Finite Group Databasebeta

← #31906545

Subgroups — #31906545

Order 18176.

686 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 131 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#22586654C212yesmaximal, complemented
2#22586268C214yesmaximal, complemented
3C71×#2309C211yesmaximal, characteristic
4#17986052—22no—
5#17985725—22no—
6#17985723C2214yes—
7C23×D4×C71C411yescharacteristic
8C2×#117×C71C411yescharacteristic
9C2×#117×C71C2211yescharacteristic
10#390×C71—22no—
11#390×C71—22no—
12#379×C71C412yes—
13#379×C71C2212yes—
14C4⋊(C71⋊C8)—42no—
15C22⋊(C71⋊C8)—24no—
16C4×C71⋊C8—42no—
17C25×C71C2×C412yes—
18C22×D4×C71—21no—
19C22×D4×C71—21no—
20C22×D4×C71—22no—
21C22×D4×C71—42no—
22C22×D4×C71—42no—
23C22×D4×C71C814yescomplemented
24C23×C4×C71C2×C411yescharacteristic
25#117×C71—42no—
26#117×C71—42no—
27C2×C4⋊C4×C71—21no—
28C2×C4⋊C4×C71D412yes—
29C2×C22⋊C4×C71—24no—
30C2×C22⋊C4×C71—24no—
31C2×C22⋊C4×C71C2×C412yes—
32C2×C22⋊C4×C71C2×C411yescharacteristic
33C2×C22⋊C4×C71C2311yescharacteristic
34C2×C42×C71—21no—
35C2×C42×C71D412yes—
36C2×C71⋊C8—44no—
37C24×C71—22no—
38C24×C71—22no—
39C24×C71—42no—
40C24×C71—42no—
41C24×C71—84no—
42C24×C71C2×C814yes—
43C24×C71C22×C411yescharacteristic
44C2×D4×C71—24no—
45C2×D4×C71—24no—
46C2×D4×C71—42no—
47C2×D4×C71—42no—
48C2×D4×C71—48no—
49C2×D4×C71—48no—
50C2×D4×C71—42no—
51C2×D4×C71—42no—
52C22×C4×C71—22no—
53C22×C4×C71—22no—
54C22×C4×C71—22no—
55C22×C4×C71C22⋊C411yescharacteristic
56C22×C4×C71C22⋊C412yes—
57C22×C4×C71C22⋊C411yescharacteristic
58C22×C4×C71C2×C812yes—
59C22×C4×C71M1612yes—
60C22×C4×C71C2×D412yes—
61C4⋊C4×C71—42no—
62C4⋊C4×C71—42no—
63C22⋊C4×C71—41no—
64C22⋊C4×C71—41no—
65C22⋊C4×C71—42no—
66C22⋊C4×C71—84no—
67C22⋊C4×C71—84no—
68C42×C71—22no—
69C42×C71—22no—
70C42×C71—42no—
71C23×C71—21no—
72C23×C71—21no—
73C23×C71—24no—
74C23×C71—42no—
75C23×C71—44no—
76C23×C71—44no—
77C23×C71—44no—
78C23×C71—44no—
79C23×C71—42no—
80C23×C71—42no—
81C23×C71—84no—
82C23×C71—84no—
83C23×C71—88no—
84C23×C71—88no—
85C23×C71C2×C22⋊C411yescharacteristic
86C23×C71C22×C811yescharacteristic
87C23×C71C2×M1611yescharacteristic
88D4×C71—48no—
89D4×C71—48no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—24no—
93C2×C4×C71—24no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—44no—
99C2×C4×C71—42no—
100C2×C4×C71C22⋊C812yes—
101C2×C4×C71C22⋊C812yes—
102C2×C4×C71#9912yes—
103C2×C4×C71#10012yes—
104C71⋊C8—84no—
105C22×C71—22no—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71—41no—
109C22×C71—41no—
110C22×C71—42no—
111C22×C71—44no—
112C22×C71—44no—
113C22×C71—42no—
114C22×C71—44no—
115C22×C71—84no—
116C22×C71—88no—
117C22×C71—88no—
118C22×C71—84no—
119C22×C71C2×C22⋊C811yescharacteristic
120C22×C71C2×#9911yescharacteristic
121C22×C71C2×#10011yescharacteristic
122C4×C71—22no—
123C4×C71—42no—
124C4×C71—42no—
125C4×C71—82no—
126C4×C71#32314yes—
127#11427—711nomaximal, Sylow
128C2×C71—21no—
129C2×C71—21no—
130C2×C71—41no—
131C2×C71—41no—
132C2×C71—84no—
133C2×C71C2×#32311yescharacteristic
134C2×C71#212212yes—
135#2309D7111yescharacteristic
136#1458—712no—
137#1316—714no—
138C71#1142711yescharacteristic, complemented, Sylow
139C23×D4Dic7111yescharacteristic
140C2×#117Dic7111yescharacteristic
141C2×#117C2×D7111yescharacteristic
142#431—1422no—
143#390—22no—
144#390—22no—
145#379Dic7112yes—
146#379C2×D7112yes—
147#324—1422no—
148#323—714no—
149C25C2×Dic7112yes—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4C71⋊C814yescomplemented
156C23×C4C2×Dic7111yescharacteristic
157#117—42no—
158#117—42no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4C22⋊D7112yes—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4C2×Dic7112yes—
164C2×C22⋊C4C2×Dic7111yescharacteristic
165C2×C22⋊C4C22×D7111yescharacteristic
166C2×C42—21no—
167C2×C42C22⋊D7112yes—
168C4⋊C8—2842no—
169C22⋊C8—1424no—
170C4×C8—2842no—
171C24—22no—
172C24—22no—
173C24—42no—
174C24—42no—
175C24—84no—
176C24C2×C71⋊C814yes—
177C24C22×Dic7111yescharacteristic
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—48no—
183C2×D4—48no—
184C2×D4—42no—
185C2×D4—42no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4C22⋊Dic7111yescharacteristic
190C22×C4C22⋊Dic7112yes—
191C22×C4C22⋊Dic7111yescharacteristic
192C22×C4C2×C71⋊C812yes—
193C22×C4#1176584912yes—
194C22×C4C2×C22⋊D7112yes—
195C2×C8—2844no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—42no—
201C22⋊C4—84no—
202C22⋊C4—84no—
203C42—22no—
204C42—22no—
205C42—42no—
206C23—21no—
207C23—21no—
208C23—24no—
209C23—42no—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—42no—
215C23—42no—
216C23—84no—
217C23—84no—
218C23—88no—
219C23—88no—
220C23C2×C22⋊Dic7111yescharacteristic
221C23C22×C71⋊C811yescharacteristic
222C23C2×#1176584911yescharacteristic
223D4—48no—
224D4—48no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4C22⋊(C71⋊C8)12yes—
236C2×C4C22⋊(C71⋊C8)12yes—
237C2×C4#1455604312yes—
238C2×C4#1455604512yes—
239C8—5684no—
240C22—22no—
241C22—21no—
242C22—21no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—44no—
247C22—44no—
248C22—42no—
249C22—44no—
250C22—84no—
251C22—88no—
252C22—88no—
253C22—84no—
254C22C2×C22⋊(C71⋊C8)11yescharacteristic
255C22C2×#1455604311yescharacteristic
256C22C2×#1455604511yescharacteristic
257C4—22no—
258C4—42no—
259C4—42no—
260C4—82no—
261C4#1798572314yes—
262C2—21no—
263C2—21no—
264C2—41no—
265C2—41no—
266C2—84no—
267C2C2×#1798572311yescharacteristic
268C2#2258999012yes—
subgroupG/H
centerC22C2×#14556043
commutatorC23×C71C22×C8
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2309C2
socleC22×C71C2×#99
prime radicalC23×D4×C71C4
cosocleC2×C22⋊C4×C71C23
hypercenter#2309D71
hypocenterC71#11427
LCSG ▷ C23×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
1. #22586654C2semidirect
2. #22586268C2semidirect
23. C22×D4×C71C8semidirect
138. C71#11427semidirect
155. C22×D4C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225866542
#225862682
C71×#23092
#1142771