Finite Group Databasebeta

← #33060838

Subgroups — #33060838

Order 18688.

910 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 191 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#22623741C211yesmaximal, characteristic
2#22620879C212yesmaximal
3#22608904C218yesmaximal
4#22608429C212yesmaximal
5#22608255C211yesmaximal, characteristic
6C73×#2567C211yesmaximal, characteristic
7Q8×Dic146—22no—
8#17989316—22no—
9#17989313—24no—
10#17989218—21no—
11#17989218C2214yes—
12#17989144—22no—
13#17989144C2212yes—
14#17989065—21no—
15#17988966—22no—
16#17988937—22no—
17#17988794—24no—
18#17988743C2218yes—
19#17988640C2212yes—
20#17988323C2218yes—
21#17988313C2214yes—
22C2×C4⋊Q8×C73C2211yescharacteristic
23C2×C4×Q8×C73C2211yescharacteristic
24#395×C73C2214yes—
25#391×C73—21no—
26#389×C73—22no—
27#384×C73—21no—
28C4×C4⋊C4×C73C2211yescharacteristic
29C2×#11924018—28no—
30#14556591—22no—
31#14556591—22no—
32#14556591D412yes—
33#14556591D412yes—
34#14556590—24no—
35#14556590—28no—
36#14556583—24no—
37#14556583—22no—
38#14556582—24no—
39#14556582—28no—
40#14556570—22no—
41#14556570—24no—
42#14556570C2314yes—
43Q8×Dic73—24no—
44Q8×Dic73—24no—
45C4×Dic146—24no—
46C4×Dic146—24no—
47C4×Dic146—22no—
48C4×Dic146D414yes—
49#14556550—24no—
50#14556550—22no—
51C4⋊#10596831—22no—
52#14556515—28no—
53C22⋊#10596831C2314yes—
54C4×#10596831—22no—
55C22×Q8×C73D411yescharacteristic
56C22×Q8×C73C2312yes—
57C4⋊Q8×C73—22no—
58C4⋊Q8×C73—22no—
59C4×Q8×C73—22no—
60C4×Q8×C73—22no—
61C2×C4⋊C4×C73—21no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C4⋊C4×C73—24no—
65C2×C4⋊C4×C73D411yescharacteristic
66C2×C4⋊C4×C73C2312yes—
67C2×C42×C73—21no—
68C2×C42×C73D412yes—
69C2×C42×C73C2311yescharacteristic
70#95×C73—24no—
71#95×C73C2312yes—
72C2×Dic146—22no—
73C2×Dic146—24no—
74C2×Dic146C2×D414yes—
75#11924018—88no—
76C2×#10596831—24no—
77C4⋊Dic73—24no—
78C4⋊Dic73—24no—
79C4⋊Dic73—24no—
80C4⋊Dic73—22no—
81C4⋊Dic73—22no—
82C4⋊Dic73—22no—
83C4⋊Dic73C2×D414yes—
84#11924007—28no—
85#11924007—24no—
86#11924007—24no—
87C22⋊Dic73—24no—
88C4×Dic73—28no—
89C4×Dic73—24no—
90C2×Q8×C73—22no—
91C2×Q8×C73—22no—
92C2×Q8×C73—24no—
93C2×Q8×C73—82no—
94C2×Q8×C73Q1614yescomplemented
95C22×C4×C73—22no—
96C22×C4×C73—22no—
97C22×C4×C73C2×D411yescharacteristic
98C22×C4×C73C2×D412yes—
99C22×C4×C73C2×D412yes—
100C22×C4×C73C2×D411yescharacteristic
101C22×C4×C73C2411yescharacteristic
102C4⋊C4×C73—24no—
103C4⋊C4×C73—22no—
104C4⋊C4×C73—24no—
105C4⋊C4×C73—41no—
106C4⋊C4×C73—42no—
107C4⋊C4×C73—41no—
108C4⋊C4×C73—84no—
109C42×C73—22no—
110C42×C73—22no—
111C42×C73—42no—
112C42×C73C2×D412yes—
113C42×C73C2×D412yes—
114C23×C73C22×D411yescharacteristic
115Q8×C73—28no—
116Q8×C73—48no—
117C2×C4×C73—22no—
118C2×C4×C73—22no—
119C2×C4×C73—22no—
120C2×C4×C73—24no—
121C2×C4×C73—22no—
122C2×C4×C73—41no—
123C2×C4×C73—42no—
124C2×C4×C73—42no—
125C2×C4×C73—42no—
126C2×C4×C73—44no—
127C2×C4×C73—42no—
128C2×C4×C73#12011yescharacteristic
129C2×C4×C73#12012yes—
130C2×C4×C73#12011yescharacteristic
131C2×C4×C73C2×Q1614yes—
132C2×C4×C73C2×Q1612yes—
133C2×C4×C73#13712yes—
134C2×C4×C73C22×D412yes—
135C2×Dic73—24no—
136C2×Dic73—24no—
137Dic146—44no—
138Dic146—44no—
139Dic146—84no—
140#10596831—44no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73C2×#12011yescharacteristic
144C22×C73C22×Q1611yescharacteristic
145C22×C73C2×#13711yescharacteristic
146C4×C73—24no—
147C4×C73—21no—
148C4×C73—22no—
149C4×C73—44no—
150C4×C73—42no—
151C4×C73—82no—
152C4×C73#45114yes—
153C4×C73#45612yes—
154Dic73—44no—
155Dic73—44no—
156#18404—731nomaximal, Sylow
157C2×C73—21no—
158C2×C73—21no—
159C2×C73C2×#45111yescharacteristic
160C2×C73C2×#45611yescharacteristic
161C2×C73#302711yescharacteristic
162#3502—731no—
163#3209—732no—
164#2567D7311yescharacteristic
165#1605—738no—
166#1503—732no—
167#1461—731no—
168C73#1840411yescharacteristic, complemented, Sylow
169Q82—1462no—
170#557—1464no—
171#557—1462no—
172#544—734no—
173#544—1461no—
174#533—732no—
175#533—1462no—
176C2×C4⋊Q8C2×D7311yescharacteristic
177#519—1461no—
178C2×C4×Q8C2×D7311yescharacteristic
179#494—1462no—
180#487—1462no—
181#462—1464no—
182#451—738no—
183#431—732no—
184#395C2×D7314yes—
185#391—21no—
186#389—22no—
187#384—21no—
188C4×C4⋊C4C2×D7311yescharacteristic
189#328—738no—
190#324—734no—
191C22×Q8C22⋊D7311yescharacteristic
192C22×Q8C22×D7312yes—
193C2×Q16—1468no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—732no—
197C4⋊Q8—732no—
198C4⋊Q8—1462no—
199C4⋊Q8—1462no—
200C4⋊Q8—1468no—
201C4⋊Q8—1464no—
202#125—1464no—
203#125—1468no—
204#125—1464no—
205#125—1462no—
206C22⋊Q8—734no—
207C22⋊Q8—1462no—
208C22⋊Q8—1464no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—734no—
212C4×Q8—1464no—
213C4×Q8—1464no—
214C4×Q8—1464no—
215C4×Q8—1464no—
216C4×Q8—1462no—
217#117—1462no—
218#117—1464no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4C22⋊D7311yescharacteristic
224C2×C4⋊C4C22×D7312yes—
225C2×C42—21no—
226C2×C42C22⋊D7312yes—
227C2×C42C22×D7311yescharacteristic
228C4⋊C8—1462no—
229#103—1468no—
230C22⋊C8—734no—
231C4×C8—1462no—
232#95—24no—
233#95C22×D7312yes—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—82no—
238C2×Q8—734no—
239C2×Q8—1464no—
240C2×Q8—1462no—
241C2×Q8#1192401814yescomplemented
242C22×C4—22no—
243C22×C4—22no—
244C22×C4C2×C22⋊D7311yescharacteristic
245C22×C4C2×C22⋊D7312yes—
246C22×C4C2×C22⋊D7312yes—
247C22×C4C2×C22⋊D7311yescharacteristic
248C22×C4C23×D7311yescharacteristic
249Q16—5848no—
250C2×C8—1464no—
251C4⋊C4—24no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—84no—
258C4⋊C4—734no—
259C4⋊C4—1468no—
260C4⋊C4—1464no—
261C4⋊C4—1464no—
262C4⋊C4—1464no—
263C4⋊C4—1464no—
264C4⋊C4—1462no—
265C4⋊C4—1462no—
266C4⋊C4—1462no—
267C4⋊C4—1464no—
268C22⋊C4—1464no—
269C42—22no—
270C42—22no—
271C42—42no—
272C42—1468no—
273C42—1464no—
274C42D4×D7312yes—
275C42D4×D7312yes—
276C23C22×C22⋊D7311yescharacteristic
277Q8—28no—
278Q8—48no—
279Q8—2924no—
280Q8—2924no—
281Q8—5844no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—1464no—
294C2×C4—1464no—
295C2×C4#1455655911yescharacteristic
296C2×C4#1455655912yes—
297C2×C4#1455655911yescharacteristic
298C2×C4C2×#1192401814yes—
299C2×C4C2×#1192401812yes—
300C2×C4#1455662112yes—
301C2×C4C2×D4×D7312yes—
302C8—2924no—
303C22—21no—
304C22—21no—
305C22C2×#1455655911yescharacteristic
306C22C22×#1192401811yescharacteristic
307C22C2×#1455662111yescharacteristic
308C4—21no—
309C4—22no—
310C4—24no—
311C4—44no—
312C4—42no—
313C4—82no—
314C4—2924no—
315C4—2924no—
316C4#1798874214yes—
317C4#1798876912yes—
318C2—21no—
319C2—21no—
320C2C2×#1798874211yescharacteristic
321C2C2×#1798876911yescharacteristic
322C2#2261933211yescharacteristic
subgroupG/H
centerC22C2×#14556559
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2567C2
socleC22×C73C2×#120
prime radicalC23×C73C22×D4
hypercenter#2567D73
hypocenterC73#18404
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2567

Complements

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

NH ≅ G/Ntype
94. C2×Q8×C73Q16semidirect
168. C73#18404semidirect
241. C2×Q8#11924018semidirect

Maximal subgroups

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

subgroupindex
#226237412
#226208792
#226089042
#226084292
#226082552
C73×#25672
#1840473