Finite Group Databasebeta

← #11736537

Subgroups — #11736537

Order 896.

376 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 141 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#90214C211yesmaximal, characteristic
2#90210C211yesmaximal, characteristic, complemented
3C7×#424C211yesmaximal, characteristic, complemented
4#89905C211yesmaximal, characteristic
5#89893C211yesmaximal, characteristic, complemented
6#89885C211yesmaximal, characteristic
7C2×#6391C211yesmaximal, characteristic, complemented
8#89623C212yesmaximal, complemented
9#89619C212yesmaximal, complemented
10#89614C212yesmaximal, complemented
11#89601C212yesmaximal, complemented
12C2×#1157C2211yescharacteristic
13C2×C7×M16C2211yescharacteristic
14C22×C7×C8C2211yescharacteristic
15C7×#117C2211yescharacteristic
16#6489C2211yescharacteristic
17C2×C4⋊Dic7C2211yescharacteristic
18C2×C4⋊Dic7C2211yescharacteristic
19#6455—21no—
20D7×C4⋊C4—21no—
21#6453—21no—
22#6451—21no—
23(C22⋊D7)⋊C4—22no—
24#6441—22no—
25C4×D28C2212yes—
26#6433C2212yes—
27#6432C2212yes—
28C4×Dic14C2212yes—
29C7×C4⋊C8C2212yes—
30C7×C4⋊C8C2212yes—
31#6395C2211yescharacteristic
32#6395C2212yes—
33#6395C2211yescharacteristic
34#6392C2212yes—
35#6391C2212yes—
36#6391C2212yes—
37#6391C2212yes—
38#6390C2211yescharacteristic
39#6390C2212yes—
40#6390C2211yescharacteristic
41#3084—71nomaximal, Sylow
42C22×C4×C7C2311yescharacteristic
43C2×C22⋊D7—21no—
44C22×Dic7—21no—
45C22×Dic7—21no—
46#1157—22no—
47#1157Q814yescomplemented
48C2×D28C2311yescharacteristic
49C2×C4×D7—21no—
50C2×C4×D7—22no—
51C2×Dic14—22no—
52C2×Dic14C2311yescharacteristic
53C7×M16—22no—
54C2×C7×C8—22no—
55C2×C7×C8C2311yescharacteristic
56C2×C7×C8C2311yescharacteristic
57C2×C7×C8C2312yes—
58C7×C4⋊C4D411yescharacteristic
59C7×C4⋊C4D411yescharacteristic
60C7×C22⋊C4D412yes—
61C42×C7C2312yes—
62C22⋊Dic7—21no—
63(C2×D7)⋊C4—22no—
64C4⋊Dic7—22no—
65C4⋊Dic7—21no—
66C4⋊Dic7—21no—
67C4⋊Dic7C2312yes—
68C4⋊Dic7C2311yescharacteristic
69C4⋊Dic7C2312yes—
70C4⋊Dic7C2311yescharacteristic
71Dic7⋊C4—22no—
72Dic7⋊C4—22no—
73Dic7⋊C4—22no—
74C4×Dic7—22no—
75C4×Dic7—21no—
76#544—71no—
77#520—71no—
78#479—72no—
79#478—72no—
80#476—72no—
81#475—72no—
82#428—71no—
83C2×#106—71no—
84#424D711yescharacteristic, complemented
85#417—71no—
86#416—71no—
87C23×C7C2×D411yescharacteristic
88C22×D7—21no—
89C2×C4×C7C2×D412yes—
90C2×C4×C7C2×D412yes—
91C2×C4×C7C2×D411yescharacteristic
92C2×C4×C7C2×Q812yes—
93C2×C4×C7#4112yes—
94C2×C4×C7C2411yescharacteristic
95C22⋊D7—24no—
96C2×Dic7—21no—
97C2×Dic7—21no—
98C2×Dic7—22no—
99C2×Dic7—21no—
100C2×Dic7—22no—
101C2×Dic7—81no—
102C2×Dic7—81no—
103D28—21no—
104D28C2×Q812yes—
105C4×D7—24no—
106C4×D7—82no—
107Dic14—21no—
108Dic14—82no—
109Dic14C2×Q812yes—
110C7×C8—21no—
111C7×C8—21no—
112C7×C8—22no—
113C2×#41—71no—
114C2×M16C2×D711yescharacteristic
115C22×C8C2×D711yescharacteristic
116C4⋊Q8—72no—
117C4⋊Q8—141no—
118#125—72no—
119#125—141no—
120C22⋊Q8—142no—
121C4×Q8—72no—
122C4×D4—72no—
123C4×D4—142no—
124#117—71no—
125#117—141no—
126#117C2×D711yescharacteristic
127C2×C4⋊C4—71no—
128C2×C4⋊C4—71no—
129C2×C4⋊C4—141no—
130#107—72no—
131#106—72no—
132#106—72no—
133#106—72no—
134C4⋊C8C2×D712yes—
135C4⋊C8C2×D712yes—
136#103—71no—
137#103—71no—
138#103—72no—
139#102—71no—
140#102—71no—
141#102—72no—
142C22×C7—21no—
143C22×C7—21no—
144C22×C7C22⋊Q812yes—
145C22×C7C22×D411yescharacteristic
146C2×D7—22no—
147C2×D7—41no—
148C2×D7—41no—
149C4×C7—22no—
150C4×C7—22no—
151C4×C7C22⋊Q812yes—
152C4×C7C22×Q811yescharacteristic
153C4×C7C2×#4111yescharacteristic
154Dic7—22no—
155Dic7—41no—
156Dic7—42no—
157Dic7—42no—
158Dic7—41no—
159#41—74no—
160#41—142no—
161C2×Q8—71no—
162C2×Q8—142no—
163C2×D4—71no—
164C2×D4—141no—
165C22×C4—141no—
166C22×C4—141no—
167C22×C4—141no—
168C22×C4—142no—
169C22×C4C22×D711yescharacteristic
170M16—22no—
171C2×C8—22no—
172C2×C8C22×D711yescharacteristic
173C2×C8C22×D711yescharacteristic
174C2×C8C22×D712yes—
175C4⋊C4—72no—
176C4⋊C4—71no—
177C4⋊C4—71no—
178C4⋊C4—72no—
179C4⋊C4—142no—
180C4⋊C4—142no—
181C4⋊C4—142no—
182C4⋊C4—141no—
183C4⋊C4—141no—
184C4⋊C4—142no—
185C4⋊C4D2811yescharacteristic
186C4⋊C4D2811yescharacteristic
187C22⋊C4—141no—
188C22⋊C4—142no—
189C22⋊C4D2812yes—
190C42—141no—
191C42—142no—
192C42C22×D712yes—
193C2×C7—22no—
194C2×C7C2×C22⋊Q811yescharacteristic
195C2×C7#57711yescharacteristic
196C2×C7#57711yescharacteristic
197D7—42no—
198C23—141no—
199C23C2×D2811yescharacteristic
200Q8—72no—
201Q8—141no—
202Q8—562no—
203D4—72no—
204D4—144no—
205D4—141no—
206C2×C4—141no—
207C2×C4—144no—
208C2×C4—141no—
209C2×C4—142no—
210C2×C4—141no—
211C2×C4—142no—
212C2×C4—562no—
213C2×C4—561no—
214C2×C4—561no—
215C2×C4C2×D2812yes—
216C2×C4C2×D2812yes—
217C2×C4C2×D2811yescharacteristic
218C2×C4#115912yes—
219C2×C4Q8×D712yes—
220C2×C4C23×D711yescharacteristic
221C8—21no—
222C8—21no—
223C8—22no—
224C7#308411yescharacteristic, complemented, Sylow
225C22—21no—
226C22—21no—
227C22—142no—
228C22—281no—
229C22—281no—
230C22#646012yes—
231C22C22×D2811yescharacteristic
232C4—22no—
233C4—22no—
234C4—142no—
235C4—282no—
236C4—282no—
237C4—281no—
238C4—281no—
239C4#646012yes—
240C4C2×#115911yescharacteristic
241C4C2×Q8×D711yescharacteristic
242C2—22no—
243C2—282no—
244C2C2×#646011yescharacteristic
245C2#9044211yescharacteristic
246C2#9044211yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#424C2
socleC22×C7C22×D4
prime radicalC2×#1157C22
hypercenter#424D7
hypocenterC7#3084
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #90210C2semidirect
3. C7×#424C2semidirect
5. #89893C2semidirect
7. C2×#6391C2semidirect
8. #89623C2semidirect
9. #89619C2semidirect
10. #89614C2semidirect
11. #89601C2semidirect
47. #1157Q8semidirect
84. #424D7semidirect
224. C7#3084semidirect

Maximal subgroups

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

subgroupindex
#902142
#902102
C7×#4242
#899052
#898932
#898852
C2×#63912
#896232
#896192
#896142
#896012
#30847