Finite Group Databasebeta

← #32983959

Subgroups — #32983959

Order 18688.

522 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 111 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#22618972C211yesmaximal, characteristic
2#22610011C212yesmaximal, complemented
3#22608269C211yesmaximal, characteristic
4#22607837C212yesmaximal, complemented
5C73×#2487C211yesmaximal, characteristic
6#17988665—24no—
7#17988640—21no—
8#17988551—21no—
9#17988551C2212yes—
10#17988551C2212yes—
11#17988356—24no—
12#17988313—21no—
13#17988311—21no—
14C2×C4×D4×C73C2211yescharacteristic
15C22×C4⋊C4×C73C2211yescharacteristic
16#400×C73—21no—
17#399×C73—21no—
18#389×C73—21no—
19#386×C73C412yes—
20#385×C73C412yes—
21#380×C73—21no—
22#380×C73C2211yescharacteristic
23C2×#11924012—22no—
24C2×#11924012—22no—
25C2×#11924012D414yes—
26C4⋊#10596831—22no—
27C22⋊#10596831—22no—
28C22⋊#10596831—22no—
29C22⋊#10596831—22no—
30C22⋊#10596831—22no—
31C22⋊#10596831—22no—
32C4×#10596831—22no—
33C22×D4×C73C2×C411yescharacteristic
34C23×C4×C73C2×C412yes—
35C23×C4×C73D411yescharacteristic
36C23×C4×C73C2311yescharacteristic
37C4×D4×C73—22no—
38C4×D4×C73—22no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73—22no—
42C2×C4⋊C4×C73—21no—
43C2×C4⋊C4×C73—22no—
44C2×C4⋊C4×C73—22no—
45C2×C4⋊C4×C73D411yescharacteristic
46C2×C22⋊C4×C73—22no—
47C2×C22⋊C4×C73C2×C411yescharacteristic
48C2×C22⋊C4×C73C2×C412yes—
49C2×C22⋊C4×C73D411yescharacteristic
50C2×C42×C73D411yescharacteristic
51#95×C73—22no—
52#95×C73—22no—
53#95×C73—21no—
54#11924012—44no—
55#11924012—44no—
56#11924012—82no—
57C2×#10596831—24no—
58C2×#10596831—42no—
59C24×C73C22⋊C411yescharacteristic
60C24×C73C22×C411yescharacteristic
61C2×D4×C73—22no—
62C2×D4×C73—22no—
63C2×D4×C73—41no—
64C22×C4×C73—21no—
65C22×C4×C73—21no—
66C22×C4×C73—22no—
67C22×C4×C73—22no—
68C22×C4×C73—24no—
69C22×C4×C73—42no—
70C22×C4×C73—42no—
71C22×C4×C73—82no—
72C22×C4×C73C22⋊C411yescharacteristic
73C22×C4×C73C22⋊C411yescharacteristic
74C22×C4×C73C22⋊C411yescharacteristic
75C22×C4×C73C2×D412yes—
76C22×C4×C73C2×D411yescharacteristic
77C22×C4×C73C2×D411yescharacteristic
78C22×C4×C73#4112yes—
79C4⋊C4×C73—22no—
80C4⋊C4×C73—42no—
81C4⋊C4×C73—41no—
82C4⋊C4×C73—42no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73—82no—
85C22⋊C4×C73—22no—
86C22⋊C4×C73—22no—
87C22⋊C4×C73—42no—
88C22⋊C4×C73—82no—
89C42×C73—21no—
90C42×C73—21no—
91C23×C73—21no—
92C23×C73—21no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—41no—
97C23×C73—41no—
98C23×C73—41no—
99C23×C73—41no—
100C23×C73C2×C22⋊C411yescharacteristic
101C23×C73C4×D412yes—
102D4×C73—44no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—22no—
106C2×C4×C73—24no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—41no—
110C2×C4×C73—42no—
111C2×C4×C73—42no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—42no—
115C2×C4×C73—42no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—81no—
121C2×C4×C73—84no—
122C2×C4×C73#10112yes—
123C2×C4×C73#12011yescharacteristic
124C2×C4×C73#12112yes—
125C2×C4×C73#12311yescharacteristic
126#10596831—44no—
127#10596831—82no—
128C22×C73—21no—
129C22×C73—21no—
130C22×C73—22no—
131C22×C73—22no—
132C22×C73—22no—
133C22×C73—42no—
134C22×C73—42no—
135C22×C73—42no—
136C22×C73—41no—
137C22×C73—41no—
138C22×C73—41no—
139C22×C73—81no—
140C22×C73#38611yescharacteristic
141C22×C73C2×#10111yescharacteristic
142C22×C73#41311yescharacteristic
143C4×C73—22no—
144C4×C73—21no—
145C4×C73—22no—
146C4×C73—41no—
147C4×C73—41no—
148C4×C73—82no—
149C4×C73—82no—
150C4×C73#45812yes—
151#9510—731nomaximal, Sylow
152C2×C73—21no—
153C2×C73—21no—
154C2×C73—42no—
155C2×C73—81no—
156C2×C73#189011yescharacteristic
157C2×C73#190111yescharacteristic
158C2×C73#191411yescharacteristic
159#2975—731no—
160#2487D7311yescharacteristic
161#1821—732no—
162#1465—731no—
163#1305—732no—
164C73#951011yescharacteristic, complemented, Sylow
165C2×C4×D4C2×D7311yescharacteristic
166C22×C4⋊C4C2×D7311yescharacteristic
167#436—1464no—
168#431—1461no—
169#407—732no—
170#407—732no—
171#407—1461no—
172#400—21no—
173#399—21no—
174#389—21no—
175#386Dic7312yes—
176#385Dic7312yes—
177#380—21no—
178#380C2×D7311yescharacteristic
179#343—1464no—
180#324—1461no—
181#323—1461no—
182C22×D4C2×Dic7311yescharacteristic
183C23×C4C2×Dic7312yes—
184C23×C4C22⋊D7311yescharacteristic
185C23×C4C22×D7311yescharacteristic
186C2×M16—734no—
187C2×M16—1462no—
188C2×M16—1462no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C22⋊D7311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4C2×Dic7311yescharacteristic
200C2×C22⋊C4C2×Dic7312yes—
201C2×C22⋊C4C22⋊D7311yescharacteristic
202C2×C42C22⋊D7311yescharacteristic
203C4⋊C8—1462no—
204C22⋊C8—1462no—
205C22⋊C8—1462no—
206C22⋊C8—1462no—
207C22⋊C8—1462no—
208C22⋊C8—1462no—
209C4×C8—1462no—
210#95—22no—
211#95—22no—
212#95—21no—
213C24C22⋊Dic7311yescharacteristic
214C24C22×Dic7311yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4C22⋊Dic7311yescharacteristic
227C22×C4C22⋊Dic7311yescharacteristic
228C22×C4C22⋊Dic7311yescharacteristic
229C22×C4D4×D7312yes—
230C22×C4C2×C22⋊D7311yescharacteristic
231C22×C4C2×C22⋊D7311yescharacteristic
232C22×C4#1192402712yes—
233M16—2924no—
234M16—2924no—
235M16—5842no—
236C2×C8—2922no—
237C2×C8—1464no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—82no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—42no—
247C22⋊C4—82no—
248C42—21no—
249C42—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23C2×C22⋊Dic7311yescharacteristic
260C23D4×Dic7312yes—
261D4—44no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—84no—
281C2×C4#1455651012yes—
282C2×C4#1455655911yescharacteristic
283C2×C4#1455656312yes—
284C2×C4#1455657211yescharacteristic
285C8—2924no—
286C8—5842no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22#1798846711yescharacteristic
300C22C2×#1455651011yescharacteristic
301C22#1798857111yescharacteristic
302C4—22no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—82no—
308C4—82no—
309C4#1798877612yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2#2261034511yescharacteristic
315C2#2261040011yescharacteristic
316C2#2261047911yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC24×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2487C2
socleC22×C73#386
prime radicalC22×D4×C73C2×C4
cosocleC23×C4×C73C23
hypercenter#2487D73
hypocenterC73#9510
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. #22610011C2semidirect
4. #22607837C2semidirect
164. C73#9510semidirect

Maximal subgroups

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

subgroupindex
#226189722
#226100112
#226082692
#226078372
C73×#24872
#951073