Finite Group Databasebeta

← #31906823

Subgroups — #31906823

Order 18176.

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

normal subgroups only

This group has 670 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#22586674C212yesmaximal, complemented
2#22586268C214yesmaximal, complemented
3C71×#2309C211yesmaximal, characteristic
4#17986056—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—
16#14556039—42no—
17C25×C71C2×C412yes—
18C22×D4×C71—24no—
19C22×D4×C71—21no—
20C22×D4×C71—21no—
21C22×D4×C71—41no—
22C22×D4×C71—41no—
23C22×D4×C71—42no—
24C23×C4×C71C2×C411yescharacteristic
25#117×C71—41no—
26#117×C71—41no—
27#117×C71—42no—
28C2×C4⋊C4×C71—21no—
29C2×C4⋊C4×C71D412yes—
30C2×C22⋊C4×C71—24no—
31C2×C22⋊C4×C71—24no—
32C2×C22⋊C4×C71C2×C412yes—
33C2×C22⋊C4×C71C2×C411yescharacteristic
34C2×C22⋊C4×C71C2311yescharacteristic
35C2×C42×C71—21no—
36C2×C42×C71D412yes—
37C2×C71⋊C8—44no—
38C24×C71—22no—
39C24×C71—22no—
40C24×C71—42no—
41C24×C71—42no—
42C24×C71—84no—
43C24×C71M1614yes—
44C24×C71C22×C411yescharacteristic
45C2×D4×C71—24no—
46C2×D4×C71—24no—
47C2×D4×C71—48no—
48C2×D4×C71—48no—
49C2×D4×C71—42no—
50C2×D4×C71—42no—
51C2×D4×C71—41no—
52C2×D4×C71—41no—
53C2×D4×C71—42no—
54C22×C4×C71—22no—
55C22×C4×C71—21no—
56C22×C4×C71—21no—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71C22⋊C411yescharacteristic
60C22×C4×C71C22⋊C412yes—
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C2×D412yes—
63C4⋊C4×C71—42no—
64C4⋊C4×C71—42no—
65C22⋊C4×C71—41no—
66C22⋊C4×C71—41no—
67C22⋊C4×C71—42no—
68C22⋊C4×C71—84no—
69C22⋊C4×C71—84no—
70C42×C71—22no—
71C42×C71—22no—
72C42×C71—42no—
73C23×C71—21no—
74C23×C71—21no—
75C23×C71—24no—
76C23×C71—42no—
77C23×C71—44no—
78C23×C71—44no—
79C23×C71—44no—
80C23×C71—44no—
81C23×C71—42no—
82C23×C71—42no—
83C23×C71—84no—
84C23×C71—84no—
85C23×C71—88no—
86C23×C71—88no—
87C23×C71C2×C22⋊C411yescharacteristic
88C23×C71C2×M1612yes—
89D4×C71—44no—
90D4×C71—44no—
91D4×C71—48no—
92C2×C4×C71—24no—
93C2×C4×C71—24no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—44no—
103C2×C4×C71—42no—
104C2×C4×C71#9911yescharacteristic
105C2×C4×C71#9911yescharacteristic
106C2×C4×C71#10012yes—
107C71⋊C8—84no—
108C22×C71—22no—
109C22×C71—22no—
110C22×C71—42no—
111C22×C71—42no—
112C22×C71—44no—
113C22×C71—44no—
114C22×C71—42no—
115C22×C71—44no—
116C22×C71—84no—
117C22×C71—88no—
118C22×C71—88no—
119C22×C71—84no—
120C22×C71#40711yescharacteristic
121C22×C71C2×#9911yescharacteristic
122C22×C71C2×#10011yescharacteristic
123C4×C71—21no—
124C4×C71—21no—
125C4×C71—22no—
126C4×C71—42no—
127C4×C71—42no—
128C4×C71—82no—
129#11503—711nomaximal, Sylow
130C2×C71—21no—
131C2×C71—21no—
132C2×C71—42no—
133C2×C71—84no—
134C2×C71#146211yescharacteristic
135C2×C71#212212yes—
136#2309D7111yescharacteristic
137#1464—712no—
138#1316—714no—
139C71#1150311yescharacteristic, complemented, Sylow
140C23×D4Dic7111yescharacteristic
141C2×#117Dic7111yescharacteristic
142C2×#117C2×D7111yescharacteristic
143#432—1422no—
144#390—22no—
145#390—22no—
146#379Dic7112yes—
147#379C2×D7112yes—
148#324—1422no—
149#323—714no—
150C25C2×Dic7112yes—
151C22×D4—24no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4—42no—
157C23×C4C2×Dic7111yescharacteristic
158#117—41no—
159#117—41no—
160#117—42no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C22⋊D7112yes—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4C2×Dic7112yes—
166C2×C22⋊C4C2×Dic7111yescharacteristic
167C2×C22⋊C4C22×D7111yescharacteristic
168C2×C42—21no—
169C2×C42C22⋊D7112yes—
170C4⋊C8—2842no—
171C22⋊C8—1424no—
172#97—2842no—
173C24—22no—
174C24—22no—
175C24—42no—
176C24—42no—
177C24—84no—
178C24#1176584914yes—
179C24C22×Dic7111yescharacteristic
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—48no—
183C2×D4—48no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—42no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4C22⋊Dic7111yescharacteristic
195C22×C4C22⋊Dic7112yes—
196C22×C4C22⋊Dic7111yescharacteristic
197C22×C4C2×C22⋊D7112yes—
198C2×C8—2844no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—42no—
204C22⋊C4—84no—
205C22⋊C4—84no—
206C42—22no—
207C42—22no—
208C42—42no—
209C23—21no—
210C23—21no—
211C23—24no—
212C23—42no—
213C23—44no—
214C23—44no—
215C23—44no—
216C23—44no—
217C23—42no—
218C23—42no—
219C23—84no—
220C23—84no—
221C23—88no—
222C23—88no—
223C23C2×C22⋊Dic7111yescharacteristic
224C23C2×#1176584912yes—
225D4—44no—
226D4—44no—
227D4—48no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4#1455604311yescharacteristic
241C2×C4#1455604311yescharacteristic
242C2×C4#1455604512yes—
243C8—5684no—
244C22—22no—
245C22—22no—
246C22—42no—
247C22—42no—
248C22—44no—
249C22—44no—
250C22—42no—
251C22—44no—
252C22—84no—
253C22—88no—
254C22—88no—
255C22—84no—
256C22#1798596311yescharacteristic
257C22C2×#1455604311yescharacteristic
258C22C2×#1455604511yescharacteristic
259C4—21no—
260C4—21no—
261C4—22no—
262C4—42no—
263C4—42no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—42no—
268C2—84no—
269C2#2258666711yescharacteristic
270C2#2258999012yes—
subgroupG/H
centerC22C2×#14556043
commutatorC24×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2309C2
socleC22×C71C2×#99
prime radicalC23×D4×C71C4
cosocleC2×C22⋊C4×C71C23
hypercenter#2309D71
hypocenterC71#11503
LCSG ▷ C24×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. #22586674C2semidirect
2. #22586268C2semidirect
139. C71#11503semidirect

Maximal subgroups

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

subgroupindex
#225866742
#225862682
C71×#23092
#1150371