Finite Group Databasebeta

← #32980923

Subgroups — #32980923

Order 18688.

578 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 115 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×#17988982C211yesmaximal, characteristic
2#22610903C212yesmaximal, complemented
3C2×#17988327C211yesmaximal, characteristic
4#22607812C212yesmaximal, complemented
5C73×#2379C211yesmaximal, characteristic
6C22×#11924018—21no—
7C22×#11924016—21no—
8C2×#14556591C2211yescharacteristic
9#17988982—22no—
10#17988982C414yescomplemented
11C2×#14556502C2212yes—
12C2×#14556502C2211yescharacteristic
13#17988349—22no—
14#17988349—22no—
15#17988327—22no—
16C2×#124×C73C2211yescharacteristic
17#391×C73—21no—
18#388×C73—22no—
19#386×C73—21no—
20#385×C73—22no—
21#382×C73C2212yes—
22C22×Dic146—21no—
23C2×#11924018—24no—
24C2×#11924018—42no—
25C2×#11924018—42no—
26C2×#11924016—24no—
27C2×#11924016—42no—
28C2×#11924016—42no—
29C22×#10596831—21no—
30C22×#10596831—22no—
31#14556591—21no—
32#14556591—22no—
33#14556591C2×C412yescomplemented
34C2×C4⋊Dic73—22no—
35#14556502—21no—
36#14556502—22no—
37#14556502—22no—
38#14556502C2×C412yescomplemented
39C22×Q8×C73D411yescharacteristic
40C22×D4×C73D411yescharacteristic
41C23×C4×C73—21no—
42#124×C73—22no—
43#124×C73C2×C412yes—
44#124×C73D412yes—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73D412yes—
47C2×C4⋊C4×C73D412yes—
48C2×C22⋊C4×C73—22no—
49C2×C22⋊C4×C73—22no—
50C2×C22⋊C4×C73—21no—
51C2×C42×C73—21no—
52C2×C42×C73C2311yescharacteristic
53#95×C73—22no—
54#95×C73—22no—
55#95×C73—22no—
56C2×Dic146—22no—
57C2×Dic146—41no—
58C2×Dic146—41no—
59C2×Dic146—41no—
60C2×Dic146—41no—
61C2×Dic146—41no—
62C22×Dic73—41no—
63#11924018—48no—
64#11924016—48no—
65C2×#10596831—22no—
66C2×#10596831—22no—
67C2×#10596831—22no—
68C2×#10596831—24no—
69C2×#10596831—42no—
70C2×#10596831—42no—
71C4⋊Dic73—24no—
72C4⋊Dic73—42no—
73C24×C73—21no—
74C2×Q8×C73—21no—
75C2×Q8×C73—41no—
76C2×Q8×C73—41no—
77C2×Q8×C73C22⋊C412yes—
78C2×D4×C73—21no—
79C2×D4×C73—41no—
80C2×D4×C73—41no—
81C2×D4×C73C22⋊C412yes—
82C22×C4×C73—21no—
83C22×C4×C73—22no—
84C22×C4×C73—22no—
85C22×C4×C73—81no—
86C22×C4×C73—82no—
87C22×C4×C73C2×D411yescharacteristic
88C22×C4×C73C2×D412yes—
89C4⋊C4×C73—42no—
90C4⋊C4×C73—42no—
91C4⋊C4×C73—81no—
92C22⋊C4×C73—24no—
93C22⋊C4×C73—24no—
94C22⋊C4×C73—81no—
95C22⋊C4×C73—82no—
96C42×C73—81no—
97C42×C73C22×C411yescharacteristic
98C42×C73C2×D411yescharacteristic
99C42×C73#4112yes—
100C23×C73—22no—
101C23×C73—41no—
102C23×C73—41no—
103C23×C73—42no—
104C23×C73—42no—
105C23×C73#12011yescharacteristic
106Q8×C73—42no—
107Q8×C73—81no—
108D4×C73—42no—
109D4×C73—81no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—22no—
113C2×C4×C73—22no—
114C2×C4×C73—22no—
115C2×C4×C73—41no—
116C2×C4×C73—41no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—42no—
122C2×C4×C73—42no—
123C2×C4×C73—84no—
124C2×C4×C73C2×C22⋊C411yescharacteristic
125C2×C4×C73C4×D412yes—
126C2×C4×C73#12112yes—
127C2×C4×C73#12311yescharacteristic
128C2×Dic73—42no—
129C2×Dic73—81no—
130C2×Dic73—81no—
131Dic146—42no—
132Dic146—42no—
133Dic146—81no—
134Dic146—81no—
135#10596831—24no—
136#10596831—44no—
137C22×C73—44no—
138C22×C73—42no—
139C22×C73—81no—
140C22×C73—81no—
141C22×C73—82no—
142C22×C73#38611yescharacteristic
143C22×C73#45311yescharacteristic
144C22×C73#45311yescharacteristic
145C22×C73#45512yes—
146C22×C73#45612yes—
147C4×C73—22no—
148C4×C73—22no—
149C4×C73—21no—
150C4×C73—21no—
151C4×C73—42no—
152C4×C73—82no—
153C4×C73—82no—
154Dic73—82no—
155#9003—731nomaximal, Sylow
156C2×C73—82no—
157C2×C73#165911yescharacteristic
158C2×C73#166011yescharacteristic
159C2×C73#166112yes—
160C2×C73#188212yes—
161C2×C73#188711yescharacteristic
162C2×#497—731no—
163#2379D7311yescharacteristic
164#1993—732no—
165C2×#329—731no—
166#1295—732no—
167C73#900311yescharacteristic, complemented, Sylow
168C22×Q16—1461no—
169C22×SD16—1461no—
170C2×C4⋊Q8—731no—
171C2×#124C2×D7311yescharacteristic
172#497—734no—
173#497—1462no—
174C2×#97—731no—
175C2×#97—732no—
176#391—21no—
177#388—22no—
178#386—21no—
179#385—22no—
180#382C2×D7312yes—
181#340—1462no—
182#340—1462no—
183#329—1462no—
184C22×Q8—1461no—
185C22×Q8C22⋊D7311yescharacteristic
186C22×D4C22⋊D7311yescharacteristic
187C23×C4—21no—
188C2×Q16—2922no—
189C2×Q16—2922no—
190C2×Q16—1464no—
191C2×SD16—2922no—
192C2×SD16—2922no—
193C2×SD16—1464no—
194C22×C8—1462no—
195C22×C8—1461no—
196C4⋊Q8—732no—
197C4⋊Q8—1461no—
198C4⋊Q8—1462no—
199#124—22no—
200#124C4×D7312yes—
201#124D29212yes—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—1462no—
204C2×C4⋊C4C22⋊D7312yes—
205C2×C4⋊C4C22⋊D7312yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42C22×D7311yescharacteristic
211#97—732no—
212#97—1462no—
213#97—1461no—
214#97—1462no—
215#95—22no—
216#95—22no—
217#95—22no—
218C24—21no—
219C2×Q8—21no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—2921no—
223C2×Q8—2921no—
224C2×Q8—2921no—
225C2×Q8—2921no—
226C2×Q8—2921no—
227C2×Q8—1462no—
228C2×Q8#1192400512yes—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4#1192400512yes—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—81no—
237C22×C4—82no—
238C22×C4—2921no—
239C22×C4C2×C22⋊D7311yescharacteristic
240C22×C4C2×C22⋊D7312yes—
241Q16—2928no—
242SD16—2928no—
243C2×C8—2922no—
244C2×C8—2922no—
245C2×C8—1464no—
246C2×C8—1462no—
247C2×C8—1462no—
248C2×C8—1462no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—81no—
252C4⋊C4—2922no—
253C4⋊C4—1464no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—81no—
257C22⋊C4—82no—
258C42—81no—
259C42C2×C4×D7311yescharacteristic
260C42C2×D29211yescharacteristic
261C42#1192402812yes—
262C23—22no—
263C23—41no—
264C23—41no—
265C23—42no—
266C23—42no—
267C23#1455656011yescharacteristic
268Q8—42no—
269Q8—81no—
270Q8—2922no—
271Q8—2922no—
272Q8—5841no—
273Q8—5841no—
274D4—42no—
275D4—81no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4—2922no—
291C2×C4—5841no—
292C2×C4—5841no—
293C2×C4C2×#1192400511yescharacteristic
294C2×C4C4×C22⋊D7312yes—
295C2×C4#1455656512yes—
296C2×C4#1455657411yescharacteristic
297C8—2924no—
298C8—1464no—
299C22—44no—
300C22—42no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22#1798846611yescharacteristic
305C22#1798875511yescharacteristic
306C22#1798875511yescharacteristic
307C22#1798876512yes—
308C22#1798877012yes—
309C4—22no—
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—82no—
315C4—82no—
316C4—5842no—
317C2—82no—
318C2#2260923711yescharacteristic
319C2#2260924211yescharacteristic
320C2#2260924812yes—
321C2#2261030012yes—
322C2#2261032911yescharacteristic
subgroupG/H
centerC23#14556560
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2379C2
socleC23×C73#120
prime radicalC22×D4×C73D4
cosocleC2×C42×C73C23
hypercenter#2379D73
hypocenterC73#9003
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2379

Complements

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

NH ≅ G/Ntype
2. #22610903C2semidirect
4. #22607812C2semidirect
10. #17988982C4semidirect
33. #14556591C2×C4semidirect
38. #14556502C2×C4semidirect
167. C73#9003semidirect

Maximal subgroups

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

subgroupindex
C2×#179889822
#226109032
C2×#179883272
#226078122
C73×#23792
#900373