Finite Group Databasebeta

← #11736887

Subgroups — #11736887

Order 896.

386 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 123 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#90242C211yesmaximal, characteristic
2#90233C211yesmaximal, characteristic, complemented
3C7×C8×D4C211yesmaximal, characteristic, complemented
4#89888C212yesmaximal, complemented
5#89885C212yesmaximal, complemented
6#89621C211yesmaximal, characteristic, complemented
7#89618C211yesmaximal, characteristic, complemented
8#89505C212yesmaximal, complemented
9#89495C212yesmaximal, complemented
10C4×#1132C211yesmaximal, characteristic, complemented
11#89453C211yesmaximal, characteristic, complemented
12C2×#1157C2212yes—
13C22×C7×C8C2212yes—
14C4×C7×D4C2211yescharacteristic
15#6501—21no—
16#6499—21no—
17#6489C2212yes—
18#6487C2212yes—
19#6487C2212yes—
20C2×Dic28C2212yes—
21C2×#1132C2211yescharacteristic
22D7×C4⋊C4—21no—
23#6450—21no—
24#6446—22no—
25#6442—22no—
26C4×D28C2211yescharacteristic
27#6432C2211yescharacteristic
28#6432C2211yescharacteristic
29C4×Dic14C2211yescharacteristic
30C7×C4⋊C8C2211yescharacteristic
31C7×C22⋊C8C2212yes—
32C4×C7×C8C2211yescharacteristic
33#6395C2212yes—
34#6395C2211yescharacteristic
35#6392C2212yes—
36#6391C2211yescharacteristic
37#6390C2212yes—
38#6390C2212yes—
39#6390C2211yescharacteristic
40#6390C2212yes—
41#3302—71nomaximal, Sylow
42C2×C7×D4D411yescharacteristic
43C22×C4×C7C2312yes—
44C2×C22⋊D7—22no—
45#1157—24no—
46#1157—42no—
47C2×D28C2311yescharacteristic
48C2×C4×D7—21no—
49C2×C4×D7—22no—
50C2×Dic14—21no—
51C2×Dic14C2311yescharacteristic
52C2×Dic14C2312yes—
53C2×C7×C8—42no—
54C2×C7×C8C2311yescharacteristic
55C2×C7×C8C2311yescharacteristic
56C2×C7×C8C2312yes—
57C7×C4⋊C4D411yescharacteristic
58C7×C22⋊C4D412yes—
59C42×C7C2311yescharacteristic
60C22⋊Dic7—22no—
61C22⋊Dic7—22no—
62C22⋊Dic7—22no—
63(C2×D7)⋊C4—21no—
64C4⋊Dic7—21no—
65C4⋊Dic7—21no—
66C4⋊Dic7—21no—
67C4⋊Dic7C2311yescharacteristic
68C4⋊Dic7C2312yes—
69C4⋊Dic7C2311yescharacteristic
70Dic7⋊C4—21no—
71Dic7⋊C4—22no—
72Dic7⋊C4—22no—
73C4×Dic7—21no—
74C4×Dic7—22no—
75Dic28—42no—
76#1132—22no—
77#556—71no—
78#547—71no—
79#487—71no—
80#485—72no—
81#474—71no—
82#467—72no—
83#462—71no—
84#452—72no—
85C4×SD16—71no—
86C8×D4D711yescharacteristic, complemented
87#416—72no—
88C23×C7C2×D412yes—
89C22×D7—21no—
90C7×D4—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7C2×D412yes—
94C2×C4×C7C2×D411yescharacteristic
95C2×C4×C7C2×D411yescharacteristic
96C2×C4×C7C2411yescharacteristic
97C22⋊D7—44no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—22no—
101C2×Dic7—21no—
102C2×Dic7—22no—
103D28—21no—
104D28#4112yes—
105C4×D7—44no—
106C4×D7—42no—
107Dic14—21no—
108Dic14—22no—
109Dic14—21no—
110Dic14—22no—
111Dic14—42no—
112C7×C8—21no—
113C7×C8—22no—
114C7×C8—21no—
115C2×#41—72no—
116C2×Q16—72no—
117C2×SD16—71no—
118C22×C8C2×D712yes—
119C4⋊Q8—71no—
120C4⋊Q8—71no—
121#126—142no—
122#124—141no—
123#123—142no—
124C22⋊Q8—72no—
125C22⋊Q8—72no—
126#121—141no—
127C4×Q8—71no—
128C4×Q8—141no—
129C4×D4—71no—
130C4×D4C2×D711yescharacteristic
131#117—72no—
132C2×C4⋊C4—141no—
133#107—72no—
134#106—71no—
135C4⋊C8C2×D711yescharacteristic
136#103—72no—
137#103—72no—
138#103—71no—
139#103—72no—
140#102—71no—
141#102—72no—
142C22⋊C8C2×D712yes—
143C4×C8C2×D711yescharacteristic
144C22×C7—22no—
145C22×C7—22no—
146C22×C7—22no—
147C22×C7C22×D411yescharacteristic
148C2×D7—22no—
149C2×D7—41no—
150C2×D7—41no—
151C4×C7—22no—
152C4×C7—21no—
153C4×C7#13512yes—
154C4×C7C2×#4111yescharacteristic
155C4×C7H(5,𝔽2)11yescharacteristic
156Dic7—42no—
157Dic7—41no—
158Dic7—42no—
159Dic7—41no—
160Dic7—41no—
161#41—144no—
162#41—282no—
163C2×Q8—72no—
164C2×Q8—71no—
165C2×Q8—141no—
166C2×D4—71no—
167C2×D4—142no—
168C2×D4D2811yescharacteristic
169C22×C4—141no—
170C22×C4—142no—
171C22×C4C22×D712yes—
172Q16—282no—
173SD16—142no—
174C2×C8—42no—
175C2×C8C22×D711yescharacteristic
176C2×C8C22×D711yescharacteristic
177C2×C8C22×D712yes—
178C4⋊C4—71no—
179C4⋊C4—72no—
180C4⋊C4—71no—
181C4⋊C4—142no—
182C4⋊C4—141no—
183C4⋊C4—142no—
184C4⋊C4—141no—
185C4⋊C4—141no—
186C4⋊C4—141no—
187C4⋊C4D2811yescharacteristic
188C22⋊C4—142no—
189C22⋊C4—142no—
190C22⋊C4—142no—
191C22⋊C4—141no—
192C22⋊C4D2812yes—
193C42—142no—
194C42—141no—
195C42C22×D711yescharacteristic
196C2×C7—42no—
197C2×C7#54611yescharacteristic
198C2×C7C2×#13511yescharacteristic
199C2×C7#57811yescharacteristic
200D7—42no—
201C23—141no—
202C23C2×D2812yes—
203Q8—141no—
204Q8—142no—
205Q8—141no—
206Q8—142no—
207Q8—282no—
208D4—22no—
209D4—72no—
210D4—141no—
211D4—284no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—141no—
215C2×C4—142no—
216C2×C4—141no—
217C2×C4—141no—
218C2×C4—142no—
219C2×C4—284no—
220C2×C4—282no—
221C2×C4C2×D2812yes—
222C2×C4C2×D2811yescharacteristic
223C2×C4C2×D2811yescharacteristic
224C2×C4C23×D711yescharacteristic
225C8—21no—
226C8—22no—
227C8—21no—
228C7#330211yescharacteristic, complemented, Sylow
229C22—22no—
230C22—22no—
231C22—22no—
232C22—142no—
233C22—281no—
234C22—281no—
235C22C22×D2811yescharacteristic
236C4—21no—
237C4—22no—
238C4—281no—
239C4—282no—
240C4—281no—
241C4—282no—
242C4—281no—
243C4#646712yes—
244C4#654811yescharacteristic
245C4D7×#4111yescharacteristic
246C2—42no—
247C2—282no—
248C2#9024511yescharacteristic
249C2C2×#646711yescharacteristic
250C2#9044411yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×C8×D4C2
socleC22×C7C22×D4
prime radical#90242C2
hypercenterC8×D4D7
hypocenterC7#3302
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
2. #90233C2semidirect
3. C7×C8×D4C2semidirect
4. #89888C2semidirect
5. #89885C2semidirect
6. #89621C2semidirect
7. #89618C2semidirect
8. #89505C2semidirect
9. #89495C2semidirect
10. C4×#1132C2semidirect
11. #89453C2semidirect
86. C8×D4D7semidirect
228. C7#3302semidirect

Maximal subgroups

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

subgroupindex
#902422
#902332
C7×C8×D42
#898882
#898852
#896212
#896182
#895052
#894952
C4×#11322
#894532
#33027