Finite Group Databasebeta

← #34010093

Subgroups — #34010093

Order 18688.

434 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 80 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#22625179C212yesmaximal
2#22624873C212yesmaximal, complemented
3#22624871C212yesmaximal, complemented
4#22612554C211yesmaximal, characteristic
5#17989557—22no—
6#17989432—22no—
7#17989432C2214yes—
8C2×#14556550C411yescharacteristic
9C22×C4⋊Dic73C2211yescharacteristic
10#17988486—22no—
11#17988462—22no—
12#17988457C411yescharacteristic
13#17988457C2211yescharacteristic
14#17988453C411yescharacteristic
15#17988453C2211yescharacteristic
16C2×#117×C73C411yescharacteristic
17C2×#11924037—42no—
18C22⋊#10596833—24no—
19C22⋊#10596833—42no—
20C23×Dic73D412yes—
21#14556550—42no—
22C2×C4⋊Dic73—22no—
23C2×C4⋊Dic73—22no—
24C2×C4⋊Dic73—42no—
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×C4⋊Dic73D412yes—
29C2×C4⋊Dic73C2311yescharacteristic
30C2×C22⋊Dic73—21no—
31C2×C4×Dic73—21no—
32#14556498—22no—
33#14556498—22no—
34C23×C4×C73C2×C411yescharacteristic
35#117×C73—22no—
36#117×C73—22no—
37C2×C4⋊C4×C73—21no—
38C2×C22⋊C4×C73—21no—
39C2×C42×C73C2×C411yescharacteristic
40C2×C42×C73C2×C411yescharacteristic
41#11924037—84no—
42C2×#10596833—44no—
43C22×Dic73—22no—
44C22×Dic73—22no—
45C22×Dic73—21no—
46C22×Dic73—21no—
47C22×Dic73—82no—
48C22×Dic73C22⋊C412yes—
49C22×Dic73C2×D412yes—
50C4⋊Dic73—41no—
51C4⋊Dic73—42no—
52C4⋊Dic73—41no—
53C4⋊Dic73—41no—
54C4⋊Dic73—41no—
55C4⋊Dic73—82no—
56C4⋊Dic73—82no—
57C22⋊Dic73—42no—
58C4×Dic73—42no—
59C24×C73C22⋊C411yescharacteristic
60C22×C4×C73—22no—
61C22×C4×C73—22no—
62C22×C4×C73—21no—
63C22×C4×C73—21no—
64C22×C4×C73—21no—
65C22×C4×C73—21no—
66C22×C4×C73C22⋊C411yescharacteristic
67C22×C4×C73C22×C411yescharacteristic
68C4⋊C4×C73—24no—
69C22⋊C4×C73—24no—
70C42×C73—21no—
71C42×C73—21no—
72C42×C73M1612yescomplemented
73C42×C73M1612yescomplemented
74C23×C73—21no—
75C23×C73—21no—
76C23×C73—41no—
77C23×C73—41no—
78C23×C73#9912yes—
79C23×C73C2×C22⋊C411yescharacteristic
80C2×C4×C73—24no—
81C2×C4×C73—24no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—22no—
87C2×C4×C73—22no—
88C2×C4×C73—41no—
89C2×C4×C73—41no—
90C2×C4×C73—41no—
91C2×C4×C73—41no—
92C2×C4×C73#10112yes—
93C2×C4×C73C2×M1611yescharacteristic
94C2×C4×C73C2×M1611yescharacteristic
95C2×Dic73—22no—
96C2×Dic73—22no—
97C2×Dic73—44no—
98C2×Dic73—42no—
99C2×Dic73—42no—
100C2×Dic73—42no—
101C2×Dic73—42no—
102C2×Dic73—41no—
103C2×Dic73—41no—
104C2×Dic73—82no—
105#10596833—84no—
106C22×C73—21no—
107C22×C73—21no—
108C22×C73—22no—
109C22×C73—22no—
110C22×C73—22no—
111C22×C73—42no—
112C22×C73—42no—
113C22×C73#40711yescharacteristic
114C22×C73C2×#9911yescharacteristic
115C22×C73C2×#10111yescharacteristic
116C4×C73—22no—
117C4×C73—21no—
118C4×C73—21no—
119C4×C73—42no—
120C4×C73—42no—
121Dic73—42no—
122Dic73—42no—
123Dic73—81no—
124Dic73—81no—
125#11514—731nomaximal, Sylow
126C2×C73—21no—
127C2×C73—21no—
128C2×C73—42no—
129C2×C73#146311yescharacteristic
130C2×C73#212711yescharacteristic
131C2×C73#213311yescharacteristic
132#2317—731no—
133#1463—732no—
134#1325—732no—
135#1324—732no—
136C73#1151411yescharacteristic, complemented, Sylow
137C2×#117—731no—
138C2×#117#6440811yescharacteristic
139C22×C4⋊C4—731no—
140#407—1462no—
141#389—1462no—
142#385—1462no—
143#384—731no—
144#384—731no—
145#382—731no—
146#382—731no—
147#324—734no—
148#324—1462no—
149C23×C4—732no—
150C23×C4C2×#6440811yescharacteristic
151C2×M16—2922no—
152#117—22no—
153#117—22no—
154#117—2922no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—2922no—
157C2×C4⋊C4—732no—
158C2×C4⋊C4—731no—
159C2×C4⋊C4—731no—
160C2×C4⋊C4—731no—
161C2×C4⋊C4—731no—
162C2×C4⋊C4—1462no—
163C2×C4⋊C4—1462no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—1461no—
166C2×C42—1461no—
167C2×C42C2×#6440811yescharacteristic
168C2×C42C2×#6440811yescharacteristic
169C22⋊C8—2922no—
170C22⋊C8—1464no—
171#95—1462no—
172#95—1462no—
173C24C22⋊#6440811yescharacteristic
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—5842no—
181C22×C4—732no—
182C22×C4—732no—
183C22×C4—1461no—
184C22×C4—1461no—
185C22×C4—1462no—
186C22×C4—1462no—
187C22×C4C22⋊#6440811yescharacteristic
188C22×C4C22×#6440811yescharacteristic
189M16—5844no—
190C2×C8—2924no—
191C4⋊C4—24no—
192C4⋊C4—2921no—
193C4⋊C4—2921no—
194C4⋊C4—2922no—
195C4⋊C4—2921no—
196C4⋊C4—2921no—
197C4⋊C4—5842no—
198C4⋊C4—5842no—
199C22⋊C4—24no—
200C22⋊C4—2922no—
201C42—21no—
202C42—21no—
203C42—2922no—
204C42#1192403712yescomplemented
205C42#1192403712yescomplemented
206C23—21no—
207C23—21no—
208C23—41no—
209C23—41no—
210C23#1455665012yes—
211C23C2×C22⋊#6440811yescharacteristic
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—2924no—
225C2×C4—2922no—
226C2×C4—2922no—
227C2×C4—2922no—
228C2×C4—2922no—
229C2×C4—2921no—
230C2×C4—2921no—
231C2×C4—5842no—
232C2×C4—1462no—
233C2×C4—1462no—
234C2×C4#1455665212yes—
235C2×C4C2×#1192403711yescharacteristic
236C2×C4C2×#1192403711yescharacteristic
237C8—5844no—
238C22—21no—
239C22—21no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22#1798955711yescharacteristic
246C22C2×#1455665011yescharacteristic
247C22C2×#1455665211yescharacteristic
248C4—21no—
249C4—21no—
250C4—22no—
251C4—42no—
252C4—42no—
253C4—2922no—
254C4—2922no—
255C4—5841no—
256C4—5841no—
257C2—21no—
258C2—21no—
259C2—42no—
260C2#2262517911yescharacteristic
261C2#2262620811yescharacteristic
262C2#2262622811yescharacteristic
subgroupG/H
centerC22C2×#14556650
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC2×#117#64408
hypocenterC73#11514
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22624873C2semidirect
3. #22624871C2semidirect
72. C42×C73M16semidirect
73. C42×C73M16semidirect
136. C73#11514semidirect
204. C42#11924037semidirect
205. C42#11924037semidirect

Maximal subgroups

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

subgroupindex
#226251792
#226248732
#226248712
#226125542
#1151473