Finite Group Databasebeta

← #34005256

Subgroups — #34005256

Order 18688.

630 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 140 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#22625787C212yesmaximal
2#22624809C214yesmaximal, complemented
3C2×#17988456C211yesmaximal, characteristic
4C2×#14556658C2214yes—
5C2×C22⋊#10596833—22no—
6#17989453—22no—
7C22×C4⋊Dic73C411yescharacteristic
8C22×#11924007—22no—
9C22×C4×Dic73C2211yescharacteristic
10#17988456—22no—
11#17988456C412yes—
12#17988456C2212yes—
13C22×C4⋊C4×C73C411yescharacteristic
14C22×#10596833—24no—
15#14556658—28no—
16C22⋊#10596833—44no—
17C23×Dic73—21no—
18C23×Dic73D412yes—
19C2×C4⋊Dic73—21no—
20C2×C4⋊Dic73—42no—
21C2×C4⋊Dic73C2×C412yes—
22C2×#11924007—24no—
23C2×#11924007—24no—
24C2×#11924007—48no—
25C2×C4×Dic73—24no—
26C2×C4×Dic73C2×C411yescharacteristic
27C2×C4×Dic73D412yes—
28C2×C4×Dic73C2311yescharacteristic
29C23×C4×C73—21no—
30C23×C4×C73C2×C411yescharacteristic
31C2×C4⋊C4×C73—21no—
32C2×C4⋊C4×C73—42no—
33C2×C4⋊C4×C73C2×C412yes—
34C2×#10596833—44no—
35C2×#10596833—44no—
36C2×#10596833—44no—
37C22×Dic73—24no—
38C22×Dic73—24no—
39C22×Dic73—22no—
40C22×Dic73—44no—
41C22×Dic73—42no—
42C22×Dic73C22⋊C412yes—
43C22×Dic73SD1614yes—
44C22×Dic73Q1614yes—
45C22×Dic73C2×D412yes—
46C4⋊Dic73—42no—
47C4⋊Dic73—42no—
48#11924007—48no—
49#11924007—48no—
50C4×Dic73—22no—
51C4×Dic73—24no—
52C4×Dic73#4114yes—
53C24×C73C22⋊C411yescharacteristic
54C22×C4×C73—22no—
55C22×C4×C73—22no—
56C22×C4×C73—22no—
57C22×C4×C73—22no—
58C22×C4×C73—44no—
59C22×C4×C73—42no—
60C22×C4×C73C22⋊C411yescharacteristic
61C22×C4×C73C22×C411yescharacteristic
62C4⋊C4×C73—42no—
63C4⋊C4×C73—42no—
64C23×C73—21no—
65C23×C73—21no—
66C23×C73—22no—
67C23×C73#10012yes—
68C23×C73#10314yes—
69C23×C73C2×C22⋊C411yescharacteristic
70C2×C4×C73—28no—
71C2×C4×C73—21no—
72C2×C4×C73—21no—
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73—42no—
76C2×C4×C73—44no—
77C2×C4×C73—84no—
78C2×C4×C73#10112yes—
79C2×C4×C73#11712yes—
80C2×Dic73—22no—
81C2×Dic73—22no—
82C2×Dic73—28no—
83C2×Dic73—28no—
84C2×Dic73—42no—
85C2×Dic73—44no—
86C2×Dic73—84no—
87C2×Dic73#10314yes—
88C2×Dic73#12314yes—
89C2×Dic73C2×SD1612yes—
90C2×Dic73C2×Q1612yes—
91C2×Dic73AGL(1,ℤ/8)14yes—
92#10596833—84no—
93C22×C73—24no—
94C22×C73—22no—
95C22×C73—22no—
96C22×C73—24no—
97C22×C73#33314yes—
98C22×C73#38111yescharacteristic
99C22×C73C2×#10011yescharacteristic
100C22×C73C2×#10111yescharacteristic
101C22×C73C2×#10312yes—
102C22×C73#41812yes—
103C4×C73—22no—
104C4×C73—22no—
105C4×C73—82no—
106Dic73—24no—
107Dic73—82no—
108Dic73#48114yes—
109Dic73#48414yes—
110#8725—731nomaximal, Sylow
111C2×C73—24no—
112C2×C73C2×#33311yescharacteristic
113C2×C73#154912yes—
114C2×C73#155511yescharacteristic
115C2×C73#182111yescharacteristic
116C2×C73#182412yes—
117C2×#384—731no—
118#1797—732no—
119#1296—734no—
120C73#872511yescharacteristic, complemented, Sylow
121C22×C4⋊C4—731no—
122C22×C4⋊C4—1462no—
123C22×C4⋊C4#6440811yescharacteristic
124C22×C42—731no—
125C2×C4⋊C8—734no—
126C2×C22⋊C8—1462no—
127#384—732no—
128#384—732no—
129#384—1462no—
130#336—1462no—
131C23×C4—21no—
132C23×C4—732no—
133C23×C4—1461no—
134C23×C4C2×#6440811yescharacteristic
135C22×C8—1464no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—2922no—
139C2×C4⋊C4—2928no—
140C2×C4⋊C4—732no—
141C2×C4⋊C4—1464no—
142C2×C4⋊C4—1464no—
143C2×C4⋊C4—1461no—
144C2×C4⋊C4C2×#6440812yes—
145C2×C42—731no—
146C2×C42—732no—
147C2×C42—731no—
148C2×C42—1464no—
149C4⋊C8—1468no—
150C22⋊C8—2924no—
151C24C22⋊#6440811yescharacteristic
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—2924no—
159C22×C4—2922no—
160C22×C4—732no—
161C22×C4—734no—
162C22×C4—734no—
163C22×C4—732no—
164C22×C4—1462no—
165C22×C4—1464no—
166C22×C4—1464no—
167C22×C4C22⋊#6440811yescharacteristic
168C22×C4C22×#6440811yescharacteristic
169C2×C8—2924no—
170C2×C8—2924no—
171C2×C8—2924no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—2922no—
175C4⋊C4—2922no—
176C4⋊C4—2928no—
177C4⋊C4—2928no—
178C42—734no—
179C42—1464no—
180C42—1462no—
181C23—21no—
182C23—21no—
183C23—22no—
184C23#1455665112yes—
185C23#1455665414yes—
186C23C2×C22⋊#6440811yescharacteristic
187C2×C4—28no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—84no—
195C2×C4—2922no—
196C2×C4—2924no—
197C2×C4—5844no—
198C2×C4—734no—
199C2×C4—732no—
200C2×C4—732no—
201C2×C4—734no—
202C2×C4—734no—
203C2×C4—1468no—
204C2×C4—1468no—
205C2×C4—1462no—
206C2×C4—1462no—
207C2×C4#1455665212yes—
208C2×C4#1455667012yes—
209C8—5844no—
210C22—22no—
211C22—24no—
212C22—24no—
213C22—22no—
214C22#1798944714yes—
215C22#1798953011yescharacteristic
216C22C2×#1455665111yescharacteristic
217C22C2×#1455665211yescharacteristic
218C22C2×#1455665412yes—
219C22#1798957912yes—
220C4—22no—
221C4—22no—
222C4—82no—
223C4—5842no—
224C4—734no—
225C4—734no—
226C4—1464no—
227C2—24no—
228C2C2×#1798944711yescharacteristic
229C2#2262544612yes—
230C2#2262545811yescharacteristic
231C2#2262583011yescharacteristic
232C2#2262583512yes—
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#8725
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22624809C2semidirect
120. C73#8725semidirect

Maximal subgroups

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

subgroupindex
#226257872
#226248092
C2×#179884562
#872573