Finite Group Databasebeta

← #31889210

Subgroups — #31889210

Order 18176.

738 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 151 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#22588336C211yesmaximal, characteristic, complemented
2#22587021C212yesmaximal, complemented
3#22586765C212yesmaximal
4#22586664C211yesmaximal, characteristic, complemented
5C2×#556×C71C211yesmaximal, characteristic
6#17986052C2212yes—
7#17986023C2212yes—
8C2×#14556047—21no—
9#17985768—21no—
10#17985748—24no—
11D4⋊(C71⋊C8)—24no—
12#17985725—22no—
13#556×C71—24no—
14#556×C71—24no—
15C2×C4⋊Q8×C71C2211yescharacteristic
16C2×#126×C71—22no—
17C2×#124×C71C412yes—
18C2×C22⋊Q8×C71—21no—
19C2×C4×Q8×C71C412yes—
20C2×C4×Q8×C71C2211yescharacteristic
21C2×C4×D4×C71C2211yescharacteristic
22C2×#117×C71—21no—
23C2×#11765849—22no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26C4⋊(C71⋊C8)—22no—
27#14556047—42no—
28C22⋊(C71⋊C8)—22no—
29C4×C71⋊C8—22no—
30C22×Q8×C71C2×C412yes—
31C22×Q8×C71D411yescharacteristic
32C22×D4×C71C2×C411yescharacteristic
33C23×C4×C71—21no—
34C4⋊Q8×C71—22no—
35C4⋊Q8×C71D414yes—
36#126×C71—216no—
37#124×C71—24no—
38#124×C71—24no—
39C22⋊Q8×C71—28no—
40C4×Q8×C71—24no—
41C4×Q8×C71—24no—
42C4×Q8×C71—22no—
43C4×Q8×C71—22no—
44C4×D4×C71—22no—
45C4×D4×C71D414yes—
46#117×C71—28no—
47C2×C4⋊C4×C71—22no—
48C2×C4⋊C4×C71—21no—
49C2×C4⋊C4×C71—21no—
50C2×C4⋊C4×C71C2×C411yescharacteristic
51C2×C4⋊C4×C71D411yescharacteristic
52C2×C4⋊C4×C71D412yes—
53C2×C22⋊C4×C71—22no—
54C2×C22⋊C4×C71—22no—
55C2×C22⋊C4×C71—21no—
56C2×C42×C71—22no—
57C2×C42×C71C2×C412yes—
58C2×C42×C71C2311yescharacteristic
59#11765849—42no—
60#11765849—42no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—42no—
63C24×C71—21no—
64C2×Q8×C71—22no—
65C2×Q8×C71—22no—
66C2×Q8×C71—22no—
67C2×Q8×C71—21no—
68C2×Q8×C71—21no—
69C2×Q8×C71—44no—
70C2×Q8×C71C22⋊C414yes—
71C2×D4×C71—21no—
72C2×D4×C71—42no—
73C2×D4×C71C22⋊C412yes—
74C22×C4×C71—21no—
75C22×C4×C71—22no—
76C22×C4×C71—24no—
77C22×C4×C71—42no—
78C22×C4×C71—41no—
79C22×C4×C71—41no—
80C22×C4×C71C22⋊C412yes—
81C22×C4×C71C22⋊C412yes—
82C22×C4×C71C22×C411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85C4⋊C4×C71—28no—
86C4⋊C4×C71—21no—
87C4⋊C4×C71—24no—
88C4⋊C4×C71—24no—
89C4⋊C4×C71—22no—
90C4⋊C4×C71—24no—
91C4⋊C4×C71C22⋊C412yes—
92C22⋊C4×C71—28no—
93C22⋊C4×C71—28no—
94C22⋊C4×C71—24no—
95C42×C71—28no—
96C42×C71—22no—
97C42×C71—22no—
98C42×C71C2×D412yes—
99C42×C71C2×D412yes—
100C23×C71—22no—
101C23×C71—42no—
102C23×C71—42no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71C2×C22⋊C411yescharacteristic
106Q8×C71—44no—
107Q8×C71—44no—
108Q8×C71—42no—
109Q8×C71—42no—
110D4×C71—24no—
111D4×C71—42no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—24no—
116C2×C4×C71—22no—
117C2×C4×C71—22no—
118C2×C4×C71—22no—
119C2×C4×C71—41no—
120C2×C4×C71—44no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—44no—
124C2×C4×C71—44no—
125C2×C4×C71—42no—
126C2×C4×C71C2×C22⋊C411yescharacteristic
127C2×C4×C71C2×C22⋊C411yescharacteristic
128C2×C4×C71#12012yes—
129C2×C4×C71#12012yes—
130C2×C4×C71#13512yes—
131C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C71#13711yescharacteristic
133C71⋊C8—82no—
134C71⋊C8—82no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71—42no—
138C22×C71—42no—
139C22×C71—42no—
140C22×C71—82no—
141C22×C71—82no—
142C22×C71#37911yescharacteristic
143C22×C71#41611yescharacteristic
144C22×C71#41711yescharacteristic
145C4×C71—22no—
146C4×C71—21no—
147C4×C71—42no—
148C4×C71—44no—
149C4×C71—42no—
150C4×C71—42no—
151C4×C71#41312yes—
152C4×C71#44912yes—
153C4×C71#45212yes—
154#8394—711nomaximal, Sylow
155C2×C71—21no—
156C2×C71—21no—
157C2×C71—82no—
158C2×C71#178511yescharacteristic
159C2×C71#178811yescharacteristic
160C2×C71#179611yescharacteristic
161C2×#556D7111yescharacteristic
162#1801—711no—
163#1548—712no—
164#1486—712no—
165#1461—711no—
166C71#839411yescharacteristic, complemented, Sylow
167#556—24no—
168#556—24no—
169C2×C4⋊Q8C2×D7111yescharacteristic
170C2×#126—22no—
171C2×#124Dic7112yes—
172C2×C22⋊Q8—21no—
173C2×C4×Q8Dic7112yes—
174C2×C4×Q8C2×D7111yescharacteristic
175C2×C4×D4C2×D7111yescharacteristic
176C2×#117—21no—
177#431—712no—
178#423—712no—
179C2×#101—1421no—
180#343—1421no—
181#333—1424no—
182D4⋊C8—1424no—
183#324—1422no—
184C22×Q8C2×Dic7112yes—
185C22×Q8C22⋊D7111yescharacteristic
186C22×D4C2×Dic7111yescharacteristic
187C23×C4—21no—
188C2×M16—1422no—
189C4⋊Q8—22no—
190C4⋊Q8C22⋊D7114yes—
191#126—216no—
192#124—24no—
193#124—24no—
194C22⋊Q8—28no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×D4—22no—
200C4×D4C22⋊D7114yes—
201#117—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×Dic7111yescharacteristic
206C2×C4⋊C4C22⋊D7111yescharacteristic
207C2×C4⋊C4C22⋊D7112yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C42—22no—
212C2×C42C2×Dic7112yes—
213C2×C42C22×D7111yescharacteristic
214C4⋊C8—1422no—
215C4⋊C8—1422no—
216C4⋊C8—1422no—
217#101—2842no—
218C22⋊C8—1422no—
219C4×C8—1422no—
220C24—21no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—44no—
227C2×Q8C22⋊Dic7114yes—
228C2×D4—21no—
229C2×D4—42no—
230C2×D4C22⋊Dic7112yes—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4C22⋊Dic7112yes—
238C22×C4C22⋊Dic7112yes—
239C22×C4C22×Dic7111yescharacteristic
240C22×C4C2×C22⋊D7111yescharacteristic
241C22×C4C2×C22⋊D7111yescharacteristic
242M16—2842no—
243M16—2842no—
244C2×C8—2842no—
245C2×C8—2842no—
246C4⋊C4—28no—
247C4⋊C4—21no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4C22⋊Dic7112yes—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—24no—
256C42—28no—
257C42—22no—
258C42—22no—
259C42C2×C22⋊D7112yes—
260C42C2×C22⋊D7112yes—
261C23—22no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic7111yescharacteristic
267Q8—44no—
268Q8—42no—
269Q8—42no—
270Q8—44no—
271D4—24no—
272D4—42no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4C2×C22⋊Dic7111yescharacteristic
288C2×C4C2×C22⋊Dic7111yescharacteristic
289C2×C4#1455609712yes—
290C2×C4#1455609712yes—
291C2×C4#1455614812yes—
292C2×C4#1455615311yescharacteristic
293C2×C4#1455615811yescharacteristic
294C8—5682no—
295C8—5682no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—82no—
303C22#1798585511yescharacteristic
304C22#1798599511yescharacteristic
305C22#1798600011yescharacteristic
306C4—21no—
307C4—22no—
308C4—44no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4#1798598312yes—
313C4#1798614412yes—
314C4#1798616112yes—
315C2—21no—
316C2—21no—
317C2—82no—
318C2#2258826811yescharacteristic
319C2#2258828011yescharacteristic
320C2#2258831511yescharacteristic
subgroupG/H
centerC22#17985855
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#556×C71C2
socleC22×C71#379
prime radicalC22×D4×C71C2×C4
cosocleC2×C42×C71C23
hypercenterC2×#556D71
hypocenterC71#8394
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #22588336C2semidirect
2. #22587021C2semidirect
4. #22586664C2semidirect
166. C71#8394semidirect

Maximal subgroups

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

subgroupindex
#225883362
#225870212
#225867652
#225866642
C2×#556×C712
#839471