Finite Group Databasebeta

← #11736664

Subgroups — #11736664

Order 896.

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

normal subgroups only

This group has 418 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#90216C211yesmaximal, characteristic
2#90195C211yesmaximal, characteristic, complemented
3C7×#428C211yesmaximal, characteristic, complemented
4#89903C211yesmaximal, characteristic, complemented
5#89901C211yesmaximal, characteristic, complemented
6#89770C211yesmaximal, characteristic, complemented
7C2×#6383C211yesmaximal, characteristic, complemented
8#89657C212yesmaximal, complemented
9#89654C212yesmaximal, complemented
10#89638C212yesmaximal, complemented
11#89635C212yesmaximal, complemented
12C2×#1157C2211yescharacteristic
13C22×D28C2211yescharacteristic
14C2×C7×M16C2211yescharacteristic
15C7×#117C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C4×C22⋊D7—22no—
18#6489C2211yescharacteristic
19C22×C7⋊C8C2211yescharacteristic
20#6458C2212yes—
21#6458C2212yes—
22#6456—21no—
23#6455C2212yescomplemented
24D7×C4⋊C4C2212yescomplemented
25#6450—21no—
26#6445—22no—
27#6438—21no—
28#6437—21no—
29C7×#107C2212yescomplemented
30C7×#106C2212yescomplemented
31#6395C2212yes—
32#6395C2212yes—
33(C2×D7)⋊C8C2212yescomplemented
34(C2×D7)⋊C8C2212yescomplemented
35#6383C2212yes—
36#6383C2212yes—
37#6382C2211yescharacteristic, complemented
38#6382C2211yescharacteristic, complemented
39#6381C2211yescharacteristic, complemented
40#6381C2211yescharacteristic, complemented
41#3096—71nomaximal, Sylow
42C23×D7—21no—
43C22×C4×C7—21no—
44C22×C4×C7C2311yescharacteristic
45C2×C22⋊D7D412yes—
46#1157—22no—
47#1157—22no—
48C2×D28—22no—
49C2×D28—22no—
50C2×D28—21no—
51C2×D28—22no—
52C2×D28—22no—
53C2×D28D411yescharacteristic
54C2×D28C2312yes—
55C2×C4×D7—22no—
56C2×C4×D7C2312yes—
57C2×Dic14D411yescharacteristic
58C7×M16—22no—
59C2×C7×C8C2312yes—
60C7×C4⋊C4—21no—
61C7×C4⋊C4C2311yescharacteristic
62C7×C4⋊C4C2312yes—
63C7×C4⋊C4C2311yescharacteristic
64C7×C22⋊C4—21no—
65C42×C7—21no—
66C22⋊Dic7—21no—
67(C2×D7)⋊C4—22no—
68(C2×D7)⋊C4—22no—
69(C2×D7)⋊C4—22no—
70C4⋊Dic7C2311yescharacteristic
71C4⋊Dic7C2311yescharacteristic
72Dic7⋊C4—22no—
73C4×Dic7—21no—
74C4×Dic7—22no—
75C2×C7⋊C8—22no—
76C2×C7⋊C8C2311yescharacteristic
77C2×C7⋊C8C2311yescharacteristic
78#535—71no—
79#520—71no—
80#482—72no—
81#482—72no—
82#481—72no—
83#480—72no—
84#428D711yescharacteristic, complemented
85#427—71no—
86#418—71no—
87C2×#102—71no—
88#408—71no—
89C23×C7C2×D411yescharacteristic
90C22×D7—22no—
91C22×D7—42no—
92C22×D7—81no—
93C22×D7—81no—
94C22×D7C2×D412yes—
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—81no—
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7#4112yes—
101C2×C4×C7#4112yes—
102C2×C4×C7C2411yescharacteristic
103C22⋊D7—24no—
104C2×Dic7—21no—
105C2×Dic7—21no—
106C2×Dic7C2×D412yes—
107D28—22no—
108D28—22no—
109D28—22no—
110D28—42no—
111D28—82no—
112D28—82no—
113C4×D7—22no—
114C4×D7—22no—
115C4×D7—82no—
116Dic14—22no—
117C7×C8—22no—
118C7⋊C8—21no—
119C7⋊C8—21no—
120C2×#41—71no—
121C22×D4—71no—
122C2×M16C2×D711yescharacteristic
123C22×C8—71no—
124#127—141no—
125#124—141no—
126#121—72no—
127#121—72no—
128#120—142no—
129C4×Q8—141no—
130C4×D4—141no—
131C4×D4—142no—
132#117—71no—
133#117—72no—
134#117C2×D711yescharacteristic
135C2×C4⋊C4—72no—
136C2×C4⋊C4C2×D711yescharacteristic
137#107—71no—
138#107—71no—
139#107C2×D712yescomplemented
140#106—71no—
141#106—71no—
142#106C2×D712yescomplemented
143#102—72no—
144#102—72no—
145#102—72no—
146#102—72no—
147C22⋊C8—72no—
148C22⋊C8—72no—
149C22×C7—21no—
150C22×C7—21no—
151C22×C7#12312yes—
152C22×C7C22×D411yescharacteristic
153C2×D7—22no—
154C2×D7—22no—
155C2×D7—22no—
156C2×D7—42no—
157C2×D7—42no—
158C2×D7—42no—
159C2×D7—84no—
160C4×C7—42no—
161C4×C7—41no—
162C4×C7—41no—
163C4×C7#12312yes—
164C4×C7C2×#4111yescharacteristic
165C4×C7C2×#4111yescharacteristic
166Dic7—22no—
167Dic7—41no—
168Dic7—41no—
169C24—141no—
170#41—142no—
171#41—142no—
172C2×Q8—71no—
173C2×D4—72no—
174C2×D4—71no—
175C2×D4—72no—
176C2×D4—142no—
177C2×D4—142no—
178C2×D4—142no—
179C2×D4—141no—
180C2×D4—142no—
181C22×C4—21no—
182C22×C4—72no—
183C22×C4—142no—
184C22×C4C22×D711yescharacteristic
185M16—22no—
186C2×C8—71no—
187C2×C8—71no—
188C2×C8—142no—
189C2×C8C22×D712yes—
190C4⋊C4—21no—
191C4⋊C4—71no—
192C4⋊C4—71no—
193C4⋊C4—142no—
194C4⋊C4C22×D712yes—
195C4⋊C4C22×D711yescharacteristic
196C4⋊C4C22×D711yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—142no—
199C22⋊C4—142no—
200C22⋊C4—142no—
201C22⋊C4—141no—
202C42—21no—
203C42—142no—
204C42—141no—
205C2×C7—22no—
206C2×C7C2×#12311yescharacteristic
207C2×C7#57611yescharacteristic
208C2×C7#57711yescharacteristic
209D7—42no—
210D7—82no—
211C23—72no—
212C23—142no—
213C23—282no—
214C23—561no—
215C23—561no—
216C23D4×D711yescharacteristic
217Q8—142no—
218D4—144no—
219D4—142no—
220D4—142no—
221D4—142no—
222D4—282no—
223D4—562no—
224D4—562no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—72no—
229C2×C4—81no—
230C2×C4—142no—
231C2×C4—142no—
232C2×C4—141no—
233C2×C4—141no—
234C2×C4—562no—
235C2×C4#115712yes—
236C2×C4D4×D711yescharacteristic
237C2×C4#116112yes—
238C2×C4C23×D711yescharacteristic
239C8—22no—
240C8—141no—
241C8—141no—
242C7#309611yescharacteristic, complemented, Sylow
243C22—21no—
244C22—21no—
245C22—142no—
246C22—142no—
247C22—142no—
248C22—282no—
249C22—282no—
250C22—282no—
251C22—564no—
252C22#645712yes—
253C22C2×D4×D711yescharacteristic
254C4—42no—
255C4—41no—
256C4—41no—
257C4—142no—
258C4—281no—
259C4—281no—
260C4#645712yes—
261C4C2×#115711yescharacteristic
262C4C2×#116111yescharacteristic
263C2—22no—
264C2—282no—
265C2—562no—
266C2C2×#645711yescharacteristic
267C2#9046511yescharacteristic
268C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radical#90216C2
hypercenter#428D7
hypocenterC7#3096
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #90195C2semidirect
3. C7×#428C2semidirect
4. #89903C2semidirect
5. #89901C2semidirect
6. #89770C2semidirect
7. C2×#6383C2semidirect
8. #89657C2semidirect
9. #89654C2semidirect
10. #89638C2semidirect
11. #89635C2semidirect
23. #6455C22semidirect
24. D7×C4⋊C4C22semidirect
29. C7×#107C22semidirect
30. C7×#106C22semidirect
33. (C2×D7)⋊C8C22semidirect
34. (C2×D7)⋊C8C22semidirect
37. #6382C22semidirect
38. #6382C22semidirect
39. #6381C22semidirect
40. #6381C22semidirect
84. #428D7semidirect
139. #107C2×D7semidirect
142. #106C2×D7semidirect
242. C7#3096semidirect

Maximal subgroups

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

subgroupindex
#902162
#901952
C7×#4282
#899032
#899012
#897702
C2×#63832
#896572
#896542
#896382
#896352
#30967