Finite Group Databasebeta

← #31907100

Subgroups — #31907100

Order 18176.

424 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 87 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#22593513C211yesmaximal, characteristic
2#22586658C211yesmaximal, characteristic
3#22586283C214yesmaximal, complemented
4C71×#1465C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic
6C2×C22⋊(C71⋊C8)—21no—
7#17985911—21no—
8#17985911—21no—
9#17985902C412yes—
10#17985864C2212yes—
11#17985859—21no—
12#17985859—21no—
13C4×C4⋊Dic71C411yescharacteristic
14#17985725C2212yes—
15#17985723—21no—
16C2×C4×D4×C71C2211yescharacteristic
17#407×C71—21no—
18#324×C71C2212yes—
19#323×C71—21no—
20C22×Dic142—21no—
21C22×C71⋊C8—41no—
22#14556128—41no—
23#14556128—41no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71C2×C412yes—
27C2×C22⋊Dic71—22no—
28C2×C22⋊Dic71—22no—
29C2×C4×Dic71C2×C412yes—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—41no—
32C71⋊#95—21no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C22×D4×C71C2×C411yescharacteristic
36C23×C4×C71D411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C2×M16×C71—41no—
39C4×D4×C71—41no—
40C4×D4×C71—41no—
41C2×C4⋊C4×C71C2311yescharacteristic
42C2×C22⋊C4×C71D411yescharacteristic
43C2×C22⋊C4×C71D411yescharacteristic
44C2×C42×C71C2×C411yescharacteristic
45C22⋊C8×C71—22no—
46C22⋊C8×C71—41no—
47C2×Dic142—42no—
48C2×Dic142—82no—
49C22×Dic71—22no—
50C22×Dic71—22no—
51C2×C71⋊C8—42no—
52C2×C71⋊C8—82no—
53C4⋊Dic71—42no—
54C4⋊Dic71—41no—
55C4⋊Dic71—41no—
56C4⋊Dic71—42no—
57C22⋊Dic71—82no—
58C22⋊Dic71—82no—
59C4×Dic71—24no—
60C24×C71C22⋊C411yescharacteristic
61C24×C71C22⋊C411yescharacteristic
62C2×D4×C71—22no—
63C2×D4×C71—22no—
64C2×D4×C71—41no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—42no—
70C22×C4×C71—42no—
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22×C411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75C22×C4×C71C2×D411yescharacteristic
76M16×C71—82no—
77C2×C8×C71—42no—
78C4⋊C4×C71—41no—
79C22⋊C4×C71—41no—
80C22⋊C4×C71—41no—
81C42×C71—21no—
82C42×C71—21no—
83C23×C71—21no—
84C23×C71—21no—
85C23×C71—21no—
86C23×C71—21no—
87C23×C71—21no—
88C23×C71—41no—
89C23×C71—41no—
90C23×C71—41no—
91C23×C71—41no—
92C23×C71#10112yes—
93C23×C71C2×C22⋊C411yescharacteristic
94D4×C71—44no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—22no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71—81no—
107C2×C4×C71—81no—
108C2×C4×C71#9911yescharacteristic
109C2×C4×C71#9911yescharacteristic
110C2×C4×C71#13112yes—
111C2×Dic71—42no—
112C2×Dic71—42no—
113C2×Dic71—42no—
114C2×Dic71—42no—
115C2×Dic71—42no—
116C2×Dic71—42no—
117Dic142—84no—
118C8×C71—82no—
119C71⋊C8—82no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71—22no—
123C22×C71—42no—
124C22×C71—42no—
125C22×C71—41no—
126C22×C71—41no—
127C22×C71—41no—
128C22×C71—41no—
129C22×C71—41no—
130C22×C71—81no—
131C22×C71—81no—
132C22×C71#40811yescharacteristic
133C22×C71C2×#9911yescharacteristic
134C22×C71C2×#10111yescharacteristic
135C4×C71—21no—
136C4×C71—22no—
137C4×C71—41no—
138C4×C71—41no—
139C4×C71—41no—
140C4×C71#35412yes—
141Dic71—82no—
142Dic71—82no—
143#11569—711nomaximal, Sylow
144C2×C71—21no—
145C2×C71—21no—
146C2×C71—42no—
147C2×C71—81no—
148C2×C71#147211yescharacteristic
149C2×C71C2×#35411yescharacteristic
150C2×C71#213411yescharacteristic
151#2563—711no—
152#1465D7111yescharacteristic
153#1459—711no—
154#1324—714no—
155C71#1156911yescharacteristic, complemented, Sylow
156C2×C4⋊Q8—711no—
157C2×C4×D4C2×D7111yescharacteristic
158#407—21no—
159C2×C22⋊C8—1421no—
160#393—1421no—
161#393—1421no—
162#390—712no—
163#382—712no—
164#380—1421no—
165#380—1421no—
166C4×C4⋊C4—711no—
167#324—712no—
168#324C2×D7112yes—
169#323—21no—
170#323—1421no—
171C22×Q8—1421no—
172C22×D4C4×D7111yescharacteristic
173C23×C4D28411yescharacteristic
174C23×C4C22⋊D7111yescharacteristic
175C2×M16—41no—
176C22×C8—2841no—
177C4⋊Q8—2841no—
178C4⋊Q8—2841no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—712no—
182C2×C4⋊C4—1421no—
183C2×C4⋊C4—1421no—
184C2×C4⋊C4C22×D7111yescharacteristic
185C2×C22⋊C4—1422no—
186C2×C22⋊C4—1422no—
187C2×C22⋊C4D28411yescharacteristic
188C2×C22⋊C4C22⋊D7111yescharacteristic
189C2×C42—712no—
190C2×C42C4×D7111yescharacteristic
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—2841no—
194C22⋊C8—1422no—
195#95—1421no—
196#95—1422no—
197#95—1422no—
198C24(C2×D71)⋊C411yescharacteristic
199C24(C2×D71)⋊C411yescharacteristic
200C2×Q8—2842no—
201C2×Q8—5682no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—1422no—
212C22×C4—1422no—
213C22×C4(C2×D71)⋊C411yescharacteristic
214C22×C4(C2×D71)⋊C411yescharacteristic
215C22×C4C2×C4×D7111yescharacteristic
216C22×C4C2×C22⋊D7111yescharacteristic
217C22×C4C2×D28411yescharacteristic
218M16—82no—
219C2×C8—42no—
220C2×C8—2842no—
221C2×C8—5682no—
222C4⋊C4—41no—
223C4⋊C4—2842no—
224C4⋊C4—2842no—
225C4⋊C4—2841no—
226C4⋊C4—2841no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—5682no—
230C22⋊C4—5682no—
231C42—21no—
232C42—21no—
233C42—1424no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#1455604612yes—
244C23C2×(C2×D71)⋊C411yescharacteristic
245Q8—5684no—
246D4—44no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—2842no—
261C2×C4—2842no—
262C2×C4—2842no—
263C2×C4—2842no—
264C2×C4—2842no—
265C2×C4—2842no—
266C2×C4#1455604211yescharacteristic
267C2×C4#1455604211yescharacteristic
268C2×C4#1455613612yes—
269C8—82no—
270C8—5682no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22#1798596711yescharacteristic
284C22C2×#1455604211yescharacteristic
285C22C2×#1455604611yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—5682no—
292C4—5682no—
293C4#1798579312yes—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2#2258670511yescharacteristic
299C2C2×#1798579311yescharacteristic
300C2#2259003111yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1465C2
socleC22×C71C2×#99
prime radicalC22×D4×C71C2×C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1465D71
hypocenterC71#11569
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #22586283C2semidirect
155. C71#11569semidirect

Maximal subgroups

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

subgroupindex
#225935132
#225866582
#225862832
C71×#14652
#1156971