Finite Group Databasebeta

← #11738521

Subgroups — #11738521

Order 896.

398 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 117 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#90425C211yesmaximal, characteristic, complemented
2#90246C211yesmaximal, characteristic, complemented
3C7×#476C211yesmaximal, characteristic, complemented
4#89865C211yesmaximal, characteristic, complemented
5C4×#1141C211yesmaximal, characteristic, complemented
6#89639C212yesmaximal, complemented
7#89636C212yesmaximal, complemented
8#89613C211yesmaximal, characteristic, complemented
9#89606C211yesmaximal, characteristic, complemented
10#89553C212yesmaximal, complemented
11#89544C212yesmaximal, complemented
12C2×#1159C2212yes—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×D4C2211yescharacteristic
15#6510—22no—
16C2×#1141C2211yescharacteristic
17#6493—21no—
18C2×C4⋊Dic7—21no—
19C2×#1132C2212yes—
20C2×C8×D7C2212yes—
21#6460C2212yes—
22#6458—21no—
23#6458C2212yes—
24#6455—21no—
25#6455C2212yes—
26#6449—22no—
27C4×D28C2211yescharacteristic
28C4×D28C2211yescharacteristic
29#6432C2211yescharacteristic
30C7×#106C2212yes—
31C7×C4⋊C8C2211yescharacteristic
32C7×#102C2212yes—
33#6406C2211yescharacteristic
34(C2×D7)⋊C8C2212yes—
35#6390C2212yes—
36#6384C2212yes—
37#6384C2212yes—
38#6383C2211yescharacteristic
39#6381C2211yescharacteristic
40C4×C7⋊C8C2211yescharacteristic
41#3303—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×D4C2311yescharacteristic
44C22×C4×C7—21no—
45C2×C22⋊D7—22no—
46C22×Dic7—22no—
47#1159—24no—
48#1159—42no—
49C2×D28—22no—
50C2×D28D411yescharacteristic
51C2×D28C2311yescharacteristic
52C2×C4×D7—21no—
53C2×C4×D7C2312yes—
54C2×Dic14C2312yes—
55C2×C7×C8C2312yes—
56C7×C4⋊C4—21no—
57C7×C4⋊C4C2312yes—
58C7×C4⋊C4C2311yescharacteristic
59C7×C22⋊C4—21no—
60C42×C7C2311yescharacteristic
61#1141—22no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—21no—
65(C2×D7)⋊C4—22no—
66(C2×D7)⋊C4D412yes—
67C4⋊Dic7—21no—
68C4⋊Dic7—21no—
69C4⋊Dic7—22no—
70C4⋊Dic7D411yescharacteristic
71C4⋊Dic7C2311yescharacteristic
72C4×Dic7—22no—
73C2×C7⋊C8C2311yescharacteristic
74C2×C7⋊C8C2311yescharacteristic
75#1132—42no—
76C8×D7—42no—
77#555—71no—
78#547—71no—
79#487—71no—
80#485—72no—
81#476D711yescharacteristic, complemented
82#465—72no—
83#461—71no—
84#449—72no—
85C4×D8—71no—
86C8×D4—71no—
87#416—72no—
88C23×C7—21no—
89C22×D7—21no—
90C22×D7C2×D412yes—
91C7×Q8—81no—
92C7×D4—21no—
93C7×D4#4112yes—
94C2×C4×C7—22no—
95C2×C4×C7—21no—
96C2×C4×C7—42no—
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2411yescharacteristic
100C22⋊D7—44no—
101C2×Dic7—21no—
102C2×Dic7—22no—
103C2×Dic7—44no—
104C2×Dic7C2×D412yes—
105D28—22no—
106D28—21no—
107D28—21no—
108D28—82no—
109C4×D7—22no—
110C4×D7—22no—
111C4×D7—42no—
112Dic14—22no—
113Dic14—22no—
114C7×C8—22no—
115C7⋊C8—21no—
116C7⋊C8—21no—
117C2×#41—72no—
118C2×SD16—72no—
119C2×D8—71no—
120C22×C8—72no—
121C4⋊Q8—71no—
122C4⋊Q8C2×D711yescharacteristic
123#124—142no—
124#123—142no—
125C22⋊Q8—72no—
126#121—72no—
127#121—141no—
128#121—141no—
129C4×D4—71no—
130C4×D4—71no—
131C4×D4C2×D711yescharacteristic
132#117—72no—
133#117—141no—
134C2×C4⋊C4—141no—
135#107—71no—
136#106C2×D712yes—
137C4⋊C8C2×D711yescharacteristic
138#103—72no—
139#103—72no—
140#103—72no—
141#102—71no—
142#102—71no—
143#102C2×D712yes—
144C22⋊C8—72no—
145C4×C8—71no—
146C22×C7—22no—
147C22×C7—41no—
148C22×C7—41no—
149C22×C7C22×D411yescharacteristic
150C2×D7—22no—
151C2×D7—22no—
152C2×D7—22no—
153C2×D7—41no—
154C2×D7—41no—
155C2×D7—41no—
156C4×C7—21no—
157C4×C7—41no—
158C4×C7—42no—
159C4×C7#13512yes—
160C4×C7C2×#4111yescharacteristic
161C4×C7H(5,𝔽2)11yescharacteristic
162Dic7—22no—
163Dic7—42no—
164Dic7—41no—
165#41—144no—
166#41—282no—
167C2×Q8—21no—
168C2×Q8—72no—
169C2×D4—71no—
170C2×D4—71no—
171C2×D4—142no—
172C2×D4—142no—
173C2×D4C22×D711yescharacteristic
174C22×C4—21no—
175C22×C4—72no—
176C22×C4—142no—
177C22×C4—141no—
178SD16—282no—
179D8—142no—
180C2×C8—71no—
181C2×C8—71no—
182C2×C8—282no—
183C2×C8C22×D712yes—
184C4⋊C4—21no—
185C4⋊C4—71no—
186C4⋊C4—71no—
187C4⋊C4—142no—
188C4⋊C4—141no—
189C4⋊C4—141no—
190C4⋊C4C22×D711yescharacteristic
191C4⋊C4C22×D712yes—
192C22⋊C4—21no—
193C22⋊C4—72no—
194C22⋊C4—142no—
195C22⋊C4—142no—
196C22⋊C4—142no—
197C22⋊C4—141no—
198C42—142no—
199C42C22×D711yescharacteristic
200C2×C7—42no—
201C2×C7#54611yescharacteristic
202C2×C7C2×#13511yescharacteristic
203C2×C7#57711yescharacteristic
204D7—42no—
205D7—81no—
206C23—21no—
207C23—72no—
208C23—141no—
209Q8—81no—
210Q8—142no—
211Q8—142no—
212D4—21no—
213D4—142no—
214D4—141no—
215D4—141no—
216D4—284no—
217D4—562no—
218D4#116112yes—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—72no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—141no—
226C2×C4—142no—
227C2×C4—284no—
228C2×C4—282no—
229C2×C4D4×D711yescharacteristic
230C2×C4D4×D711yescharacteristic
231C2×C4C23×D711yescharacteristic
232C8—22no—
233C8—141no—
234C8—141no—
235C7#330311yescharacteristic, complemented, Sylow
236C22—22no—
237C22—41no—
238C22—41no—
239C22—142no—
240C22—142no—
241C22—142no—
242C22—281no—
243C22—281no—
244C22—281no—
245C22C2×D4×D711yescharacteristic
246C4—21no—
247C4—41no—
248C4—42no—
249C4—142no—
250C4—281no—
251C4—282no—
252C4#647912yes—
253C4C2×#116111yescharacteristic
254C4#655311yescharacteristic
255C2—42no—
256C2—282no—
257C2—561no—
258C2#9032111yescharacteristic
259C2C2×#647911yescharacteristic
260C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#476C2
socleC22×C7C22×D4
hypercenter#476D7
hypocenterC7#3303
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #90425C2semidirect
2. #90246C2semidirect
3. C7×#476C2semidirect
4. #89865C2semidirect
5. C4×#1141C2semidirect
6. #89639C2semidirect
7. #89636C2semidirect
8. #89613C2semidirect
9. #89606C2semidirect
10. #89553C2semidirect
11. #89544C2semidirect
81. #476D7semidirect
235. C7#3303semidirect

Maximal subgroups

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

subgroupindex
#904252
#902462
C7×#4762
#898652
C4×#11412
#896392
#896362
#896132
#896062
#895532
#895442
#33037