Finite Group Databasebeta

← #32999670

Subgroups — #32999670

Order 18688.

298 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 69 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#22622820C211yesmaximal, characteristic, complemented
2#22612155C211yesmaximal, characteristic
3#22611971C211yesmaximal, characteristic
4#22609292C211yesmaximal, characteristic
5#22607987C211yesmaximal, characteristic, complemented
6#22607911C211yesmaximal, characteristic, complemented
7C73×#1332C211yesmaximal, characteristic, complemented
8#17989316—21no—
9C2×#14556534—21no—
10#17988992C2211yescharacteristic
11#17988988—21no—
12#17988964C2211yescharacteristic
13#17988896—22no—
14#17988872—21no—
15#17988788—21no—
16C8×Dic146C2211yescharacteristic
17Q16×Dic73—21no—
18C4⋊#11924003C2211yescharacteristic
19#17988389—22no—
20#17988387—21no—
21#17988330—21no—
22C4×Q16×C73C2211yescharacteristic
23C4×C16×C73C2211yescharacteristic
24#335×C73C2211yescharacteristic
25C2×#11924017—41no—
26C2×D584—21no—
27C2×D584—21no—
28#14556591—21no—
29#14556591—41no—
30#14556588—21no—
31#14556582—22no—
32#14556582—42no—
33Q8×Dic73—22no—
34C4×Dic146—41no—
35C4×Dic146D411yescharacteristic
36#14556534—42no—
37C2×#11924003—21no—
38#14556525—21no—
39#14556525D412yes—
40#14556522—22no—
41C4⋊#10596831D411yescharacteristic
42#14556520—21no—
43#14556515—22no—
44#14556513—41no—
45C8×Dic73—21no—
46C2×Q16×C73D411yescharacteristic
47C4×Q8×C73—21no—
48C2×C16×C73—21no—
49#107×C73D411yescharacteristic
50C4⋊C8×C73—21no—
51#103×C73—21no—
52C4×C8×C73C2311yescharacteristic
53C2×Dic146—41no—
54C2×Dic146D811yescharacteristic
55C2×D292—41no—
56C2×D292—41no—
57C2×D292—41no—
58#11924017—82no—
59D584—42no—
60D584—41no—
61D584—41no—
62C2×#10596831—21no—
63C4⋊Dic73—22no—
64C4⋊Dic73—42no—
65C4⋊Dic73—42no—
66C4⋊Dic73—41no—
67C4⋊Dic73D811yescharacteristic
68#11924007—21no—
69#11924007—42no—
70C4×Dic73—21no—
71C4×Dic73—42no—
72#11924003—41no—
73C2×Q8×C73—21no—
74Q16×C73—21no—
75Q16×C73SD1612yescomplemented
76C2×C8×C73—41no—
77C2×C8×C73C2×D411yescharacteristic
78C2×C8×C73C2×D411yescharacteristic
79C4⋊C4×C73—21no—
80C4⋊C4×C73—41no—
81C42×C73—41no—
82C42×C73C2×D411yescharacteristic
83C16×C73—41no—
84C22×D73—81no—
85Q8×C73—22no—
86Q8×C73—41no—
87C2×C4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73#12011yescharacteristic
90C2×C4×C73C2×D811yescharacteristic
91C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
92C2×Dic73—21no—
93C2×Dic73—21no—
94C2×Dic73—42no—
95D292—81no—
96D292—81no—
97D292—81no—
98D292—81no—
99D292—161no—
100Dic146—21no—
101Dic146—21no—
102Dic146—81no—
103Dic146—81no—
104C8×C73—21no—
105C8×C73—81no—
106C8×C73C2×SD1611yescharacteristic
107C8×C73#13711yescharacteristic
108#10596831—41no—
109C22×C73#44711yescharacteristic
110C2×D73—161no—
111C2×D73—161no—
112C2×D73—161no—
113C4×C73—21no—
114C4×C73—42no—
115C4×C73—81no—
116C4×C73#44811yescharacteristic
117C4×C73#45411yescharacteristic
118C4×C73#50811yescharacteristic
119C4×C73#50911yescharacteristic
120Dic73—41no—
121Dic73—41no—
122Dic73—82no—
123#12353—731nomaximal, Sylow
124C2×C73#162711yescharacteristic
125C2×C73#218611yescharacteristic
126C2×C73#219311yescharacteristic
127D73—321no—
128#3383—731no—
129#2240—731no—
130#2207—731no—
131#1668—731no—
132#1364—731no—
133Q8⋊C16—731no—
134#1332D7311yescharacteristic, complemented
135C73#1235311yescharacteristic, complemented, Sylow
136#557—1461no—
137C2×SD32—1461no—
138#500—731no—
139#499—1461no—
140#493—731no—
141#479—1462no—
142#475—1461no—
143#460—1461no—
144C8×Q8—731no—
145C4×Q16—1461no—
146C4×Q16C2×D7311yescharacteristic
147C4⋊C16—731no—
148#357—1462no—
149#357—1461no—
150C4×C16C2×D7311yescharacteristic
151#335C2×D7311yescharacteristic
152#331—1461no—
153C2×Q16D29211yescharacteristic
154C2×SD16—2921no—
155C2×D8—1461no—
156C2×D8—1461no—
157C4⋊Q8—2921no—
158C4⋊Q8—1461no—
159#127—1461no—
160#125—2922no—
161#125—1462no—
162C4×Q8—21no—
163C4×Q8—2921no—
164C4×Q8—731no—
165C4×Q8—1462no—
166SD32—2922no—
167C2×C16—21no—
168C2×C16—1461no—
169#107—732no—
170#107—1461no—
171#107D29211yescharacteristic
172#106—1462no—
173C4⋊C8—21no—
174C4⋊C8—731no—
175C4⋊C8—1461no—
176#103—21no—
177#103—1462no—
178#102—2921no—
179C4×C8—1461no—
180C4×C8C22×D7311yescharacteristic
181C2×Q8—21no—
182C2×Q8—2921no—
183C2×Q8—731no—
184C2×D4—2921no—
185C2×D4—2921no—
186C2×D4—2921no—
187Q16—21no—
188Q16#1192401712yescomplemented
189SD16—5842no—
190D8—2922no—
191D8—2921no—
192D8—2921no—
193C2×C8—41no—
194C2×C8—1461no—
195C2×C8D4×D7311yescharacteristic
196C2×C8C2×D29211yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—41no—
199C4⋊C4—2922no—
200C4⋊C4—2922no—
201C4⋊C4—2922no—
202C4⋊C4—2921no—
203C4⋊C4—731no—
204C4⋊C4—1462no—
205C4⋊C4—1461no—
206C42—41no—
207C42—2922no—
208C42—1461no—
209C42D4×D7311yescharacteristic
210C16—41no—
211C16—2921no—
212C23—5841no—
213Q8—22no—
214Q8—41no—
215Q8—5841no—
216Q8—5841no—
217Q8—1461no—
218Q8—1461no—
219D4—5841no—
220D4—5841no—
221D4—5841no—
222D4—5841no—
223D4—11681no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—2922no—
227C2×C4—1461no—
228C2×C4—1461no—
229C2×C4#1455656111yescharacteristic
230C2×C4D8×D7311yescharacteristic
231C2×C4#1455661811yescharacteristic
232C8—21no—
233C8—81no—
234C8—2921no—
235C8C2×#1192401711yescharacteristic
236C8#1455662411yescharacteristic
237C22—11681no—
238C22—11681no—
239C22—11681no—
240C22#1798872211yescharacteristic
241C4—21no—
242C4—42no—
243C4—81no—
244C4—2921no—
245C4—2921no—
246C4—5842no—
247C4#1798872811yescharacteristic
248C4#1798876111yescharacteristic
249C4#1798902011yescharacteristic
250C4#1798902411yescharacteristic
251C2—23361no—
252C2#2260903711yescharacteristic
253C2#2261185711yescharacteristic
254C2#2261189611yescharacteristic
subgroupG/H
centerC22#17988722
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#1332C2
socleC22×C73#447
prime radical#17988964C22
hypercenter#1332D73
hypocenterC73#12353
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1332

Complements

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

NH ≅ G/Ntype
1. #22622820C2semidirect
5. #22607987C2semidirect
6. #22607911C2semidirect
7. C73×#1332C2semidirect
75. Q16×C73SD16semidirect
134. #1332D73semidirect
135. C73#12353semidirect
188. Q16#11924017semidirect

Maximal subgroups

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

subgroupindex
#226228202
#226121552
#226119712
#226092922
#226079872
#226079112
C73×#13322
#1235373