Finite Group Databasebeta

← #32998442

Subgroups — #32998442

Order 18688.

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

normal subgroups only

This group has 398 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#22611847C212yesmaximal, complemented
2#22611254C211yesmaximal, characteristic, complemented
3#22607940C212yesmaximal, complemented
4#22607871C211yesmaximal, characteristic, complemented
5C73×#3245C211yesmaximal, characteristic
6C22×Dic292C2211yescharacteristic
7C2×#14556535—22no—
8C2×#14556516—21no—
9#17988493—21no—
10#17988390—22no—
11C22⋊#11924003C2212yes—
12#17988350C2211yescharacteristic
13#532×C73—21no—
14#474×C73—22no—
15#466×C73—21no—
16#466×C73C2212yes—
17#459×C73—22no—
18C2×C4⋊C8×C73C2211yescharacteristic
19C22×Dic146—21no—
20C2×Dic292—22no—
21C2×Dic292—42no—
22C2×C4⋊Dic73—21no—
23#14556535—44no—
24C2×#11924003—22no—
25#14556516—42no—
26#14556498—41no—
27#14556498—41no—
28C2×#41×C73—41no—
29C2×D8×C73—21no—
30C2×D8×C73D412yes—
31C22×C8×C73D411yescharacteristic
32C22×C8×C73C2311yescharacteristic
33C4⋊Q8×C73—21no—
34#127×C73—21no—
35#124×C73—41no—
36#121×C73—22no—
37#121×C73—41no—
38C4×D4×C73—22no—
39C4×D4×C73—41no—
40C2×C42×C73D411yescharacteristic
41#107×C73—21no—
42#107×C73D412yes—
43C4⋊C8×C73—22no—
44#102×C73—22no—
45#102×C73—22no—
46C2×Dic146—22no—
47C2×Dic146—41no—
48C2×Dic146—41no—
49C2×Dic146—41no—
50C2×Dic146—41no—
51C2×Dic146—41no—
52C22×Dic73—41no—
53C22×Dic73—41no—
54Dic292—24no—
55Dic292—42no—
56C4⋊Dic73—41no—
57C4⋊Dic73—41no—
58#11924003—42no—
59#41×C73—82no—
60#41×C73—82no—
61C2×Q8×C73—41no—
62C2×D4×C73—22no—
63C2×D4×C73—41no—
64C2×D4×C73—41no—
65C2×D4×C73—42no—
66C22×C4×C73—42no—
67C22×C4×C73D811yescharacteristic
68C22×C4×C73D811yescharacteristic
69C22×C4×C73C2×D411yescharacteristic
70D8×C73—42no—
71D8×C73—81no—
72C2×C8×C73—21no—
73C2×C8×C73—41no—
74C2×C8×C73—41no—
75C2×C8×C73C2×D412yes—
76C4⋊C4×C73—22no—
77C4⋊C4×C73—41no—
78C22⋊C4×C73—42no—
79C22⋊C4×C73—42no—
80C42×C73—21no—
81C42×C73—21no—
82C23×C73—42no—
83C23×C73C2×D811yescharacteristic
84Q8×C73—82no—
85D4×C73—42no—
86D4×C73—42no—
87D4×C73—82no—
88D4×C73—82no—
89D4×C73—82no—
90D4×C73—82no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—41no—
97C2×C4×C73—42no—
98C2×C4×C73—84no—
99C2×C4×C73SD3212yescomplemented
100C2×C4×C73#12011yescharacteristic
101C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
102C2×Dic73—42no—
103C2×Dic73—81no—
104C2×Dic73—81no—
105C2×Dic73—81no—
106C2×Dic73—81no—
107C2×Dic73—81no—
108Dic146—42no—
109Dic146—42no—
110Dic146—81no—
111Dic146—81no—
112C8×C73—22no—
113C8×C73—41no—
114C22×C73—21no—
115C22×C73—21no—
116C22×C73—82no—
117C22×C73—82no—
118C22×C73—82no—
119C22×C73#44711yescharacteristic
120C22×C73C2×SD3211yescharacteristic
121C22×C73#50911yescharacteristic
122C4×C73—21no—
123C4×C73—22no—
124C4×C73—41no—
125C4×C73—82no—
126C4×C73#45411yescharacteristic
127C4×C73B(4,𝔽2)11yescharacteristic
128Dic73—82no—
129Dic73—161no—
130#12161—731nomaximal, Sylow
131C2×C73—21no—
132C2×C73—21no—
133C2×C73—162no—
134C2×C73#159811yescharacteristic
135C2×C73#218711yescharacteristic
136C2×C73#219111yescharacteristic
137#3245D7311yescharacteristic
138#2185—732no—
139#2056—731no—
140#1347—732no—
141#1322—731no—
142C73#1216111yescharacteristic, complemented, Sylow
143C22×Q16—731no—
144#532—21no—
145C2×SD32—1462no—
146#474—22no—
147#466—21no—
148#466C2×D7312yes—
149#459—22no—
150C2×C4⋊C8C2×D7311yescharacteristic
151C2×#103—1461no—
152#391—1461no—
153#358—1462no—
154C22⋊C16—732no—
155#340—731no—
156C2×#41—41no—
157C22×Q8—1461no—
158C2×Q16—2922no—
159C2×Q16—1462no—
160C2×D8—21no—
161C2×D8C22⋊D7312yes—
162C22×C8C22⋊D7311yescharacteristic
163C22×C8C22×D7311yescharacteristic
164C4⋊Q8—21no—
165#127—21no—
166#124—41no—
167#121—22no—
168#121—41no—
169C4×D4—22no—
170C4×D4—41no—
171C2×C4⋊C4—1461no—
172C2×C42C22⋊D7311yescharacteristic
173SD32—2924no—
174C2×C16—1462no—
175#107—21no—
176#107C22⋊D7312yes—
177C4⋊C8—22no—
178#103—2922no—
179#102—22no—
180#102—22no—
181#95—2921no—
182#95—2921no—
183#41—82no—
184#41—82no—
185C2×Q8—41no—
186C2×Q8—2921no—
187C2×Q8—2921no—
188C2×Q8—2921no—
189C2×Q8—2921no—
190C2×Q8—2921no—
191C2×Q8—1462no—
192C2×D4—22no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—42no—
196C22×C4—42no—
197C22×C4—2921no—
198C22×C4—2921no—
199C22×C4#1192401311yescharacteristic
200C22×C4#1192401311yescharacteristic
201C22×C4C2×C22⋊D7311yescharacteristic
202Q16—2922no—
203Q16—1464no—
204D8—42no—
205D8—81no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8C2×C22⋊D7312yes—
210C4⋊C4—22no—
211C4⋊C4—41no—
212C4⋊C4—2921no—
213C4⋊C4—2921no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—21no—
217C42—21no—
218C16—2922no—
219C23—42no—
220C23C2×#1192401311yescharacteristic
221Q8—82no—
222Q8—2922no—
223Q8—2922no—
224Q8—5841no—
225Q8—5841no—
226D4—42no—
227D4—42no—
228D4—82no—
229D4—82no—
230D4—82no—
231D4—82no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—84no—
240C2×C4—2922no—
241C2×C4—5841no—
242C2×C4—5841no—
243C2×C4—5841no—
244C2×C4—5841no—
245C2×C4—5841no—
246C2×C4#1455653512yescomplemented
247C2×C4#1455656011yescharacteristic
248C2×C4#1455661711yescharacteristic
249C8—22no—
250C8—41no—
251C22—21no—
252C22—21no—
253C22—82no—
254C22—82no—
255C22—82no—
256C22#1798872111yescharacteristic
257C22C2×#1455653511yescharacteristic
258C22#1798902311yescharacteristic
259C4—22no—
260C4—21no—
261C4—41no—
262C4—82no—
263C4—5842no—
264C4—11681no—
265C4#1798876011yescharacteristic
266C4#1798877411yescharacteristic
267C2—21no—
268C2—21no—
269C2—162no—
270C2#2260886611yescharacteristic
271C2#2261186011yescharacteristic
272C2#2261188211yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
Fitting, prime radicalC73×#3245C2
socleC22×C73#447
hypercenter#3245D73
hypocenterC73#12161
LCSG ▷ C22×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #22611847C2semidirect
2. #22611254C2semidirect
3. #22607940C2semidirect
4. #22607871C2semidirect
99. C2×C4×C73SD32semidirect
142. C73#12161semidirect
246. C2×C4#14556535semidirect

Maximal subgroups

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

subgroupindex
#226118472
#226112542
#226079402
#226078712
C73×#32452
#1216173