Finite Group Databasebeta

← #31915331

Subgroups — #31915331

Order 18176.

356 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 97 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×#17985821C211yesmaximal, characteristic, complemented
2C2×#17985813C211yesmaximal, characteristic
3#22590055C211yesmaximal, characteristic
4#22589214C211yesmaximal, characteristic
5#22588226C211yesmaximal, characteristic, complemented
6#22586419C211yesmaximal, characteristic, complemented
7C71×#1386C211yesmaximal, characteristic, complemented
8C22×D568C2211yescharacteristic
9C2×#14556068C2211yescharacteristic
10C2×#14556064C2211yescharacteristic
11C2×#14556048—21no—
12#17985955—21no—
13C2×#14556038C2211yescharacteristic
14#17985900—21no—
15#17985821—22no—
16#17985813—22no—
17#17985799C414yescomplemented
18#17985758—21no—
19C2×M32×C71C2211yescharacteristic
20#427×C71C2211yescharacteristic
21C2×#106×C71C2211yescharacteristic
22C22×D284—21no—
23C2×D568—21no—
24C2×D568—21no—
25C2×D568—21no—
26C2×D568—21no—
27C2×D568—21no—
28C2×D568C2×C412yes—
29C2×#11765849—21no—
30C22×C71⋊C8D411yescharacteristic
31C2×(C2×D71)⋊C4—42no—
32C2×C4×Dic71D411yescharacteristic
33#14556068—21no—
34#14556068—21no—
35C2×C71⋊C16C2×C412yes—
36#14556064—21no—
37#14556064D412yes—
38#14556048—42no—
39#14556039—41no—
40#14556038—21no—
41#14556038—21no—
42C4×C71⋊C8—41no—
43C22×C8×C71C2311yescharacteristic
44#117×C71—21no—
45C2×C4⋊C4×C71—21no—
46M32×C71—21no—
47M32×C71—21no—
48C2×C16×C71—21no—
49#107×C71C2×C412yes—
50#106×C71—21no—
51#106×C71—21no—
52#106×C71—21no—
53C23×D71—41no—
54C2×D284—22no—
55C2×D284—41no—
56C2×D284—41no—
57C2×D284—41no—
58C2×D284—41no—
59C2×D284—41no—
60C22×Dic71—21no—
61D568—22no—
62D568—22no—
63D568—41no—
64D568—41no—
65#11765849—22no—
66#11765849—41no—
67C2×C71⋊C8—21no—
68C2×C71⋊C8—21no—
69C2×C71⋊C8—41no—
70C2×C71⋊C8SD1612yes—
71(C2×D71)⋊C4—84no—
72C4×Dic71—21no—
73C4×Dic71—21no—
74C71⋊C16—22no—
75C22×C4×C71—41no—
76C22×C4×C71C2×D411yescharacteristic
77C2×C8×C71C22×C411yescharacteristic
78C2×C8×C71C2×D411yescharacteristic
79C2×C8×C71#4111yescharacteristic
80C2×C8×C71#4111yescharacteristic
81C2×C8×C71#4111yescharacteristic
82C2×C8×C71#4111yescharacteristic
83C4⋊C4×C71—22no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—41no—
86C22⋊C4×C71—41no—
87C42×C71—41no—
88C16×C71—41no—
89C23×C71AGL(1,ℤ/8)11yescharacteristic
90C22×D71—42no—
91C22×D71—82no—
92C22×D71—82no—
93C22×D71—81no—
94C22×D71—81no—
95C2×C4×C71—42no—
96C2×C4×C71—82no—
97C2×C4×C71—81no—
98C2×C4×C71C4×D411yescharacteristic
99C2×C4×C71#12111yescharacteristic
100C2×C4×C71#12311yescharacteristic
101C2×C4×C71C2×SD1611yescharacteristic
102C2×C4×C71#13511yescharacteristic
103C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
104C2×Dic71—42no—
105C2×Dic71—41no—
106D284—42no—
107D284—42no—
108D284—81no—
109D284—81no—
110C8×C71#11711yescharacteristic
111C8×C71C4×D411yescharacteristic
112C8×C71#12411yescharacteristic
113C8×C71#12611yescharacteristic
114C71⋊C8—22no—
115C71⋊C8—42no—
116C22×C71—22no—
117C22×C71#44211yescharacteristic
118C22×C71#46011yescharacteristic
119C22×C71#48211yescharacteristic
120C2×D71—84no—
121C2×D71—82no—
122C2×D71—162no—
123C2×D71—161no—
124C2×D71—161no—
125C4×C71—82no—
126C4×C71—161no—
127C4×C71#38811yescharacteristic
128C4×C71C4×SD1611yescharacteristic
129C4×C71#46311yescharacteristic
130C4×C71#48111yescharacteristic
131Dic71—81no—
132#13142—711nomaximal, Sylow
133C2×C71—21no—
134C2×C71—21no—
135C2×C71#192611yescharacteristic
136C2×C71AGL(1,ℤ/16)12yes—
137D71—162no—
138C2×#365—711no—
139C2×#361—711no—
140#2140—711no—
141#1972—711no—
142#1773—711no—
143#1386D7111yescharacteristic, complemented
144#1375—711no—
145C71#1314211yescharacteristic, complemented, Sylow
146C22×D8—711no—
147C2×M32—711no—
148C2×M32C2×D7111yescharacteristic
149C2×#108—711no—
150#427C2×D7111yescharacteristic
151C2×#106C2×D7111yescharacteristic
152C2×#102—1421no—
153#405—1421no—
154C2×#97—711no—
155#390—1421no—
156#365—1422no—
157#361—1422no—
158#357—714no—
159#339—1421no—
160C22×D4—1421no—
161C2×D8—712no—
162C2×D8—1421no—
163C2×D8—1421no—
164C2×D8—1421no—
165C2×D8—1421no—
166C2×D8—1421no—
167C2×M16—1421no—
168C22×C8—711no—
169C22×C8C22×D7111yescharacteristic
170#117—21no—
171C2×C4⋊C4—21no—
172C2×C22⋊C4—2842no—
173C2×C42—711no—
174M32—21no—
175M32—21no—
176M32—1421no—
177M32—1421no—
178C2×C16—21no—
179C2×C16—712no—
180#108—712no—
181#108—1421no—
182#107C4×D7112yes—
183#106—21no—
184#106—21no—
185#106—21no—
186#102—2842no—
187#97—2841no—
188#97—1421no—
189#97—1421no—
190C4×C8—2841no—
191C24—2841no—
192C2×D4—2841no—
193C2×D4—2841no—
194C2×D4—2841no—
195C2×D4—2841no—
196C2×D4—2841no—
197C2×D4—1422no—
198C22×C4—41no—
199C22×C4—1421no—
200C22×C4D4×D7111yescharacteristic
201D8—2841no—
202D8—2841no—
203D8—1422no—
204D8—1422no—
205M16—2841no—
206M16—1422no—
207C2×C8—2841no—
208C2×C8—712no—
209C2×C8—1421no—
210C2×C8—1421no—
211C2×C8C2×C4×D7111yescharacteristic
212C2×C8D4×D7111yescharacteristic
213C2×C8#1176586411yescharacteristic
214C2×C8#1176586511yescharacteristic
215C2×C8#1176586511yescharacteristic
216C2×C8#1176586611yescharacteristic
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—5684no—
222C42—41no—
223C42—1421no—
224C42—1421no—
225C16—41no—
226C16—1422no—
227C23—2842no—
228C23—5682no—
229C23—5682no—
230C23—5681no—
231C23—5681no—
232C23#1455615211yescharacteristic
233D4—2842no—
234D4—2842no—
235D4—5681no—
236D4—5681no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—81no—
240C2×C4—2842no—
241C2×C4—2841no—
242C2×C4(C22⋊D71)⋊C411yescharacteristic
243C2×C4#1455610111yescharacteristic
244C2×C4#1455611311yescharacteristic
245C2×C4SD16×D7111yescharacteristic
246C2×C4#1455615111yescharacteristic
247C2×C4#1455615511yescharacteristic
248C8—2842no—
249C8—1422no—
250C8#1455608511yescharacteristic
251C8#1455609011yescharacteristic
252C8#1455611411yescharacteristic
253C8#1455612311yescharacteristic
254C22—22no—
255C22—5684no—
256C22—5682no—
257C22—11361no—
258C22—11361no—
259C22—11362no—
260C22#1798610811yescharacteristic
261C22#1798619811yescharacteristic
262C22#1798632611yescharacteristic
263C4—82no—
264C4—161no—
265C4—5681no—
266C4#1798588811yescharacteristic
267C4#1798608411yescharacteristic
268C4#1798621711yescharacteristic
269C4#1798631911yescharacteristic
270C2—21no—
271C2—21no—
272C2—11362no—
273C2#2258896511yescharacteristic
274C2#2259023512yes—
subgroupG/H
centerC22#17986108
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
FittingC71×#1386C2
socleC22×C71#442
prime radicalC22×D568C22
cosocleC22×C8×C71C23
hypercenter#1386D71
hypocenterC71#13142
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
1. C2×#17985821C2semidirect
5. #22588226C2semidirect
6. #22586419C2semidirect
7. C71×#1386C2semidirect
17. #17985799C4semidirect
143. #1386D71semidirect
145. C71#13142semidirect

Maximal subgroups

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

subgroupindex
C2×#179858212
C2×#179858132
#225900552
#225892142
#225882262
#225864192
C71×#13862
#1314271