Finite Group Databasebeta

← #32999583

Subgroups — #32999583

Order 18688.

306 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 71 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#22612156C211yesmaximal, characteristic, complemented
2#22611971C211yesmaximal, characteristic, complemented
3#22609286C211yesmaximal, characteristic
4#22607986C211yesmaximal, characteristic, complemented
5#22607895C211yesmaximal, characteristic, complemented
6D4⋊#11924003C211yesmaximal, characteristic, complemented
7C73×#3342C211yesmaximal, characteristic, complemented
8C2×#14556534—21no—
9#17988964C2211yescharacteristic
10#17988782—21no—
11C4⋊#11924003C2211yescharacteristic
12#17988387—21no—
13#17988387—22no—
14C4×#11924003C2211yescharacteristic
15#17988340C2211yescharacteristic
16#17988330—21no—
17#556×C73—21no—
18#500×C73C2211yescharacteristic
19#485×C73—22no—
20#475×C73—21no—
21#467×C73—21no—
22C4×Q16×C73C2211yescharacteristic
23C8×D4×C73C2211yescharacteristic
24#427×C73—21no—
25C2×D584—21no—
26C2×D584—21no—
27C2×#11924013—41no—
28#14556588—21no—
29#14556534—42no—
30C2×#11924003—21no—
31C2×#11924003—21no—
32C4⋊#10596831—21no—
33#14556511—41no—
34C2×Q16×C73D411yescharacteristic
35C22×C8×C73—21no—
36C4⋊Q8×C73—21no—
37#126×C73—42no—
38#124×C73—41no—
39C22⋊Q8×C73—22no—
40C4×Q8×C73—21no—
41C4×Q8×C73—41no—
42C4×D4×C73D411yescharacteristic
43#117×C73—22no—
44#107×C73—21no—
45#107×C73D411yescharacteristic
46#107×C73D412yes—
47#106×C73—22no—
48C4⋊C8×C73D411yescharacteristic
49#103×C73—22no—
50#103×C73—21no—
51C22⋊C8×C73—21no—
52C4×C8×C73C2311yescharacteristic
53C2×D292—41no—
54C2×D292—41no—
55C2×D292—41no—
56D584—42no—
57D584—41no—
58D584—41no—
59#11924013—82no—
60C2×#10596831—41no—
61#11924003—41no—
62#11924003—41no—
63C2×Q8×C73—21no—
64C2×Q8×C73—41no—
65C2×D4×C73D811yescharacteristic
66C22×C4×C73—21no—
67Q16×C73—21no—
68Q16×C73—21no—
69C2×C8×C73—21no—
70C2×C8×C73—41no—
71C2×C8×C73—41no—
72C2×C8×C73C2×D411yescharacteristic
73C2×C8×C73C2×D411yescharacteristic
74C4⋊C4×C73—21no—
75C4⋊C4×C73—22no—
76C4⋊C4×C73—42no—
77C4⋊C4×C73—41no—
78C4⋊C4×C73—41no—
79C4⋊C4×C73D811yescharacteristic
80C22⋊C4×C73—21no—
81C22⋊C4×C73—42no—
82C22⋊C4×C73—42no—
83C42×C73—42no—
84C42×C73—41no—
85C42×C73C2×D411yescharacteristic
86C23×C73—21no—
87C22×D73—81no—
88Q8×C73—41no—
89Q8×C73—41no—
90Q8×C73—81no—
91Q8×C73—81no—
92D4×C73—21no—
93D4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—41no—
96C2×C4×C73—41no—
97C2×C4×C73—41no—
98C2×C4×C73—42no—
99C2×C4×C73—41no—
100C2×C4×C73#12011yescharacteristic
101C2×C4×C73C2×D811yescharacteristic
102C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
103D292—81no—
104D292—81no—
105D292—81no—
106D292—81no—
107D292—161no—
108C8×C73—21no—
109C8×C73—41no—
110C8×C73#13511yescharacteristic
111C8×C73AGL(1,ℤ/8)11yescharacteristic
112#10596831—81no—
113C22×C73—41no—
114C22×C73—41no—
115C22×C73—41no—
116C22×C73#44711yescharacteristic
117C2×D73—161no—
118C2×D73—161no—
119C2×D73—161no—
120C4×C73—21no—
121C4×C73—41no—
122C4×C73—81no—
123C4×C73—82no—
124C4×C73—81no—
125C4×C73#44911yescharacteristic
126C4×C73#45411yescharacteristic
127C4×C73#50811yescharacteristic
128C4×C73#50911yescharacteristic
129#12341—731nomaximal, Sylow
130C2×C73—81no—
131C2×C73#162511yescharacteristic
132C2×C73#218611yescharacteristic
133C2×C73#219311yescharacteristic
134D73—321no—
135#3342D7311yescharacteristic, complemented
136#2240—731no—
137#2207—731no—
138#1667—731no—
139#1364—731no—
140#1332—731no—
141D4⋊C16—731no—
142C73#1234111yescharacteristic, complemented, Sylow
143#556—21no—
144C2×SD32—1461no—
145#500C2×D7311yescharacteristic
146#493—731no—
147#485—22no—
148#475—21no—
149#467—21no—
150#459—1461no—
151C4×Q16C2×D7311yescharacteristic
152C8×D4C2×D7311yescharacteristic
153#427—21no—
154C4⋊C16—731no—
155#357—1461no—
156#357—1462no—
157C4×C16—731no—
158#335—731no—
159#331—1461no—
160C2×Q16C22⋊D7311yescharacteristic
161C2×D8—2921no—
162C2×D8—1461no—
163C2×D8—1461no—
164C22×C8—21no—
165C4⋊Q8—21no—
166#127—1461no—
167#126—42no—
168#124—41no—
169C22⋊Q8—22no—
170C4×Q8—21no—
171C4×Q8—41no—
172C4×D4C22⋊D7311yescharacteristic
173#117—22no—
174SD32—2922no—
175C2×C16—1461no—
176C2×C16—1461no—
177#107—21no—
178#107C22⋊D7311yescharacteristic
179#107C22⋊D7312yes—
180#106—22no—
181C4⋊C8—1461no—
182C4⋊C8C22⋊D7311yescharacteristic
183#103—22no—
184#103—21no—
185#102—2921no—
186C22⋊C8—21no—
187C4×C8C22×D7311yescharacteristic
188C2×Q8—21no—
189C2×Q8—41no—
190C2×D4—2921no—
191C2×D4—2921no—
192C2×D4—2921no—
193C2×D4#1192401311yescharacteristic
194C22×C4—21no—
195Q16—21no—
196Q16—21no—
197D8—2922no—
198D8—2921no—
199D8—2921no—
200D8—5842no—
201C2×C8—21no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—2921no—
205C2×C8C2×C22⋊D7311yescharacteristic
206C2×C8C2×C22⋊D7311yescharacteristic
207C4⋊C4—21no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4#1192401311yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—42no—
217C42—41no—
218C42C2×C22⋊D7311yescharacteristic
219C16—2921no—
220C16—2921no—
221C23—21no—
222C23—5841no—
223Q8—41no—
224Q8—41no—
225Q8—81no—
226Q8—81no—
227D4—21no—
228D4—21no—
229D4—5841no—
230D4—5841no—
231D4—5841no—
232D4—5841no—
233D4—11681no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4#1455656011yescharacteristic
241C2×C4C2×#1192401311yescharacteristic
242C2×C4#1455661711yescharacteristic
243C8—21no—
244C8—41no—
245C8—5841no—
246C8#1455661111yescharacteristic
247C8#1455661611yescharacteristic
248C22—41no—
249C22—41no—
250C22—41no—
251C22—11681no—
252C22—11681no—
253C22—11681no—
254C22#1798872111yescharacteristic
255C4—21no—
256C4—41no—
257C4—81no—
258C4—82no—
259C4—81no—
260C4#1798873211yescharacteristic
261C4#1798876011yescharacteristic
262C4#1798901711yescharacteristic
263C4#1798902211yescharacteristic
264C2—81no—
265C2—23361no—
266C2#2260902011yescharacteristic
267C2#2261185411yescharacteristic
268C2#2261189511yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#3342C2
socleC22×C73#447
prime radical#22609286C2
hypercenter#3342D73
hypocenterC73#12341
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #3342

Complements

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

NH ≅ G/Ntype
1. #22612156C2semidirect
2. #22611971C2semidirect
4. #22607986C2semidirect
5. #22607895C2semidirect
6. D4⋊#11924003C2semidirect
7. C73×#3342C2semidirect
135. #3342D73semidirect
142. C73#12341semidirect

Maximal subgroups

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

subgroupindex
#226121562
#226119712
#226092862
#226079862
#226078952
D4⋊#119240032
C73×#33422
#1234173