Finite Group Databasebeta

← #31900686

Subgroups — #31900686

Order 18176.

414 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 77 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#22588440C212yesmaximal
2#22588415C211yesmaximal, characteristic
3#22586248C211yesmaximal, characteristic
4#22586193C212yesmaximal
5C71×#2607C211yesmaximal, characteristic
6C2×#14556087C2211yescharacteristic
7#17985963C2212yes—
8#17985862C2211yescharacteristic
9#17985765—21no—
10#17985765—21no—
11#17985759—22no—
12#17985759—22no—
13#399×C71—21no—
14#393×C71—21no—
15#391×C71—22no—
16#387×C71—22no—
17#381×C71C2211yescharacteristic
18C4×C22⋊C4×C71C2212yes—
19C2×#11765849—22no—
20#14556087—22no—
21#14556087—22no—
22C2×C4⋊Dic71Q812yescomplemented
23C2×C22⋊Dic71—21no—
24C2×C22⋊Dic71—21no—
25C2×C4×Dic71—21no—
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—22no—
28C71⋊#95—22no—
29C22×Q8×C71—21no—
30C23×C4×C71C2311yescharacteristic
31C2×C4⋊C4×C71—22no—
32C2×C4⋊C4×C71—21no—
33C2×C22⋊C4×C71—22no—
34C2×C22⋊C4×C71D411yescharacteristic
35C2×C22⋊C4×C71D411yescharacteristic
36C2×C42×C71D412yes—
37C2×C42×C71D412yes—
38#95×C71—21no—
39#95×C71—22no—
40#95×C71—21no—
41#95×C71—22no—
42#95×C71—22no—
43C22×Dic71—22no—
44C22×Dic71—41no—
45#11765849—42no—
46#11765849—42no—
47C2×C71⋊C8—42no—
48C4⋊Dic71—24no—
49C22⋊Dic71—41no—
50C22⋊Dic71—41no—
51C22⋊Dic71—41no—
52C22⋊Dic71—41no—
53C4×Dic71—42no—
54C24×C71C2×D411yescharacteristic
55C2×Q8×C71—44no—
56C2×Q8×C71—81no—
57C22×C4×C71—22no—
58C22×C4×C71—21no—
59C22×C4×C71—22no—
60C22×C4×C71—21no—
61C22×C4×C71—41no—
62C22×C4×C71—41no—
63C22×C4×C71C2×D412yes—
64C22×C4×C71C2×Q811yescharacteristic
65C22×C4×C71#4112yes—
66C22×C4×C71#4111yescharacteristic
67C4⋊C4×C71—44no—
68C4⋊C4×C71—81no—
69C22⋊C4×C71—44no—
70C22⋊C4×C71—41no—
71C22⋊C4×C71—41no—
72C42×C71—42no—
73C42×C71—42no—
74C23×C71—21no—
75C23×C71—21no—
76C23×C71—42no—
77C23×C71#12011yescharacteristic
78C23×C71C22⋊Q811yescharacteristic
79C23×C71#12411yescharacteristic
80Q8×C71—84no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—24no—
84C2×C4×C71—41no—
85C2×C4×C71—41no—
86C2×C4×C71—42no—
87C2×C4×C71—42no—
88C2×C4×C71—42no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—42no—
94C2×C4×C71—41no—
95C2×C4×C71—41no—
96C2×C4×C71C22⋊Q812yes—
97C2×C4×C71#12412yes—
98C2×Dic71—42no—
99C2×Dic71—82no—
100C2×Dic71—81no—
101C2×Dic71—82no—
102C71⋊C8—82no—
103C22×C71—22no—
104C22×C71—21no—
105C22×C71—21no—
106C22×C71—22no—
107C22×C71—22no—
108C22×C71—81no—
109C22×C71—81no—
110C22×C71#39311yescharacteristic
111C22×C71#45811yescharacteristic
112C22×C71#45811yescharacteristic
113C4×C71—22no—
114C4×C71—22no—
115C4×C71—44no—
116C4×C71—82no—
117C4×C71—81no—
118C4×C71—81no—
119Dic71—82no—
120Dic71—161no—
121#10387—711nomaximal, Sylow
122C2×C71—21no—
123C2×C71—21no—
124C2×C71—41no—
125C2×C71—41no—
126C2×C71#203011yescharacteristic
127C2×C71#203112yes—
128#2607D7111yescharacteristic
129#1826—712no—
130#1820—711no—
131#1306—711no—
132#1284—712no—
133C71#1038711yescharacteristic, complemented, Sylow
134C2×#117—711no—
135#407—712no—
136#399—21no—
137#393—21no—
138#391—22no—
139#387—22no—
140#381—711no—
141#381C2×D7111yescharacteristic
142C4×C22⋊C4C2×D7112yes—
143#342—1421no—
144#342—1421no—
145#339—1422no—
146#339—1422no—
147C22×Q8—21no—
148C23×C4C22×D7111yescharacteristic
149C2×M16—1422no—
150#117—1422no—
151#117—1422no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—712no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—1421no—
157C2×C22⋊C4—1421no—
158C2×C22⋊C4C22⋊D7111yescharacteristic
159C2×C22⋊C4C22⋊D7111yescharacteristic
160C2×C42—1421no—
161C2×C42C22⋊D7112yes—
162C2×C42C22⋊D7112yes—
163C22⋊C8—1422no—
164C22⋊C8—1422no—
165#95—21no—
166#95—22no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—1422no—
171C24C2×C22⋊D7111yescharacteristic
172C2×Q8—44no—
173C2×Q8—81no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—2841no—
181C22×C4—1422no—
182C22×C4C2×C22⋊D7112yes—
183C22×C4Q8×D7111yescharacteristic
184C22×C4#1176586612yes—
185C22×C4#1176586611yescharacteristic
186M16—2842no—
187M16—2842no—
188C2×C8—2842no—
189C4⋊C4—44no—
190C4⋊C4—81no—
191C4⋊C4—1424no—
192C22⋊C4—44no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—2841no—
196C22⋊C4—2841no—
197C22⋊C4—2841no—
198C22⋊C4—2841no—
199C42—42no—
200C42—42no—
201C42—2842no—
202C23—21no—
203C23—21no—
204C23—42no—
205C23#1455609711yescharacteristic
206C23#1455610411yescharacteristic
207C23#1455611711yescharacteristic
208Q8—84no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—2842no—
225C2×C4—5682no—
226C2×C4—5681no—
227C2×C4—5682no—
228C2×C4#1455610412yes—
229C2×C4#1455611712yes—
230C8—5682no—
231C22—22no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—81no—
237C22—81no—
238C22#1798591011yescharacteristic
239C22#1798618911yescharacteristic
240C22#1798618911yescharacteristic
241C4—22no—
242C4—22no—
243C4—44no—
244C4—81no—
245C4—82no—
246C4—81no—
247C4—5682no—
248C4—11361no—
249C2—21no—
250C2—21no—
251C2—41no—
252C2—41no—
253C2#2258950611yescharacteristic
254C2#2258950912yes—
subgroupG/H
centerC22#17985910
commutator, cosocleC23×C4×C71C23
FrattiniC23×C4C22×D71
FittingC71×#2607C2
socleC22×C71#393
prime radicalC24×C71C2×D4
hypercenter#2607D71
hypocenterC71#10387
LCSG ▷ C23×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2607

Complements

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

NH ≅ G/Ntype
22. C2×C4⋊Dic71Q8semidirect
133. C71#10387semidirect

Maximal subgroups

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

subgroupindex
#225884402
#225884152
#225862482
#225861932
C71×#26072
#1038771