Finite Group Databasebeta

← #32989237

Subgroups — #32989237

Order 18688.

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

normal subgroups only

This group has 542 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#22618184C211yesmaximal, characteristic
2#22610005C212yesmaximal
3#22607835C212yesmaximal
4#22607835C211yesmaximal, characteristic
5C73×#1819C211yesmaximal, characteristic
6#17988499—24no—
7#17988499—22no—
8#17988443C2212yes—
9#17988443C2212yes—
10#17988354—24no—
11C22×C22⋊C4×C73C2211yescharacteristic
12#379×C73C2211yescharacteristic
13#379×C73C2211yescharacteristic
14#342×C73—22no—
15C22×Dic146—22no—
16C2×C22⋊Dic73—22no—
17C2×C22⋊Dic73—22no—
18C2×C22⋊Dic73—22no—
19C2×C22⋊Dic73—22no—
20C2×C22⋊Dic73—22no—
21C2×C22⋊Dic73D414yes—
22#14556498—26no—
23C25×C73C2311yescharacteristic
24C23×C4×C73Q812yescomplemented
25C2×C22⋊C4×C73—22no—
26C2×C22⋊C4×C73—21no—
27C2×C22⋊C4×C73—22no—
28C2×C22⋊C4×C73—21no—
29C2×C22⋊C4×C73—21no—
30C2×C22⋊C4×C73—22no—
31C2×C22⋊C4×C73—22no—
32C2×C22⋊C4×C73D412yes—
33C2×Dic146—84no—
34C2×Dic146—82no—
35C22×Dic73—24no—
36C22×Dic73—42no—
37C22⋊Dic73—44no—
38C22⋊Dic73—44no—
39C22⋊Dic73—82no—
40C22⋊Dic73—82no—
41C22⋊Dic73—82no—
42C22⋊Dic73—82no—
43C24×C73—42no—
44C24×C73—82no—
45C24×C73C2×D412yes—
46C24×C73C2×D411yescharacteristic
47C24×C73C2×Q811yescharacteristic
48C24×C73#4112yes—
49C24×C73#4111yescharacteristic
50C22×C4×C73—22no—
51C22×C4×C73—22no—
52C22×C4×C73—22no—
53C22×C4×C73—22no—
54C22×C4×C73—41no—
55C22×C4×C73—82no—
56C22⋊C4×C73—41no—
57C22⋊C4×C73—42no—
58C22⋊C4×C73—41no—
59C22⋊C4×C73—42no—
60C22⋊C4×C73—42no—
61C22⋊C4×C73—82no—
62C22⋊C4×C73—82no—
63C23×C73—22no—
64C23×C73—24no—
65C23×C73—42no—
66C23×C73—41no—
67C23×C73—42no—
68C23×C73—42no—
69C23×C73—42no—
70C23×C73—41no—
71C23×C73—42no—
72C23×C73—42no—
73C23×C73—42no—
74C23×C73—42no—
75C23×C73—88no—
76C23×C73#12011yescharacteristic
77C23×C73C22⋊Q812yes—
78C23×C73C22⋊Q811yescharacteristic
79C23×C73#12412yes—
80C23×C73#12411yescharacteristic
81C2×C4×C73—44no—
82C2×C4×C73—42no—
83C2×C4×C73—42no—
84C2×C4×C73—82no—
85C2×C4×C73—82no—
86C2×C4×C73—82no—
87C2×C4×C73—81no—
88C2×C4×C73—82no—
89C2×Dic73—44no—
90C2×Dic73—44no—
91C2×Dic73—44no—
92C2×Dic73—82no—
93C2×Dic73—82no—
94C2×Dic73—82no—
95Dic146—88no—
96C22×C73—22no—
97C22×C73—22no—
98C22×C73—21no—
99C22×C73—22no—
100C22×C73—21no—
101C22×C73—22no—
102C22×C73—22no—
103C22×C73—22no—
104C22×C73—22no—
105C22×C73—22no—
106C22×C73—22no—
107C22×C73—44no—
108C22×C73—84no—
109C22×C73—82no—
110C22×C73—82no—
111C22×C73—84no—
112C22×C73#39311yescharacteristic
113C22×C73#45812yes—
114C4×C73—82no—
115C4×C73—82no—
116C4×C73—161no—
117Dic73—84no—
118Dic73—162no—
119#10391—731nomaximal, Sylow
120C2×C73—22no—
121C2×C73—42no—
122C2×C73—41no—
123C2×C73—42no—
124C2×C73—41no—
125C2×C73#203011yescharacteristic
126C2×C73#203212yes—
127#2840—731no—
128#1819—732no—
129#1819D7311yescharacteristic
130#1304—732no—
131#1304—731no—
132C73#1039111yescharacteristic, complemented, Sylow
133C22×C22⋊C4C2×D7311yescharacteristic
134#393—1464no—
135#393—1462no—
136#379—732no—
137#379—732no—
138#379C2×D7311yescharacteristic
139#379C2×D7311yescharacteristic
140#342—22no—
141#342—1464no—
142C25C22×D7311yescharacteristic
143C22×Q8—1462no—
144C23×C4Dic14612yescomplemented
145C2×C22⋊C4—22no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—734no—
153C2×C22⋊C4—1462no—
154C2×C22⋊C4—1462no—
155C2×C22⋊C4—1462no—
156C2×C22⋊C4—1462no—
157C2×C22⋊C4—1462no—
158C2×C22⋊C4C22⋊D7312yes—
159#95—1466no—
160C24—42no—
161C24—82no—
162C24D4×D7312yes—
163C24C2×C22⋊D7311yescharacteristic
164C24C2×Dic14611yescharacteristic
165C24#1192402712yes—
166C24#1192402811yescharacteristic
167C2×Q8—5844no—
168C2×Q8—5842no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—82no—
175C22×C4—2922no—
176C22×C4—1464no—
177C22⋊C4—41no—
178C22⋊C4—42no—
179C22⋊C4—41no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C22⋊C4—2924no—
185C22⋊C4—2924no—
186C22⋊C4—5842no—
187C22⋊C4—5842no—
188C22⋊C4—5842no—
189C22⋊C4—5842no—
190C23—22no—
191C23—24no—
192C23—42no—
193C23—41no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—41no—
202C23—88no—
203C23#1455655911yescharacteristic
204C23#1455656812yes—
205C23#1455657011yescharacteristic
206C23#1455657712yes—
207C23#1455657811yescharacteristic
208Q8—5848no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—2924no—
218C2×C4—2924no—
219C2×C4—2924no—
220C2×C4—5842no—
221C2×C4—5842no—
222C2×C4—5842no—
223C22—22no—
224C22—21no—
225C22—22no—
226C22—22no—
227C22—21no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—44no—
235C22—82no—
236C22—84no—
237C22—84no—
238C22—82no—
239C22#1798849911yescharacteristic
240C22#1798877612yes—
241C4—82no—
242C4—82no—
243C4—161no—
244C4—5844no—
245C4—11682no—
246C2—22no—
247C2—41no—
248C2—42no—
249C2—42no—
250C2—41no—
251C2#2261109811yescharacteristic
252C2#2261110812yes—
subgroupG/H
centerC22#17988499
commutator, prime radical, cosocleC25×C73C23
FrattiniC25C22×D73
FittingC73×#1819C2
socleC22×C73#393
hypercenter#1819D73
hypocenterC73#10391
LCSG ▷ C25×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
24. C23×C4×C73Q8semidirect
132. C73#10391semidirect
144. C23×C4Dic146semidirect

Maximal subgroups

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

subgroupindex
#226181842
#226100052
#226078352
C73×#18192
#1039173