Finite Group Databasebeta

← #31907099

Subgroups — #31907099

Order 18176.

424 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 95 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#22586678C211yesmaximal, characteristic
2#22586658C211yesmaximal, characteristic
3#22586284C214yesmaximal, complemented
4C71×#2563C211yesmaximal, characteristic
5#17985963—21no—
6C2×C22⋊(C71⋊C8)—21no—
7#17985725C2212yes—
8#17985725C2212yes—
9#17985723—21no—
10#17985723—21no—
11C2×C4⋊Q8×C71C411yescharacteristic
12C2×C4×D4×C71C2211yescharacteristic
13#393×C71—21no—
14#393×C71—21no—
15#390×C71C412yes—
16#382×C71C2212yes—
17#380×C71—21no—
18#380×C71—21no—
19C4×C4⋊C4×C71C411yescharacteristic
20C2×#11765849—41no—
21C22×C71⋊C8—41no—
22C22⋊(C71⋊C8)—22no—
23C22⋊(C71⋊C8)—22no—
24C22⋊(C71⋊C8)—41no—
25C22⋊(C71⋊C8)—41no—
26C22×Q8×C71—21no—
27C22×D4×C71C2×C411yescharacteristic
28C23×C4×C71D411yescharacteristic
29C23×C4×C71D411yescharacteristic
30C4⋊Q8×C71—41no—
31C4⋊Q8×C71—41no—
32C4×D4×C71—41no—
33C4×D4×C71—41no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71C2×C412yes—
37C2×C4⋊C4×C71C2311yescharacteristic
38C2×C22⋊C4×C71—22no—
39C2×C22⋊C4×C71—22no—
40C2×C22⋊C4×C71D411yescharacteristic
41C2×C22⋊C4×C71D411yescharacteristic
42C2×C42×C71C2×C412yes—
43C2×C42×C71C2×C411yescharacteristic
44#95×C71—21no—
45#95×C71—22no—
46#95×C71—22no—
47#11765849—82no—
48C2×C71⋊C8—42no—
49C2×C71⋊C8—42no—
50C2×C71⋊C8—82no—
51C24×C71C22⋊C411yescharacteristic
52C24×C71C22⋊C411yescharacteristic
53C2×Q8×C71—42no—
54C2×Q8×C71—82no—
55C2×D4×C71—22no—
56C2×D4×C71—22no—
57C2×D4×C71—41no—
58C22×C4×C71—21no—
59C22×C4×C71—21no—
60C22×C4×C71—21no—
61C22×C4×C71—21no—
62C22×C4×C71—22no—
63C22×C4×C71—22no—
64C22×C4×C71—42no—
65C22×C4×C71—42no—
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22×C411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70C22×C4×C71C2×D411yescharacteristic
71C4⋊C4×C71—41no—
72C4⋊C4×C71—42no—
73C4⋊C4×C71—42no—
74C4⋊C4×C71—41no—
75C4⋊C4×C71—41no—
76C22⋊C4×C71—41no—
77C22⋊C4×C71—41no—
78C22⋊C4×C71—82no—
79C22⋊C4×C71—82no—
80C42×C71—21no—
81C42×C71—21no—
82C42×C71—24no—
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
94Q8×C71—84no—
95D4×C71—44no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—22no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—42no—
110C2×C4×C71—42no—
111C2×C4×C71—42no—
112C2×C4×C71—42no—
113C2×C4×C71—81no—
114C2×C4×C71—81no—
115C2×C4×C71#9911yescharacteristic
116C2×C4×C71#9911yescharacteristic
117C2×C4×C71#13112yes—
118C71⋊C8—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—82no—
141C4×C71—82no—
142C4×C71#35412yes—
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#2563D7111yescharacteristic
152#1465—711no—
153#1459—711no—
154#1324—714no—
155C71#1156911yescharacteristic, complemented, Sylow
156C2×C4⋊Q8Dic7111yescharacteristic
157C2×C4×D4C2×D7111yescharacteristic
158#407—1421no—
159C2×C22⋊C8—1421no—
160#393—21no—
161#393—21no—
162#390Dic7112yes—
163#382C2×D7112yes—
164#380—21no—
165#380—21no—
166C4×C4⋊C4Dic7111yescharacteristic
167#324—712no—
168#324—712no—
169#323—1421no—
170#323—1421no—
171C22×Q8—21no—
172C22×D4C2×Dic7111yescharacteristic
173C23×C4C22⋊D7111yescharacteristic
174C23×C4C22⋊D7111yescharacteristic
175C2×M16—2841no—
176C22×C8—2841no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C2×Dic7112yes—
184C2×C4⋊C4C22×D7111yescharacteristic
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4C22⋊D7111yescharacteristic
188C2×C22⋊C4C22⋊D7111yescharacteristic
189C2×C42C2×Dic7112yes—
190C2×C42C2×Dic7111yescharacteristic
191C22⋊C8—2841no—
192C22⋊C8—2841no—
193C22⋊C8—1422no—
194C22⋊C8—1422no—
195#95—21no—
196#95—22no—
197#95—22no—
198C24C22⋊Dic7111yescharacteristic
199C24C22⋊Dic7111yescharacteristic
200C2×Q8—42no—
201C2×Q8—82no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4C22⋊Dic7111yescharacteristic
214C22×C4C22⋊Dic7111yescharacteristic
215C22×C4C22×Dic7111yescharacteristic
216C22×C4C2×C22⋊D7111yescharacteristic
217C22×C4C2×C22⋊D7111yescharacteristic
218M16—5682no—
219C2×C8—2842no—
220C2×C8—2842no—
221C2×C8—5682no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C42—21no—
232C42—21no—
233C42—24no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#1455604712yes—
244C23C2×C22⋊Dic7111yescharacteristic
245Q8—84no—
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—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4#1455604311yescharacteristic
267C2×C4#1455604311yescharacteristic
268C2×C4#1455613512yes—
269C8—5682no—
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#1798596611yescharacteristic
284C22C2×#1455604311yescharacteristic
285C22C2×#1455604711yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—82no—
293C4#1798579412yes—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2#2258670411yescharacteristic
299C2C2×#1798579411yescharacteristic
300C2#2259003211yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2563C2
socleC22×C71C2×#99
prime radicalC22×D4×C71C2×C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2563D71
hypocenterC71#11569
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #22586284C2semidirect
155. C71#11569semidirect

Maximal subgroups

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

subgroupindex
#225866782
#225866582
#225862842
C71×#25632
#1156971