Finite Group Databasebeta

← #11739031

Subgroups — #11739031

Order 896.

396 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 115 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#90421C211yesmaximal, characteristic, complemented
2#90409C211yesmaximal, characteristic, complemented
3C7×#486C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic, complemented
5#89844C211yesmaximal, characteristic, complemented
6#89558C212yesmaximal, complemented
7#89557C212yesmaximal, complemented
8#89538C212yesmaximal, complemented
9#89536C212yesmaximal, complemented
10C4×D56C211yesmaximal, characteristic, complemented
11#89451C211yesmaximal, characteristic, complemented
12C2×#1159—21no—
13C7×C4⋊Q8C2211yescharacteristic
14C7×#127C2211yescharacteristic
15#6509—22no—
16#6503C2212yes—
17#6502—22no—
18#6501—21no—
19D4×Dic7C2212yes—
20C2×#1141C2212yes—
21C2×D56C2211yescharacteristic
22#6460—21no—
23#6456C2212yes—
24D7×C4⋊C4—21no—
25#6453C2212yes—
26C4×D28C2211yescharacteristic
27C4×D28C2211yescharacteristic
28C4×Dic14C2211yescharacteristic
29C7×#102C2212yes—
30C7×#102C2212yes—
31C4×C7×C8C2211yescharacteristic
32#6406C2212yes—
33#6395C2211yescharacteristic
34#6395C2211yescharacteristic
35#6392C2211yescharacteristic
36#6389C2212yes—
37Dic7⋊C8C2212yes—
38#6383C2212yes—
39#6382C2212yes—
40C4⋊(C7⋊C8)C2211yescharacteristic
41#3366—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×D4—21no—
44C2×C7×D4—21no—
45C2×C7×D4C2312yes—
46C2×C22⋊D7—22no—
47C2×C22⋊D7—22no—
48C22×Dic7—22no—
49#1159—44no—
50C2×D28C2311yescharacteristic
51C2×D28C2311yescharacteristic
52C2×C4×D7—21no—
53C2×C4×D7—21no—
54C2×C4×D7—22no—
55C2×Dic14D411yescharacteristic
56C2×C7×C8C2311yescharacteristic
57C2×C7×C8C2311yescharacteristic
58C7×C4⋊C4—21no—
59C7×C4⋊C4C2312yes—
60C42×C7C2311yescharacteristic
61C22⋊Dic7—22no—
62#1141—24no—
63(C2×D7)⋊C4—21no—
64(C2×D7)⋊C4—21no—
65(C2×D7)⋊C4—22no—
66C4⋊Dic7—22no—
67C4⋊Dic7D411yescharacteristic
68C4⋊Dic7C2311yescharacteristic
69C4⋊Dic7C2311yescharacteristic
70Dic7⋊C4—22no—
71Dic7⋊C4D412yes—
72C4×Dic7C2312yes—
73C2×C7⋊C8C2312yes—
74D56—22no—
75#554—71no—
76#550—71no—
77#490—72no—
78#486D711yescharacteristic, complemented
79#478—72no—
80#476—71no—
81#459—71no—
82#459—72no—
83#446—71no—
84#442—72no—
85C4×D8—71no—
86C23×C7—22no—
87C22×D7—21no—
88C22×D7—21no—
89C7×Q8—81no—
90C7×D4—22no—
91C7×D4—22no—
92C7×D4—42no—
93C7×D4—81no—
94C2×C4×C7—22no—
95C2×C4×C7C2×D411yescharacteristic
96C2×C4×C7C2×D411yescharacteristic
97C2×C4×C7C2411yescharacteristic
98C22⋊D7—44no—
99C22⋊D7—82no—
100C2×Dic7—21no—
101C2×Dic7—21no—
102C2×Dic7—44no—
103C2×Dic7C2×D412yes—
104C2×Dic7C2×D412yes—
105D28—21no—
106D28—21no—
107D28—21no—
108D28#4112yescomplemented
109C4×D7—42no—
110C4×D7—42no—
111C4×D7—82no—
112Dic14—22no—
113C7×C8—21no—
114C7×C8—21no—
115C7⋊C8—22no—
116C2×#41—141no—
117C2×D8—72no—
118C2×D8—71no—
119C4⋊Q8C2×D711yescharacteristic
120#127—72no—
121#127C2×D711yescharacteristic
122#125—72no—
123C22⋊Q8—141no—
124C22⋊Q8—142no—
125#121—141no—
126#121—142no—
127C4×Q8—71no—
128C4×D4—72no—
129C4×D4—72no—
130C4×D4—71no—
131C4×D4—71no—
132C2×C4⋊C4—141no—
133#107—71no—
134#106—72no—
135C4⋊C8—72no—
136C4⋊C8—71no—
137#102—72no—
138#102—71no—
139#102—71no—
140#102—72no—
141#102C2×D712yes—
142#102C2×D712yes—
143#97—72no—
144C4×C8C2×D711yescharacteristic
145C22×C7—42no—
146C22×C7—42no—
147C22×C7—42no—
148C22×C7C22×D411yescharacteristic
149C2×D7—22no—
150C2×D7—41no—
151C2×D7—41no—
152C2×D7—41no—
153C2×D7—41no—
154C2×D7—41no—
155C4×C7—21no—
156C4×C7—42no—
157C4×C7AGL(1,ℤ/8)12yes—
158C4×C7C2×#4111yescharacteristic
159C4×C72−1+411yescharacteristic
160Dic7—22no—
161Dic7—22no—
162Dic7—41no—
163Dic7—41no—
164#41—284no—
165C2×Q8—21no—
166C2×Q8—71no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—71no—
170C2×D4—71no—
171C2×D4—142no—
172C2×D4—142no—
173C2×D4C22×D712yes—
174C22×C4—142no—
175C22×C4—142no—
176C22×C4—141no—
177C22×C4—141no—
178D8—142no—
179D8—144no—
180C2×C8—72no—
181C2×C8C22×D711yescharacteristic
182C2×C8C22×D711yescharacteristic
183C4⋊C4—21no—
184C4⋊C4—72no—
185C4⋊C4—71no—
186C4⋊C4—71no—
187C4⋊C4—71no—
188C4⋊C4—142no—
189C4⋊C4—142no—
190C4⋊C4C22×D712yes—
191C22⋊C4—142no—
192C22⋊C4—142no—
193C22⋊C4—141no—
194C22⋊C4—141no—
195C42—72no—
196C42C22×D711yescharacteristic
197C2×C7—82no—
198C2×C7#54811yescharacteristic
199C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
200C2×C7#57611yescharacteristic
201D7—42no—
202D7—81no—
203C23—22no—
204C23—141no—
205C23—141no—
206Q8—81no—
207Q8—142no—
208D4—22no—
209D4—22no—
210D4—42no—
211D4—72no—
212D4—81no—
213D4—141no—
214D4—141no—
215D4—141no—
216D4—284no—
217D4—562no—
218C2×C4—22no—
219C2×C4—72no—
220C2×C4—72no—
221C2×C4—141no—
222C2×C4—141no—
223C2×C4—284no—
224C2×C4—282no—
225C2×C4—282no—
226C2×C4—562no—
227C2×C4D4×D711yescharacteristic
228C2×C4D4×D711yescharacteristic
229C2×C4C23×D711yescharacteristic
230C8—21no—
231C8—21no—
232C8—142no—
233C7#336611yescharacteristic, complemented, Sylow
234C22—42no—
235C22—42no—
236C22—42no—
237C22—142no—
238C22—281no—
239C22—281no—
240C22—281no—
241C22—281no—
242C22—281no—
243C22C2×D4×D711yescharacteristic
244C4—21no—
245C4—42no—
246C4—142no—
247C4—142no—
248C4—281no—
249C4—281no—
250C4#647712yes—
251C4D7×#4111yescharacteristic
252C4#655411yescharacteristic
253C2—82no—
254C2—282no—
255C2—561no—
256C2#9036811yescharacteristic
257C2#9044811yescharacteristic
258C2C2×#647711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#486C2
socleC22×C7C22×D4
hypercenter#486D7
hypocenterC7#3366
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #90421C2semidirect
2. #90409C2semidirect
3. C7×#486C2semidirect
4. #89856C2semidirect
5. #89844C2semidirect
6. #89558C2semidirect
7. #89557C2semidirect
8. #89538C2semidirect
9. #89536C2semidirect
10. C4×D56C2semidirect
11. #89451C2semidirect
78. #486D7semidirect
108. D28#41semidirect
233. C7#3366semidirect

Maximal subgroups

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

subgroupindex
#904212
#904092
C7×#4862
#898562
#898442
#895582
#895572
#895382
#895362
C4×D562
#894512
#33667