Finite Group Databasebeta

← #31890365

Subgroups — #31890365

Order 18176.

460 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#17986071C211yesmaximal, characteristic
2#22588376C212yesmaximal, complemented
3#22586678C211yesmaximal, characteristic
4#22586216C212yesmaximal, complemented
5C71×#1465C211yesmaximal, characteristic
6C22×#11765849C411yescharacteristic
7C23×C71⋊C8C2211yescharacteristic
8#17986071—22no—
9#17986071—22no—
10C2×#14556063—22no—
11C2×C4⋊(C71⋊C8)C411yescharacteristic
12#17985963C2212yes—
13C2×C22⋊(C71⋊C8)C411yescharacteristic
14C2×C22⋊(C71⋊C8)C2211yescharacteristic
15C2×#14556039C411yescharacteristic
16#17985763—22no—
17#17985725—21no—
18#17985723—21no—
19C2×C4×D4×C71C2211yescharacteristic
20#407×C71C2212yes—
21#324×C71—21no—
22#323×C71—21no—
23C2×#11765849—21no—
24C2×#11765849—21no—
25C2×#11765849—22no—
26C2×#11765849—42no—
27C22×C71⋊C8—42no—
28C22×C71⋊C8C2×C411yescharacteristic
29C22×C71⋊C8C2×C411yescharacteristic
30C22×C71⋊C8C2×C411yescharacteristic
31C22×C71⋊C8C2×C411yescharacteristic
32C22×C71⋊C8D412yes—
33C22×C71⋊C8D412yes—
34#14556063—44no—
35C4⋊(C71⋊C8)—22no—
36C22⋊(C71⋊C8)—22no—
37C22⋊(C71⋊C8)—22no—
38C22⋊(C71⋊C8)—22no—
39C22⋊(C71⋊C8)—22no—
40#14556039—21no—
41#14556039—21no—
42C22×D4×C71Q811yescharacteristic
43C23×C4×C71C2×C411yescharacteristic
44C23×C4×C71C2311yescharacteristic
45C2×M16×C71—22no—
46C4×D4×C71—22no—
47C4×D4×C71—22no—
48C2×C4⋊C4×C71D411yescharacteristic
49C2×C22⋊C4×C71D411yescharacteristic
50C2×C22⋊C4×C71Q811yescharacteristic
51C2×C42×C71C2×C411yescharacteristic
52C22⋊C8×C71—22no—
53C22⋊C8×C71—22no—
54#11765849—42no—
55#11765849—42no—
56#11765849—42no—
57#11765849—42no—
58C2×C71⋊C8—24no—
59C2×C71⋊C8—22no—
60C2×C71⋊C8—21no—
61C2×C71⋊C8—21no—
62C2×C71⋊C8—41no—
63C2×C71⋊C8—42no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—42no—
67C2×C71⋊C8C22⋊C411yescharacteristic
68C2×C71⋊C8C22⋊C412yes—
69C2×C71⋊C8C22⋊C411yescharacteristic
70C24×C71C4⋊C411yescharacteristic
71C24×C71C2×Q811yescharacteristic
72C2×D4×C71—22no—
73C2×D4×C71—42no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—41no—
77C22×C4×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71—41no—
80C22×C4×C71C4⋊C411yescharacteristic
81C22×C4×C71C4⋊C411yescharacteristic
82C22×C4×C71C4⋊C411yescharacteristic
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C2×D412yes—
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71#4112yes—
87M16×C71—42no—
88M16×C71—42no—
89C2×C8×C71—42no—
90C4⋊C4×C71—22no—
91C22⋊C4×C71—22no—
92C22⋊C4×C71—22no—
93C42×C71—21no—
94C42×C71—21no—
95C23×C71—22no—
96C23×C71—21no—
97C23×C71—22no—
98C23×C71—41no—
99C23×C71—41no—
100C23×C71—41no—
101C23×C71—41no—
102C23×C71C2×C4⋊C411yescharacteristic
103C23×C71C22⋊Q812yes—
104D4×C71—44no—
105C2×C4×C71—22no—
106C2×C4×C71—21no—
107C2×C4×C71—22no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—22no—
113C2×C4×C71—41no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71C2×C22⋊C411yescharacteristic
121C2×C4×C71#11711yescharacteristic
122C2×C4×C71#12112yes—
123C8×C71—82no—
124C71⋊C8—22no—
125C71⋊C8—21no—
126C71⋊C8—41no—
127C71⋊C8—41no—
128C71⋊C8—82no—
129C71⋊C8#9912yes—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71—22no—
133C22×C71—22no—
134C22×C71—22no—
135C22×C71—44no—
136C22×C71—41no—
137C22×C71—41no—
138C22×C71—41no—
139C22×C71—82no—
140C22×C71#38011yescharacteristic
141C22×C71#43012yes—
142C4×C71—22no—
143C4×C71—21no—
144C4×C71—41no—
145C4×C71—41no—
146C4×C71—41no—
147C4×C71C2×#9911yescharacteristic
148C4×C71#41011yescharacteristic
149#8613—711nomaximal, Sylow
150C2×C71—22no—
151C2×C71—42no—
152C2×C71—81no—
153C2×C71#179911yescharacteristic
154C2×C71#181111yescharacteristic
155C2×C71#181311yescharacteristic
156C2×#435—711no—
157#1810—712no—
158#1465—711no—
159#1465D7111yescharacteristic
160#1293—712no—
161C71#861311yescharacteristic, complemented, Sylow
162C22×M16—711no—
163C23×C8—711no—
164C2×C4×D4C2×D7111yescharacteristic
165#435—1422no—
166#435—1422no—
167C2×#108—1422no—
168C2×C4⋊C8—711no—
169#407—712no—
170#407C2×D7112yes—
171C2×C22⋊C8—711no—
172C2×C22⋊C8—711no—
173C2×#97—711no—
174#341—1422no—
175#324—21no—
176#324—1421no—
177#323—21no—
178#323—1421no—
179C22×D4Dic14211yescharacteristic
180C23×C4C4×D7111yescharacteristic
181C23×C4C22×D7111yescharacteristic
182C2×M16—22no—
183C2×M16—2842no—
184C2×M16—1422no—
185C2×M16—1421no—
186C2×M16—1421no—
187C22×C8—2842no—
188C22×C8—711no—
189C22×C8—712no—
190C22×C8—711no—
191C22×C8—711no—
192C22×C8—711no—
193C22×C8—712no—
194C4×D4—22no—
195C4×D4—22no—
196C2×C4⋊C4C22⋊D7111yescharacteristic
197C2×C22⋊C4Dic14211yescharacteristic
198C2×C22⋊C4C22⋊D7111yescharacteristic
199C2×C42C4×D7111yescharacteristic
200#108—2844no—
201C4⋊C8—1422no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—1422no—
205C22⋊C8—1422no—
206C22⋊C8—1422no—
207C22⋊C8—1422no—
208#97—1421no—
209#97—1421no—
210C24Dic71⋊C411yescharacteristic
211C24C2×Dic14211yescharacteristic
212C2×D4—22no—
213C2×D4—42no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4Dic71⋊C411yescharacteristic
221C22×C4Dic71⋊C411yescharacteristic
222C22×C4Dic71⋊C411yescharacteristic
223C22×C4C2×C4×D7111yescharacteristic
224C22×C4D4×D7112yes—
225C22×C4C2×C22⋊D7111yescharacteristic
226C22×C4#1176586412yes—
227M16—42no—
228M16—42no—
229M16—2842no—
230M16—2842no—
231M16—2842no—
232M16—2842no—
233C2×C8—42no—
234C2×C8—2841no—
235C2×C8—2841no—
236C2×C8—2842no—
237C2×C8—2842no—
238C2×C8—2841no—
239C2×C8—711no—
240C2×C8—711no—
241C2×C8—712no—
242C2×C8—1421no—
243C2×C8—1421no—
244C2×C8—1424no—
245C2×C8—1422no—
246C4⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—22no—
249C42—21no—
250C42—21no—
251C23—22no—
252C23—21no—
253C23—22no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23C2×Dic71⋊C411yescharacteristic
259C23#1455610512yes—
260D4—44no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4C22⋊C4×D7111yescharacteristic
277C2×C4#1455608411yescharacteristic
278C2×C4#1455610012yes—
279C8—82no—
280C8—2841no—
281C8—2841no—
282C8—5682no—
283C8—712no—
284C8—1421no—
285C8—1422no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—44no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—82no—
296C22#1798585711yescharacteristic
297C22#1798604712yes—
298C4—22no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4#99×D7111yescharacteristic
304C4#1798597111yescharacteristic
305C2—22no—
306C2—42no—
307C2—81no—
308C2#2258832611yescharacteristic
309C2#2258838011yescharacteristic
310C2#2258838811yescharacteristic
subgroupG/H
centerC22#17985857
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#1465C2
socleC22×C71#380
prime radicalC22×D4×C71Q8
cosocleC23×C4×C71C23
hypercenter#1465D71
hypocenterC71#8613
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #22588376C2semidirect
4. #22586216C2semidirect
161. C71#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#179860712
#225883762
#225866782
#225862162
C71×#14652
#861371