Finite Group Databasebeta

← #34005252

Subgroups — #34005252

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—44no—
25C2×#11924007—44no—
26C2×C4×Dic73—24no—
27C2×C4×Dic73C2×C411yescharacteristic
28C2×C4×Dic73D412yes—
29C2×C4×Dic73C2311yescharacteristic
30C23×C4×C73—21no—
31C23×C4×C73C2×C411yescharacteristic
32C2×C4⋊C4×C73—21no—
33C2×C4⋊C4×C73—42no—
34C2×C4⋊C4×C73C2×C412yes—
35C2×#10596833—44no—
36C2×#10596833—44no—
37C2×#10596833—44no—
38C22×Dic73—24no—
39C22×Dic73—24no—
40C22×Dic73—22no—
41C22×Dic73—42no—
42C22×Dic73—44no—
43C22×Dic73C22⋊C412yes—
44C22×Dic73D814yes—
45C22×Dic73SD1614yes—
46C22×Dic73C2×D412yes—
47C4⋊Dic73—42no—
48C4⋊Dic73—42no—
49#11924007—48no—
50#11924007—44no—
51#11924007—44no—
52C4×Dic73—22no—
53C4×Dic73—24no—
54C4×Dic73#4114yes—
55C24×C73C22⋊C411yescharacteristic
56C22×C4×C73—22no—
57C22×C4×C73—22no—
58C22×C4×C73—22no—
59C22×C4×C73—22no—
60C22×C4×C73—42no—
61C22×C4×C73—44no—
62C22×C4×C73C22⋊C411yescharacteristic
63C22×C4×C73C22×C411yescharacteristic
64C4⋊C4×C73—42no—
65C4⋊C4×C73—42no—
66C23×C73—21no—
67C23×C73—21no—
68C23×C73—22no—
69C23×C73#10012yes—
70C23×C73#10214yes—
71C23×C73C2×C22⋊C411yescharacteristic
72C2×C4×C73—28no—
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—21no—
77C2×C4×C73—42no—
78C2×C4×C73—44no—
79C2×C4×C73—84no—
80C2×C4×C73#10112yes—
81C2×C4×C73#11712yes—
82C2×Dic73—22no—
83C2×Dic73—22no—
84C2×Dic73—28no—
85C2×Dic73—28no—
86C2×Dic73—42no—
87C2×Dic73—44no—
88C2×Dic73—84no—
89C2×Dic73#10214yes—
90C2×Dic73#12314yes—
91C2×Dic73C2×D812yes—
92C2×Dic73C2×SD1612yes—
93C2×Dic73#13714yes—
94#10596833—84no—
95C22×C73—24no—
96C22×C73—22no—
97C22×C73—22no—
98C22×C73—24no—
99C22×C73#33114yes—
100C22×C73#38111yescharacteristic
101C22×C73C2×#10011yescharacteristic
102C22×C73C2×#10111yescharacteristic
103C22×C73C2×#10212yes—
104C22×C73#41912yes—
105C4×C73—22no—
106C4×C73—22no—
107C4×C73—82no—
108Dic73—24no—
109Dic73—82no—
110Dic73#48014yes—
111Dic73#48314yes—
112#8723—731nomaximal, Sylow
113C2×C73—24no—
114C2×C73C2×#33111yescharacteristic
115C2×C73#154612yes—
116C2×C73#155811yescharacteristic
117C2×C73#182111yescharacteristic
118C2×C73#182312yes—
119C2×#384—731no—
120#1797—732no—
121#1296—734no—
122C73#872311yescharacteristic, complemented, Sylow
123C22×C4⋊C4—731no—
124C22×C4⋊C4—1462no—
125C22×C4⋊C4#6440811yescharacteristic
126C22×C42—731no—
127C2×C4⋊C8—734no—
128C2×C22⋊C8—1462no—
129#384—732no—
130#384—732no—
131#384—1462no—
132#336—1462no—
133C23×C4—21no—
134C23×C4—732no—
135C23×C4—1461no—
136C23×C4C2×#6440811yescharacteristic
137C22×C8—1464no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—2922no—
141C2×C4⋊C4—2924no—
142C2×C4⋊C4—2924no—
143C2×C4⋊C4—732no—
144C2×C4⋊C4—1464no—
145C2×C4⋊C4—1464no—
146C2×C4⋊C4—1461no—
147C2×C4⋊C4C2×#6440812yes—
148C2×C42—731no—
149C2×C42—732no—
150C2×C42—731no—
151C2×C42—1464no—
152C4⋊C8—1468no—
153C22⋊C8—2924no—
154C24C22⋊#6440811yescharacteristic
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—42no—
160C22×C4—44no—
161C22×C4—2922no—
162C22×C4—2924no—
163C22×C4—732no—
164C22×C4—734no—
165C22×C4—734no—
166C22×C4—732no—
167C22×C4—1462no—
168C22×C4—1464no—
169C22×C4—1464no—
170C22×C4C22⋊#6440811yescharacteristic
171C22×C4C22×#6440811yescharacteristic
172C2×C8—2924no—
173C2×C8—2924no—
174C2×C8—2924no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—2922no—
178C4⋊C4—2922no—
179C4⋊C4—2924no—
180C4⋊C4—2924no—
181C4⋊C4—2928no—
182C42—734no—
183C42—1464no—
184C42—1462no—
185C23—21no—
186C23—21no—
187C23—22no—
188C23#1455665112yes—
189C23#1455665314yes—
190C23C2×C22⋊#6440811yescharacteristic
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—44no—
198C2×C4—84no—
199C2×C4—2922no—
200C2×C4—2924no—
201C2×C4—5844no—
202C2×C4—734no—
203C2×C4—732no—
204C2×C4—732no—
205C2×C4—734no—
206C2×C4—734no—
207C2×C4—1468no—
208C2×C4—1468no—
209C2×C4—1462no—
210C2×C4—1462no—
211C2×C4#1455665212yes—
212C2×C4#1455667012yes—
213C8—5844no—
214C22—22no—
215C22—24no—
216C22—24no—
217C22—22no—
218C22#1798944514yes—
219C22#1798953011yescharacteristic
220C22C2×#1455665111yescharacteristic
221C22C2×#1455665211yescharacteristic
222C22C2×#1455665312yes—
223C22#1798958012yes—
224C4—22no—
225C4—22no—
226C4—82no—
227C4—5842no—
228C4—734no—
229C4—734no—
230C4—1464no—
231C2—24no—
232C2C2×#1798944511yescharacteristic
233C2#2262544012yes—
234C2#2262546411yescharacteristic
235C2#2262583011yescharacteristic
236C2#2262583412yes—
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#8723
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
122. C73#8723semidirect

Maximal subgroups

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

subgroupindex
#226257872
#226248092
C2×#179884562
#872373