×Q16—41no—
52C7×M16—41no—
53C2×C7×C8C2311yescharacteristic
54C2×C7×C8C2311yescharacteristic
55C7×C4⋊C4—22no—
56C7×C4⋊C4C2312yes—
57C7×C22⋊C4—22no—
58#1143—42no—
59(C2×D7)⋊C4—22no—
60(C2×D7)⋊C4—22no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—22no—
63C4⋊Dic7C2312yes—
64Dic7⋊C4—22no—
65C4×Dic7—22no—
66#1136—41no—
67C2×C7⋊C8C2312yes—
68D56—41no—
69#525—72no—
70#485—72no—
71#482—72no—
72#470D711yescharacteristic, complemented
73#469—71no—
74#468—72no—
75#452—72no—
76#449—72no—
77#407—71no—
78C23×C7C2×D411yescharacteristic
79C22×D7—22no—
80C22×D7—41no—
81C22×D7—42no—
82C22×D7C2×D412yes—
83C7×Q8—22no—
84C7×Q8—22no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7C2×D411yescharacteristic
90C2×C4×C7C2411yescharacteristic
91C22⋊D7—44no—
92C2×Dic7—22no—
93C2×Dic7—41no—
94C2×Dic7C2×D412yes—
95D28—22no—
96D28—22no—
97D28—44no—
98D28—44no—
99C4×D7—22no—
100C4×D7—22no—
101C4×D7—42no—
102C4×D7—44no—
103C7×C8—21no—
104C7×C8—21no—
105C7⋊C8—22no—
106C2×#41—72no—
107C23×C4—71no—
108C2×Q16C2×D711yescharacteristic, complemented
109C2×SD16—72no—
110C2×D8—71no—
111C2×M16—71no—
112C2×M16C2×D711yescharacteristic
113#123—142no—
114C22⋊Q8—72no—
115C22⋊Q8C2×D712yes—
116#121—72no—
117#121—72no—
118#120—142no—
119C4×D4—142no—
120C4×D4—142no—
121#117—72no—
122#107—72no—
123#106—71no—
124#106C2×D711yescharacteristic, complemented
125#103—72no—
126#103C2×D712yescomplemented
127#102—72no—
128#102—72no—
129C22⋊C8—72no—
130C22⋊C8—72no—
131C22×C7—21no—
132C22×C7—22no—
133C22×C7C22×D411yescharacteristic
134C2×D7—22no—
135C2×D7—22no—
136C2×D7—22no—
137C2×D7—44no—
138C2×D7—42no—
139C2×D7—42no—
140C2×D7—42no—
141C4×C7—21no—
142C4×C7—42no—
143C4×C7—42no—
144C4×C7H(5,𝔽2)11yescharacteristic
145C4×C7H(5,𝔽2)11yescharacteristic
146Dic7—22no—
147Dic7—42no—
148C24—71no—
149#41—282no—
150#41—282no—
151C2×Q8C22×D712yes—
152C2×D4—72no—
153C2×D4—142no—
154C2×D4—142no—
155C2×D4—142no—
156C22×C4—71no—
157C22×C4—72no—
158C22×C4—72no—
159C22×C4—142no—
160C22×C4—281no—
161C22×C4—281no—
162C22×C4C22×D711yescharacteristic
163Q16—41no—
164SD16—282no—
165D8—281no—
166M16—41no—
167M16—281no—
168C2×C8—72no—
169C2×C8C22×D711yescharacteristic
170C2×C8C22×D711yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—72no—
173C4⋊C4—142no—
174C4⋊C4C22×D712yes—
175C22⋊C4—22no—
176C22⋊C4—142no—
177C22⋊C4—142no—
178C22⋊C4—142no—
179C22⋊C4—142no—
180C42—142no—
181C2×C7—41no—
182C2×C7#53411yescharacteristic
183C2×C7#57512yes—
184D7—42no—
185D7—82no—
186C23—72no—
187C23—142no—
188C23—281no—
189C23—282no—
190C23D4×D711yescharacteristic
191Q8—22no—
192Q8—22no—
193D4—142no—
194D4—142no—
195D4—284no—
196D4—284no—
197D4—284no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—72no—
203C2×C4—142no—
204C2×C4—142no—
205C2×C4—142no—
206C2×C4—282no—
207C2×C4—284no—
208C2×C4—281no—
209C2×C4D4×D711yescharacteristic
210C2×C4C23×D711yescharacteristic
211C8—21no—
212C8—21no—
213C8—142no—
214C7#319711yescharacteristic, complemented, Sylow
215C22—22no—
216C22—21no—
217C22—142no—
218C22—142no—
219C22—142no—
220C22—284no—
221C22—282no—
222C22—282no—
223C22—282no—
224C22C2×D4×D711yescharacteristic
225C4—21no—
226C4—42no—
227C4—42no—
228C4—142no—
229C4—282no—
230C4#654811yescharacteristic
231C4#655311yescharacteristic
232C2—41no—
233C2—282no—
234C2—562no—
235C2#9029611yescharacteristic
236C2#9046812yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#470C2
socleC22×C7C22×D4
hypercenter#470D7
hypocenterC7#3197
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #90318C2semidirect
2. C7×#470C2semidirect
3. #89959C2semidirect
4. #89907C2semidirect
5. #89898C2semidirect
6. #89821C2semidirect
7. #89638C2semidirect
8. #89587C2semidirect
9. #89583C2semidirect
12. C2×C7×Q16C22semidirect
16. C2×#1143C22semidirect
19. C2×D56C22semidirect
26. C7×#106C22semidirect
27. C7×#103C22semidirect
28. #6409C22semidirect
29. #6395C22semidirect
32. #6391C22semidirect
33. #6383C22semidirect
34. #6381C22semidirect
72. #470D7semidirect
108. C2×Q16C2×D7semidirect
124. #106C2×D7semidirect
126. #103C2×D7semidirect
214. C7#3197semidirect

Maximal subgroups

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

subgroupindex
#903182
C7×#4702
#899592
#899072
#898982
#898212
#896382
#895872
#895832
#31977