Finite Group Databasebeta

← #34009837

Subgroups — #34009837

Order 18688.

450 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 102 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#22626159C212yesmaximal
2#22624868C212yesmaximal, complemented
3#22624867C212yesmaximal, complemented
4C2×C4⋊C8×D73C211yesmaximal, characteristic
5C2×#14556666—22no—
6#17989482—22no—
7C22⋊#11924031C2214yes—
8C22×C8×D73C411yescharacteristic
9C22×C8×D73C2211yescharacteristic
10C2×C42×D73C411yescharacteristic
11C2×C4⋊#10596831C2211yescharacteristic
12C2×#14556520C411yescharacteristic
13C2×#14556520C2211yescharacteristic
14C4⋊C8×D73—22no—
15C4⋊C8×D73—22no—
16C2×C4⋊C8×C73C411yescharacteristic
17#14556666—44no—
18C2×#11924031—24no—
19C22×C4×D73C811yescharacteristic
20C22×C4×D73C811yescharacteristic
21C22×C4×D73C2×C411yescharacteristic
22C22×#10596831C2×C411yescharacteristic
23C22×#10596831C2311yescharacteristic
24C2×C8×D73—21no—
25C2×C8×D73—21no—
26C2×C8×D73—42no—
27C2×C8×D73—42no—
28C2×C8×D73D412yes—
29C2×C8×D73D412yes—
30C2×C4×Dic73C811yescharacteristic
31C2×C4×Dic73C811yescharacteristic
32C2×C4×Dic73C2×C411yescharacteristic
33C42×D73—21no—
34C42×D73—21no—
35C42×D73—21no—
36C42×D73—21no—
37C4⋊#10596831—22no—
38#14556520—22no—
39#14556520—22no—
40C22×C8×C73C2×C411yescharacteristic
41C22×C8×C73C2×C411yescharacteristic
42C2×C42×C73C2×C411yescharacteristic
43C4⋊C8×C73—22no—
44#11924031—44no—
45C23×D73C2×C811yescharacteristic
46C22×Dic73C2×C811yescharacteristic
47C22×Dic73C2×C811yescharacteristic
48C22×Dic73C2×C811yescharacteristic
49C2×C4×D73—21no—
50C2×C4×D73—21no—
51C2×C4×D73—22no—
52C2×C4×D73—22no—
53C2×C4×D73—22no—
54C2×C4×D73—22no—
55C2×C4×D73—21no—
56C2×C4×D73—21no—
57C2×C4×D73—42no—
58C2×C4×D73C22⋊C411yescharacteristic
59C2×C4×D73C22⋊C411yescharacteristic
60C2×C4×D73M1611yescharacteristic
61C2×C4×D73M1611yescharacteristic
62C2×#10596831—21no—
63C2×#10596831—41no—
64C2×#10596831—41no—
65C2×#10596831C2×D412yes—
66C8×D73—24no—
67C8×D73—42no—
68C8×D73—42no—
69C8×D73—42no—
70C4×Dic73—21no—
71C4×Dic73—21no—
72C4×Dic73—21no—
73C4×Dic73—21no—
74C4×Dic73—21no—
75C4×Dic73—21no—
76C22×C4×C73C2×C811yescharacteristic
77C22×C4×C73C2×C811yescharacteristic
78C22×C4×C73C22×C411yescharacteristic
79C2×C8×C73—21no—
80C2×C8×C73—41no—
81C2×C8×C73—41no—
82C2×C8×C73C22⋊C412yes—
83C42×C73—21no—
84C42×C73—21no—
85C23×C73C22×C811yescharacteristic
86C22×D73—21no—
87C22×D73—21no—
88C22×D73—22no—
89C22×D73—22no—
90C22×D73C22⋊C811yescharacteristic
91C22×D73C22⋊C811yescharacteristic
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—41no—
98C2×C4×C73M3212yescomplemented
99C2×C4×C73C2×C22⋊C411yescharacteristic
100C2×C4×C73C2×M1611yescharacteristic
101C2×Dic73—21no—
102C2×Dic73—21no—
103C2×Dic73—22no—
104C2×Dic73—21no—
105C2×Dic73—41no—
106C2×Dic73C22⋊C811yescharacteristic
107C2×Dic73C22⋊C811yescharacteristic
108C2×Dic73M3212yescomplemented
109C4×D73—24no—
110C4×D73—24no—
111C4×D73—22no—
112C4×D73—21no—
113C4×D73—21no—
114C4×D73—22no—
115C4×D73—42no—
116C4×D73—42no—
117C4×D73#9912yes—
118C4×D73#10112yes—
119C8×C73—22no—
120C8×C73—41no—
121#10596831—22no—
122#10596831—41no—
123C22×C73—21no—
124C22×C73—21no—
125C22×C73C2×C22⋊C811yescharacteristic
126C22×C73C2×M3211yescharacteristic
127C22×C73C2×M3211yescharacteristic
128C2×D73—24no—
129C2×D73—24no—
130C2×D73—21no—
131C2×D73—21no—
132C2×D73—21no—
133C2×D73—21no—
134C2×D73#32412yes—
135C2×D73#34912yes—
136C4×C73—22no—
137C4×C73—21no—
138C4×C73—41no—
139C4×C73C2×#9911yescharacteristic
140C4×C73C2×#10111yescharacteristic
141Dic73—22no—
142Dic73—21no—
143Dic73—41no—
144Dic73#32412yes—
145#11440—731nomaximal, Sylow
146C2×C73—21no—
147C2×C73—21no—
148C2×C73C2×#32411yescharacteristic
149C2×C73#211211yescharacteristic
150C2×C73C2×#34911yescharacteristic
151D73—22no—
152D73—22no—
153C22×C4⋊C8—731no—
154#2112—732no—
155#1322—732no—
156#1322—732no—
157C73#1144011yescharacteristic, complemented, Sylow
158C23×C8—731no—
159C23×C8—731no—
160C22×C42—731no—
161C2×M32—1462no—
162C2×C4⋊C8—731no—
163C2×C4⋊C8—731no—
164C2×C4⋊C8—731no—
165C2×C4⋊C8—1462no—
166C2×C4⋊C8—1462no—
167C2×C4⋊C8#6440811yescharacteristic
168C22⋊C16—734no—
169C22⋊C16—1462no—
170C23×C4—731no—
171C23×C4—731no—
172C23×C4—731no—
173C22×C8—2922no—
174C22×C8—2922no—
175C22×C8—731no—
176C22×C8—732no—
177C22×C8—731no—
178C22×C8—732no—
179C22×C8—1461no—
180C22×C8—1461no—
181C22×C8C2×#6440811yescharacteristic
182C22×C8C2×#6440811yescharacteristic
183C2×C42—731no—
184C2×C42—731no—
185C2×C42—731no—
186C2×C42—1461no—
187C2×C42—1461no—
188C2×C42—1461no—
189C2×C42—1461no—
190C2×C42C2×#6440811yescharacteristic
191M32—2924no—
192C2×C16—1464no—
193C4⋊C8—22no—
194C4⋊C8—1462no—
195C4⋊C8—1462no—
196C4⋊C8—1462no—
197C24—731no—
198C22×C4—2922no—
199C22×C4—731no—
200C22×C4—731no—
201C22×C4—731no—
202C22×C4—731no—
203C22×C4—731no—
204C22×C4—731no—
205C22×C4—731no—
206C22×C4—1461no—
207C22×C4—1461no—
208C22×C4—1462no—
209C22×C4—1462no—
210C22×C4—1462no—
211C22×C4—1461no—
212C22×C4—1461no—
213C22×C4—1462no—
214C22×C4#1192403611yescharacteristic
215C22×C4#1192403611yescharacteristic
216C22×C4C22×#6440811yescharacteristic
217C2×C8—21no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—2922no—
221C2×C8—2922no—
222C2×C8—2922no—
223C2×C8—2921no—
224C2×C8—2921no—
225C2×C8—732no—
226C2×C8—1464no—
227C2×C8—1461no—
228C2×C8C22⋊#6440812yes—
229C42—21no—
230C42—21no—
231C42—1461no—
232C42—1461no—
233C42—1461no—
234C42—1461no—
235C42—1461no—
236C42—1461no—
237C16—2924no—
238C23—731no—
239C23—731no—
240C23—1462no—
241C23—1462no—
242C23—1461no—
243C23—1461no—
244C23C2×#1192403611yescharacteristic
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—2922no—
252C2×C4—2922no—
253C2×C4—2921no—
254C2×C4—731no—
255C2×C4—732no—
256C2×C4—732no—
257C2×C4—732no—
258C2×C4—731no—
259C2×C4—1462no—
260C2×C4—1462no—
261C2×C4—1464no—
262C2×C4—1464no—
263C2×C4—1461no—
264C2×C4—1462no—
265C2×C4—1461no—
266C2×C4—1461no—
267C2×C4—1461no—
268C2×C4—1461no—
269C2×C4#1455666612yescomplemented
270C2×C4C2×C22⋊#6440811yescharacteristic
271C2×C4D73⋊M1611yescharacteristic
272C8—22no—
273C8—41no—
274C8—2921no—
275C8—1462no—
276C22—21no—
277C22—21no—
278C22—732no—
279C22—732no—
280C22—1464no—
281C22—1464no—
282C22—1461no—
283C22—1461no—
284C22—1461no—
285C22—1461no—
286C22#1798955511yescharacteristic
287C22#1798964311yescharacteristic
288C22C2×#1455666611yescharacteristic
289C4—22no—
290C4—21no—
291C4—41no—
292C4—2921no—
293C4—732no—
294C4—1462no—
295C4—1461no—
296C4D73⋊#9911yescharacteristic
297C4D73⋊#10111yescharacteristic
298C2—21no—
299C2—21no—
300C2—1462no—
301C2—1462no—
302C2#2262516011yescharacteristic
303C2#2262615911yescharacteristic
304C2#2262616511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC23×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×C4⋊C8×C73C4
socleC22×C73C2×C22⋊C8
prime radicalC23×D73C2×C8
cosocleC22×#10596831C23
hypercenterC2×C4⋊C8#64408
hypocenterC73#11440
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22624868C2semidirect
3. #22624867C2semidirect
98. C2×C4×C73M32semidirect
108. C2×Dic73M32semidirect
157. C73#11440semidirect
269. C2×C4#14556666semidirect

Maximal subgroups

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

subgroupindex
#226261592
#226248682
#226248672
C2×C4⋊C8×D732
#1144073