Finite Group Databasebeta

← #34005193

Subgroups — #34005193

Order 18688.

530 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 112 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#22625822C212yesmaximal
2#22624769C214yesmaximal
3C2×#17988452C211yesmaximal, characteristic
4C2×C22⋊#10596833C2214yes—
5#17989453—22no—
6#17989453—22no—
7C22×C4⋊Dic73C411yescharacteristic
8C22×C4×Dic73C2211yescharacteristic
9#17988452—22no—
10#17988452—22no—
11C2×#14556498C412yes—
12C2×#14556498C2212yes—
13C22×C4⋊C4×C73C411yescharacteristic
14C22×#10596833—24no—
15C22⋊#10596833—28no—
16C23×Dic73C2×C411yescharacteristic
17C23×Dic73C2×C412yes—
18C23×Dic73C2311yescharacteristic
19C2×C4⋊Dic73—21no—
20C2×C4⋊Dic73—21no—
21C2×C4⋊Dic73—42no—
22C2×C4×Dic73—24no—
23C2×C4×Dic73D412yes—
24C2×C4×Dic73D412yes—
25#14556498—24no—
26#14556498—24no—
27C23×C4×C73C2×C411yescharacteristic
28C23×C4×C73C2×C412yes—
29C2×C4⋊C4×C73—21no—
30C2×C4⋊C4×C73—21no—
31C2×C4⋊C4×C73—42no—
32C2×#10596833—44no—
33C2×#10596833—44no—
34C2×#10596833—44no—
35C22×Dic73—22no—
36C22×Dic73—22no—
37C22×Dic73—22no—
38C22×Dic73—22no—
39C22×Dic73—24no—
40C22×Dic73—41no—
41C22×Dic73—41no—
42C22×Dic73—42no—
43C22×Dic73—42no—
44C22×Dic73C22⋊C412yes—
45C22×Dic73C2×D412yes—
46C22×Dic73#4114yes—
47C4⋊Dic73—42no—
48C4⋊Dic73—42no—
49C4×Dic73—24no—
50C4×Dic73—24no—
51C24×C73C22×C411yescharacteristic
52C22×C4×C73—22no—
53C22×C4×C73—22no—
54C22×C4×C73—22no—
55C22×C4×C73—22no—
56C22×C4×C73—24no—
57C22×C4×C73—42no—
58C22×C4×C73—42no—
59C22×C4×C73C22⋊C411yescharacteristic
60C22×C4×C73C22⋊C411yescharacteristic
61C4⋊C4×C73—42no—
62C4⋊C4×C73—42no—
63C23×C73—21no—
64C23×C73—21no—
65C23×C73—21no—
66C23×C73—21no—
67C23×C73C2×C22⋊C411yescharacteristic
68C23×C73#11712yes—
69C23×C73#13112yes—
70C23×C73#13112yes—
71C2×C4×C73—28no—
72C2×C4×C73—21no—
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—42no—
77C2×C4×C73—42no—
78C2×C4×C73—42no—
79C2×C4×C73—84no—
80C2×C4×C73#10012yes—
81C2×C4×C73#10112yes—
82C2×Dic73—22no—
83C2×Dic73—22no—
84C2×Dic73—22no—
85C2×Dic73—22no—
86C2×Dic73—22no—
87C2×Dic73—44no—
88C2×Dic73—44no—
89C2×Dic73—42no—
90C2×Dic73—42no—
91C2×Dic73—42no—
92C2×Dic73—84no—
93C2×Dic73#12314yes—
94#10596833—84no—
95C22×C73—22no—
96C22×C73—22no—
97C22×C73—22no—
98C22×C73—22no—
99C22×C73—22no—
100C22×C73—22no—
101C22×C73#38111yescharacteristic
102C22×C73#40811yescharacteristic
103C22×C73#40811yescharacteristic
104C22×C73#41011yescharacteristic
105C22×C73#41011yescharacteristic
106C22×C73C2×#10011yescharacteristic
107C22×C73C2×#10111yescharacteristic
108C22×C73#43314yes—
109C4×C73—22no—
110C4×C73—22no—
111C4×C73—82no—
112Dic73—24no—
113Dic73—42no—
114Dic73—82no—
115#8703—731nomaximal, Sylow
116C2×C73—24no—
117C2×C73#146911yescharacteristic
118C2×C73#146911yescharacteristic
119C2×C73#147011yescharacteristic
120C2×C73#147011yescharacteristic
121C2×C73#181811yescharacteristic
122C2×C73#182011yescharacteristic
123C2×C73#182111yescharacteristic
124C2×#382—731no—
125#1817—732no—
126#1280—734no—
127C73#870311yescharacteristic, complemented, Sylow
128C22×C4⋊C4—731no—
129C22×C4⋊C4#6440811yescharacteristic
130C22×C42—731no—
131C2×C22⋊C8—734no—
132#382—1462no—
133#382—1462no—
134C2×#95—732no—
135C2×#95—732no—
136#336—1462no—
137#336—1462no—
138C23×C4—731no—
139C23×C4—732no—
140C23×C4—731no—
141C23×C4C2×#6440812yes—
142C23×C4C2×#6440811yescharacteristic
143C22×C8—1464no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—2922no—
148C2×C4⋊C4—1461no—
149C2×C4⋊C4—1461no—
150C2×C42—732no—
151C2×C42—732no—
152C2×C42—1464no—
153C22⋊C8—1468no—
154#95—1464no—
155#95—1464no—
156C24C22×#6440811yescharacteristic
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—2921no—
165C22×C4—2921no—
166C22×C4—2922no—
167C22×C4—2922no—
168C22×C4—734no—
169C22×C4—732no—
170C22×C4—732no—
171C22×C4—1462no—
172C22×C4—1462no—
173C22×C4—1462no—
174C22×C4—1462no—
175C22×C4—1464no—
176C22×C4C22⋊#6440811yescharacteristic
177C22×C4C22⋊#6440811yescharacteristic
178C2×C8—2924no—
179C2×C8—2924no—
180C2×C8—2924no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—2922no—
184C4⋊C4—2922no—
185C42—1464no—
186C42—1464no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23C2×C22⋊#6440811yescharacteristic
192C23#1455667012yes—
193C23#1455667812yes—
194C23#1455667912yes—
195C2×C4—28no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—84no—
204C2×C4—2924no—
205C2×C4—2922no—
206C2×C4—2922no—
207C2×C4—2922no—
208C2×C4—2924no—
209C2×C4—5844no—
210C2×C4—734no—
211C2×C4—1462no—
212C2×C4—1462no—
213C2×C4—1462no—
214C2×C4—1462no—
215C2×C4—1462no—
216C2×C4#1455665112yes—
217C2×C4#1455665212yes—
218C8—5844no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22#1798953011yescharacteristic
226C22#1798956111yescharacteristic
227C22#1798956211yescharacteristic
228C22#1798956711yescharacteristic
229C22#1798956811yescharacteristic
230C22C2×#1455665111yescharacteristic
231C22C2×#1455665211yescharacteristic
232C22#1798961214yes—
233C4—22no—
234C4—22no—
235C4—82no—
236C4—2922no—
237C4—5842no—
238C4—1464no—
239C2—24no—
240C2#2262520111yescharacteristic
241C2#2262520211yescharacteristic
242C2#2262520411yescharacteristic
243C2#2262520511yescharacteristic
244C2#2262582511yescharacteristic
245C2#2262582911yescharacteristic
246C2#2262583011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutator, prime radicalC24×C73C22×C4
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
cosocleC23×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#8703
LCSG ▷ C24×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
127. C73#8703semidirect

Maximal subgroups

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

subgroupindex
#226258222
#226247692
C2×#179884522
#870373