Finite Group Databasebeta

← #11738558

Subgroups — #11738558

Order 896.

332 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 113 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#90419C211yesmaximal, characteristic, complemented
2#90271C211yesmaximal, characteristic
3C7×#477C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic, complemented
5#89792C211yesmaximal, characteristic, complemented
6#89657C212yesmaximal
7#89650C212yesmaximal, complemented
8#89618C211yesmaximal, characteristic, complemented
9#89605C211yesmaximal, characteristic, complemented
10#89586C212yesmaximal, complemented
11#89572C212yesmaximal, complemented
12C7×C4⋊Q8C2211yescharacteristic
13C4×C7×Q8C2211yescharacteristic
14#6510C2211yescharacteristic
15#6509—21no—
16#6509C2211yescharacteristic
17Q8×Dic7—21no—
18#6461—21no—
19#6460—21no—
20#6458C2211yescharacteristic
21#6457—21no—
22#6455C2212yes—
23#6453C2211yescharacteristic
24#6453C2212yes—
25#6450—21no—
26C4×D28—21no—
27C4×D28C2211yescharacteristic
28#6435—21no—
29C42×D7C2211yescharacteristic
30#6432C2211yescharacteristic
31C7×#107C2212yes—
32C7×C4⋊C8C2211yescharacteristic
33C7×#103C2212yes—
34#6409C2212yes—
35#6395C2212yes—
36(C2×D7)⋊C8C2212yes—
37#6392C2212yes—
38C8×Dic7C2212yes—
39#6383C2212yes—
40#6382C2212yes—
41#6382C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3233—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46C2×D28—21no—
47C2×D28C2311yescharacteristic
48C2×C4×D7—21no—
49C2×C4×D7D411yescharacteristic
50C2×C4×D7D411yescharacteristic
51C2×C4×D7C2311yescharacteristic
52C2×Dic14—21no—
53C2×C7×C8C2312yes—
54C7×C4⋊C4—21no—
55C7×C4⋊C4—21no—
56C7×C4⋊C4C2312yes—
57C7×C4⋊C4C2311yescharacteristic
58C42×C7—21no—
59C42×C7C2311yescharacteristic
60(C2×D7)⋊C4—21no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—21no—
63(C2×D7)⋊C4—21no—
64(C2×D7)⋊C4—21no—
65(C2×D7)⋊C4—21no—
66C4⋊Dic7—21no—
67C4⋊Dic7—21no—
68C4⋊Dic7—22no—
69C4⋊Dic7C2312yes—
70C4⋊Dic7C2311yescharacteristic
71Dic7⋊C4—21no—
72Dic7⋊C4—21no—
73Dic7⋊C4—22no—
74C4×Dic7—22no—
75C4×Dic7D411yescharacteristic
76C4×Dic7D411yescharacteristic
77C4×Dic7C2311yescharacteristic
78C2×C7⋊C8C2312yes—
79#533—71no—
80#529—71no—
81#499—72no—
82#488—72no—
83#485—72no—
84#482—72no—
85#477D711yescharacteristic, complemented
86#476—71no—
87#475—71no—
88#474—71no—
89#431—71no—
90C22×D7—21no—
91C22×D7C2×D411yescharacteristic
92C7×Q8—21no—
93C7×Q8—21no—
94C7×Q8—42no—
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2411yescharacteristic
101C2×Dic7—22no—
102C2×Dic7—21no—
103C2×Dic7C2×D411yescharacteristic
104C2×Dic7C2×D411yescharacteristic
105C2×Dic7C2×D411yescharacteristic
106D28—21no—
107D28—21no—
108D28—42no—
109C4×D7—22no—
110C4×D7—21no—
111C4×D7—21no—
112C4×D7—41no—
113C4×D7—42no—
114Dic14—42no—
115C7×C8—22no—
116C7⋊C8—22no—
117C4⋊Q8—71no—
118C4⋊Q8C2×D711yescharacteristic
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—141no—
129C4×Q8—141no—
130C4×Q8C2×D711yescharacteristic
131C4×D4—71no—
132C4×D4—141no—
133#117—72no—
134#117—141no—
135C2×C42—71no—
136#107—72no—
137#107C2×D712yes—
138#106—72no—
139#106—72no—
140C4⋊C8—71no—
141C4⋊C8C2×D711yescharacteristic
142#103—72no—
143#103C2×D712yes—
144#102—72no—
145#102—72no—
146C22⋊C8—72no—
147C4×C8—72no—
148C22×C7C22×D411yescharacteristic
149C2×D7—21no—
150C2×D7—21no—
151C2×D7—21no—
152C2×D7—41no—
153C2×D7—41no—
154C2×D7—41no—
155C4×C7—21no—
156C4×C7—41no—
157C4×C7—41no—
158C4×C7—42no—
159C4×C7#13512yes—
160C4×C72−1+411yescharacteristic
161C4×C72−1+411yescharacteristic
162Dic7—21no—
163Dic7—21no—
164Dic7—21no—
165Dic7—42no—
166Dic7—41no—
167C2×Q8—21no—
168C2×Q8—141no—
169C2×Q8C22×D711yescharacteristic
170C2×D4—71no—
171C2×D4—141no—
172C22×C4—71no—
173C22×C4—71no—
174C22×C4—71no—
175C22×C4—141no—
176C2×C8—72no—
177C2×C8C22×D712yes—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—72no—
181C4⋊C4—71no—
182C4⋊C4—141no—
183C4⋊C4—141no—
184C4⋊C4—142no—
185C4⋊C4—141no—
186C4⋊C4—142no—
187C4⋊C4—141no—
188C4⋊C4C22×D712yes—
189C4⋊C4C22×D711yescharacteristic
190C22⋊C4—141no—
191C22⋊C4—141no—
192C22⋊C4—141no—
193C22⋊C4—142no—
194C22⋊C4—141no—
195C22⋊C4—141no—
196C42—21no—
197C42—71no—
198C42—71no—
199C42—71no—
200C42—142no—
201C42C22×D711yescharacteristic
202C2×C7#53611yescharacteristic
203C2×C7C2×#13511yescharacteristic
204C2×C7#57511yescharacteristic
205D7—41no—
206D7—81no—
207C23—71no—
208C23—141no—
209Q8—21no—
210Q8—21no—
211Q8—42no—
212Q8—282no—
213D4—141no—
214D4—141no—
215D4—282no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—71no—
220C2×C4—71no—
221C2×C4—71no—
222C2×C4—142no—
223C2×C4—141no—
224C2×C4—141no—
225C2×C4—142no—
226C2×C4—141no—
227C2×C4—281no—
228C2×C4—282no—
229C2×C4D4×D711yescharacteristic
230C2×C4D4×D711yescharacteristic
231C2×C4C23×D711yescharacteristic
232C8—22no—
233C8—142no—
234C7#323311yescharacteristic, complemented, Sylow
235C22—141no—
236C22—141no—
237C22—141no—
238C22—281no—
239C22—281no—
240C22—281no—
241C22C2×D4×D711yescharacteristic
242C4—21no—
243C4—41no—
244C4—42no—
245C4—41no—
246C4—141no—
247C4—141no—
248C4—141no—
249C4—281no—
250C4—282no—
251C4#648212yes—
252C4#655111yescharacteristic
253C4#655411yescharacteristic
254C2—281no—
255C2—561no—
256C2#9031911yescharacteristic
257C2C2×#648211yescharacteristic
258C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#477C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#477D7
hypocenterC7#3233
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #90419C2semidirect
3. C7×#477C2semidirect
4. #89856C2semidirect
5. #89792C2semidirect
7. #89650C2semidirect
8. #89618C2semidirect
9. #89605C2semidirect
10. #89586C2semidirect
11. #89572C2semidirect
85. #477D7semidirect
234. C7#3233semidirect

Maximal subgroups

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

subgroupindex
#904192
#902712
C7×#4772
#898562
#897922
#896572
#896502
#896182
#896052
#895862
#895722
#32337