Finite Group Databasebeta

← #31906565

Subgroups — #31906565

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#22586272C214yesmaximal, complemented
3C71×#2309C211yesmaximal, characteristic
4#17986052—22no—
5#17985725C2214yescomplemented
6#17985723—22no—
7C23×D4×C71C411yescharacteristic
8C2×#117×C71C411yescharacteristic
9C2×#117×C71C2211yescharacteristic
10#390×C71—22no—
11#390×C71C412yes—
12#390×C71C2212yes—
13#379×C71—22no—
14C4⋊(C71⋊C8)—42no—
15C22⋊(C71⋊C8)—24no—
16C4×C71⋊C8—42no—
17C25×C71—21no—
18C22×D4×C71—21no—
19C22×D4×C71—22no—
20C22×D4×C71—42no—
21C22×D4×C71—42no—
22C22×D4×C71C814yescomplemented
23C22×D4×C71C2×C412yes—
24C23×C4×C71C2×C411yescharacteristic
25#117×C71—42no—
26#117×C71—42no—
27C2×C4⋊C4×C71—21no—
28C2×C4⋊C4×C71C2×C411yescharacteristic
29C2×C4⋊C4×C71C2311yescharacteristic
30C2×C22⋊C4×C71—21no—
31C2×C22⋊C4×C71—24no—
32C2×C22⋊C4×C71—24no—
33C2×C22⋊C4×C71D412yes—
34C2×C42×C71C2×C412yes—
35C2×C42×C71D412yes—
36C2×C71⋊C8—44no—
37C24×C71—22no—
38C24×C71—24no—
39C24×C71—42no—
40C24×C71—42no—
41C24×C71—82no—
42C24×C71—82no—
43C24×C71C22⋊C411yescharacteristic
44C2×D4×C71—24no—
45C2×D4×C71—22no—
46C2×D4×C71—42no—
47C2×D4×C71—42no—
48C2×D4×C71—48no—
49C2×D4×C71—48no—
50C2×D4×C71—42no—
51C2×D4×C71—42no—
52C2×D4×C71C2×C814yes—
53C22×C4×C71—22no—
54C22×C4×C71—22no—
55C22×C4×C71—22no—
56C22×C4×C71C22⋊C412yes—
57C22×C4×C71C22⋊C411yescharacteristic
58C22×C4×C71C2×C812yes—
59C22×C4×C71M1612yes—
60C22×C4×C71C22×C411yescharacteristic
61C22×C4×C71C2×D412yes—
62C4⋊C4×C71—41no—
63C4⋊C4×C71—41no—
64C4⋊C4×C71—42no—
65C22⋊C4×C71—42no—
66C22⋊C4×C71—42no—
67C22⋊C4×C71—84no—
68C22⋊C4×C71—84no—
69C42×C71—22no—
70C42×C71—22no—
71C42×C71—42no—
72C23×C71—21no—
73C23×C71—21no—
74C23×C71—24no—
75C23×C71—41no—
76C23×C71—41no—
77C23×C71—44no—
78C23×C71—44no—
79C23×C71—48no—
80C23×C71—42no—
81C23×C71—42no—
82C23×C71—88no—
83C23×C71—88no—
84C23×C71—84no—
85C23×C71—84no—
86C23×C71C22⋊C812yes—
87C23×C71C2×C22⋊C411yescharacteristic
88D4×C71—44no—
89D4×C71—44no—
90D4×C71—48no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—24no—
94C2×C4×C71—24no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—42no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71C22⋊C812yes—
103C2×C4×C71#9912yes—
104C2×C4×C71#10012yes—
105C2×C4×C71C22×C811yescharacteristic
106C2×C4×C71C2×M1611yescharacteristic
107C71⋊C8—84no—
108C22×C71—21no—
109C22×C71—21no—
110C22×C71—22no—
111C22×C71—42no—
112C22×C71—42no—
113C22×C71—48no—
114C22×C71—41no—
115C22×C71—41no—
116C22×C71—44no—
117C22×C71—84no—
118C22×C71—816no—
119C22×C71—82no—
120C22×C71—82no—
121C22×C71C2×C22⋊C811yescharacteristic
122C22×C71C2×#9911yescharacteristic
123C22×C71C2×#10011yescharacteristic
124C4×C71—21no—
125C4×C71—21no—
126C4×C71—42no—
127C4×C71—42no—
128C4×C71—82no—
129C4×C71#32314yes—
130#11433—711nomaximal, Sylow
131C2×C71—21no—
132C2×C71—21no—
133C2×C71—42no—
134C2×C71—84no—
135C2×C71C2×#32311yescharacteristic
136C2×C71#212811yescharacteristic
137C2×C71#213411yescharacteristic
138#2309D7111yescharacteristic
139#1458—712no—
140#1318—714no—
141C71#1143311yescharacteristic, complemented, Sylow
142C23×D4Dic7111yescharacteristic
143C2×#117Dic7111yescharacteristic
144C2×#117C2×D7111yescharacteristic
145#431—1422no—
146#390—22no—
147#390Dic7112yes—
148#390C2×D7112yes—
149#379—22no—
150#324—714no—
151#323—1422no—
152C25—21no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4C71⋊C814yescomplemented
158C22×D4C2×Dic7112yes—
159C23×C4C2×Dic7111yescharacteristic
160#117—42no—
161#117—42no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C2×Dic7111yescharacteristic
164C2×C4⋊C4C22×D7111yescharacteristic
165C2×C22⋊C4—21no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4C22⋊D7112yes—
169C2×C42C2×Dic7112yes—
170C2×C42C22⋊D7112yes—
171C4⋊C8—2842no—
172C22⋊C8—1424no—
173C4×C8—2842no—
174C24—22no—
175C24—24no—
176C24—42no—
177C24—42no—
178C24—82no—
179C24—82no—
180C24C22⋊Dic7111yescharacteristic
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—48no—
186C2×D4—48no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4C2×C71⋊C814yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4C22⋊Dic7112yes—
194C22×C4C22⋊Dic7111yescharacteristic
195C22×C4C2×C71⋊C812yes—
196C22×C4#1176584912yes—
197C22×C4C22×Dic7111yescharacteristic
198C22×C4C2×C22⋊D7112yes—
199C2×C8—2844no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—84no—
206C22⋊C4—84no—
207C42—22no—
208C42—22no—
209C42—42no—
210C23—21no—
211C23—21no—
212C23—24no—
213C23—41no—
214C23—41no—
215C23—44no—
216C23—44no—
217C23—48no—
218C23—42no—
219C23—42no—
220C23—88no—
221C23—88no—
222C23—84no—
223C23—84no—
224C23C22⋊(C71⋊C8)12yes—
225C23C2×C22⋊Dic7111yescharacteristic
226D4—44no—
227D4—44no—
228D4—48no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4C22⋊(C71⋊C8)12yes—
241C2×C4#1455604312yes—
242C2×C4#1455604512yes—
243C2×C4C22×C71⋊C811yescharacteristic
244C2×C4C2×#1176584911yescharacteristic
245C8—5684no—
246C22—21no—
247C22—21no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—48no—
252C22—41no—
253C22—41no—
254C22—44no—
255C22—84no—
256C22—816no—
257C22—82no—
258C22—82no—
259C22C2×C22⋊(C71⋊C8)11yescharacteristic
260C22C2×#1455604311yescharacteristic
261C22C2×#1455604511yescharacteristic
262C4—21no—
263C4—21no—
264C4—42no—
265C4—42no—
266C4—82no—
267C4#1798572314yes—
268C2—21no—
269C2—21no—
270C2—42no—
271C2—84no—
272C2C2×#1798572311yescharacteristic
273C2#2259001211yescharacteristic
274C2#2259003211yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC2×C4×C71C22×C8
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2309C2
socleC22×C71C2×#99
prime radicalC23×D4×C71C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2309D71
hypocenterC71#11433
LCSG ▷ C2×C4×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. #22586272C2semidirect
5. #17985725C22semidirect
22. C22×D4×C71C8semidirect
141. C71#11433semidirect
157. C22×D4C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225866542
#225862722
C71×#23092
#1143371