Finite Group Databasebeta

← #32983696

Subgroups — #32983696

Order 18688.

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

normal subgroups only

This group has 536 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#22619016C211yesmaximal, characteristic, complemented
2#22614930C211yesmaximal, characteristic
3#22608644C212yesmaximal
4#22607786C212yesmaximal, complemented
5C73×#1467C211yesmaximal, characteristic, complemented
6C2×#14556588C2211yescharacteristic
7C2×#14556580C412yes—
8#17988718—24no—
9#17988644—21no—
10#17988644C2212yes—
11#17988640—21no—
12#17988504—21no—
13#17988504—21no—
14#17988489—21no—
15#17988489—21no—
16#17988488C412yes—
17C4×C4⋊Dic73C2211yescharacteristic
18#17988330—24no—
19C2×C4×Q8×C73C2211yescharacteristic
20#432×C73C2212yes—
21#324×C73—21no—
22#324×C73—21no—
23C22×D292—21no—
24C22×D292—21no—
25C22×D292C2×C412yes—
26#14556588—22no—
27#14556588—22no—
28#14556580—44no—
29C2×C4⋊Dic73—21no—
30C2×#11924005—22no—
31C2×#11924005—22no—
32C2×#11924005—22no—
33C2×#11924005—22no—
34C2×C4×Dic73C2×C412yes—
35C4⋊#10596831—22no—
36C4⋊#10596831—22no—
37C4⋊#10596831D414yes—
38C22⋊#10596831—22no—
39#14556502—22no—
40#14556502—22no—
41C4×#10596831—22no—
42#14556498—21no—
43C22×Q8×C73C2×C411yescharacteristic
44C4×Q8×C73—22no—
45C4×Q8×C73—22no—
46C2×C4⋊C4×C73C2×C411yescharacteristic
47C2×C4⋊C4×C73D411yescharacteristic
48C2×C4⋊C4×C73D411yescharacteristic
49C2×C42×C73D411yescharacteristic
50C2×C42×C73D411yescharacteristic
51C2×C42×C73C2311yescharacteristic
52C4⋊C8×C73—22no—
53C22⋊C8×C73—22no—
54#97×C73—22no—
55C23×D73—22no—
56C2×D292—24no—
57C2×D292—42no—
58C2×D292—42no—
59C2×D292—42no—
60C2×D292—42no—
61C2×D292—41no—
62C2×D292—42no—
63C2×D292—41no—
64C2×D292—82no—
65C22×Dic73—22no—
66C2×#10596831—24no—
67C4⋊Dic73—42no—
68#11924005—44no—
69#11924005—44no—
70#11924005—82no—
71#11924005—82no—
72C4×Dic73—42no—
73C2×Q8×C73—22no—
74C2×Q8×C73—22no—
75C2×Q8×C73—41no—
76C22×C4×C73C22⋊C411yescharacteristic
77C22×C4×C73C22⋊C411yescharacteristic
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22×C411yescharacteristic
81C22×C4×C73C2×D411yescharacteristic
82C22×C4×C73C2×D411yescharacteristic
83C2×C8×C73—42no—
84C4⋊C4×C73—22no—
85C4⋊C4×C73—22no—
86C4⋊C4×C73—22no—
87C42×C73—21no—
88C42×C73—21no—
89C42×C73—21no—
90C42×C73—21no—
91C42×C73C2×D412yes—
92C42×C73#4112yes—
93C23×C73C2×C22⋊C411yescharacteristic
94C22×D73—42no—
95C22×D73—42no—
96C22×D73—42no—
97C22×D73—42no—
98C22×D73—42no—
99C22×D73—42no—
100C22×D73—42no—
101Q8×C73—44no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—22no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73#10012yes—
116C2×C4×C73C4×D412yes—
117C2×C4×C73#12011yescharacteristic
118C2×C4×C73#12112yes—
119C2×C4×C73#12311yescharacteristic
120C2×Dic73—42no—
121C2×Dic73—42no—
122C2×Dic73—42no—
123D292—84no—
124D292—82no—
125D292—84no—
126D292—82no—
127C8×C73—82no—
128#10596831—44no—
129C22×C73—21no—
130C22×C73—21no—
131C22×C73#38611yescharacteristic
132C22×C73C2×#10011yescharacteristic
133C22×C73#41311yescharacteristic
134C2×D73—82no—
135C2×D73—82no—
136C2×D73—82no—
137C2×D73—82no—
138C2×D73—82no—
139C2×D73—82no—
140C2×D73—82no—
141C4×C73—21no—
142C4×C73—22no—
143C4×C73—22no—
144C4×C73—22no—
145C4×C73—41no—
146C4×C73—41no—
147C4×C73—41no—
148C4×C73#45412yes—
149Dic73—82no—
150#9458—731nomaximal, Sylow
151C2×C73—21no—
152C2×C73—21no—
153C2×C73#188811yescharacteristic
154C2×C73#190011yescharacteristic
155C2×C73#191011yescharacteristic
156D73—162no—
157#2983—731no—
158#2548—731no—
159#1555—732no—
160#1467D7311yescharacteristic, complemented
161#1285—732no—
162C73#945811yescharacteristic, complemented, Sylow
163C2×#127—731no—
164C2×#124—732no—
165C2×C4×Q8C2×D7311yescharacteristic
166#446—1464no—
167#432—732no—
168#432—1461no—
169#432C2×D7312yes—
170#431—1461no—
171#394—1461no—
172#394—1461no—
173#390—732no—
174#390—1461no—
175#390—1461no—
176C4×C4⋊C4—731no—
177#331—1464no—
178#324—21no—
179#324—21no—
180C22×Q8C4×D7311yescharacteristic
181C22×D4—732no—
182C22×D4—1461no—
183C22×D4—1461no—
184#127—1462no—
185#127—1462no—
186#124—2924no—
187C4×Q8—22no—
188C4×Q8—22no—
189C2×C4⋊C4—1461no—
190C2×C4⋊C4C4×D7311yescharacteristic
191C2×C4⋊C4C22⋊D7311yescharacteristic
192C2×C4⋊C4C22⋊D7311yescharacteristic
193C2×C22⋊C4—1462no—
194C2×C22⋊C4—1462no—
195C2×C22⋊C4—1462no—
196C2×C22⋊C4—1462no—
197C2×C42—732no—
198C2×C42D29211yescharacteristic
199C2×C42D29211yescharacteristic
200C2×C42C22×D7311yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—734no—
203C4⋊C8—1462no—
204C4⋊C8—1462no—
205C22⋊C8—22no—
206C22⋊C8—1462no—
207#97—22no—
208#97—1462no—
209#97—1462no—
210C4×C8—1462no—
211#95—1461no—
212C24—1462no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×D4—2922no—
217C2×D4—2922no—
218C2×D4—2922no—
219C2×D4—2922no—
220C2×D4—2921no—
221C2×D4—2922no—
222C2×D4—2921no—
223C2×D4—5842no—
224C2×D4—1464no—
225C22×C4—1462no—
226C22×C4#1192400511yescharacteristic
227C22×C4#1192400511yescharacteristic
228C22×C4#1192400511yescharacteristic
229C22×C4#1192400511yescharacteristic
230C22×C4C2×C4×D7311yescharacteristic
231C22×C4C2×C22⋊D7311yescharacteristic
232C22×C4C2×D29211yescharacteristic
233C2×C8—42no—
234C2×C8—1464no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—2922no—
239C22⋊C4—2924no—
240C22⋊C4—2924no—
241C22⋊C4—5842no—
242C22⋊C4—5842no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—2922no—
248C42D4×D7312yes—
249C42#1192402712yes—
250C23—2922no—
251C23—2922no—
252C23—2922no—
253C23—2922no—
254C23—2922no—
255C23—2922no—
256C23—2922no—
257C23C2×#1192400511yescharacteristic
258Q8—44no—
259D4—5844no—
260D4—5842no—
261D4—5844no—
262D4—5842no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—2922no—
277C2×C4—2922no—
278C2×C4—2922no—
279C2×C4#1455650712yes—
280C2×C4#1455655112yes—
281C2×C4#1455656111yescharacteristic
282C2×C4#1455656312yes—
283C2×C4#1455657511yescharacteristic
284C8—82no—
285C8—2924no—
286C22—21no—
287C22—21no—
288C22—5842no—
289C22—5842no—
290C22—5842no—
291C22—5842no—
292C22—5842no—
293C22—5842no—
294C22—5842no—
295C22#1798846411yescharacteristic
296C22C2×#1455650711yescharacteristic
297C22#1798856911yescharacteristic
298C4—22no—
299C4—21no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—5842no—
306C4#1798876212yes—
307C2—21no—
308C2—21no—
309C2—11682no—
310C2#2261033211yescharacteristic
311C2#2261039211yescharacteristic
312C2#2261045611yescharacteristic
subgroupG/H
centerC22#17988464
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1467C2
socleC22×C73#386
prime radicalC2×#14556588C22
cosocleC2×C42×C73C23
hypercenter#1467D73
hypocenterC73#9458
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
1. #22619016C2semidirect
4. #22607786C2semidirect
5. C73×#1467C2semidirect
160. #1467D73semidirect
162. C73#9458semidirect

Maximal subgroups

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

subgroupindex
#226190162
#226149302
#226086442
#226077862
C73×#14672
#945873