Finite Group Databasebeta

← #11737002

Subgroups — #11737002

Order 896.

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

normal subgroups only

This group has 420 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#90246C211yesmaximal, characteristic, complemented
2#90242C211yesmaximal, characteristic, complemented
3C7×#436C211yesmaximal, characteristic, complemented
4#89906C212yesmaximal, complemented
5#89905C212yesmaximal, complemented
6#89618C211yesmaximal, characteristic, complemented
7#89613C211yesmaximal, characteristic, complemented
8#89506C212yesmaximal, complemented
9#89494C212yesmaximal, complemented
10C4×D56C211yesmaximal, characteristic, complemented
11#89453C211yesmaximal, characteristic, complemented
12C2×#1159—21no—
13C2×#1157C2212yes—
14C2×C7×M16C2212yes—
15C4×C7×D4C2211yescharacteristic
16#6501—21no—
17#6493C2212yes—
18C2×C4⋊Dic7C2212yes—
19#6487C2212yes—
20C2×D56C2211yescharacteristic
21C2×#1132C2212yes—
22#6460—21no—
23D7×C4⋊C4—21no—
24#6449—22no—
25#6446—22no—
26C4×D28C2211yescharacteristic
27C4×D28C2211yescharacteristic
28#6432C2211yescharacteristic
29#6432C2211yescharacteristic
30C7×C4⋊C8C2211yescharacteristic
31C7×C22⋊C8C2212yes—
32C4×C7×C8C2211yescharacteristic
33#6395C2212yes—
34#6395C2211yescharacteristic
35#6395C2211yescharacteristic
36#6392C2212yes—
37#6392C2211yescharacteristic
38#6390C2212yes—
39#6390C2212yes—
40#6390C2212yes—
41#3330—71nomaximal, Sylow
42C2×C7×D4D411yescharacteristic
43C22×C4×C7C2312yes—
44C2×C22⋊D7—22no—
45C2×C22⋊D7—22no—
46C22×Dic7—22no—
47#1159—44no—
48#1157—24no—
49#1157—42no—
50C2×D28C2311yescharacteristic
51C2×D28C2311yescharacteristic
52C2×C4×D7—21no—
53C2×C4×D7—21no—
54C2×C4×D7—22no—
55C2×Dic14—21no—
56C2×Dic14C2312yes—
57C7×M16—42no—
58C2×C7×C8C2311yescharacteristic
59C2×C7×C8C2311yescharacteristic
60C2×C7×C8C2312yes—
61C7×C4⋊C4D411yescharacteristic
62C7×C22⋊C4D412yes—
63C42×C7C2311yescharacteristic
64C22⋊Dic7—22no—
65(C2×D7)⋊C4—21no—
66(C2×D7)⋊C4—21no—
67(C2×D7)⋊C4—22no—
68C4⋊Dic7—21no—
69C4⋊Dic7—21no—
70C4⋊Dic7—21no—
71C4⋊Dic7—22no—
72C4⋊Dic7C2312yes—
73C4⋊Dic7C2311yescharacteristic
74C4⋊Dic7C2311yescharacteristic
75Dic7⋊C4—22no—
76D56—22no—
77#1132—42no—
78#547—71no—
79#547—71no—
80#487—71no—
81#484—72no—
82#474—71no—
83#468—72no—
84#461—71no—
85#449—72no—
86C4×D8—71no—
87#436D711yescharacteristic, complemented
88#417—72no—
89C23×C7C2×D412yes—
90C22×D7—21no—
91C22×D7—21no—
92C7×D4—22no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7C2×D412yes—
96C2×C4×C7C2×D411yescharacteristic
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2411yescharacteristic
99C22⋊D7—44no—
100C22⋊D7—44no—
101C2×Dic7—21no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—21no—
105C2×Dic7—44no—
106D28—21no—
107D28—21no—
108D28—21no—
109D28#4112yescomplemented
110C4×D7—44no—
111C4×D7—42no—
112C4×D7—42no—
113Dic14—22no—
114Dic14—22no—
115Dic14—42no—
116C7×C8—21no—
117C7×C8—22no—
118C7×C8—21no—
119C2×#41—72no—
120C2×#41—141no—
121C2×SD16—72no—
122C2×D8—71no—
123C2×M16C2×D712yes—
124C4⋊Q8—71no—
125C4⋊Q8—71no—
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—71no—
138#107—72no—
139C4⋊C8C2×D711yescharacteristic
140#103—72no—
141#103—72no—
142#103—72no—
143#102—71no—
144#102—72no—
145#102—71no—
146C22⋊C8C2×D712yes—
147C4×C8C2×D711yescharacteristic
148C22×C7—22no—
149C22×C7—22no—
150C22×C7—22no—
151C22×C7C22×D411yescharacteristic
152C2×D7—22no—
153C2×D7—41no—
154C2×D7—41no—
155C2×D7—41no—
156C2×D7—41no—
157C2×D7—41no—
158C4×C7—22no—
159C4×C7—21no—
160C4×C7AGL(1,ℤ/8)12yes—
161C4×C7C2×#4111yescharacteristic
162C4×C7H(5,𝔽2)11yescharacteristic
163Dic7—42no—
164Dic7—42no—
165Dic7—41no—
166Dic7—41no—
167#41—144no—
168#41—282no—
169#41—284no—
170C2×Q8—72no—
171C2×Q8—141no—
172C2×D4—71no—
173C2×D4—71no—
174C2×D4—142no—
175C2×D4—142no—
176C2×D4D2811yescharacteristic
177C22×C4—142no—
178C22×C4—141no—
179C22×C4—142no—
180C22×C4—141no—
181C22×C4C22×D712yes—
182SD16—282no—
183D8—142no—
184M16—42no—
185C2×C8C22×D711yescharacteristic
186C2×C8C22×D711yescharacteristic
187C2×C8C22×D712yes—
188C4⋊C4—71no—
189C4⋊C4—72no—
190C4⋊C4—71no—
191C4⋊C4—142no—
192C4⋊C4—142no—
193C4⋊C4—141no—
194C4⋊C4—141no—
195C4⋊C4—141no—
196C4⋊C4D2811yescharacteristic
197C22⋊C4—142no—
198C22⋊C4—141no—
199C22⋊C4—141no—
200C22⋊C4—142no—
201C22⋊C4D2812yes—
202C42C22×D711yescharacteristic
203C2×C7—42no—
204C2×C7#54611yescharacteristic
205C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
206C2×C7#57811yescharacteristic
207D7—42no—
208D7—81no—
209C23—141no—
210C23—141no—
211C23C2×D2812yes—
212Q8—142no—
213Q8—142no—
214Q8—282no—
215D4—22no—
216D4—72no—
217D4—141no—
218D4—141no—
219D4—141no—
220D4—284no—
221D4—284no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—141no—
225C2×C4—141no—
226C2×C4—142no—
227C2×C4—142no—
228C2×C4—284no—
229C2×C4—284no—
230C2×C4—282no—
231C2×C4—282no—
232C2×C4C2×D2811yescharacteristic
233C2×C4C2×D2812yes—
234C2×C4C2×D2811yescharacteristic
235C2×C4C23×D711yescharacteristic
236C8—21no—
237C8—22no—
238C8—21no—
239C7#333011yescharacteristic, complemented, Sylow
240C22—22no—
241C22—22no—
242C22—22no—
243C22—142no—
244C22—281no—
245C22—281no—
246C22—281no—
247C22—281no—
248C22—281no—
249C22C22×D2811yescharacteristic
250C4—21no—
251C4—22no—
252C4—281no—
253C4—282no—
254C4—281no—
255C4—282no—
256C4#647112yes—
257C4#654811yescharacteristic
258C4D7×#4111yescharacteristic
259C2—42no—
260C2—282no—
261C2—561no—
262C2#9024511yescharacteristic
263C2C2×#647111yescharacteristic
264C2#9044411yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#436C2
socleC22×C7C22×D4
hypercenter#436D7
hypocenterC7#3330
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #90246C2semidirect
2. #90242C2semidirect
3. C7×#436C2semidirect
4. #89906C2semidirect
5. #89905C2semidirect
6. #89618C2semidirect
7. #89613C2semidirect
8. #89506C2semidirect
9. #89494C2semidirect
10. C4×D56C2semidirect
11. #89453C2semidirect
87. #436D7semidirect
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
#902462
#902422
C7×#4362
#899062
#899052
#896182
#896132
#895062
#894942
C4×D562
#894532
#33307