Finite Group Databasebeta

← #11736720

Subgroups — #11736720

Order 896.

332 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 121 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#90170C211yesmaximal, characteristic
2#90161C211yesmaximal, characteristic, complemented
3C7×#431C211yesmaximal, characteristic, complemented
4#89623C211yesmaximal, characteristic, complemented
5#89622C211yesmaximal, characteristic, complemented
6#89618C211yesmaximal, characteristic, complemented
7#89614C211yesmaximal, characteristic, complemented
8#89509C212yesmaximal
9#89495C212yesmaximal, complemented
10#89469C212yesmaximal, complemented
11#89455C212yesmaximal, complemented
12C2×C42×C7C2211yescharacteristic
13#6493C2211yescharacteristic
14#6492—21no—
15C4×C22⋊D7—21no—
16#6489C2212yes—
17#6487—21no—
18#6487—21no—
19#6487C2211yescharacteristic
20#6439—21no—
21#6438C2211yescharacteristic
22C4×D28C2211yescharacteristic
23#6435—21no—
24#6433C2212yes—
25#6433C2211yescharacteristic
26#6432C2211yescharacteristic
27#6432C2211yescharacteristic
28C4×Dic14—21no—
29C4×Dic14—21no—
30C4×Dic14C2211yescharacteristic
31C7×C4⋊C8C2211yescharacteristic
32C7×C4⋊C8C2211yescharacteristic
33C7×C22⋊C8C2212yes—
34C4×C7×C8C2212yes—
35#6395C2212yes—
36#6395C2212yes—
37#6392C2212yes—
38#6392C2212yes—
39#6391C2212yes—
40#6391C2212yes—
41#6390C2212yes—
42#6390C2212yes—
43#3233—71nomaximal, Sylow
44C22×C4×C7D411yescharacteristic
45C22×C4×C7D411yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—21no—
48C2×D28C2311yescharacteristic
49C2×C4×D7—21no—
50C2×Dic14—21no—
51C2×Dic14—21no—
52C2×Dic14C2311yescharacteristic
53C2×C7×C8C2312yes—
54C2×C7×C8C2312yes—
55C42×C7D411yescharacteristic
56C42×C7D411yescharacteristic
57C42×C7C2311yescharacteristic
58C42×C7C2311yescharacteristic
59C22⋊Dic7—22no—
60C22⋊Dic7—21no—
61(C2×D7)⋊C4—21no—
62(C2×D7)⋊C4—21no—
63(C2×D7)⋊C4—21no—
64(C2×D7)⋊C4—21no—
65C4⋊Dic7—21no—
66C4⋊Dic7—21no—
67C4⋊Dic7C2311yescharacteristic
68C4⋊Dic7C2312yes—
69C4⋊Dic7C2312yes—
70C4⋊Dic7C2311yescharacteristic
71Dic7⋊C4—22no—
72Dic7⋊C4—22no—
73Dic7⋊C4—21no—
74Dic7⋊C4—21no—
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77C4×Dic7—22no—
78C4×Dic7—21no—
79#533—71no—
80#529—71no—
81#499—72no—
82#488—72no—
83#485—72no—
84#482—72no—
85#477—71no—
86#476—71no—
87#475—71no—
88#474—71no—
89#431D711yescharacteristic, complemented
90C23×C7C2×D411yescharacteristic
91C22×D7—21no—
92C2×C4×C7—22no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—41no—
96C2×C4×C7C2×D411yescharacteristic
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2411yescharacteristic
102C22⋊D7—42no—
103C2×Dic7—21no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—21no—
107C2×Dic7—21no—
108D28—21no—
109D28—21no—
110C4×D7—42no—
111Dic14—21no—
112Dic14—21no—
113Dic14—42no—
114Dic14—42no—
115C7×C8—22no—
116C7×C8—22no—
117C4⋊Q8—71no—
118C4⋊Q8—71no—
119#126—141no—
120#125—71no—
121#125—72no—
122#124—71no—
123#123—141no—
124C22⋊Q8—71no—
125C22⋊Q8—141no—
126C22⋊Q8—141no—
127#121—71no—
128C4×Q8—71no—
129C4×Q8—141no—
130C4×Q8—141no—
131C4×D4—71no—
132C4×D4—141no—
133#117—72no—
134#117—141no—
135C2×C42C2×D711yescharacteristic
136#107—72no—
137#107—72no—
138#106—72no—
139#106—72no—
140C4⋊C8C2×D711yescharacteristic
141C4⋊C8C2×D711yescharacteristic
142#103—72no—
143#103—72no—
144#102—72no—
145#102—72no—
146C22⋊C8C2×D712yes—
147C4×C8C2×D712yes—
148C22×C7—21no—
149C22×C7—21no—
150C22×C7—21no—
151C22×C7C22×D411yescharacteristic
152C2×D7—41no—
153C2×D7—41no—
154C2×D7—41no—
155C4×C7—21no—
156C4×C7—21no—
157C4×C7—21no—
158C4×C7—21no—
159C4×C7#13512yes—
160C4×C72−1+411yescharacteristic
161C4×C72−1+411yescharacteristic
162Dic7—41no—
163Dic7—42no—
164Dic7—41no—
165Dic7—42no—
166Dic7—41no—
167C2×Q8—71no—
168C2×Q8—141no—
169C2×Q8—141no—
170C2×D4—71no—
171C2×D4—141no—
172C22×C4—141no—
173C22×C4D2811yescharacteristic
174C22×C4D2811yescharacteristic
175C22×C4C22×D711yescharacteristic
176C2×C8C22×D712yes—
177C2×C8C22×D712yes—
178C4⋊C4—72no—
179C4⋊C4—72no—
180C4⋊C4—71no—
181C4⋊C4—71no—
182C4⋊C4—141no—
183C4⋊C4—141no—
184C4⋊C4—141no—
185C4⋊C4—141no—
186C4⋊C4—142no—
187C4⋊C4—142no—
188C4⋊C4—141no—
189C4⋊C4—141no—
190C22⋊C4—142no—
191C22⋊C4—141no—
192C22⋊C4—141no—
193C22⋊C4—141no—
194C22⋊C4—141no—
195C22⋊C4—141no—
196C42—142no—
197C42—141no—
198C42D2811yescharacteristic
199C42D2811yescharacteristic
200C42C22×D711yescharacteristic
201C42C22×D711yescharacteristic
202C2×C7—41no—
203C2×C7#53611yescharacteristic
204C2×C7C2×#13511yescharacteristic
205C2×C7#57511yescharacteristic
206D7—81no—
207C23—141no—
208C23C2×D2811yescharacteristic
209Q8—141no—
210Q8—141no—
211Q8—282no—
212Q8—282no—
213D4—141no—
214D4—141no—
215D4—282no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—141no—
221C2×C4—141no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4—141no—
225C2×C4—282no—
226C2×C4C2×D2811yescharacteristic
227C2×C4C2×D2811yescharacteristic
228C2×C4C2×D2811yescharacteristic
229C2×C4C2×D2811yescharacteristic
230C2×C4C2×D2811yescharacteristic
231C2×C4C23×D711yescharacteristic
232C8—22no—
233C8—22no—
234C7#323311yescharacteristic, complemented, Sylow
235C22—21no—
236C22—21no—
237C22—21no—
238C22—281no—
239C22—281no—
240C22—281no—
241C22C22×D2811yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—281no—
247C4—282no—
248C4—281no—
249C4—282no—
250C4—281no—
251C4#646712yes—
252C4#655411yescharacteristic
253C4#655411yescharacteristic
254C2—41no—
255C2—561no—
256C2#9021711yescharacteristic
257C2C2×#646711yescharacteristic
258C2#9043911yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#431C2
socleC22×C7C22×D4
prime radical#6493C22
hypercenter#431D7
hypocenterC7#3233
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #90161C2semidirect
3. C7×#431C2semidirect
4. #89623C2semidirect
5. #89622C2semidirect
6. #89618C2semidirect
7. #89614C2semidirect
9. #89495C2semidirect
10. #89469C2semidirect
11. #89455C2semidirect
89. #431D7semidirect
234. C7#3233semidirect

Maximal subgroups

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

subgroupindex
#901702
#901612
C7×#4312
#896232
#896222
#896182
#896142
#895092
#894952
#894692
#894552
#32337