Finite Group Databasebeta

← #31907142

Subgroups — #31907142

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#22592657C211yesmaximal, characteristic
2#22586678C211yesmaximal, characteristic
3#22586287C214yesmaximal, complemented
4C71×#1460C211yesmaximal, characteristic
5C22×C4⋊Dic71C411yescharacteristic
6#17985963—21no—
7#17985938—21no—
8#17985935—21no—
9#17985897—21no—
10#17985879C412yes—
11#17985871C2212yes—
12#17985859—21no—
13#17985859C411yescharacteristic
14#17985725—21no—
15#17985723C2212yes—
16C2×C4×D4×C71C2211yescharacteristic
17#431×C71—21no—
18#323×C71—21no—
19#323×C71C2212yes—
20C23×Dic71C2×C412yes—
21C2×#11765849—41no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—41no—
26C2×C4⋊Dic71—41no—
27C2×Dic71⋊C4—22no—
28C2×C22⋊Dic71—22no—
29C2×C22⋊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×C71C2×C411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C4×D4×C71—41no—
39C4×D4×C71—41no—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C22⋊C4×C71D411yescharacteristic
42C2×C22⋊C4×C71C2311yescharacteristic
43C2×C42×C71D411yescharacteristic
44C4⋊C8×C71—41no—
45C22⋊C8×C71—22no—
46C4×C8×C71—41no—
47C22×Dic71—22no—
48C22×Dic71—22no—
49C22×Dic71—42no—
50C22×Dic71—82no—
51#11765849—82no—
52C2×C71⋊C8—42no—
53C4⋊Dic71—42no—
54C4⋊Dic71—42no—
55C4⋊Dic71—41no—
56C4⋊Dic71—41no—
57Dic71⋊C4—82no—
58C22⋊Dic71—24no—
59C22⋊Dic71—82no—
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
76C2×C8×C71—42no—
77C4⋊C4×C71—41no—
78C22⋊C4×C71—41no—
79C22⋊C4×C71—41no—
80C42×C71—21no—
81C42×C71—21no—
82C23×C71—21no—
83C23×C71—21no—
84C23×C71—21no—
85C23×C71—21no—
86C23×C71—21no—
87C23×C71—41no—
88C23×C71—41no—
89C23×C71—41no—
90C23×C71—41no—
91C23×C71C2×C22⋊C411yescharacteristic
92C23×C71#13112yes—
93D4×C71—44no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—22no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—42no—
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#9911yescharacteristic
108C2×C4×C71#9911yescharacteristic
109C2×C4×C71#10112yes—
110C2×Dic71—42no—
111C2×Dic71—42no—
112C2×Dic71—42no—
113C2×Dic71—42no—
114C2×Dic71—42no—
115C2×Dic71—42no—
116C2×Dic71—84no—
117C8×C71—82no—
118C71⋊C8—82no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71—22no—
122C22×C71—42no—
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#40811yescharacteristic
132C22×C71C2×#9911yescharacteristic
133C22×C71C2×#10111yescharacteristic
134C4×C71—21no—
135C4×C71—22no—
136C4×C71—41no—
137C4×C71—41no—
138C4×C71—41no—
139C4×C71#35212yes—
140Dic71—82no—
141Dic71—82no—
142#11579—711nomaximal, Sylow
143C2×C71—21no—
144C2×C71—21no—
145C2×C71—42no—
146C2×C71—81no—
147C2×C71#147211yescharacteristic
148C2×C71C2×#35211yescharacteristic
149C2×C71#212211yescharacteristic
150#2487—711no—
151#1465—711no—
152#1460D7111yescharacteristic
153#1326—714no—
154C71#1157911yescharacteristic, complemented, Sylow
155C2×C4×D4C2×D7111yescharacteristic
156C22×C4⋊C4—711no—
157#431—21no—
158#407—1421no—
159#400—1421no—
160#399—1421no—
161#389—1421no—
162#386—712no—
163#385—712no—
164#380—711no—
165#380—1421no—
166#324—1421no—
167#323—21no—
168#323—712no—
169#323C2×D7112yes—
170C22×D4C4×D7111yescharacteristic
171C23×C4—712no—
172C23×C4C4×D7111yescharacteristic
173C23×C4D28411yescharacteristic
174C2×M16—2841no—
175C4×D4—41no—
176C4×D4—41no—
177C2×C4⋊C4—2841no—
178C2×C4⋊C4—2841no—
179C2×C4⋊C4—1421no—
180C2×C4⋊C4—1421no—
181C2×C4⋊C4—1421no—
182C2×C4⋊C4—1422no—
183C2×C4⋊C4D28411yescharacteristic
184C2×C22⋊C4—712no—
185C2×C22⋊C4—1422no—
186C2×C22⋊C4C22⋊D7111yescharacteristic
187C2×C22⋊C4C22×D7111yescharacteristic
188C2×C42C22⋊D7111yescharacteristic
189C4⋊C8—41no—
190C22⋊C8—22no—
191C22⋊C8—2841no—
192C22⋊C8—1422no—
193C4×C8—41no—
194#95—1421no—
195#95—1422no—
196#95—1422no—
197C24(C2×D71)⋊C411yescharacteristic
198C24C2×C4×D7111yescharacteristic
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—41no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—2842no—
209C22×C4—5682no—
210C22×C4—1422no—
211C22×C4—1422no—
212C22×C4(C2×D71)⋊C411yescharacteristic
213C22×C4(C2×D71)⋊C411yescharacteristic
214C22×C4(C2×D71)⋊C411yescharacteristic
215C22×C4C2×C22⋊D7111yescharacteristic
216C22×C4C2×D28411yescharacteristic
217M16—5682no—
218C2×C8—42no—
219C2×C8—2842no—
220C4⋊C4—41no—
221C4⋊C4—2842no—
222C4⋊C4—2842no—
223C4⋊C4—2841no—
224C4⋊C4—2841no—
225C4⋊C4—5682no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—5682no—
229C22⋊C4—1424no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23C2×(C2×D71)⋊C411yescharacteristic
242C23#1455613412yes—
243D4—44no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—81no—
257C2×C4—2842no—
258C2×C4—2842no—
259C2×C4—2842no—
260C2×C4—2842no—
261C2×C4—2842no—
262C2×C4—2842no—
263C2×C4—5684no—
264C2×C4#1455604211yescharacteristic
265C2×C4#1455604211yescharacteristic
266C2×C4#1455604612yes—
267C8—82no—
268C8—5682no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22#1798596511yescharacteristic
282C22C2×#1455604211yescharacteristic
283C22C2×#1455604611yescharacteristic
284C4—21no—
285C4—22no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—5682no—
290C4—5682no—
291C4#1798579012yes—
292C2—21no—
293C2—21no—
294C2—42no—
295C2—81no—
296C2#2258670311yescharacteristic
297C2C2×#1798579011yescharacteristic
298C2#2258998911yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC24×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#1460C2
socleC22×C71C2×#99
prime radicalC22×D4×C71C2×C4
cosocleC2×C22⋊C4×C71C23
hypercenter#1460D71
hypocenterC71#11579
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #22586287C2semidirect
154. C71#11579semidirect

Maximal subgroups

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

subgroupindex
#225926572
#225866782
#225862872
C71×#14602
#1157971