Finite Group Databasebeta

← #31885675

Subgroups — #31885675

Order 18176.

874 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 139 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#22588102C212yesmaximal, complemented
2#22586161C214yesmaximal, complemented
3C2×#390×C71C211yesmaximal, characteristic
4C2×C22⋊(C71⋊C8)—22no—
5C2×#14556039—22no—
6#17985754C2214yescomplemented
7C23×D4×C71C411yescharacteristic
8C22×C4⋊C4×C71C411yescharacteristic
9C22×C22⋊C4×C71—22no—
10C22×C42×C71C2211yescharacteristic
11#390×C71—22no—
12#390×C71C412yes—
13#390×C71C2212yes—
14C22×C71⋊C8—24no—
15C22⋊(C71⋊C8)—28no—
16#14556039—44no—
17C25×C71—21no—
18C22×D4×C71—24no—
19C22×D4×C71—21no—
20C22×D4×C71—42no—
21C22×D4×C71—42no—
22C22×D4×C71C2×C412yes—
23C23×C4×C71—21no—
24C23×C4×C71C2×C411yescharacteristic
25C23×C4×C71D412yes—
26C2×C4⋊C4×C71—21no—
27C2×C4⋊C4×C71—42no—
28C2×C4⋊C4×C71C2×C412yes—
29C2×C22⋊C4×C71—24no—
30C2×C22⋊C4×C71—24no—
31C2×C22⋊C4×C71—48no—
32C2×C42×C71—42no—
33C2×C42×C71C2×C411yescharacteristic
34C2×C42×C71D412yes—
35C2×C42×C71C2311yescharacteristic
36C2×C71⋊C8—28no—
37C2×C71⋊C8—44no—
38C2×C71⋊C8—44no—
39C24×C71—24no—
40C24×C71—22no—
41C24×C71—44no—
42C24×C71—42no—
43C24×C71—42no—
44C24×C71—44no—
45C24×C71C22⋊C411yescharacteristic
46C2×D4×C71—24no—
47C2×D4×C71—22no—
48C2×D4×C71—42no—
49C2×D4×C71—42no—
50C2×D4×C71—416no—
51C2×D4×C71—42no—
52C2×D4×C71—42no—
53C2×D4×C71M1614yes—
54C22×C4×C71—22no—
55C22×C4×C71—22no—
56C22×C4×C71—22no—
57C22×C4×C71—24no—
58C22×C4×C71—24no—
59C22×C4×C71—22no—
60C22×C4×C71—42no—
61C22×C4×C71—44no—
62C22×C4×C71—42no—
63C22×C4×C71C22⋊C412yes—
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C22×C411yescharacteristic
66C22×C4×C71C2×D412yes—
67C4⋊C4×C71—42no—
68C4⋊C4×C71—42no—
69C22⋊C4×C71—48no—
70C22⋊C4×C71—48no—
71C42×C71—24no—
72C42×C71—41no—
73C42×C71—41no—
74C23×C71—21no—
75C23×C71—21no—
76C23×C71—22no—
77C23×C71—24no—
78C23×C71—48no—
79C23×C71—416no—
80C23×C71—42no—
81C23×C71—42no—
82C23×C71—44no—
83C23×C71—44no—
84C23×C71—84no—
85C23×C71—88no—
86C23×C71—84no—
87C23×C71#10012yes—
88C23×C71#10414yes—
89C23×C71C2×C22⋊C411yescharacteristic
90D4×C71—44no—
91D4×C71—48no—
92D4×C71—44no—
93C2×C4×C71—28no—
94C2×C4×C71—21no—
95C2×C4×C71—22no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71—22no—
103C2×C4×C71—48no—
104C2×C4×C71—42no—
105C2×C4×C71—44no—
106C2×C4×C71—84no—
107C2×C4×C71#9912yes—
108C2×C4×C71C2×M1612yes—
109C2×C4×C71AGL(1,ℤ/8)14yes—
110C71⋊C8—48no—
111C22×C71—22no—
112C22×C71—22no—
113C22×C71—24no—
114C22×C71—24no—
115C22×C71—44no—
116C22×C71—48no—
117C22×C71—816no—
118C22×C71—82no—
119C22×C71—84no—
120C22×C71—82no—
121C22×C71#32914yes—
122C22×C71#40711yescharacteristic
123C22×C71C2×#9911yescharacteristic
124C22×C71C2×#10011yescharacteristic
125C22×C71#41812yes—
126C22×C71C2×#10412yes—
127C4×C71—22no—
128C4×C71—22no—
129C4×C71—24no—
130C4×C71—42no—
131C4×C71—82no—
132#7626—711nomaximal, Sylow
133C2×C71—24no—
134C2×C71—84no—
135C2×C71#146211yescharacteristic
136C2×C71#149012yes—
137C2×C71#150712yes—
138C2×C71C2×#32911yescharacteristic
139C2×C71#155511yescharacteristic
140C2×#390D7111yescharacteristic
141#1749—712no—
142#1270—714no—
143C71#762611yescharacteristic, complemented, Sylow
144C23×D4Dic7111yescharacteristic
145C22×C4⋊C4Dic7111yescharacteristic
146C22×C22⋊C4—22no—
147C22×C42C2×D7111yescharacteristic
148C2×C22⋊C8—1422no—
149C2×#97—1422no—
150#390—22no—
151#390Dic7112yes—
152#390C2×D7112yes—
153#336—714no—
154C25—21no—
155C22×D4—24no—
156C22×D4—21no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4C2×Dic7112yes—
160C23×C4—21no—
161C23×C4C2×Dic7111yescharacteristic
162C23×C4C22⋊D7112yes—
163C22×C8—1424no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4C2×Dic7112yes—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—48no—
170C2×C42—42no—
171C2×C42C2×Dic7111yescharacteristic
172C2×C42C22⋊D7112yes—
173C2×C42C22×D7111yescharacteristic
174C22⋊C8—1428no—
175#97—2844no—
176C24—24no—
177C24—22no—
178C24—44no—
179C24—42no—
180C24—42no—
181C24—44no—
182C24C22⋊Dic7111yescharacteristic
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—416no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4#1176584914yes—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4C22⋊Dic7112yes—
201C22×C4C22⋊Dic7111yescharacteristic
202C22×C4C22×Dic7111yescharacteristic
203C22×C4C2×C22⋊D7112yes—
204C2×C8—2844no—
205C2×C8—2844no—
206C2×C8—1428no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C22⋊C4—48no—
210C22⋊C4—48no—
211C42—24no—
212C42—41no—
213C42—41no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—24no—
218C23—48no—
219C23—416no—
220C23—42no—
221C23—42no—
222C23—44no—
223C23—44no—
224C23—84no—
225C23—88no—
226C23—84no—
227C23#1455604512yes—
228C23#1455605514yes—
229C23C2×C22⋊Dic7111yescharacteristic
230D4—44no—
231D4—48no—
232D4—44no—
233C2×C4—28no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—48no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—84no—
247C2×C4#1455604312yes—
248C2×C4C2×#1176584912yes—
249C2×C4#1455615414yes—
250C8—2848no—
251C22—22no—
252C22—24no—
253C22—22no—
254C22—24no—
255C22—44no—
256C22—48no—
257C22—816no—
258C22—82no—
259C22—82no—
260C22—84no—
261C22#1798573914yes—
262C22#1798596311yescharacteristic
263C22C2×#1455604311yescharacteristic
264C22C2×#1455604511yescharacteristic
265C22#1798600412yes—
266C22C2×#1455605512yes—
267C4—22no—
268C4—22no—
269C4—24no—
270C4—42no—
271C4—82no—
272C2—24no—
273C2—84no—
274C2#2258666711yescharacteristic
275C2#2258678312yes—
276C2#2258685612yes—
277C2C2×#1798573911yescharacteristic
278C2#2258705411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#390×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC23×D4×C71C4
cosocleC2×C42×C71C23
hypercenterC2×#390D71
hypocenterC71#7626
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. #22588102C2semidirect
2. #22586161C2semidirect
6. #17985754C22semidirect
143. C71#7626semidirect

Maximal subgroups

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

subgroupindex
#225881022
#225861612
C2×#390×C712
#762671