Finite Group Databasebeta

← #32999534

Subgroups — #32999534

Order 18688.

306 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 75 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#22612183C211yesmaximal, characteristic
2#22611981C211yesmaximal, characteristic
3#22609301C211yesmaximal, characteristic
4#22607980C211yesmaximal, characteristic
5Q8⋊#11924003C211yesmaximal, characteristic
6#22607895C211yesmaximal, characteristic
7C73×#3395C211yesmaximal, characteristic
8C2×#14556537—21no—
9C2×#14556537—22no—
10#17988967C2211yescharacteristic
11#17988872—21no—
12C4⋊#11924003C2211yescharacteristic
13#17988390—21no—
14C4×#11924003C2211yescharacteristic
15#17988340C2211yescharacteristic
16#17988334—21no—
17#557×C73—21no—
18#494×C73—21no—
19#494×C73C2211yescharacteristic
20#462×C73—22no—
21#462×C73—21no—
22C8×Q8×C73C2211yescharacteristic
23C4×Q16×C73—21no—
24C4×Q16×C73C2211yescharacteristic
25C2×Dic292—21no—
26C2×Dic292—21no—
27#14556591—21no—
28#14556537—42no—
29#14556537—44no—
30C2×#11924003—21no—
31C2×#11924003—21no—
32#14556522—41no—
33C4⋊#10596831—21no—
34#14556515—41no—
35C2×Q16×C73—22no—
36C2×Q16×C73—21no—
37C2×Q16×C73D411yescharacteristic
38C2×Q16×C73D412yes—
39C4⋊Q8×C73—22no—
40C4⋊Q8×C73—21no—
41#125×C73—42no—
42#125×C73—41no—
43C4×Q8×C73—22no—
44C4×Q8×C73—21no—
45C4×Q8×C73—41no—
46C4×Q8×C73D411yescharacteristic
47#107×C73D411yescharacteristic
48C4⋊C8×C73—21no—
49C4⋊C8×C73D411yescharacteristic
50#103×C73—22no—
51#103×C73—21no—
52C4×C8×C73—21no—
53C4×C8×C73C2311yescharacteristic
54C2×Dic146—41no—
55Dic292—42no—
56Dic292—41no—
57Dic292—41no—
58C2×#10596831—41no—
59C4⋊Dic73—41no—
60C4⋊Dic73—41no—
61#11924003—22no—
62#11924003—41no—
63C2×Q8×C73—21no—
64C2×Q8×C73—22no—
65C2×Q8×C73D811yescharacteristic
66Q16×C73—21no—
67Q16×C73—21no—
68Q16×C73—42no—
69Q16×C73—41no—
70Q16×C73—41no—
71Q16×C73—82no—
72C2×C8×C73—21no—
73C2×C8×C73C2×D411yescharacteristic
74C2×C8×C73C2×D411yescharacteristic
75C4⋊C4×C73—21no—
76C4⋊C4×C73—21no—
77C4⋊C4×C73—42no—
78C4⋊C4×C73—42no—
79C4⋊C4×C73—42no—
80C4⋊C4×C73—41no—
81C4⋊C4×C73—41no—
82C4⋊C4×C73—41no—
83C4⋊C4×C73D811yescharacteristic
84C42×C73—21no—
85C42×C73—42no—
86C42×C73—41no—
87C42×C73C2×D411yescharacteristic
88Q8×C73—21no—
89Q8×C73—41no—
90Q8×C73—42no—
91Q8×C73—42no—
92Q8×C73—41no—
93Q8×C73Q3212yescomplemented
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—41no—
97C2×C4×C73—42no—
98C2×C4×C73—41no—
99C2×C4×C73#12011yescharacteristic
100C2×C4×C73C2×D811yescharacteristic
101C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
102C2×Dic73—81no—
103Dic146—81no—
104Dic146—81no—
105C8×C73—21no—
106C8×C73—41no—
107C8×C73#13511yescharacteristic
108C8×C73AGL(1,ℤ/8)11yescharacteristic
109#10596831—81no—
110C22×C73#44711yescharacteristic
111C4×C73—22no—
112C4×C73—21no—
113C4×C73—41no—
114C4×C73—81no—
115C4×C73—82no—
116C4×C73—81no—
117C4×C73#44911yescharacteristic
118C4×C73#45311yescharacteristic
119C4×C73C2×Q3211yescharacteristic
120C4×C73#51011yescharacteristic
121Dic73—161no—
122#12334—731nomaximal, Sylow
123C2×C73#162111yescharacteristic
124C2×C73#218811yescharacteristic
125C2×C73#219511yescharacteristic
126#3395D7311yescharacteristic
127#2247—731no—
128#2209—731no—
129#1670—731no—
130#1362—731no—
131Q8⋊C16—731no—
132#1332—731no—
133C73#1233411yescharacteristic, complemented, Sylow
134#557—21no—
135C2×Q32—1461no—
136C2×Q32—1462no—
137#494—21no—
138#494—731no—
139#494C2×D7311yescharacteristic
140#475—1461no—
141#462—22no—
142#462—21no—
143C8×Q8C2×D7311yescharacteristic
144C4×Q16—21no—
145C4×Q16C2×D7311yescharacteristic
146C4⋊C16—731no—
147#358—1461no—
148C4×C16—731no—
149#335—731no—
150#332—1461no—
151C2×Q16—22no—
152C2×Q16—21no—
153C2×Q16—1461no—
154C2×Q16—1461no—
155C2×Q16C22⋊D7311yescharacteristic
156C2×Q16C22⋊D7312yes—
157C4⋊Q8—22no—
158C4⋊Q8—21no—
159C4⋊Q8—1461no—
160#125—42no—
161#125—41no—
162C4×Q8—22no—
163C4×Q8—21no—
164C4×Q8—41no—
165C4×Q8C22⋊D7311yescharacteristic
166Q32—2922no—
167Q32—2924no—
168C2×C16—1461no—
169C2×C16—1461no—
170#107C22⋊D7311yescharacteristic
171#106—2921no—
172C4⋊C8—21no—
173C4⋊C8—1461no—
174C4⋊C8C22⋊D7311yescharacteristic
175#103—22no—
176#103—21no—
177#103—2921no—
178C4×C8—21no—
179C4×C8C22×D7311yescharacteristic
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—2921no—
183C2×Q8#1192401311yescharacteristic
184Q16—21no—
185Q16—21no—
186Q16—42no—
187Q16—41no—
188Q16—41no—
189Q16—82no—
190Q16—2922no—
191Q16—2921no—
192Q16—2921no—
193C2×C8—21no—
194C2×C8—2921no—
195C2×C8C2×C22⋊D7311yescharacteristic
196C2×C8C2×C22⋊D7311yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—2921no—
206C4⋊C4—2921no—
207C4⋊C4#1192401311yescharacteristic
208C42—21no—
209C42—42no—
210C42—41no—
211C42C2×C22⋊D7311yescharacteristic
212C16—2921no—
213C16—1462no—
214Q8—21no—
215Q8—41no—
216Q8—42no—
217Q8—42no—
218Q8—41no—
219Q8—5841no—
220Q8—5841no—
221Q8#1455653712yescomplemented
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—5841no—
228C2×C4#1455656011yescharacteristic
229C2×C4C2×#1192401311yescharacteristic
230C2×C4#1455661711yescharacteristic
231C8—21no—
232C8—41no—
233C8—5841no—
234C8#1455661111yescharacteristic
235C8#1455661611yescharacteristic
236C22#1798872111yescharacteristic
237C4—22no—
238C4—21no—
239C4—41no—
240C4—81no—
241C4—82no—
242C4—81no—
243C4—11681no—
244C4#1798873211yescharacteristic
245C4#1798875511yescharacteristic
246C4C2×#1455653711yescharacteristic
247C4#1798902811yescharacteristic
248C2#2260899611yescharacteristic
249C2#2261186511yescharacteristic
250C2#2261190911yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#3395C2
socle, prime radicalC22×C73#447
hypercenter#3395D73
hypocenterC73#12334
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3395

Complements

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

NH ≅ G/Ntype
93. Q8×C73Q32semidirect
133. C73#12334semidirect
221. Q8#14556537semidirect

Maximal subgroups

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

subgroupindex
#226121832
#226119812
#226093012
#226079802
Q8⋊#119240032
#226078952
C73×#33952
#1233473