Finite Group Databasebeta

← #11740995

Subgroups — #11740995

Order 896.

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

normal subgroups only

This group has 420 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
1C7×D4×Q8C211yesmaximal, characteristic, complemented
2#90246C211yesmaximal, characteristic
3#89999C212yesmaximal, complemented
4C2×#6409C212yesmaximal, complemented
5#89855C211yesmaximal, characteristic, complemented
6#89853C211yesmaximal, characteristic, complemented
7#89820C212yesmaximal, complemented
8#89815C212yesmaximal, complemented
9C4×#1143C211yesmaximal, characteristic, complemented
10#89792C211yesmaximal, characteristic, complemented
11D4×C7⋊C8C211yesmaximal, characteristic, complemented
12C22×C7×Q8C2212yes—
13C2×#1159—21no—
14C7×C4⋊Q8—21no—
15C7×C4⋊Q8C2211yescharacteristic
16C7×C22⋊Q8—22no—
17C7×C22⋊Q8C2212yes—
18C4×C7×Q8C2211yescharacteristic
19C4×C7×D4—21no—
20C4×C7×D4C2211yescharacteristic
21C2×#1143C2211yescharacteristic
22C2×#1143C2212yes—
23#6493C2212yes—
24C2×C4⋊Dic7C2212yes—
25C22×C7⋊C8C2212yes—
26#6460—21no—
27#6449—22no—
28C4×D28C2211yescharacteristic
29#6432C2211yescharacteristic
30#6409C2211yescharacteristic
31#6409C2212yes—
32#6409C2212yes—
33#6409C2212yes—
34#6409C2212yes—
35C22⋊(C7⋊C8)C2212yes—
36#6383C2211yescharacteristic
37#6382C2212yes—
38#6382C2211yescharacteristic
39C4⋊(C7⋊C8)C2211yescharacteristic
40C4×C7⋊C8C2211yescharacteristic
41#3298—71nomaximal, Sylow
42C2×C7×Q8—22no—
43C2×C7×Q8—24no—
44C2×C7×Q8—42no—
45C2×C7×Q8C2312yes—
46C2×C7×Q8C2311yescharacteristic
47C2×C7×D4D411yescharacteristic
48C22×C4×C7—22no—
49C22×C4×C7C2312yes—
50C2×C22⋊D7—22no—
51C22×Dic7—22no—
52#1159—44no—
53C2×D28C2311yescharacteristic
54C2×C4×D7—21no—
55C2×Dic14—21no—
56C7×C4⋊C4—21no—
57C7×C4⋊C4—21no—
58C7×C4⋊C4—21no—
59C7×C4⋊C4—22no—
60C7×C4⋊C4D411yescharacteristic
61C7×C4⋊C4C2311yescharacteristic
62C7×C22⋊C4—22no—
63C7×C22⋊C4D412yes—
64C42×C7—21no—
65C42×C7C2311yescharacteristic
66#1143—22no—
67#1143—42no—
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—22no—
70C4⋊Dic7—21no—
71C4⋊Dic7—22no—
72C4⋊Dic7C2312yes—
73C4⋊Dic7C2311yescharacteristic
74C2×C7⋊C8—42no—
75C2×C7⋊C8C2311yescharacteristic
76C2×C7⋊C8C2312yes—
77C2×C7⋊C8C2311yescharacteristic
78D4×Q8D711yescharacteristic, complemented
79#547—71no—
80#487—71no—
81#483—72no—
82#475—71no—
83#465—72no—
84#461—71no—
85#448—72no—
86C4×SD16—71no—
87C8×D4—71no—
88C2×#103—72no—
89C23×C7C2×D412yes—
90C22×D7—21no—
91C7×Q8—21no—
92C7×Q8—22no—
93C7×Q8—22no—
94C7×Q8—44no—
95C7×Q8#4112yescomplemented
96C7×D4SD1614yescomplemented
97C2×C4×C7—21no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7—44no—
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2×D412yes—
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2411yescharacteristic
107C22⋊D7—44no—
108C2×Dic7—22no—
109C2×Dic7—21no—
110C2×Dic7—44no—
111D28—21no—
112D28—21no—
113C4×D7—42no—
114Dic14—42no—
115C7⋊C8—21no—
116C7⋊C8—21no—
117C7⋊C8—22no—
118C2×#41—141no—
119C22×Q8C2×D712yes—
120C2×SD16—71no—
121C2×SD16—72no—
122C22×C8—72no—
123C4⋊Q8—21no—
124C4⋊Q8—71no—
125C4⋊Q8C2×D711yescharacteristic
126#123—142no—
127C22⋊Q8—22no—
128C22⋊Q8—141no—
129C22⋊Q8C2×D712yes—
130#121—72no—
131C4×Q8C2×D711yescharacteristic
132C4×D4—21no—
133C4×D4—71no—
134C4×D4C2×D711yescharacteristic
135C2×C4⋊C4—72no—
136#106—71no—
137#106—72no—
138C4⋊C8—71no—
139#103—71no—
140#103—72no—
141#103—72no—
142#103—72no—
143#103—72no—
144#102—71no—
145C22⋊C8—72no—
146C4×C8—71no—
147C22×C7—22no—
148C22×C7—22no—
149C22×C7C2×SD1614yes—
150C22×C7C22×D411yescharacteristic
151C2×D7—41no—
152C2×D7—41no—
153C2×D7—41no—
154C4×C7—22no—
155C4×C7—22no—
156C4×C7—21no—
157C4×C7—42no—
158C4×C7—41no—
159C4×C7C2×SD1612yes—
160C4×C7C2×#4111yescharacteristic
161C4×C7H(5,𝔽2)11yescharacteristic
162Dic7—42no—
163Dic7—41no—
164#41—284no—
165C2×Q8—22no—
166C2×Q8—24no—
167C2×Q8—42no—
168C2×Q8—141no—
169C2×Q8C22×D712yes—
170C2×Q8C22×D711yescharacteristic
171C2×D4—71no—
172C2×D4—142no—
173C2×D4C22⋊D711yescharacteristic
174C22×C4—22no—
175C22×C4—142no—
176C22×C4—141no—
177C22×C4C22×D712yes—
178SD16—142no—
179SD16—282no—
180C2×C8—71no—
181C2×C8—71no—
182C2×C8—72no—
183C2×C8—282no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—22no—
188C4⋊C4—71no—
189C4⋊C4—72no—
190C4⋊C4—142no—
191C4⋊C4—141no—
192C4⋊C4C22⋊D711yescharacteristic
193C4⋊C4C22×D711yescharacteristic
194C22⋊C4—22no—
195C22⋊C4—141no—
196C22⋊C4—142no—
197C22⋊C4C22⋊D712yes—
198C42—21no—
199C42C22×D711yescharacteristic
200C2×C7—24no—
201C2×C7#54611yescharacteristic
202C2×C7C22×SD1611yescharacteristic
203C2×C7#57811yescharacteristic
204D7—81no—
205C23—141no—
206C23C2×C22⋊D712yes—
207Q8—21no—
208Q8—22no—
209Q8—22no—
210Q8—44no—
211Q8—282no—
212Q8#115912yescomplemented
213D4—141no—
214D4—141no—
215D4—284no—
216D4#114314yescomplemented
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—44no—
223C2×C4—141no—
224C2×C4—142no—
225C2×C4—284no—
226C2×C4—282no—
227C2×C4C2×C22⋊D711yescharacteristic
228C2×C4C2×C22⋊D712yes—
229C2×C4C2×C22⋊D711yescharacteristic
230C2×C4C23×D711yescharacteristic
231C8—141no—
232C8—141no—
233C8—142no—
234C7#329811yescharacteristic, complemented, Sylow
235C22—22no—
236C22—22no—
237C22—281no—
238C22—281no—
239C22—281no—
240C22C2×#114314yes—
241C22C22×C22⋊D711yescharacteristic
242C4—22no—
243C4—22no—
244C4—21no—
245C4—42no—
246C4—41no—
247C4—281no—
248C4—282no—
249C4C2×#114312yes—
250C4C2×#115911yescharacteristic
251C4#654811yescharacteristic
252C2—24no—
253C2—561no—
254C2#9049711yescharacteristic
255C2C22×#114311yescharacteristic
256C2#9052911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×D4×Q8C2
socleC22×C7C22×D4
prime radical#90246C2
hypercenterD4×Q8D7
hypocenterC7#3298
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C7×D4×Q8C2semidirect
3. #89999C2semidirect
4. C2×#6409C2semidirect
5. #89855C2semidirect
6. #89853C2semidirect
7. #89820C2semidirect
8. #89815C2semidirect
9. C4×#1143C2semidirect
10. #89792C2semidirect
11. D4×C7⋊C8C2semidirect
78. D4×Q8D7semidirect
95. C7×Q8#41semidirect
96. C7×D4SD16semidirect
212. Q8#1159semidirect
216. D4#1143semidirect
234. C7#3298semidirect

Maximal subgroups

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

subgroupindex
C7×D4×Q82
#902462
#899992
C2×#64092
#898552
#898532
#898202
#898152
C4×#11432
#897922
D4×C7⋊C82
#32987