Finite Group Databasebeta

← #31908008

Subgroups — #31908008

Order 18176.

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

normal subgroups only

This group has 482 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
1C4×#14556043C211yesmaximal, characteristic
2#22586681C211yesmaximal, characteristic
3#22586340C214yesmaximal, complemented
4C71×#2486C211yesmaximal, characteristic
5#17986056—21no—
6C2×#14556043—21no—
7C4×C22⋊Dic71—21no—
8#17985766C2212yes—
9#17985723—21no—
10#17985723C2212yes—
11C2×#123×C71—21no—
12C2×C4×D4×C71C2211yescharacteristic
13C22×C4⋊C4×C71C411yescharacteristic, complemented
14#398×C71C412yescomplemented
15#397×C71C2212yes—
16#389×C71—21no—
17#381×C71—21no—
18#380×C71—21no—
19#380×C71C411yescharacteristic, complemented
20C2×C22⋊Dic71—22no—
21C2×C4×Dic71—41no—
22C4⋊(C71⋊C8)—41no—
23#14556043—24no—
24C22⋊(C71⋊C8)—22no—
25#14556039—41no—
26C71⋊#95—41no—
27C22×D4×C71D411yescharacteristic
28C23×C4×C71—21no—
29C23×C4×C71C2×C411yescharacteristic
30C23×C4×C71D411yescharacteristic
31#123×C71—44no—
32C4×D4×C71—41no—
33C4×D4×C71—41no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—41no—
38C2×C4⋊C4×C71—41no—
39C2×C4⋊C4×C71C2×C412yes—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71D411yescharacteristic
44C2×C22⋊C4×C71C2311yescharacteristic
45C2×C42×C71D411yescharacteristic
46#95×C71—22no—
47#95×C71—22no—
48#95×C71C2×C412yes—
49C22×Dic71—42no—
50C2×C71⋊C8—42no—
51C22⋊Dic71—24no—
52C22⋊Dic71—42no—
53C4×Dic71—82no—
54C24×C71C22⋊C411yescharacteristic
55C24×C71C2×D411yescharacteristic
56C2×D4×C71—22no—
57C2×D4×C71—22no—
58C2×D4×C71—41no—
59C22×C4×C71—21no—
60C22×C4×C71—21no—
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71—22no—
64C22×C4×C71—22no—
65C22×C4×C71—42no—
66C22×C4×C71—42no—
67C22×C4×C71—42no—
68C22×C4×C71—42no—
69C22×C4×C71—42no—
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22×C411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75C4⋊C4×C71—41no—
76C4⋊C4×C71—42no—
77C4⋊C4×C71—42no—
78C4⋊C4×C71—44no—
79C4⋊C4×C71—41no—
80C4⋊C4×C71—41no—
81C22⋊C4×C71—41no—
82C22⋊C4×C71—41no—
83C22⋊C4×C71—42no—
84C22⋊C4×C71—44no—
85C42×C71—21no—
86C42×C71—21no—
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—42no—
113C2×C4×C71—42no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—81no—
119C2×C4×C71—84no—
120C2×C4×C71#9911yescharacteristic
121C2×C4×C71#9911yescharacteristic
122C2×Dic71—44no—
123C2×Dic71—82no—
124C2×Dic71—82no—
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—82no—
147C4×C71—82no—
148C4×C71#35212yes—
149Dic71—84no—
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#2486D7111yescharacteristic
159C4×#99—711no—
160#1466—711no—
161#1344—714no—
162C71#1177311yescharacteristic, complemented, Sylow
163C2×#123—21no—
164C2×C4×D4C2×D7111yescharacteristic
165C22×C4⋊C4Dic7111yescharacteristic, complemented
166#432—1421no—
167C2×#99—1421no—
168#398Dic7112yescomplemented
169#397C2×D7112yes—
170#389—21no—
171#381—21no—
172#380—21no—
173#380Dic7111yescharacteristic, complemented
174C4×C22⋊C4—1421no—
175#342—712no—
176#323—712no—
177#323—1421no—
178C22×D4C22⋊D7111yescharacteristic
179C23×C4—21no—
180C23×C4C2×Dic7111yescharacteristic
181C23×C4C22⋊D7111yescharacteristic
182#123—44no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×Dic7112yes—
191C2×C4⋊C4C2×Dic7111yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—1422no—
195C2×C22⋊C4C22⋊D7111yescharacteristic
196C2×C22⋊C4C22×D7111yescharacteristic
197C2×C42—2841no—
198C2×C42C22⋊D7111yescharacteristic
199C4⋊C8—2841no—
200#99—1424no—
201C22⋊C8—1422no—
202#97—2841no—
203#95—22no—
204#95—22no—
205#95—2841no—
206#95C2×Dic7112yes—
207C24C22⋊Dic7111yescharacteristic
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—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—2842no—
224C22×C4C22⋊Dic7111yescharacteristic
225C22×C4C22⋊Dic7111yescharacteristic
226C22×C4C22⋊Dic7111yescharacteristic
227C22×C4C22×Dic7111yescharacteristic
228C22×C4C2×C22⋊D7111yescharacteristic
229C2×C8—2842no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C22⋊C4—2842no—
241C22⋊C4—1424no—
242C42—21no—
243C42—21no—
244C42—5682no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#1455605512yes—
254C23C2×C22⋊Dic7111yescharacteristic
255C23#1455615412yes—
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—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—84no—
278C2×C4—2844no—
279C2×C4—5682no—
280C2×C4—5682no—
281C2×C4#1455604311yescharacteristic
282C2×C4#1455604311yescharacteristic
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×#1455604311yescharacteristic
297C22#1798600411yescharacteristic
298C22C2×#1455605511yescharacteristic
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4—82no—
306C4—5684no—
307C4#1798578912yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2258685611yescharacteristic
313C2C2×#1798578911yescharacteristic
314C2#2259001511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2486C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenter#2486D71
hypocenterC71#11773
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #22586340C2semidirect
13. C22×C4⋊C4×C71C4semidirect
14. #398×C71C4semidirect
19. #380×C71C4semidirect
162. C71#11773semidirect
165. C22×C4⋊C4Dic71semidirect
168. #398Dic71semidirect
173. #380Dic71semidirect

Maximal subgroups

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

subgroupindex
C4×#145560432
#225866812
#225863402
C71×#24862
#1177371