Finite Group Databasebeta

← #32983439

Subgroups — #32983439

Order 18688.

606 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 103 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#22615089C211yesmaximal, characteristic
2#22608450C212yesmaximal, complemented
3#22608258C211yesmaximal, characteristic
4#22607817C212yesmaximal, complemented
5C73×#2312C211yesmaximal, characteristic
6C2×#14556570C2211yescharacteristic
7#17988551C2212yes—
8#17988512C412yescomplemented
9#17988499C412yescomplemented
10#17988481—22no—
11#17988450—21no—
12#17988443—21no—
13C4×C22⋊Dic73C2211yescharacteristic
14#17988323—24no—
15#17988311—22no—
16C22×C22⋊C4×C73C2211yescharacteristic
17#388×C73—24no—
18#380×C73—21no—
19#379×C73—21no—
20C4×C22⋊C4×C73C2212yes—
21C22×Dic146C2×C411yescharacteristic
22C2×#11924012—22no—
23#14556570—22no—
24#14556570—22no—
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×#11924007—21no—
27C2×C22⋊Dic73—22no—
28C2×C22⋊Dic73—22no—
29C2×C22⋊Dic73—21no—
30C2×C22⋊Dic73—21no—
31C2×C4×Dic73—21no—
32C22⋊#10596831—22no—
33C22⋊#10596831—22no—
34#14556498—22no—
35#14556498—22no—
36#14556498C2×C412yes—
37C25×C73D411yescharacteristic
38C23×C4×C73D411yescharacteristic
39C23×C4×C73C2311yescharacteristic
40C2×C4⋊C4×C73—22no—
41C2×C22⋊C4×C73—22no—
42C2×C22⋊C4×C73—22no—
43C2×C22⋊C4×C73—22no—
44C2×C22⋊C4×C73—22no—
45C2×C22⋊C4×C73—22no—
46C2×C22⋊C4×C73C2×C412yes—
47C2×C22⋊C4×C73D412yes—
48C2×C42×C73—22no—
49#95×C73—22no—
50#95×C73D414yes—
51C2×Dic146—22no—
52C2×Dic146—81no—
53C22×Dic73—22no—
54C22×Dic73—21no—
55C22×Dic73—21no—
56#11924012—82no—
57C2×#10596831—42no—
58C4⋊Dic73—22no—
59#11924007—42no—
60C22⋊Dic73—42no—
61C22⋊Dic73—44no—
62C22⋊Dic73—41no—
63C22⋊Dic73—41no—
64C22⋊Dic73—82no—
65C4×Dic73—81no—
66C24×C73—21no—
67C24×C73—21no—
68C24×C73—42no—
69C24×C73—42no—
70C24×C73—81no—
71C24×C73C22⋊C412yes—
72C24×C73C2×D411yescharacteristic
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—24no—
76C22×C4×C73—42no—
77C22×C4×C73—81no—
78C22×C4×C73C22⋊C412yes—
79C22×C4×C73C22×C411yescharacteristic
80C22×C4×C73C2×D412yes—
81C22×C4×C73C2×D411yescharacteristic
82C22×C4×C73#4112yes—
83C4⋊C4×C73—82no—
84C22⋊C4×C73—24no—
85C22⋊C4×C73—24no—
86C22⋊C4×C73—44no—
87C22⋊C4×C73—82no—
88C22⋊C4×C73—82no—
89C42×C73—82no—
90C23×C73—22no—
91C23×C73—22no—
92C23×C73—22no—
93C23×C73—22no—
94C23×C73—22no—
95C23×C73—41no—
96C23×C73—42no—
97C23×C73—42no—
98C23×C73—42no—
99C23×C73—42no—
100C23×C73—42no—
101C23×C73—42no—
102C23×C73—81no—
103C23×C73—81no—
104C23×C73—88no—
105C23×C73C2×C22⋊C411yescharacteristic
106C23×C73#12011yescharacteristic
107C23×C73#12311yescharacteristic
108C2×C4×C73—22no—
109C2×C4×C73—42no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73—42no—
113C2×C4×C73—42no—
114C2×C4×C73—42no—
115C2×C4×C73—44no—
116C2×C4×C73—44no—
117C2×C4×C73—44no—
118C2×C4×C73—81no—
119C2×C4×C73C4×D412yes—
120C2×C4×C73#12112yes—
121C2×Dic73—42no—
122C2×Dic73—42no—
123C2×Dic73—42no—
124C2×Dic73—41no—
125C2×Dic73—41no—
126C2×Dic73—42no—
127C2×Dic73—42no—
128Dic146—82no—
129#10596831—82no—
130C22×C73—21no—
131C22×C73—21no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73—44no—
138C22×C73—44no—
139C22×C73—42no—
140C22×C73—42no—
141C22×C73—42no—
142C22×C73—82no—
143C22×C73—84no—
144C22×C73—84no—
145C22×C73#38611yescharacteristic
146C22×C73#42112yes—
147C22×C73#45512yes—
148C4×C73—22no—
149C4×C73—22no—
150C4×C73—42no—
151C4×C73—84no—
152Dic73—82no—
153Dic73—81no—
154Dic73—81no—
155#9411—731nomaximal, Sylow
156C2×C73—22no—
157C2×C73—22no—
158C2×C73—41no—
159C2×C73—82no—
160C2×C73#188511yescharacteristic
161C2×C73#189711yescharacteristic
162C2×C73#191311yescharacteristic
163#2561—731no—
164#2312D7311yescharacteristic
165#1508—732no—
166#1462—731no—
167#1297—732no—
168C73#941111yescharacteristic, complemented, Sylow
169C2×C22⋊Q8—731no—
170C22×C22⋊C4C2×D7311yescharacteristic
171#407—732no—
172#397—732no—
173#393—732no—
174#388—24no—
175#388—1462no—
176#381—1461no—
177#380—21no—
178#379—21no—
179#379—1461no—
180C4×C22⋊C4—731no—
181C4×C22⋊C4C2×D7312yes—
182#328—1464no—
183#323—1462no—
184C25C22⋊D7311yescharacteristic
185C22×Q8—731no—
186C23×C4C22⋊D7311yescharacteristic
187C23×C4C22×D7311yescharacteristic
188C2×M16—1462no—
189C22⋊Q8—1462no—
190C22⋊Q8—1462no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—731no—
193C2×C4⋊C4—1461no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—1462no—
200C2×C22⋊C4—1461no—
201C2×C22⋊C4—1461no—
202C2×C22⋊C4—1462no—
203C2×C22⋊C4C4×D7312yes—
204C2×C22⋊C4D29212yes—
205C2×C42—22no—
206C2×C42—1461no—
207C22⋊C8—1462no—
208C22⋊C8—1462no—
209#95—22no—
210#95—732no—
211#95—1462no—
212#95—1462no—
213#95C22⋊D7314yes—
214C24—21no—
215C24—21no—
216C24—42no—
217C24—42no—
218C24—81no—
219C24#1192400512yes—
220C24C2×C22⋊D7311yescharacteristic
221C2×Q8—5841no—
222C2×Q8—1462no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—81no—
228C22×C4—1462no—
229C22×C4—1461no—
230C22×C4—1461no—
231C22×C4#1192400512yes—
232C22×C4C2×C4×D7311yescharacteristic
233C22×C4C2×C22⋊D7312yes—
234C22×C4C2×D29211yescharacteristic
235C22×C4#1192402812yes—
236M16—5842no—
237C2×C8—2922no—
238C4⋊C4—82no—
239C4⋊C4—2922no—
240C4⋊C4—1462no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—44no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—2922no—
247C22⋊C4—2924no—
248C22⋊C4—2921no—
249C22⋊C4—2921no—
250C22⋊C4—5842no—
251C42—82no—
252C42—5841no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—81no—
266C23—81no—
267C23—88no—
268C23C2×#1192400511yescharacteristic
269C23#1455656011yescharacteristic
270C23#1455657411yescharacteristic
271Q8—5842no—
272C2×C4—22no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—81no—
283C2×C4—2922no—
284C2×C4—2922no—
285C2×C4—2922no—
286C2×C4—2921no—
287C2×C4—2921no—
288C2×C4—2922no—
289C2×C4—2922no—
290C2×C4C4×C22⋊D7312yes—
291C2×C4#1455656512yes—
292C8—5842no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—44no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—84no—
307C22—84no—
308C22#1798846611yescharacteristic
309C22#1798860512yes—
310C22#1798876512yes—
311C4—22no—
312C4—22no—
313C4—42no—
314C4—84no—
315C4—5842no—
316C4—5841no—
317C4—5841no—
318C2—22no—
319C2—22no—
320C2—41no—
321C2—82no—
322C2#2261031911yescharacteristic
323C2#2261037911yescharacteristic
324C2#2261047311yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2312C2
socleC22×C73#386
prime radicalC25×C73D4
cosocleC23×C4×C73C23
hypercenter#2312D73
hypocenterC73#9411
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #22608450C2semidirect
4. #22607817C2semidirect
8. #17988512C4semidirect
9. #17988499C4semidirect
168. C73#9411semidirect

Maximal subgroups

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

subgroupindex
#226150892
#226084502
#226082582
#226078172
C73×#23122
#941173