Finite Group Databasebeta

← #32977748

Subgroups — #32977748

Order 18688.

782 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#17989294C211yesmaximal, characteristic
2#22608617C212yesmaximal, complemented
3#22608359C212yesmaximal, complemented
4#22608252C211yesmaximal, characteristic
5C73×#1800C211yesmaximal, characteristic
6#17989294—24no—
7#17989294—24no—
8C2×#14556591C2211yescharacteristic
9C2×#14556582—21no—
10C2×#14556570C412yes—
11C2×#14556568—22no—
12C2×C4×Dic146C2211yescharacteristic
13C2×D4×Dic73—21no—
14C22×C4⋊Dic73C412yes—
15#17988640C2212yes—
16#17988332—24no—
17#17988317—24no—
18#17988311—22no—
19C2×C4×D4×C73C2211yescharacteristic
20#423×C73C2212yes—
21C2×#100×C73—21no—
22#343×C73—21no—
23C22×Dic146—21no—
24C22×Dic146C2×C411yescharacteristic
25C23×Dic73—22no—
26#14556591—22no—
27#14556591D414yes—
28#14556582—28no—
29#14556570—24no—
30#14556570—24no—
31#14556568—216no—
32C4×Dic146—22no—
33C4×Dic146D414yes—
34D4×Dic73—28no—
35C2×C4⋊Dic73—24no—
36C2×C4⋊Dic73—24no—
37C2×C4⋊Dic73—22no—
38C2×C4⋊Dic73—21no—
39C2×C4⋊Dic73C2×C412yes—
40C2×C4⋊Dic73C2×C411yescharacteristic
41C2×#11924007—22no—
42C2×#11924007—21no—
43C2×C22⋊Dic73—22no—
44C2×C4×Dic73—21no—
45C4⋊#10596831—22no—
46C22⋊#10596831—22no—
47C4×#10596831—22no—
48C22×D4×C73D411yescharacteristic
49C23×C4×C73C2×C412yes—
50C2×M16×C73—22no—
51C4×D4×C73—22no—
52C4×D4×C73—22no—
53C2×C4⋊C4×C73D411yescharacteristic
54C2×C22⋊C4×C73D412yes—
55C2×C42×C73C2311yescharacteristic
56C4⋊C8×C73—22no—
57C4⋊C8×C73—22no—
58#100×C73—42no—
59C2×Dic146—21no—
60C2×Dic146—24no—
61C2×Dic146—42no—
62C2×Dic146—82no—
63C2×Dic146C22⋊C412yes—
64C22×Dic73—21no—
65C22×Dic73—21no—
66C22×Dic73—22no—
67C22×Dic73—28no—
68C22×Dic73—84no—
69C2×#10596831—42no—
70C4⋊Dic73—28no—
71C4⋊Dic73—21no—
72C4⋊Dic73—22no—
73C4⋊Dic73—24no—
74C4⋊Dic73C22⋊C412yes—
75C4⋊Dic73C22⋊C414yes—
76#11924007—28no—
77#11924007—24no—
78C22⋊Dic73—28no—
79C4×Dic73—24no—
80C24×C73C22⋊C412yes—
81C2×D4×C73—22no—
82C2×D4×C73—22no—
83C2×D4×C73SD1612yes—
84C2×D4×C73Q1612yes—
85C22×C4×C73—22no—
86C22×C4×C73—22no—
87C22×C4×C73—44no—
88C22×C4×C73C22⋊C412yes—
89C22×C4×C73C22×C411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91C22×C4×C73C2×D411yescharacteristic
92M16×C73—42no—
93M16×C73—42no—
94C2×C8×C73—42no—
95C4⋊C4×C73—22no—
96C22⋊C4×C73—24no—
97C42×C73C2×D412yes—
98C42×C73C2×D412yes—
99C23×C73—22no—
100C23×C73—22no—
101C23×C73—42no—
102C23×C73—42no—
103C23×C73#10314yes—
104C23×C73C2×C22⋊C411yescharacteristic
105D4×C73—24no—
106D4×C73—24no—
107C2×C4×C73—22no—
108C2×C4×C73—22no—
109C2×C4×C73—41no—
110C2×C4×C73—44no—
111C2×C4×C73—44no—
112C2×C4×C73—42no—
113C2×C4×C73C2×C22⋊C411yescharacteristic
114C2×C4×C73C2×C22⋊C411yescharacteristic
115C2×C4×C73#12012yes—
116C2×C4×C73#12012yes—
117C2×C4×C73C2×SD1611yescharacteristic
118C2×C4×C73C2×Q1611yescharacteristic
119C2×C4×C73#13712yes—
120C2×Dic73—22no—
121C2×Dic73—22no—
122C2×Dic73—24no—
123C2×Dic73—42no—
124C2×Dic73—44no—
125C2×Dic73—42no—
126C2×Dic73—88no—
127Dic146—24no—
128Dic146—42no—
129Dic146—84no—
130C8×C73—82no—
131#10596831—82no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73—24no—
135C22×C73—44no—
136C22×C73—44no—
137C22×C73—42no—
138C22×C73—42no—
139C22×C73#37911yescharacteristic
140C22×C73C2×#10311yescharacteristic
141C22×C73#41911yescharacteristic
142C4×C73—22no—
143C4×C73—21no—
144C4×C73—42no—
145C4×C73#41312yes—
146C4×C73#44812yes—
147C4×C73#45112yes—
148Dic73—44no—
149Dic73—42no—
150Dic73—42no—
151#8395—731nomaximal, Sylow
152C2×C73—21no—
153C2×C73—21no—
154C2×C73—44no—
155C2×C73#178511yescharacteristic
156C2×C73#178711yescharacteristic
157C2×C73#179511yescharacteristic
158C2×#554—731no—
159#1800D7311yescharacteristic
160#1549—732no—
161#1487—732no—
162#1460—731no—
163C73#839511yescharacteristic, complemented, Sylow
164#554—1464no—
165#554—1464no—
166C2×C4⋊Q8—731no—
167C2×#125—1461no—
168C2×C22⋊Q8—732no—
169C2×C22⋊Q8—1462no—
170C2×C4×Q8—731no—
171C2×C4×D4—1461no—
172C2×C4×D4C2×D7311yescharacteristic
173C22×C4⋊C4—732no—
174#431—732no—
175#423C2×D7312yes—
176C2×#100—21no—
177#343—21no—
178#332—1464no—
179Q8⋊C8—1464no—
180#323—1462no—
181C22×Q8—731no—
182C22×Q8—1461no—
183C22×D4C22⋊D7311yescharacteristic
184C23×C4—1462no—
185C23×C4C4×D7312yes—
186C2×M16—22no—
187C4⋊Q8—734no—
188C4⋊Q8—1462no—
189#125—1468no—
190C22⋊Q8—14616no—
191C22⋊Q8—1464no—
192C22⋊Q8—1464no—
193C4×Q8—734no—
194C4×Q8—1462no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—1468no—
198C2×C4⋊C4—731no—
199C2×C4⋊C4—732no—
200C2×C4⋊C4—1464no—
201C2×C4⋊C4—1464no—
202C2×C4⋊C4—1462no—
203C2×C4⋊C4—1462no—
204C2×C4⋊C4—1461no—
205C2×C4⋊C4—1461no—
206C2×C4⋊C4C22⋊D7311yescharacteristic
207C2×C22⋊C4—1462no—
208C2×C22⋊C4D29212yes—
209C2×C42—1461no—
210C2×C42C22×D7311yescharacteristic
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—1462no—
214#100—42no—
215C22⋊C8—1462no—
216C4×C8—1462no—
217C24#1192400512yes—
218C2×Q8—2922no—
219C2×Q8—5842no—
220C2×Q8—732no—
221C2×Q8—1461no—
222C2×Q8—1464no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4#1192401612yes—
226C2×D4#1192401812yes—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—44no—
230C22×C4—5844no—
231C22×C4—1461no—
232C22×C4—1461no—
233C22×C4—1462no—
234C22×C4—1468no—
235C22×C4#1192400512yes—
236C22×C4C2×C4×D7311yescharacteristic
237C22×C4C2×C22⋊D7311yescharacteristic
238C22×C4C2×D29211yescharacteristic
239M16—42no—
240M16—42no—
241C2×C8—42no—
242C2×C8—2922no—
243C4⋊C4—22no—
244C4⋊C4—732no—
245C4⋊C4—734no—
246C4⋊C4—1468no—
247C4⋊C4—1468no—
248C4⋊C4—1461no—
249C4⋊C4—1462no—
250C4⋊C4—1464no—
251C4⋊C4—1464no—
252C22⋊C4—24no—
253C22⋊C4—1468no—
254C42—1464no—
255C42D4×D7312yes—
256C42D4×D7312yes—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23#1455651414yes—
262C23C2×#1192400511yescharacteristic
263Q8—2922no—
264Q8—5844no—
265Q8—1464no—
266D4—24no—
267D4—24no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—2922no—
275C2×C4—2924no—
276C2×C4—2922no—
277C2×C4—5848no—
278C2×C4—1462no—
279C2×C4—1462no—
280C2×C4—1464no—
281C2×C4C22⋊C4×D7311yescharacteristic
282C2×C4C22⋊C4×D7311yescharacteristic
283C2×C4#1455655912yes—
284C2×C4#1455656112yes—
285C2×C4C2×#1192401611yescharacteristic
286C2×C4C2×#1192401811yescharacteristic
287C2×C4#1455662112yes—
288C8—82no—
289C8—5842no—
290C22—21no—
291C22—21no—
292C22—24no—
293C22—44no—
294C22—44no—
295C22—42no—
296C22—42no—
297C22#1798844211yescharacteristic
298C22C2×#1455651411yescharacteristic
299C22#1798859511yescharacteristic
300C4—21no—
301C4—22no—
302C4—42no—
303C4—2922no—
304C4—2922no—
305C4—2924no—
306C4#1798856812yes—
307C4#1798872512yes—
308C4#1798874212yes—
309C2—21no—
310C2—21no—
311C2—44no—
312C2#2260985611yescharacteristic
313C2#2260986511yescharacteristic
314C2#2260990211yescharacteristic
subgroupG/H
centerC22#17988442
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1800C2
socleC22×C73#379
prime radicalC22×D4×C73D4
cosocleC2×C42×C73C23
hypercenter#1800D73
hypocenterC73#8395
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #22608617C2semidirect
3. #22608359C2semidirect
163. C73#8395semidirect

Maximal subgroups

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

subgroupindex
C2×#179892942
#226086172
#226083592
#226082522
C73×#18002
#839573