Finite Group Databasebeta

← #31914010

Subgroups — #31914010

Order 18176.

458 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 117 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×#106×D71C211yesmaximal, characteristic
2C2×#17985822C211yesmaximal, characteristic, complemented
3C2×#17985821C211yesmaximal, characteristic, complemented
4#22589950C211yesmaximal, characteristic
5#22588375C211yesmaximal, characteristic
6#22586440C211yesmaximal, characteristic, complemented
7C71×#1383C211yesmaximal, characteristic, complemented
8C22×C8×D71C2211yescharacteristic
9C2×C4⋊C4×D71—21no—
10C2×#14556068C2211yescharacteristic
11C2×#14556064C2211yescharacteristic
12C2×#14556060C2211yescharacteristic
13C2×#14556059—21no—
14#106×D71—22no—
15#106×D71—22no—
16#17985962—21no—
17#17985822—22no—
18#17985821—22no—
19(C2×D71)⋊C16C414yescomplemented
20#17985763—21no—
21C2×M32×C71C2211yescharacteristic
22C2×#108×C71C2211yescharacteristic
23C2×#106×C71C2211yescharacteristic
24C22×C4×D71—41no—
25C22×C4×D71D411yescharacteristic
26C2×#11765849—21no—
27C22×C71⋊C8D411yescharacteristic
28C2×C8×D71—21no—
29C2×C8×D71—21no—
30C2×C8×D71—21no—
31C2×C8×D71—21no—
32C2×C8×D71C2×C412yes—
33C2×C8×D71Q812yes—
34C2×C4⋊Dic71—21no—
35C2×Dic71⋊C4—41no—
36C4⋊C4×D71—24no—
37C4⋊C4×D71—42no—
38#14556068—21no—
39#14556068—21no—
40C2×C71⋊C16C2×C412yescomplemented
41#14556064—21no—
42#14556064—21no—
43#14556060—21no—
44#14556060D412yes—
45#14556059—24no—
46(C2×D71)⋊C8—42no—
47C2×M16×C71—21no—
48C22×C8×C71C2311yescharacteristic
49C2×C4⋊C4×C71—21no—
50M32×C71—21no—
51M32×C71—21no—
52C2×C16×C71C2×C412yescomplemented
53#108×C71—21no—
54#108×C71—21no—
55#106×C71—21no—
56#106×C71D412yes—
57C23×D71SD1611yescharacteristic
58C22×Dic71—41no—
59C22×Dic71SD1611yescharacteristic
60C2×C4×D71—21no—
61C2×C4×D71—21no—
62C2×C4×D71—21no—
63C2×C4×D71—21no—
64C2×C4×D71—44no—
65C2×C4×D71—84no—
66C2×C4×D71C4⋊C412yes—
67C2×C4×D71SD1612yes—
68#11765849—41no—
69#11765849—41no—
70C2×C71⋊C8—21no—
71C2×C71⋊C8—21no—
72C2×C71⋊C8—41no—
73C2×C71⋊C8C4⋊C412yes—
74C8×D71—22no—
75C8×D71—22no—
76C8×D71—22no—
77C8×D71—22no—
78C4⋊Dic71—22no—
79C4⋊Dic71—41no—
80Dic71⋊C4—44no—
81C71⋊C16—22no—
82C22×C4×C71—41no—
83C22×C4×C71C2×D411yescharacteristic
84M16×C71—41no—
85M16×C71—41no—
86C2×C8×C71—41no—
87C2×C8×C71C22×C411yescharacteristic
88C2×C8×C71C2×D411yescharacteristic
89C2×C8×C71C2×D411yescharacteristic
90C2×C8×C71C2×Q811yescharacteristic
91C2×C8×C71#4111yescharacteristic
92C2×C8×C71#4111yescharacteristic
93C4⋊C4×C71—22no—
94C4⋊C4×C71—41no—
95C16×C71—22no—
96C23×C71C2×SD1611yescharacteristic
97C22×D71—21no—
98C22×D71—21no—
99C22×D71—24no—
100C22×D71#10612yes—
101C2×C4×C71—42no—
102C2×C4×C71—82no—
103C2×C4×C71C2×C4⋊C411yescharacteristic
104C2×C4×C71#12011yescharacteristic
105C2×C4×C71#12311yescharacteristic
106C2×C4×C71C2×SD1611yescharacteristic
107C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
108C2×C4×C71#13711yescharacteristic
109C2×Dic71—21no—
110C2×Dic71—21no—
111C2×Dic71—42no—
112C2×Dic71—82no—
113C2×Dic71#10612yes—
114C4×D71—22no—
115C4×D71—22no—
116C4×D71—22no—
117C4×D71—22no—
118C4×D71—88no—
119C8×C71—81no—
120C8×C71C4×D411yescharacteristic
121C8×C71C4×D411yescharacteristic
122C8×C71C22⋊Q811yescharacteristic
123C8×C71C22⋊Q811yescharacteristic
124C71⋊C8—22no—
125C71⋊C8—81no—
126C22×C71—22no—
127C22×C71C2×#10611yescharacteristic
128C22×C71#45011yescharacteristic
129C22×C71#48311yescharacteristic
130C2×D71—24no—
131C2×D71—24no—
132C2×D71—22no—
133C2×D71—22no—
134C2×D71#36514yescomplemented
135C4×C71—82no—
136C4×C71#38511yescharacteristic
137C4×C71#42811yescharacteristic
138C4×C71#44811yescharacteristic
139C4×C71#48111yescharacteristic
140Dic71—22no—
141Dic71—82no—
142#12929—711nomaximal, Sylow
143C2×C71—21no—
144C2×C71—21no—
145C2×C71#185311yescharacteristic
146C2×C71C2×#36512yes—
147D71—24no—
148C22×#106—711no—
149C2×#365—711no—
150C2×#365—711no—
151#2112—711no—
152#1810—711no—
153#1383—711no—
154#1383D7111yescharacteristic, complemented
155C71#1292911yescharacteristic, complemented, Sylow
156C23×C8—711no—
157C22×C4⋊C4—1421no—
158C2×M32—711no—
159C2×M32C2×D7111yescharacteristic
160C2×#108—711no—
161C2×#108C2×D7111yescharacteristic
162C2×#106—711no—
163C2×#106—1422no—
164C2×#106—1421no—
165C2×#106—1422no—
166C2×#106C2×D7111yescharacteristic
167#407—1421no—
168#365—1422no—
169#365—1422no—
170C22⋊C16—714no—
171#341—1421no—
172C23×C4—2841no—
173C23×C4—711no—
174C2×M16—21no—
175C2×M16—1421no—
176C22×C8—711no—
177C22×C8—712no—
178C22×C8—712no—
179C22×C8—1421no—
180C22×C8—1421no—
181C22×C8—1421no—
182C22×C8—1421no—
183C22×C8C22×D7111yescharacteristic
184C2×C4⋊C4—21no—
185C2×C4⋊C4—2842no—
186C2×C4⋊C4—2841no—
187C2×C4⋊C4—1424no—
188C2×C4⋊C4—1421no—
189M32—21no—
190M32—21no—
191M32—1421no—
192M32—1421no—
193C2×C16—712no—
194C2×C16C4×D7112yescomplemented
195#108—21no—
196#108—21no—
197#108—1421no—
198#108—1421no—
199#106—21no—
200#106—712no—
201#106—1424no—
202#106—1421no—
203#106D28412yes—
204C22⋊C8—2842no—
205C24—711no—
206C22×C4—41no—
207C22×C4—2844no—
208C22×C4—2841no—
209C22×C4—5684no—
210C22×C4—711no—
211C22×C4—712no—
212C22×C4—712no—
213C22×C4—1421no—
214C22×C4—1421no—
215C22×C4—1421no—
216C22×C4—1421no—
217C22×C4D4×D7111yescharacteristic
218M16—41no—
219M16—41no—
220M16—2841no—
221M16—2841no—
222C2×C8—41no—
223C2×C8—2841no—
224C2×C8—712no—
225C2×C8—1421no—
226C2×C8—1421no—
227C2×C8—1422no—
228C2×C8—1422no—
229C2×C8—1422no—
230C2×C8—1422no—
231C2×C8C2×C4×D7111yescharacteristic
232C2×C8D4×D7111yescharacteristic
233C2×C8C2×D28411yescharacteristic
234C2×C8Q8×D7111yescharacteristic
235C2×C8#1176586411yescharacteristic
236C2×C8#1176586511yescharacteristic
237C4⋊C4—22no—
238C4⋊C4—41no—
239C4⋊C4—2844no—
240C4⋊C4—2841no—
241C4⋊C4—1422no—
242C16—22no—
243C16—1422no—
244C23—712no—
245C23—1424no—
246C23—1421no—
247C23—1421no—
248C23SD16×D7111yescharacteristic
249C2×C4—42no—
250C2×C4—82no—
251C2×C4—2842no—
252C2×C4—5688no—
253C2×C4—5682no—
254C2×C4—712no—
255C2×C4—1421no—
256C2×C4—1421no—
257C2×C4—1422no—
258C2×C4—1422no—
259C2×C4—1422no—
260C2×C4—1422no—
261C2×C4C4⋊C4×D7111yescharacteristic
262C2×C4#1455609811yescharacteristic
263C2×C4#1455611011yescharacteristic
264C2×C4SD16×D7111yescharacteristic
265C2×C4#1455615211yescharacteristic
266C2×C4#1455616011yescharacteristic
267C8—81no—
268C8—5681no—
269C8—1422no—
270C8(C22⋊D71)⋊C411yescharacteristic
271C8C4×D28411yescharacteristic
272C8#1455610611yescharacteristic
273C8#1455610811yescharacteristic
274C22—22no—
275C22—714no—
276C22—1424no—
277C22—1424no—
278C22—1422no—
279C22—1422no—
280C22#106×D7111yescharacteristic
281C22#1798615111yescharacteristic
282C22#1798632811yescharacteristic
283C4—82no—
284C4—5682no—
285C4—1422no—
286C4#1798587311yescharacteristic
287C4#1798604011yescharacteristic
288C4#1798613911yescharacteristic
289C4#1798631611yescharacteristic
290C2—21no—
291C2—21no—
292C2—1424no—
293C2#2258853411yescharacteristic
294C2#365×D7112yes—
subgroupG/H
centerC22#106×D71
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
FittingC71×#1383C2
socleC22×C71C2×#106
prime radicalC23×D71SD16
cosocleC22×C8×C71C23
hypercenter#1383D71
hypocenterC71#12929
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1383

Complements

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

NH ≅ G/Ntype
2. C2×#17985822C2semidirect
3. C2×#17985821C2semidirect
6. #22586440C2semidirect
7. C71×#1383C2semidirect
19. (C2×D71)⋊C16C4semidirect
40. C2×C71⋊C16C2×C4semidirect
52. C2×C16×C71C2×C4semidirect
134. C2×D71#365semidirect
154. #1383D71semidirect
155. C71#12929semidirect
194. C2×C16C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#106×D712
C2×#179858222
C2×#179858212
#225899502
#225883752
#225864402
C71×#13832
#1292971