Finite Group Databasebeta

← #32999263

Subgroups — #32999263

Order 18688.

354 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 67 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#22621241C211yesmaximal, characteristic
2#22612116C212yesmaximal, complemented
3#22607939C212yesmaximal, complemented
4#22607871C211yesmaximal, characteristic, complemented
5C73×#2087C211yesmaximal, characteristic, complemented
6#17989137—21no—
7#17988868—22no—
8#17988823—21no—
9#17988823C2212yes—
10#17988783—22no—
11#17988414—22no—
12#17988387—22no—
13C22⋊#11924003C2212yes—
14C2×#107×C73C2211yescharacteristic
15C2×C4⋊C8×C73C2211yescharacteristic
16#389×C73—21no—
17#340×C73—21no—
18#340×C73C2211yescharacteristic
19C2×#11924028—41no—
20C2×D584—21no—
21C2×D584D412yes—
22#14556591—21no—
23#14556588—21no—
24#14556579—41no—
25#14556565—22no—
26#14556565—41no—
27C4×D292—22no—
28C4×D292—41no—
29C2×#11924003—22no—
30#14556525—21no—
31#14556525D412yes—
32#14556513—22no—
33#14556513—22no—
34C22×C8×C73D411yescharacteristic
35C22×C8×C73C2311yescharacteristic
36C2×C4⋊C4×C73—21no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73—41no—
39C2×C42×C73D411yescharacteristic
40#107×C73—22no—
41C4⋊C8×C73—22no—
42#95×C73—41no—
43#11924028—82no—
44#11924028—82no—
45C2×Dic146—41no—
46C2×D292—22no—
47C2×D292—41no—
48C2×D292—41no—
49C2×C22⋊D73—42no—
50C2×C4×D73—42no—
51D584—42no—
52D584—81no—
53C4⋊Dic73—22no—
54C4⋊Dic73—41no—
55#11924005—42no—
56#11924005—42no—
57#11924003—42no—
58C22×C4×C73—41no—
59C22×C4×C73—41no—
60C22×C4×C73D811yescharacteristic
61C22×C4×C73D811yescharacteristic
62C22×C4×C73C2×D411yescharacteristic
63C2×C8×C73—21no—
64C2×C8×C73—41no—
65C2×C8×C73—41no—
66C2×C8×C73C2×D412yes—
67C4⋊C4×C73—22no—
68C4⋊C4×C73—41no—
69C4⋊C4×C73—41no—
70C4⋊C4×C73—41no—
71C4⋊C4×C73—82no—
72C42×C73—21no—
73C42×C73—21no—
74C23×C73C2×D811yescharacteristic
75C22×D73—42no—
76C2×C4×C73—21no—
77C2×C4×C73—21no—
78C2×C4×C73—21no—
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73—41no—
82C2×C4×C73—42no—
83C2×C4×C73—81no—
84C2×C4×C73—81no—
85C2×C4×C73—81no—
86C2×C4×C73—81no—
87C2×C4×C73—81no—
88C2×C4×C73SD3212yes—
89C2×C4×C73#12011yescharacteristic
90C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
91C22⋊D73—82no—
92C22⋊D73—82no—
93C2×Dic73—42no—
94D292—42no—
95D292—42no—
96D292—82no—
97D292—82no—
98C4×D73—84no—
99Dic146—82no—
100C8×C73—22no—
101C8×C73—41no—
102C22×C73—21no—
103C22×C73—21no—
104C22×C73#44711yescharacteristic
105C22×C73C2×SD3211yescharacteristic
106C22×C73#50911yescharacteristic
107C2×D73—82no—
108C2×D73—82no—
109C2×D73—82no—
110C4×C73—21no—
111C4×C73—22no—
112C4×C73—41no—
113C4×C73—82no—
114C4×C73—161no—
115C4×C73#45611yescharacteristic
116C4×C73#45811yescharacteristic
117Dic73—82no—
118#12291—731nomaximal, Sylow
119C2×C73—21no—
120C2×C73—21no—
121C2×C73#161011yescharacteristic
122C2×C73#218711yescharacteristic
123C2×C73#219011yescharacteristic
124D73—162no—
125#3245—731no—
126#2233—732no—
127#2087D7311yescharacteristic, complemented
128#1347—732no—
129#1322—731no—
130C73#1229111yescharacteristic, complemented, Sylow
131#532—1461no—
132#474—1462no—
133#466—732no—
134#466—1461no—
135#459—1462no—
136C2×#107C2×D7311yescharacteristic
137C2×C4⋊C8C2×D7311yescharacteristic
138#389—21no—
139#367—1462no—
140#357—1462no—
141C22⋊C16—732no—
142#340—21no—
143#340C2×D7311yescharacteristic
144C2×#41—2921no—
145C2×D8—732no—
146C2×D8—1461no—
147C22×C8C22⋊D7311yescharacteristic
148C22×C8C22×D7311yescharacteristic
149C4⋊Q8—1461no—
150#127—1461no—
151#124—2921no—
152#121—2921no—
153#121—1462no—
154C4×D4—2921no—
155C4×D4—1462no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C42C22⋊D7311yescharacteristic
160C2×C16—1462no—
161#107—22no—
162#107—732no—
163#107—1461no—
164C4⋊C8—22no—
165#102—1462no—
166#102—1462no—
167#95—41no—
168#41—5842no—
169#41—5842no—
170C2×Q8—2921no—
171C2×D4—2921no—
172C2×D4—2921no—
173C2×D4—2922no—
174C2×D4—1462no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—2922no—
178C22×C4#1192401311yescharacteristic
179C22×C4#1192401311yescharacteristic
180C22×C4C2×C22⋊D7311yescharacteristic
181D8—2922no—
182D8—5841no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8D4×D7312yes—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—82no—
192C4⋊C4—2921no—
193C4⋊C4—1462no—
194C22⋊C4—2922no—
195C22⋊C4—2922no—
196C42—21no—
197C42—21no—
198C16—2922no—
199C23—2922no—
200C23C2×#1192401311yescharacteristic
201Q8—5842no—
202D4—2922no—
203D4—2922no—
204D4—5842no—
205D4—5842no—
206D4—5842no—
207D4—5842no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—2922no—
221C2×C4—5844no—
222C2×C4#1455653412yes—
223C2×C4#1455655911yescharacteristic
224C2×C4#1455661611yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—5842no—
230C22—5842no—
231C22—5842no—
232C22#1798872011yescharacteristic
233C22C2×#1455653411yescharacteristic
234C22#1798902211yescharacteristic
235C4—21no—
236C4—22no—
237C4—41no—
238C4—82no—
239C4—161no—
240C4—5842no—
241C4#1798876911yescharacteristic
242C4#1798877611yescharacteristic
243C2—21no—
244C2—21no—
245C2—11682no—
246C2#2260893311yescharacteristic
247C2#2261185911yescharacteristic
248C2#2261187611yescharacteristic
subgroupG/H
centerC22#17988720
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC73×#2087C2
socleC22×C73#447
prime radical#22621241C2
hypercenter#2087D73
hypocenterC73#12291
LCSG ▷ C22×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2087

Complements

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

NH ≅ G/Ntype
2. #22612116C2semidirect
3. #22607939C2semidirect
4. #22607871C2semidirect
5. C73×#2087C2semidirect
127. #2087D73semidirect
130. C73#12291semidirect

Maximal subgroups

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

subgroupindex
#226212412
#226121162
#226079392
#226078712
C73×#20872
#1229173