Finite Group Databasebeta

← #31908007

Subgroups — #31908007

Order 18176.

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

normal subgroups only

This group has 482 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#22592648C211yesmaximal, characteristic
2#22586681C211yesmaximal, characteristic
3#22586339C214yesmaximal, complemented
4C4×#99×C71C211yesmaximal, characteristic
5C2×#14556109—21no—
6C22×C4⋊Dic71C411yescharacteristic, complemented
7#17986056—21no—
8#17985929C412yescomplemented
9#17985924C2212yes—
10#17985897—21no—
11#17985860—21no—
12#17985859—21no—
13#17985859C411yescharacteristic, complemented
14#17985723—21no—
15#17985723C2212yes—
16C2×C4×D4×C71C2211yescharacteristic
17C2×#99×C71—21no—
18C4×C22⋊C4×C71—21no—
19#342×C71C2212yes—
20C23×Dic71—21no—
21#14556109—44no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—41no—
25C2×C4⋊Dic71—41no—
26C2×C4⋊Dic71C2×C412yes—
27C2×Dic71⋊C4—22no—
28C2×C22⋊Dic71—21no—
29C2×C22⋊Dic71—22no—
30C4⋊(C71⋊C8)—41no—
31C22⋊(C71⋊C8)—22no—
32#14556039—41no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95C2×C412yes—
36C22×D4×C71D411yescharacteristic
37C23×C4×C71C2×C411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C4×D4×C71—41no—
40C4×D4×C71—41no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71D411yescharacteristic
44C2×C22⋊C4×C71C2311yescharacteristic
45C2×C42×C71—41no—
46C2×C42×C71D411yescharacteristic
47#99×C71—24no—
48#95×C71—41no—
49C22×Dic71—22no—
50C22×Dic71—22no—
51C22×Dic71—42no—
52C22×Dic71—42no—
53C22×Dic71—42no—
54C2×C71⋊C8—42no—
55C4⋊Dic71—42no—
56C4⋊Dic71—41no—
57C4⋊Dic71—41no—
58C4⋊Dic71—42no—
59Dic71⋊C4—44no—
60C22⋊Dic71—42no—
61C22⋊Dic71—44no—
62C24×C71C22⋊C411yescharacteristic
63C24×C71C2×D411yescharacteristic
64C2×D4×C71—22no—
65C2×D4×C71—22no—
66C2×D4×C71—41no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—42no—
72C22×C4×C71—42no—
73C22×C4×C71—42no—
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22×C411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C4⋊C4×C71—41no—
80C22⋊C4×C71—24no—
81C22⋊C4×C71—41no—
82C22⋊C4×C71—41no—
83C22⋊C4×C71—42no—
84C42×C71—21no—
85C42×C71—21no—
86C42×C71—82no—
87C23×C71—21no—
88C23×C71—21no—
89C23×C71—21no—
90C23×C71—21no—
91C23×C71—41no—
92C23×C71—41no—
93C23×C71—41no—
94C23×C71—41no—
95C23×C71#10412yes—
96C23×C71C2×C22⋊C411yescharacteristic
97C23×C71AGL(1,ℤ/8)12yes—
98D4×C71—44no—
99C2×C4×C71—22no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—41no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—44no—
113C2×C4×C71—81no—
114C2×C4×C71—82no—
115C2×C4×C71—82no—
116C2×C4×C71#9911yescharacteristic
117C2×C4×C71#9911yescharacteristic
118C2×Dic71—42no—
119C2×Dic71—42no—
120C2×Dic71—42no—
121C2×Dic71—42no—
122C2×Dic71—42no—
123C2×Dic71—42no—
124C2×Dic71—84no—
125C71⋊C8—82no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71—22no—
129C22×C71—22no—
130C22×C71—42no—
131C22×C71—42no—
132C22×C71—42no—
133C22×C71—41no—
134C22×C71—41no—
135C22×C71—41no—
136C22×C71—41no—
137C22×C71—81no—
138C22×C71C2×#9911yescharacteristic
139C22×C71#41811yescharacteristic
140C22×C71C2×#10411yescharacteristic
141C4×C71—21no—
142C4×C71—41no—
143C4×C71—41no—
144C4×C71—41no—
145C4×C71—41no—
146C4×C71—84no—
147C4×C71#35212yes—
148Dic71—82no—
149Dic71—82no—
150#11773—711nomaximal, Sylow
151C2×C71—21no—
152C2×C71—21no—
153C2×C71—42no—
154C2×C71—42no—
155C2×C71#150711yescharacteristic
156C2×C71C2×#35211yescharacteristic
157C2×C71#212911yescharacteristic
158#2486—711no—
159C4×#99D7111yescharacteristic
160#1466—711no—
161#1344—714no—
162C71#1177311yescharacteristic, complemented, Sylow
163C2×#123—1421no—
164C2×C4×D4C2×D7111yescharacteristic
165C22×C4⋊C4—711no—
166#432—1421no—
167C2×#99—21no—
168#398—712no—
169#397—712no—
170#389—1421no—
171#381—1421no—
172#380—711no—
173#380—1421no—
174C4×C22⋊C4—21no—
175#342C2×D7112yes—
176#323—712no—
177#323—1421no—
178C22×D4C22⋊D7111yescharacteristic
179C23×C4—1421no—
180C23×C4C4×D7111yescharacteristic
181C23×C4C22⋊D7111yescharacteristic
182#123—2844no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—2841no—
186C2×C4⋊C4—2841no—
187C2×C4⋊C4—712no—
188C2×C4⋊C4—1422no—
189C2×C4⋊C4—1421no—
190C2×C4⋊C4—1421no—
191C2×C4⋊C4C4×D7111yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—1421no—
194C2×C22⋊C4—1422no—
195C2×C22⋊C4D28411yescharacteristic
196C2×C22⋊C4C22×D7111yescharacteristic
197C2×C42—41no—
198C2×C42D28411yescharacteristic
199C4⋊C8—2841no—
200#99—24no—
201C22⋊C8—1422no—
202#97—2841no—
203#95—41no—
204#95—712no—
205#95—1422no—
206#95—1422no—
207C24(C2×D71)⋊C411yescharacteristic
208C24C2×C22⋊D7111yescharacteristic
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—2842no—
220C22×C4—2842no—
221C22×C4—2842no—
222C22×C4—1422no—
223C22×C4—1422no—
224C22×C4(C2×D71)⋊C411yescharacteristic
225C22×C4(C2×D71)⋊C411yescharacteristic
226C22×C4(C2×D71)⋊C411yescharacteristic
227C22×C4C2×C4×D7111yescharacteristic
228C22×C4C2×D28411yescharacteristic
229C2×C8—2842no—
230C4⋊C4—41no—
231C4⋊C4—2842no—
232C4⋊C4—2844no—
233C4⋊C4—2842no—
234C4⋊C4—2841no—
235C4⋊C4—2841no—
236C22⋊C4—24no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—2842no—
241C22⋊C4—2844no—
242C42—21no—
243C42—21no—
244C42—82no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#1455605612yes—
254C23C2×(C2×D71)⋊C411yescharacteristic
255C23#1455615312yes—
256D4—44no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—44no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—2842no—
275C2×C4—2842no—
276C2×C4—2842no—
277C2×C4—2842no—
278C2×C4—2842no—
279C2×C4—2842no—
280C2×C4—5684no—
281C2×C4#1455604211yescharacteristic
282C2×C4#1455604211yescharacteristic
283C8—5682no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22C2×#1455604211yescharacteristic
297C22#1798600311yescharacteristic
298C22C2×#1455605611yescharacteristic
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—84no—
305C4—5682no—
306C4—5682no—
307C4#1798578812yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2258685511yescharacteristic
313C2C2×#1798578811yescharacteristic
314C2#2259001411yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC4×#99×C71C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenterC4×#99D71
hypocenterC71#11773
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #22586339C2semidirect
6. C22×C4⋊Dic71C4semidirect
8. #17985929C4semidirect
13. #17985859C4semidirect
162. C71#11773semidirect

Maximal subgroups

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

subgroupindex
#225926482
#225866812
#225863392
C4×#99×C712
#1177371