Finite Group Databasebeta

← #32999694

Subgroups — #32999694

Order 18688.

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

normal subgroups only

This group has 282 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
2#22612158C211yesmaximal, characteristic
3#22609298C211yesmaximal, characteristic
4#22607985C211yesmaximal, characteristic
5#22607909C211yesmaximal, characteristic
6#22607895C211yesmaximal, characteristic
7C73×#2209C211yesmaximal, characteristic
8#17989316—21no—
9#17988992C2211yescharacteristic
10#17988988—21no—
11#17988896—22no—
12#17988872—21no—
13#17988869—21no—
14C8×Dic146C2211yescharacteristic
15Q16×Dic73—21no—
16#17988391—22no—
17C4×#11924003C2211yescharacteristic
18#17988340C2211yescharacteristic
19#17988333—21no—
20C2×Q32×C73—21no—
21#494×C73C2211yescharacteristic
22C4×Q16×C73C2211yescharacteristic
23C4⋊C16×C73C2211yescharacteristic
24#358×C73—21no—
25#14556591—21no—
26#14556591—41no—
27#14556582—22no—
28#14556582—42no—
29Q8×Dic73—22no—
30C4×Dic146—41no—
31C4×Dic146D411yescharacteristic
32C2×#11924003—21no—
33#14556525—21no—
34#14556525D412yes—
35#14556522—22no—
36#14556522—41no—
37C4⋊#10596831—21no—
38C4⋊#10596831D411yescharacteristic
39#14556520—21no—
40#14556515—22no—
41#14556514—41no—
42C8×Dic73—21no—
43C2×Q16×C73—21no—
44C2×Q16×C73—21no—
45C2×Q16×C73D411yescharacteristic
46C4⋊Q8×C73—21no—
47C4×Q8×C73—21no—
48Q32×C73—42no—
49C2×C16×C73—21no—
50#107×C73D411yescharacteristic
51#103×C73—21no—
52C4×C8×C73C2311yescharacteristic
53C2×Dic146—41no—
54C2×Dic146D811yescharacteristic
55C2×#10596831—21no—
56C2×#10596831—41no—
57C4⋊Dic73—22no—
58C4⋊Dic73—42no—
59C4⋊Dic73—42no—
60C4⋊Dic73—41no—
61C4⋊Dic73D811yescharacteristic
62#11924007—21no—
63#11924007—42no—
64C4×Dic73—21no—
65C4×Dic73—42no—
66#11924003—41no—
67C2×Q8×C73—21no—
68C2×Q8×C73—41no—
69Q16×C73—21no—
70Q16×C73—42no—
71Q16×C73—41no—
72Q16×C73—41no—
73Q16×C73SD1612yes—
74C2×C8×C73C2×D411yescharacteristic
75C2×C8×C73C2×D411yescharacteristic
76C4⋊C4×C73—21no—
77C4⋊C4×C73—41no—
78C4⋊C4×C73—41no—
79C4⋊C4×C73—41no—
80C42×C73—41no—
81C42×C73C2×D411yescharacteristic
82C16×C73—41no—
83Q8×C73—22no—
84Q8×C73—41no—
85Q8×C73—81no—
86Q8×C73—81no—
87C2×C4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73—81no—
90C2×C4×C73#12011yescharacteristic
91C2×C4×C73C2×D811yescharacteristic
92C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
93C2×Dic73—21no—
94C2×Dic73—21no—
95C2×Dic73—42no—
96Dic146—21no—
97Dic146—21no—
98Dic146—81no—
99Dic146—81no—
100C8×C73—21no—
101C8×C73C2×SD1611yescharacteristic
102C8×C73#13711yescharacteristic
103#10596831—41no—
104#10596831—81no—
105C22×C73#44711yescharacteristic
106C4×C73—21no—
107C4×C73—42no—
108C4×C73—81no—
109C4×C73—161no—
110C4×C73#44811yescharacteristic
111C4×C73#45411yescharacteristic
112C4×C73#50811yescharacteristic
113C4×C73#51011yescharacteristic
114Dic73—41no—
115Dic73—41no—
116Dic73—82no—
117#12357—731nomaximal, Sylow
118C2×C73#162711yescharacteristic
119C2×C73#218911yescharacteristic
120C2×C73#219511yescharacteristic
121#3383—731no—
122#2241—731no—
123#2209D7311yescharacteristic
124#1670—731no—
125#1364—731no—
126Q8⋊C16—731no—
127#1332—731no—
128C73#1235711yescharacteristic, complemented, Sylow
129#557—1461no—
130C2×Q32—21no—
131#500—731no—
132#499—1461no—
133#494C2×D7311yescharacteristic
134#479—1462no—
135#475—1461no—
136#475—1461no—
137C8×Q8—731no—
138C4×Q16—1461no—
139C4×Q16C2×D7311yescharacteristic
140C4⋊C16C2×D7311yescharacteristic
141#358—21no—
142#358—1462no—
143C4×C16—731no—
144#335—731no—
145#332—1461no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16C22⋊D7311yescharacteristic
149C4⋊Q8—21no—
150C4⋊Q8—2921no—
151C4⋊Q8—1461no—
152#125—2922no—
153#125—1462no—
154C4×Q8—21no—
155C4×Q8—2921no—
156C4×Q8—731no—
157C4×Q8—1462no—
158Q32—42no—
159C2×C16—21no—
160C2×C16—1461no—
161#107—732no—
162#107—1461no—
163#107C22⋊D7311yescharacteristic
164#106—2921no—
165#106—1462no—
166C4⋊C8—731no—
167C4⋊C8—1461no—
168C4⋊C8—1461no—
169#103—21no—
170#103—2921no—
171#103—1462no—
172C4×C8—1461no—
173C4×C8C22×D7311yescharacteristic
174C2×Q8—21no—
175C2×Q8—41no—
176C2×Q8—2921no—
177C2×Q8—731no—
178Q16—21no—
179Q16—42no—
180Q16—41no—
181Q16—41no—
182Q16#1192401612yes—
183C2×C8—2921no—
184C2×C8—1461no—
185C2×C8D4×D7311yescharacteristic
186C2×C8C2×C22⋊D7311yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—2922no—
192C4⋊C4—2922no—
193C4⋊C4—2922no—
194C4⋊C4—2921no—
195C4⋊C4—731no—
196C4⋊C4—1462no—
197C4⋊C4—1461no—
198C42—41no—
199C42—2922no—
200C42—1461no—
201C42D4×D7311yescharacteristic
202C16—41no—
203C16—2921no—
204Q8—22no—
205Q8—41no—
206Q8—81no—
207Q8—81no—
208Q8—5841no—
209Q8—5841no—
210Q8—1461no—
211Q8—1461no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—81no—
215C2×C4—2922no—
216C2×C4—1461no—
217C2×C4—1461no—
218C2×C4#1455655911yescharacteristic
219C2×C4D8×D7311yescharacteristic
220C2×C4#1455661511yescharacteristic
221C8—21no—
222C8—2921no—
223C8—5841no—
224C8C2×#1192401611yescharacteristic
225C8#1455662111yescharacteristic
226C22#1798871911yescharacteristic
227C4—21no—
228C4—42no—
229C4—81no—
230C4—161no—
231C4—2921no—
232C4—2921no—
233C4—5842no—
234C4#1798872511yescharacteristic
235C4#1798875811yescharacteristic
236C4#1798902011yescharacteristic
237C4#1798902911yescharacteristic
238C2#2260903311yescharacteristic
239C2#2261187011yescharacteristic
240C2#2261190611yescharacteristic
subgroupG/H
centerC22#17988719
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#2209C2
socle, prime radicalC22×C73#447
hypercenter#2209D73
hypocenterC73#12357
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2209

Complements

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

NH ≅ G/Ntype
128. C73#12357semidirect

Maximal subgroups

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

subgroupindex
#226228202
#226121582
#226092982
#226079852
#226079092
#226078952
C73×#22092
#1235773