Finite Group Databasebeta

← #32986543

Subgroups — #32986543

Order 18688.

804 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 121 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×#17988742C212yesmaximal
2C2×#17988324C212yesmaximal
3#22608201C211yesmaximal, characteristic
4#22607774C211yesmaximal, characteristic
5C73×#2659C211yesmaximal, characteristic
6C23×Dic146C2211yescharacteristic
7C22×#11924018—22no—
8#17988742—24no—
9#17988742—24no—
10C2×#14556514—22no—
11C2×C22⋊#10596831C2212yes—
12C2×#14556498—22no—
13C2×#14556498C2211yescharacteristic
14#17988324—24no—
15C2×C22⋊Q8×C73C2212yes—
16#398×C73—22no—
17#395×C73—22no—
18#380×C73—22no—
19C2×#95×C73C2211yescharacteristic
20C22×Dic146—22no—
21C22×Dic146—22no—
22C22×Dic146—21no—
23C22×Dic146—44no—
24C22×Dic146D412yes—
25C23×Dic73—21no—
26C23×Dic73—21no—
27C2×#11924018—28no—
28C2×#11924018—48no—
29C22×#10596831—22no—
30#14556514—28no—
31C22⋊#10596831—22no—
32C22⋊#10596831D414yes—
33#14556498—22no—
34#14556498—44no—
35C22×Q8×C73D412yes—
36C23×C4×C73D412yes—
37C23×C4×C73C2311yescharacteristic
38C22⋊Q8×C73—22no—
39C22⋊Q8×C73—22no—
40C22⋊Q8×C73—22no—
41C22⋊Q8×C73—22no—
42C2×C4⋊C4×C73—22no—
43C2×C4⋊C4×C73—22no—
44C2×C4⋊C4×C73—22no—
45C2×C4⋊C4×C73D412yes—
46C2×C22⋊C4×C73—22no—
47#95×C73—24no—
48#95×C73—22no—
49C2×Dic146—28no—
50C2×Dic146—24no—
51C2×Dic146—44no—
52C2×Dic146—48no—
53C2×Dic146—44no—
54C2×Dic146—44no—
55C2×Dic146—42no—
56C22×Dic73—24no—
57C22×Dic73—22no—
58C22×Dic73—22no—
59C22×Dic73—42no—
60C22×Dic73—41no—
61C22×Dic73—41no—
62C22×Dic73—42no—
63C22×Dic73—41no—
64C22×Dic73—41no—
65C22×Dic73—42no—
66#11924018—416no—
67C2×#10596831—24no—
68C2×#10596831—44no—
69C24×C73C2×D411yescharacteristic
70C2×Q8×C73—22no—
71C2×Q8×C73—22no—
72C2×Q8×C73—44no—
73C22×C4×C73—22no—
74C22×C4×C73—22no—
75C22×C4×C73—22no—
76C22×C4×C73—22no—
77C22×C4×C73—42no—
78C22×C4×C73—42no—
79C22×C4×C73—42no—
80C22×C4×C73Q1614yescomplemented
81C22×C4×C73C2×D411yescharacteristic
82C22×C4×C73C2×D412yes—
83C22×C4×C73C2×D412yes—
84C22×C4×C73#4111yescharacteristic
85C4⋊C4×C73—22no—
86C4⋊C4×C73—22no—
87C4⋊C4×C73—42no—
88C4⋊C4×C73—42no—
89C4⋊C4×C73—82no—
90C4⋊C4×C73—82no—
91C22⋊C4×C73—42no—
92C22⋊C4×C73—42no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—22no—
96C23×C73#12011yescharacteristic
97C23×C73#12111yescharacteristic
98C23×C73#12711yescharacteristic
99C23×C73C2×Q1612yes—
100C23×C73#13712yes—
101Q8×C73—44no—
102Q8×C73—82no—
103C2×C4×C73—24no—
104C2×C4×C73—24no—
105C2×C4×C73—41no—
106C2×C4×C73—42no—
107C2×C4×C73—41no—
108C2×C4×C73—42no—
109C2×C4×C73—42no—
110C2×C4×C73—44no—
111C2×C4×C73—42no—
112C2×C4×C73—42no—
113C2×C4×C73—42no—
114C2×C4×C73—42no—
115C2×C4×C73—44no—
116C2×C4×C73—82no—
117C2×C4×C73#12012yes—
118C2×C4×C73#12112yes—
119C2×Dic73—24no—
120C2×Dic73—42no—
121C2×Dic73—44no—
122C2×Dic73—42no—
123C2×Dic73—44no—
124C2×Dic73—44no—
125C2×Dic73—42no—
126C2×Dic73—82no—
127C2×Dic73—81no—
128C2×Dic73—81no—
129Dic146—28no—
130Dic146—44no—
131Dic146—48no—
132#10596831—44no—
133C22×C73—22no—
134C22×C73—22no—
135C22×C73—22no—
136C22×C73—22no—
137C22×C73—21no—
138C22×C73—21no—
139C22×C73—21no—
140C22×C73—21no—
141C22×C73#39211yescharacteristic
142C22×C73#45112yes—
143C22×C73#45511yescharacteristic
144C22×C73#45611yescharacteristic
145C22×C73B(4,𝔽2)12yes—
146C22×C73#46212yes—
147C22×C73#49411yescharacteristic
148C22×C73#49711yescharacteristic
149C4×C73—22no—
150C4×C73—22no—
151C4×C73—44no—
152C4×C73—82no—
153C4×C73—82no—
154Dic73—44no—
155Dic73—82no—
156#9967—731nomaximal, Sylow
157C2×C73—21no—
158C2×C73—22no—
159C2×C73—21no—
160C2×C73#160212yes—
161C2×C73#160512yes—
162C2×C73#200711yescharacteristic
163C2×C73#201611yescharacteristic
164C2×C73#202111yescharacteristic
165C2×#451—732no—
166#2659D7311yescharacteristic
167C2×#328—732no—
168#1446—731no—
169#1280—731no—
170C73#996711yescharacteristic, complemented, Sylow
171C23×Q8—731no—
172C22×Q16—1462no—
173C2×C22⋊Q8C2×D7312yes—
174#451—1464no—
175#451—1464no—
176C2×#103—1462no—
177C2×C22⋊C8—732no—
178#398—22no—
179#395—22no—
180#380—22no—
181C2×#95—731no—
182C2×#95—1462no—
183C2×#95C2×D7311yescharacteristic
184#328—1464no—
185C22×Q8—2924no—
186C22×Q8—732no—
187C22×Q8—1462no—
188C22×Q8—1462no—
189C22×Q8—1461no—
190C22×Q8C22⋊D7312yes—
191C23×C4—1461no—
192C23×C4—1461no—
193C23×C4C22⋊D7312yes—
194C23×C4C22×D7311yescharacteristic
195C2×Q16—2928no—
196C2×Q16—1468no—
197C22×C8—1462no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C22⋊D7312yes—
206C2×C22⋊C4—22no—
207#103—1468no—
208C22⋊C8—734no—
209C22⋊C8—1462no—
210#95—24no—
211#95—22no—
212#95—2924no—
213#95—1462no—
214C24C2×C22⋊D7311yescharacteristic
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—2924no—
219C2×Q8—2928no—
220C2×Q8—2924no—
221C2×Q8—2924no—
222C2×Q8—2922no—
223C2×Q8—1468no—
224C2×Q8—1464no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—2922no—
233C22×C4—2921no—
234C22×C4—2921no—
235C22×C4—2922no—
236C22×C4—2921no—
237C22×C4—2921no—
238C22×C4—2922no—
239C22×C4—1462no—
240C22×C4—1464no—
241C22×C4—1462no—
242C22×C4#1192401814yescomplemented
243C22×C4D4×D7311yescharacteristic
244C22×C4D4×D7312yes—
245C22×C4C2×C22⋊D7312yes—
246C22×C4#1192402711yescharacteristic
247Q16—29216no—
248C2×C8—2924no—
249C2×C8—1464no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23#1455656011yescharacteristic
262C23#1455656211yescharacteristic
263C23#1455658711yescharacteristic
264C23C2×#1192401812yes—
265C23#1455662212yes—
266Q8—44no—
267Q8—82no—
268Q8—2924no—
269Q8—2928no—
270Q8—1468no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—82no—
285C2×C4—2924no—
286C2×C4—2922no—
287C2×C4—2922no—
288C2×C4—2924no—
289C2×C4—2924no—
290C2×C4—2922no—
291C2×C4—5841no—
292C2×C4—5842no—
293C2×C4—5841no—
294C2×C4—1464no—
295C2×C4#1455655912yes—
296C2×C4#1455656312yes—
297C8—2924no—
298C22—22no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—22no—
306C22#1798849511yescharacteristic
307C22#1798874312yes—
308C22#1798876511yescharacteristic
309C22#1798877011yescharacteristic
310C22#1798877412yes—
311C22#1798879412yes—
312C22#1798896611yescharacteristic
313C22#1798897911yescharacteristic
314C4—22no—
315C4—22no—
316C4—44no—
317C4—82no—
318C4—82no—
319C4—2924no—
320C4—5842no—
321C2—21no—
322C2—21no—
323C2—22no—
324C2#2260888812yes—
325C2#2260890512yes—
326C2#2261097811yescharacteristic
327C2#2261103111yescharacteristic
328C2#2261105511yescharacteristic
subgroupG/H
centerC23#14556560
commutator, cosocleC23×C4×C73C23
FrattiniC23×C4C22×D73
FittingC73×#2659C2
socleC23×C73#120
prime radicalC24×C73C2×D4
hypercenter#2659D73
hypocenterC73#9967
LCSG ▷ C23×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
80. C22×C4×C73Q16semidirect
170. C73#9967semidirect
242. C22×C4#11924018semidirect

Maximal subgroups

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

subgroupindex
C2×#179887422
C2×#179883242
#226082012
#226077742
C73×#26592
#996773