Finite Group Databasebeta

← #31895086

Subgroups — #31895086

Order 18176.

456 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 99 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#22598743C211yesmaximal, characteristic
2#22586967C211yesmaximal, characteristic, complemented
3#22586847C212yesmaximal
4#22586233C212yesmaximal, complemented
5C71×#2371C211yesmaximal, characteristic, complemented
6#17986549C2211yescharacteristic
7#17986369C412yescomplemented
8#17986365C412yescomplemented
9#17986356—22no—
10#17986056C2212yes—
11#17985998—22no—
12C4×#11765849C2211yescharacteristic
13#17985741—22no—
14#17985732—22no—
15#17985732—22no—
16C2×#125×C71C2211yescharacteristic
17#387×C71—22no—
18#387×C71—22no—
19C4×C4⋊C4×C71—22no—
20#376×C71C2212yes—
21C2×#11765865—21no—
22C2×#11765849—21no—
23#14556128C2×C411yescharacteristic
24#14556125C2×C411yescharacteristic
25#14556116—21no—
26#14556102—22no—
27C4×D284—22no—
28C4⋊(C71⋊C8)—22no—
29#14556051—24no—
30#14556048—24no—
31C22⋊(C71⋊C8)—22no—
32#14556039—22no—
33#14556039C2×C412yes—
34C4×C71⋊C8—21no—
35#125×C71—22no—
36#125×C71C2×C412yes—
37#125×C71D412yes—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71D412yes—
41C2×C42×C71—22no—
42C2×C42×C71—22no—
43C2×C42×C71C2311yescharacteristic
44#95×C71—22no—
45#95×C71—22no—
46#95×C71—22no—
47#95×C71D412yes—
48#95×C71D412yes—
49#11765865—44no—
50C2×Dic142—21no—
51C2×D284—21no—
52C2×D284—22no—
53C2×C22⋊D71—41no—
54C2×C4×D71—41no—
55#11765849—42no—
56C2×C71⋊C8—22no—
57C2×C71⋊C8—42no—
58C4⋊Dic71—22no—
59(C2×D71)⋊C4—42no—
60C22×C4×C71—22no—
61C22×C4×C71—22no—
62C22×C4×C71—22no—
63C22×C4×C71C2×D411yescharacteristic
64C22×C4×C71C2×D412yes—
65C4⋊C4×C71—24no—
66C4⋊C4×C71—24no—
67C4⋊C4×C71—22no—
68C4⋊C4×C71—44no—
69C4⋊C4×C71C22⋊C414yes—
70C42×C71—44no—
71C42×C71—44no—
72C42×C71C22×C411yescharacteristic
73C42×C71C2×D411yescharacteristic
74C42×C71#4112yes—
75C23×C71#12011yescharacteristic
76C22×D71—41no—
77C2×C4×C71—22no—
78C2×C4×C71—24no—
79C2×C4×C71—24no—
80C2×C4×C71—42no—
81C2×C4×C71—42no—
82C2×C4×C71—42no—
83C2×C4×C71—42no—
84C2×C4×C71—42no—
85C2×C4×C71—42no—
86C2×C4×C71—42no—
87C2×C4×C71—42no—
88C2×C4×C71C2×C22⋊C411yescharacteristic
89C2×C4×C71C4×D412yes—
90C2×C4×C71#12112yes—
91C2×C4×C71#12311yescharacteristic
92C22⋊D71—82no—
93C2×Dic71—41no—
94D284—42no—
95D284—82no—
96C4×D71—82no—
97Dic142—42no—
98C71⋊C8—42no—
99C71⋊C8—82no—
100C22×C71—22no—
101C22×C71#38611yescharacteristic
102C22×C71#45412yes—
103C2×D71—81no—
104C2×D71—82no—
105C4×C71—21no—
106C4×C71—21no—
107C4×C71—22no—
108C4×C71—22no—
109C4×C71—44no—
110C4×C71—44no—
111C4×C71—82no—
112Dic71—81no—
113#9442—711nomaximal, Sylow
114C2×C71—22no—
115C2×C71#188811yescharacteristic
116C2×C71#190612yes—
117D71—161no—
118#3142—711no—
119#2371D7111yescharacteristic, complemented
120#1534—711no—
121#1505—712no—
122#1300—712no—
123C71#944211yescharacteristic, complemented, Sylow
124#532—711no—
125C2×#125C2×D7111yescharacteristic
126#491—712no—
127#490—712no—
128#488—1422no—
129#432—712no—
130#417—1422no—
131C4×M16—711no—
132#387—22no—
133#387—22no—
134C4×C4⋊C4—22no—
135#376C2×D7112yes—
136#330—1422no—
137#327—1422no—
138#327—1422no—
139C2×#41—1421no—
140C2×M16—1421no—
141C4⋊Q8—711no—
142#127—711no—
143#125—22no—
144#125C4×D7112yes—
145#125D28412yes—
146#124—1421no—
147#121—1422no—
148C4×D4—1422no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C22⋊D7112yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42C22×D7111yescharacteristic
155C4⋊C8—1422no—
156#103—1424no—
157#102—1424no—
158C22⋊C8—1422no—
159#97—712no—
160#97—1422no—
161C4×C8—1421no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95C22⋊D7112yes—
166#95C22⋊D7112yes—
167#41—2844no—
168C2×Q8—1421no—
169C2×D4—2841no—
170C2×D4—1421no—
171C2×D4—1422no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—2841no—
176C22×C4C2×C22⋊D7111yescharacteristic
177C22×C4C2×C22⋊D7112yes—
178M16—2842no—
179C2×C8—2842no—
180C2×C8—1422no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—44no—
185C4⋊C4—1422no—
186C4⋊C4(C2×D71)⋊C414yes—
187C22⋊C4—2842no—
188C42—44no—
189C42—44no—
190C42C2×C4×D7111yescharacteristic
191C42C2×D28411yescharacteristic
192C42#1176586512yes—
193C23—2841no—
194C23#1455609711yescharacteristic
195Q8—2842no—
196D4—2842no—
197D4—5682no—
198D4—5682no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—2841no—
211C2×C4—5682no—
212C2×C4C2×(C2×D71)⋊C411yescharacteristic
213C2×C4C4×C22⋊D7112yes—
214C2×C4#1455610212yes—
215C2×C4#1455611111yescharacteristic
216C8—2842no—
217C8—5682no—
218C22—22no—
219C22—5681no—
220C22—5682no—
221C22#1798587811yescharacteristic
222C22#1798617212yes—
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—44no—
229C4—82no—
230C4—5681no—
231C2—22no—
232C2—11361no—
233C2#2258874311yescharacteristic
234C2#2258884112yes—
subgroupG/H
centerC22#17985878
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2371C2
socleC22×C71#386
prime radical#17986549C22
cosocleC2×C42×C71C23
hypercenter#2371D71
hypocenterC71#9442
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2371

Complements

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

NH ≅ G/Ntype
2. #22586967C2semidirect
4. #22586233C2semidirect
5. C71×#2371C2semidirect
7. #17986369C4semidirect
8. #17986365C4semidirect
119. #2371D71semidirect
123. C71#9442semidirect

Maximal subgroups

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

subgroupindex
#225987432
#225869672
#225868472
#225862332
C71×#23712
#944271