Finite Group Databasebeta

← #32980662

Subgroups — #32980662

Order 18688.

542 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 119 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#22619016C211yesmaximal, characteristic, complemented
2C4×#14556580C211yesmaximal, characteristic
3#22608323C212yesmaximal
4#22607763C212yesmaximal, complemented
5C73×#1461C211yesmaximal, characteristic, complemented
6C2×#14556580C411yescharacteristic
7C2×C4×D292C2211yescharacteristic
8#17988718—24no—
9#17988644—22no—
10#17988640C2212yes—
11#17988494C411yescharacteristic
12#17988488C411yescharacteristic
13#17988477—22no—
14#17988452C411yescharacteristic
15C4×#11924005—22no—
16C42×Dic73C2211yescharacteristic
17#17988316—24no—
18C2×C4×Q8×C73C2211yescharacteristic
19#431×C73C2212yes—
20#324×C73—22no—
21C22×D292C2×C411yescharacteristic
22C22×C4×D73—21no—
23#14556580—22no—
24#14556580—22no—
25C4×D292—22no—
26C4×D292—22no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924005—22no—
29C2×#11924005—22no—
30C2×#11924005—21no—
31C2×C4×Dic73—21no—
32C2×C4×Dic73—21no—
33C2×C4×Dic73C2×C411yescharacteristic
34C2×C4×Dic73C2×C411yescharacteristic
35C4⋊#10596831—24no—
36C4⋊#10596831D414yes—
37C22⋊#10596831—22no—
38#14556502—24no—
39C4×#10596831—22no—
40#14556498—22no—
41#14556498—22no—
42C22×Q8×C73C2×C411yescharacteristic
43C4×Q8×C73—22no—
44C4×Q8×C73—22no—
45C2×C4⋊C4×C73C2×C411yescharacteristic
46C2×C4⋊C4×C73D412yes—
47C2×C42×C73D412yes—
48C2×C42×C73C2311yescharacteristic
49C4⋊C8×C73—22no—
50C22⋊C8×C73—22no—
51C4×C8×C73—22no—
52C23×D73—21no—
53C2×D292—22no—
54C2×D292—22no—
55C2×D292—21no—
56C2×D292M1612yes—
57C22×Dic73—21no—
58C22×Dic73—21no—
59C22×Dic73—21no—
60C2×C4×D73—44no—
61C2×C4×D73—41no—
62C2×C4×D73—41no—
63C2×#10596831—24no—
64C4⋊Dic73—22no—
65#11924005—28no—
66#11924005—44no—
67#11924005—42no—
68C4×Dic73—21no—
69C4×Dic73—22no—
70C4×Dic73—42no—
71C4×Dic73—42no—
72C4×Dic73M1612yes—
73C2×Q8×C73—22no—
74C2×Q8×C73—21no—
75C2×Q8×C73—21no—
76C2×Q8×C73—22no—
77C22×C4×C73C22⋊C412yes—
78C22×C4×C73C22⋊C412yes—
79C22×C4×C73C22×C411yescharacteristic
80C22×C4×C73C2×D411yescharacteristic
81C22×C4×C73C2×D411yescharacteristic
82C2×C8×C73—42no—
83C4⋊C4×C73—22no—
84C4⋊C4×C73—24no—
85C42×C73—22no—
86C42×C73—22no—
87C42×C73C2×D412yes—
88C42×C73#4112yes—
89C23×C73C2×C22⋊C411yescharacteristic
90C22×D73—22no—
91C22×D73—42no—
92C22×D73—42no—
93C22×D73—41no—
94C22×D73—41no—
95Q8×C73—42no—
96Q8×C73—42no—
97C2×C4×C73—22no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—22no—
101C2×C4×C73—22no—
102C2×C4×C73—41no—
103C2×C4×C73—42no—
104C2×C4×C73C4×D412yes—
105C2×C4×C73#12011yescharacteristic
106C2×C4×C73#12112yes—
107C2×C4×C73#12311yescharacteristic
108C2×C4×C73C2×M1611yescharacteristic
109C2×C4×C73C2×M1611yescharacteristic
110C2×C4×C73#13112yes—
111C2×Dic73—22no—
112C2×Dic73—22no—
113C2×Dic73—41no—
114C2×Dic73—41no—
115C2×Dic73—41no—
116C2×Dic73—41no—
117C2×Dic73—42no—
118C2×Dic73—41no—
119D292—24no—
120D292—42no—
121C4×D73—82no—
122C4×D73—82no—
123C8×C73—82no—
124#10596831—44no—
125C22×C73—21no—
126C22×C73—21no—
127C22×C73#38611yescharacteristic
128C22×C73#40711yescharacteristic
129C22×C73#40811yescharacteristic
130C2×D73—44no—
131C2×D73—42no—
132C2×D73—82no—
133C2×D73—81no—
134C2×D73—81no—
135C4×C73—22no—
136C4×C73—21no—
137C4×C73—42no—
138C4×C73—42no—
139C4×C73#43512yes—
140C4×C73#43612yes—
141C4×C73#45412yes—
142Dic73—42no—
143Dic73—42no—
144Dic73—81no—
145#8964—731nomaximal, Sylow
146C2×C73—21no—
147C2×C73—21no—
148C2×C73#187011yescharacteristic
149C2×C73#188411yescharacteristic
150C2×C73#188811yescharacteristic
151D73—82no—
152#2983—731no—
153C4×#124—731no—
154#1478—732no—
155#1461D7311yescharacteristic, complemented
156#1276—732no—
157C73#896411yescharacteristic, complemented, Sylow
158C2×#124—731no—
159C2×C4×Q8C2×D7311yescharacteristic
160C2×C4×D4—731no—
161#446—1464no—
162#432—1462no—
163#431—732no—
164#431C2×D7312yes—
165#391—731no—
166#390—731no—
167#388—1462no—
168#382—731no—
169C4×C22⋊C4—1462no—
170C43—731no—
171D4⋊C8—1464no—
172#324—22no—
173C22×Q8C4×D7311yescharacteristic
174C22×D4—731no—
175C23×C4—1461no—
176#124—1462no—
177#124—1462no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×D4—1462no—
181C4×D4—1462no—
182C2×C4⋊C4—731no—
183C2×C4⋊C4C4×D7311yescharacteristic
184C2×C4⋊C4C22⋊D7312yes—
185C2×C22⋊C4—1462no—
186C2×C22⋊C4—1461no—
187C2×C22⋊C4—1462no—
188C2×C42—731no—
189C2×C42—731no—
190C2×C42—1461no—
191C2×C42—1461no—
192C2×C42D29212yes—
193C2×C42C22×D7311yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—734no—
196C4⋊C8—1464no—
197C22⋊C8—22no—
198C22⋊C8—1462no—
199#97—1464no—
200C4×C8—22no—
201C4×C8—1462no—
202#95—1462no—
203#95—1462no—
204C24—1461no—
205C2×Q8—22no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×D4—732no—
210C2×D4—1462no—
211C2×D4—1461no—
212C2×D4—1462no—
213C22×C4—2924no—
214C22×C4—2921no—
215C22×C4—2921no—
216C22×C4—1461no—
217C22×C4—1461no—
218C22×C4—1461no—
219C22×C4#1192400512yes—
220C22×C4#1192400512yes—
221C22×C4C2×C4×D7311yescharacteristic
222C22×C4C2×C22⋊D7311yescharacteristic
223C22×C4C2×D29211yescharacteristic
224C2×C8—42no—
225C2×C8—1464no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—1462no—
229C22⋊C4—2924no—
230C22⋊C4—2922no—
231C22⋊C4—1468no—
232C42—22no—
233C42—22no—
234C42—2922no—
235C42—2922no—
236C42—732no—
237C42—1462no—
238C42—1461no—
239C42D4×D7312yes—
240C42#1192402712yes—
241C23—2922no—
242C23—2922no—
243C23—2921no—
244C23—2921no—
245C23—1462no—
246C23C2×#1192400511yescharacteristic
247Q8—42no—
248Q8—42no—
249D4—2922no—
250D4—1464no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—2921no—
259C2×C4—2921no—
260C2×C4—2921no—
261C2×C4—2921no—
262C2×C4—2922no—
263C2×C4—2921no—
264C2×C4—5842no—
265C2×C4—5842no—
266C2×C4—1462no—
267C2×C4—1462no—
268C2×C4#1455655112yes—
269C2×C4#1455656111yescharacteristic
270C2×C4#1455656312yes—
271C2×C4#1455657511yescharacteristic
272C2×C4M16×D7311yescharacteristic
273C2×C4M16×D7311yescharacteristic
274C2×C4#1455659712yes—
275C8—82no—
276C8—2924no—
277C22—21no—
278C22—21no—
279C22—2924no—
280C22—2922no—
281C22—5842no—
282C22—5841no—
283C22—5841no—
284C22#1798846411yescharacteristic
285C22#1798855011yescharacteristic
286C22#1798855311yescharacteristic
287C4—22no—
288C4—21no—
289C4—42no—
290C4—42no—
291C4—2922no—
292C4—2922no—
293C4—5841no—
294C4#1798866112yes—
295C4#1798866212yes—
296C4#1798876212yes—
297C2—21no—
298C2—21no—
299C2—5842no—
300C2#2261022611yescharacteristic
301C2#2261031211yescharacteristic
302C2#2261033211yescharacteristic
subgroupG/H
centerC2×C4#14556561
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1461C2
socleC22×C73#386
prime radicalC22×D292C2×C4
cosocleC2×C42×C73C23
hypercenter#1461D73
hypocenterC73#8964
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. #22619016C2semidirect
4. #22607763C2semidirect
5. C73×#1461C2semidirect
155. #1461D73semidirect
157. C73#8964semidirect

Maximal subgroups

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

subgroupindex
#226190162
C4×#145565802
#226083232
#226077632
C73×#14612
#896473