Finite Group Databasebeta

← #31890860

Subgroups — #31890860

Order 18176.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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#22590747C211yesmaximal, characteristic
2#22588102C211yesmaximal, characteristic
3#22586222C214yesmaximal, complemented
4C71×#1749C211yesmaximal, characteristic
5C2×#14556087C412yes—
6C2×C22⋊(C71⋊C8)—21no—
7C2×#14556039C2212yes—
8#17985869C411yescharacteristic
9#17985869C411yescharacteristic
10#17985865C2212yes—
11#17985859—21no—
12#17985859—21no—
13C4×C4⋊Dic71—21no—
14C4×C4⋊Dic71—21no—
15#17985754—21no—
16C22×C42×C71C2211yescharacteristic
17C2×C22⋊C8×C71—21no—
18C2×#97×C71C2212yes—
19#336×C71—21no—
20C22×C71⋊C8—22no—
21#14556087—44no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71C2×C412yes—
27C2×C4⋊Dic71C2×C412yes—
28C2×C22⋊Dic71—22no—
29C2×C4×Dic71—22no—
30C22⋊(C71⋊C8)—24no—
31#14556039—24no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C23×C4×C71C2×C411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C23×C4×C71D411yescharacteristic
37C22×C8×C71—22no—
38C2×C42×C71—24no—
39C2×C42×C71C2×C411yescharacteristic
40C2×C42×C71D411yescharacteristic
41C2×C42×C71D411yescharacteristic
42C2×C42×C71C2311yescharacteristic
43C22⋊C8×C71—24no—
44#97×C71—24no—
45C22×Dic71—22no—
46C22×Dic71—22no—
47C2×C71⋊C8—24no—
48C2×C71⋊C8—42no—
49C2×C71⋊C8—42no—
50C4⋊Dic71—42no—
51C4⋊Dic71—42no—
52C4⋊Dic71—42no—
53C4⋊Dic71—42no—
54C4⋊Dic71—42no—
55C4⋊Dic71—42no—
56C22⋊Dic71—44no—
57C4×Dic71—44no—
58C24×C71C22⋊C411yescharacteristic
59C22×C4×C71—22no—
60C22×C4×C71—22no—
61C22×C4×C71—22no—
62C22×C4×C71—22no—
63C22×C4×C71—22no—
64C22×C4×C71—22no—
65C22×C4×C71—22no—
66C22×C4×C71—41no—
67C22×C4×C71—41no—
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C2×C8×C71—24no—
75C2×C8×C71—42no—
76C2×C8×C71—42no—
77C42×C71—22no—
78C42×C71—22no—
79C42×C71—22no—
80C42×C71#4114yes—
81C23×C71—21no—
82C23×C71—21no—
83C23×C71—21no—
84C23×C71—21no—
85C23×C71—21no—
86C23×C71—21no—
87C23×C71#10112yes—
88C23×C71C2×C22⋊C411yescharacteristic
89C2×C4×C71—28no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—44no—
103C2×C4×C71—44no—
104C2×C4×C71#10012yes—
105C2×C4×C71#10412yes—
106C2×C4×C71#10412yes—
107C2×C4×C71#11712yes—
108C2×C4×C71#12312yes—
109C2×C4×C71#12312yes—
110C2×Dic71—42no—
111C2×Dic71—42no—
112C2×Dic71—42no—
113C2×Dic71—42no—
114C2×Dic71—42no—
115C2×Dic71—42no—
116C8×C71—44no—
117C71⋊C8—44no—
118C22×C71—22no—
119C22×C71—22no—
120C22×C71—22no—
121C22×C71—41no—
122C22×C71—41no—
123C22×C71—41no—
124C22×C71—41no—
125C22×C71#38111yescharacteristic
126C22×C71C2×#10011yescharacteristic
127C22×C71C2×#10111yescharacteristic
128C22×C71C2×#10411yescharacteristic
129C22×C71C2×#10411yescharacteristic
130C22×C71#42111yescharacteristic
131C22×C71#42111yescharacteristic
132C4×C71—22no—
133C4×C71—22no—
134C4×C71—22no—
135C4×C71—22no—
136C4×C71—22no—
137C4×C71—22no—
138Dic71—82no—
139Dic71—82no—
140#8721—711nomaximal, Sylow
141C2×C71—42no—
142C2×C71#152511yescharacteristic
143C2×C71#153111yescharacteristic
144C2×C71#153111yescharacteristic
145C2×C71#153611yescharacteristic
146C2×C71#182111yescharacteristic
147C2×C71#182611yescharacteristic
148C2×C71#182611yescharacteristic
149#2285—711no—
150#1749—711no—
151#1749D7111yescharacteristic
152#1295—714no—
153C71#872111yescharacteristic, complemented, Sylow
154C2×#117—712no—
155C22×C42C2×D7111yescharacteristic
156C2×C22⋊C8—21no—
157C2×C22⋊C8—1421no—
158C2×#97—712no—
159C2×#97C2×D7112yes—
160#384—711no—
161#384—711no—
162#382—712no—
163#380—1421no—
164#380—1421no—
165C4×C4⋊C4—1421no—
166C4×C4⋊C4—1421no—
167#336—21no—
168#336—1421no—
169C23×C4C4×D7111yescharacteristic
170C23×C4D28411yescharacteristic
171C23×C4C22⋊D7111yescharacteristic
172C22×C8—22no—
173C22×C8—1422no—
174#117—2844no—
175C2×C4⋊C4—712no—
176C2×C4⋊C4—712no—
177C2×C4⋊C4—1421no—
178C2×C4⋊C4—1421no—
179C2×C4⋊C4—1421no—
180C2×C4⋊C4—1421no—
181C2×C22⋊C4—1422no—
182C2×C42—24no—
183C2×C42—1422no—
184C2×C42C4×D7111yescharacteristic
185C2×C42D28411yescharacteristic
186C2×C42C22⋊D7111yescharacteristic
187C2×C42C22×D7111yescharacteristic
188C22⋊C8—24no—
189C22⋊C8—1424no—
190#97—24no—
191#97—1424no—
192#95—1422no—
193#95—1422no—
194C24(C2×D71)⋊C411yescharacteristic
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—1422no—
205C22×C4—1422no—
206C22×C4(C2×D71)⋊C411yescharacteristic
207C22×C4(C2×D71)⋊C411yescharacteristic
208C22×C4(C2×D71)⋊C411yescharacteristic
209C22×C4C2×C4×D7111yescharacteristic
210C22×C4C2×C22⋊D7111yescharacteristic
211C22×C4C2×D28411yescharacteristic
212C2×C8—24no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—2842no—
216C2×C8—2842no—
217C2×C8—1424no—
218C4⋊C4—2842no—
219C4⋊C4—2842no—
220C4⋊C4—2842no—
221C4⋊C4—2842no—
222C4⋊C4—2842no—
223C4⋊C4—2842no—
224C22⋊C4—2844no—
225C42—22no—
226C42—22no—
227C42—22no—
228C42—2844no—
229C42#1176586414yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23#1455604612yes—
237C23C2×(C2×D71)⋊C411yescharacteristic
238C2×C4—28no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—2842no—
254C2×C4—2842no—
255C2×C4—2842no—
256C2×C4—2842no—
257C2×C4—2842no—
258C2×C4—2842no—
259C2×C4#1455604412yes—
260C2×C4#1455605412yes—
261C2×C4#1455605612yes—
262C2×C4#1455608412yes—
263C2×C4#1455610912yes—
264C2×C4#1455611212yes—
265C8—44no—
266C8—2844no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22#1798586011yescharacteristic
275C22C2×#1455604411yescharacteristic
276C22C2×#1455604611yescharacteristic
277C22C2×#1455605411yescharacteristic
278C22C2×#1455605611yescharacteristic
279C22#1798601511yescharacteristic
280C22#1798601711yescharacteristic
281C4—22no—
282C4—22no—
283C4—22no—
284C4—22no—
285C4—22no—
286C4—22no—
287C4—5682no—
288C4—5682no—
289C2—42no—
290C2#2258693111yescharacteristic
291C2#2258695411yescharacteristic
292C2#2258695611yescharacteristic
293C2#2258697411yescharacteristic
294C2#2258841811yescharacteristic
295C2#2258843711yescharacteristic
296C2#2258844011yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1749C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1749D71
hypocenterC71#8721
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #22586222C2semidirect
153. C71#8721semidirect

Maximal subgroups

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

subgroupindex
#225907472
#225881022
#225862222
C71×#17492
#872171