Finite Group Databasebeta

← #11737900

Subgroups — #11737900

Order 896.

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

normal subgroups only

This group has 420 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#90417C211yesmaximal, characteristic, complemented
2#90242C211yesmaximal, characteristic, complemented
3C7×#461C211yesmaximal, characteristic, complemented
4#89946C212yesmaximal, complemented
5#89944C212yesmaximal, complemented
6#89853C211yesmaximal, characteristic, complemented
7C4×#1141C211yesmaximal, characteristic, complemented
8#89618C211yesmaximal, characteristic, complemented
9#89610C211yesmaximal, characteristic, complemented
10#89581C212yesmaximal, complemented
11#89575C212yesmaximal, complemented
12C2×#1161C2212yes—
13C2×#1157—21no—
14C2×C7×SD16C2212yes—
15C7×C4⋊Q8C2211yescharacteristic
16C4×C7×D4C2211yescharacteristic
17#6509C2212yes—
18#6501C2212yes—
19C2×#1141C2211yescharacteristic
20#6487—21no—
21C2×#1131C2212yes—
22#6458—21no—
23#6457—22no—
24D7×C4⋊C4—21no—
25D7×C4⋊C4C2212yes—
26#6446—22no—
27C4×D28C2211yescharacteristic
28C4×D28C2211yescharacteristic
29#6432C2211yescharacteristic
30C7×C4⋊C8C2211yescharacteristic
31C7×#103C2212yes—
32#6409C2212yes—
33#6409C2212yes—
34#6406C2211yescharacteristic
35#6395C2212yes—
36(C2×D7)⋊C8C2212yes—
37#6392C2212yes—
38#6383C2211yescharacteristic
39#6381C2211yescharacteristic
40C4×C7⋊C8C2211yescharacteristic
41#3330—71nomaximal, Sylow
42C2×C7×Q8C2312yes—
43C2×C7×D4C2311yescharacteristic
44C22×C4×C7—21no—
45C2×C22⋊D7—22no—
46#1161—24no—
47#1161—42no—
48#1157—44no—
49C2×D28—22no—
50C2×D28D411yescharacteristic
51C2×D28C2311yescharacteristic
52C2×C4×D7—22no—
53C2×C4×D7—21no—
54C2×C4×D7—22no—
55C2×C4×D7C2312yes—
56C2×Dic14—21no—
57C7×SD16—42no—
58C2×C7×C8C2312yes—
59C7×C4⋊C4—21no—
60C7×C4⋊C4—21no—
61C7×C4⋊C4C2311yescharacteristic
62C7×C22⋊C4—21no—
63C42×C7C2311yescharacteristic
64C22⋊Dic7—22no—
65#1141—22no—
66(C2×D7)⋊C4—22no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4D412yes—
69C4⋊Dic7—21no—
70C4⋊Dic7D411yescharacteristic
71C4⋊Dic7C2311yescharacteristic
72C4⋊Dic7C2312yes—
73Dic7⋊C4—22no—
74Dic7⋊C4—22no—
75C2×C7⋊C8C2311yescharacteristic
76C2×C7⋊C8C2311yescharacteristic
77#1131—42no—
78#547—71no—
79#547—71no—
80#487—71no—
81#484—72no—
82#474—71no—
83#468—72no—
84#461D711yescharacteristic, complemented
85#449—72no—
86C4×D8—71no—
87#436—71no—
88#417—72no—
89C23×C7—21no—
90C22×D7—21no—
91C22×D7C2×D412yes—
92C7×Q8—22no—
93C7×Q8—22no—
94C7×D4—21no—
95C7×D4—21no—
96C2×C4×C7—21no—
97C2×C4×C7—22no—
98C2×C4×C7—42no—
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2411yescharacteristic
102C22⋊D7—44no—
103C2×Dic7—21no—
104C2×Dic7—22no—
105C2×Dic7C2×D412yes—
106D28—22no—
107D28—21no—
108D28—44no—
109D28#4112yescomplemented
110C4×D7—22no—
111C4×D7—22no—
112C4×D7—44no—
113C4×D7—44no—
114C4×D7—42no—
115Dic14—42no—
116C7×C8—22no—
117C7⋊C8—21no—
118C7⋊C8—21no—
119C2×#41—72no—
120C2×#41—141no—
121C2×SD16C2×D712yes—
122C2×D8—71no—
123C2×M16—72no—
124C4⋊Q8—71no—
125C4⋊Q8C2×D711yescharacteristic
126#123—142no—
127#123—142no—
128C22⋊Q8—72no—
129C22⋊Q8—141no—
130#121—72no—
131#121—141no—
132C4×D4—71no—
133C4×D4—71no—
134C4×D4C2×D711yescharacteristic
135C2×C4⋊C4—72no—
136C2×C4⋊C4—141no—
137#107—72no—
138#107—71no—
139C4⋊C8C2×D711yescharacteristic
140#103—72no—
141#103—72no—
142#103C2×D712yes—
143#102—71no—
144#102—72no—
145#102—71no—
146C22⋊C8—72no—
147C4×C8—71no—
148C22×C7—41no—
149C22×C7—41no—
150C22×C7—41no—
151C22×C7C22×D411yescharacteristic
152C2×D7—22no—
153C2×D7—22no—
154C2×D7—22no—
155C2×D7—22no—
156C2×D7—41no—
157C2×D7—41no—
158C4×C7—21no—
159C4×C7—42no—
160C4×C7—41no—
161C4×C7AGL(1,ℤ/8)12yes—
162C4×C7C2×#4111yescharacteristic
163C4×C7H(5,𝔽2)11yescharacteristic
164Dic7—22no—
165Dic7—42no—
166Dic7—41no—
167#41—144no—
168#41—284no—
169#41—282no—
170C2×Q8—141no—
171C2×Q8C22×D712yes—
172C2×D4—71no—
173C2×D4—71no—
174C2×D4—142no—
175C2×D4—142no—
176C2×D4C22×D711yescharacteristic
177C22×C4—21no—
178C22×C4—72no—
179C22×C4—142no—
180C22×C4—142no—
181C22×C4—141no—
182SD16—42no—
183D8—142no—
184M16—282no—
185C2×C8—71no—
186C2×C8—71no—
187C2×C8C22×D712yes—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—71no—
191C4⋊C4—72no—
192C4⋊C4—71no—
193C4⋊C4—142no—
194C4⋊C4—142no—
195C4⋊C4—141no—
196C4⋊C4C22×D711yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—72no—
199C22⋊C4—142no—
200C22⋊C4—142no—
201C22⋊C4—141no—
202C42C22×D711yescharacteristic
203C2×C7—81no—
204C2×C7#54611yescharacteristic
205C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
206C2×C7#57811yescharacteristic
207D7—42no—
208D7—42no—
209C23—21no—
210C23—72no—
211C23—141no—
212Q8—22no—
213Q8—22no—
214Q8—282no—
215D4—21no—
216D4—21no—
217D4—72no—
218D4—142no—
219D4—141no—
220D4—284no—
221D4—284no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—72no—
226C2×C4—142no—
227C2×C4—142no—
228C2×C4—141no—
229C2×C4—142no—
230C2×C4—284no—
231C2×C4—284no—
232C2×C4—282no—
233C2×C4D4×D711yescharacteristic
234C2×C4D4×D711yescharacteristic
235C2×C4C23×D711yescharacteristic
236C8—22no—
237C8—141no—
238C8—141no—
239C7#333011yescharacteristic, complemented, Sylow
240C22—41no—
241C22—41no—
242C22—41no—
243C22—142no—
244C22—142no—
245C22—142no—
246C22—142no—
247C22—281no—
248C22—281no—
249C22C2×D4×D711yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—142no—
254C4—282no—
255C4—281no—
256C4#647712yes—
257C4#654811yescharacteristic
258C4D7×#4111yescharacteristic
259C2—81no—
260C2—282no—
261C2—282no—
262C2#9030311yescharacteristic
263C2C2×#647711yescharacteristic
264C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#461C2
socleC22×C7C22×D4
hypercenter#461D7
hypocenterC7#3330
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #90417C2semidirect
2. #90242C2semidirect
3. C7×#461C2semidirect
4. #89946C2semidirect
5. #89944C2semidirect
6. #89853C2semidirect
7. C4×#1141C2semidirect
8. #89618C2semidirect
9. #89610C2semidirect
10. #89581C2semidirect
11. #89575C2semidirect
84. #461D7semidirect
109. D28#41semidirect
239. C7#3330semidirect

Maximal subgroups

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

subgroupindex
#904172
#902422
C7×#4612
#899462
#899442
#898532
C4×#11412
#896182
#896102
#895812
#895752
#33307