Finite Group Databasebeta

← #11739323

Subgroups — #11739323

Order 896.

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

normal subgroups only

This group has 396 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#90422C211yesmaximal, characteristic, complemented
2#90365C211yesmaximal, characteristic
3C7×#497C211yesmaximal, characteristic, complemented
4#89959C212yesmaximal, complemented
5#89957C212yesmaximal, complemented
6#89941C212yesmaximal, complemented
7#89935C212yesmaximal, complemented
8#89857C211yesmaximal, characteristic, complemented
9#89828C211yesmaximal, characteristic, complemented
10#89485C211yesmaximal, characteristic, complemented
11#89481C211yesmaximal, characteristic, complemented
12C2×#1161C2211yescharacteristic
13C2×D4×D7C2211yescharacteristic
14C2×C7×Q16C2212yescomplemented
15C2×C7×SD16C2212yes—
16C7×C4⋊Q8C2211yescharacteristic, complemented
17C7×#124C2211yescharacteristic
18#6510C2211yescharacteristic
19#6509C2212yes—
20Q8×Dic7C2212yescomplemented
21C2×#1143C2212yes—
22C2×#1143C2212yescomplemented
23#6503C2211yescharacteristic
24#6461—22no—
25#6452—22no—
26#6447—22no—
27#6445—22no—
28#6437C2211yescharacteristic
29#6435C2211yescharacteristic
30#6433C2211yescharacteristic, complemented
31C7×#97C2211yescharacteristic, complemented
32#6409C2212yescomplemented
33#6406C2212yes—
34#6395C2212yes—
35#6395C2212yescomplemented
36(C2×D7)⋊C8C2212yes—
37Dic7⋊C8C2212yescomplemented
38C4×C7⋊C8C2211yescharacteristic, complemented
39#3222—71nomaximal, Sylow
40C23×D7—21no—
41C2×C7×Q8C2311yescharacteristic
42C2×C7×Q8C2312yes—
43C2×C7×D4C2311yescharacteristic
44C2×C22⋊D7—21no—
45C2×C22⋊D7—22no—
46#1161—42no—
47D4×D7—42no—
48C2×D28—21no—
49C2×D28—22no—
50C2×D28C2311yescharacteristic
51C2×D28C2311yescharacteristic
52C2×C4×D7—21no—
53C2×C4×D7C2311yescharacteristic
54C7×Q16—42no—
55C7×SD16—42no—
56C2×C7×C8C2312yes—
57C7×C4⋊C4—22no—
58C7×C22⋊C4—22no—
59C42×C7C2311yescharacteristic
60#1143—42no—
61#1143—42no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4D412yes—
65C4⋊Dic7—22no—
66C4⋊Dic7C2312yes—
67Dic7⋊C4—22no—
68Dic7⋊C4—22no—
69Dic7⋊C4D412yes—
70C4×Dic7—22no—
71C4×Dic7C2311yescharacteristic
72C2×C7⋊C8C2312yes—
73#541—71no—
74#535—71no—
75#497D711yescharacteristic, complemented
76#492—71no—
77#490—71no—
78#488—71no—
79#464—72no—
80#460—72no—
81#452—72no—
82#450—72no—
83#433—71no—
84C23×C7—21no—
85C22×D7—21no—
86C22×D7—21no—
87C22×D7—42no—
88C22×D7—82no—
89C22×D7C2×D411yescharacteristic
90C7×Q8—22no—
91C7×Q8—22no—
92C7×Q8—22no—
93C7×D4—22no—
94C2×C4×C7—22no—
95C2×C4×C7—21no—
96C2×C4×C7C2×D412yes—
97C2×C4×C7C2411yescharacteristic
98C22⋊D7—42no—
99C22⋊D7—82no—
100C2×Dic7—22no—
101C2×Dic7C2×D412yes—
102C2×Dic7C2×D411yescharacteristic
103D28—22no—
104D28—22no—
105D28—42no—
106D28—82no—
107C4×D7—22no—
108C4×D7—42no—
109C7×C8—22no—
110C7⋊C8—22no—
111C2×#41—71no—
112C22×D4—71no—
113C2×Q16C2×D712yescomplemented
114C2×SD16—72no—
115C2×SD16—72no—
116C2×SD16C2×D712yes—
117C4⋊Q8C2×D711yescharacteristic, complemented
118#127—71no—
119#127—71no—
120#126—142no—
121#125—71no—
122#125—142no—
123#124—71no—
124#124C2×D711yescharacteristic
125C22⋊Q8—72no—
126#121—142no—
127#120—142no—
128C4×Q8—72no—
129#117—71no—
130C4⋊C8—72no—
131#103—72no—
132#102—72no—
133#102—72no—
134#102—72no—
135C22⋊C8—72no—
136#97C2×D711yescharacteristic, complemented
137C4×C8—71no—
138C22×C7—41no—
139C22×C7—42no—
140C22×C7C22×D411yescharacteristic
141C2×D7—21no—
142C2×D7—22no—
143C2×D7—41no—
144C2×D7—41no—
145C2×D7—42no—
146C2×D7—42no—
147C2×D7—84no—
148C4×C7—22no—
149C4×C7—41no—
150C4×C7—42no—
151C4×C7H(5,𝔽2)12yes—
152Dic7—21no—
153Dic7—22no—
154Dic7—42no—
155C24—141no—
156#41—282no—
157C2×Q8C22×D711yescharacteristic
158C2×Q8C22×D712yes—
159C2×D4—71no—
160C2×D4—71no—
161C2×D4—141no—
162C2×D4—141no—
163C2×D4—142no—
164C2×D4—142no—
165C2×D4—282no—
166C2×D4C22×D711yescharacteristic
167C22×C4—71no—
168C22×C4—141no—
169Q16—42no—
170SD16—42no—
171SD16—282no—
172SD16—282no—
173C2×C8—72no—
174C2×C8C22×D712yes—
175C4⋊C4—22no—
176C4⋊C4—72no—
177C4⋊C4—72no—
178C4⋊C4—142no—
179C4⋊C4—142no—
180C4⋊C4—142no—
181C22⋊C4—22no—
182C22⋊C4—72no—
183C22⋊C4—142no—
184C22⋊C4—142no—
185C42—71no—
186C42—142no—
187C42C22×D711yescharacteristic
188C2×C7—81no—
189C2×C7#53511yescharacteristic
190C2×C7#57712yes—
191D7—41no—
192D7—81no—
193D7—81no—
194C23—21no—
195C23—71no—
196C23—141no—
197C23—141no—
198C23—282no—
199C23—562no—
200Q8—22no—
201Q8—22no—
202Q8—22no—
203D4—22no—
204D4—142no—
205D4—142no—
206D4—282no—
207D4—282no—
208D4—562no—
209D4—562no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—72no—
213C2×C4—71no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—282no—
217C2×C4D4×D712yes—
218C2×C4C23×D711yescharacteristic
219C8—22no—
220C8—142no—
221C7#322211yescharacteristic, complemented, Sylow
222C22—41no—
223C22—42no—
224C22—141no—
225C22—142no—
226C22—281no—
227C22—281no—
228C22—282no—
229C22—282no—
230C22—564no—
231C22C2×D4×D711yescharacteristic
232C4—22no—
233C4—41no—
234C4—42no—
235C4—141no—
236C4—142no—
237C4—282no—
238C4#654812yes—
239C2—81no—
240C2—281no—
241C2—561no—
242C2—561no—
243C2#9040611yescharacteristic
244C2#9046912yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#497C2
socleC22×C7C22×D4
prime radical#90365C2
hypercenter#497D7
hypocenterC7#3222
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #90422C2semidirect
3. C7×#497C2semidirect
4. #89959C2semidirect
5. #89957C2semidirect
6. #89941C2semidirect
7. #89935C2semidirect
8. #89857C2semidirect
9. #89828C2semidirect
10. #89485C2semidirect
11. #89481C2semidirect
14. C2×C7×Q16C22semidirect
16. C7×C4⋊Q8C22semidirect
20. Q8×Dic7C22semidirect
22. C2×#1143C22semidirect
30. #6433C22semidirect
31. C7×#97C22semidirect
32. #6409C22semidirect
35. #6395C22semidirect
37. Dic7⋊C8C22semidirect
38. C4×C7⋊C8C22semidirect
75. #497D7semidirect
113. C2×Q16C2×D7semidirect
117. C4⋊Q8C2×D7semidirect
136. #97C2×D7semidirect
221. C7#3222semidirect

Maximal subgroups

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

subgroupindex
#904222
#903652
C7×#4972
#899592
#899572
#899412
#899352
#898572
#898282
#894852
#894812
#32227