Finite Group Databasebeta

← #34013084

Subgroups — #34013084

Order 18688.

524 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 84 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×#17989493C212yesmaximal
2#22625794C212yesmaximal
3#22625025C212yesmaximal
4#22612645C211yesmaximal, characteristic
5C2×#14556650C2214yes—
6#17989529—22no—
7#17989493—24no—
8#17989465—22no—
9C2×#14556580C2211yescharacteristic
10C2×Q8×Dic73C411yescharacteristic, complemented
11C2×#14556553C2211yescharacteristic
12#17988489C2211yescharacteristic
13#17988476—22no—
14#17988461—22no—
15#17988456C411yescharacteristic, complemented
16#17988433C411yescharacteristic, complemented
17#391×C73C411yescharacteristic, complemented
18C2×C4⋊#64408—42no—
19C2×C22⋊#64408—24no—
20#14556650—24no—
21#14556650—24no—
22(C2×#64408)⋊C4—42no—
23C22×D292C2311yescharacteristic
24C22×C4×D73D412yes—
25#14556580—22no—
26#14556580D412yes—
27#14556580Q812yes—
28Q8×Dic73—42no—
29#14556553—42no—
30C2×C4⋊Dic73—21no—
31C2×C4⋊Dic73C2×C411yescharacteristic
32C2×#11924007—22no—
33C2×#11924005—22no—
34C2×#11924005—22no—
35C2×#11924005D412yes—
36C2×C4×Dic73—21no—
37C2×C4×Dic73C2×C411yescharacteristic
38C2×C4×Dic73C2×C411yescharacteristic
39#14556498—21no—
40#14556498—21no—
41C22×Q8×C73C2×C411yescharacteristic
42C2×C4⋊C4×C73C2×C411yescharacteristic
43C2×C42×C73C2×C411yescharacteristic
44#95×C73—22no—
45C22×#64408—44no—
46C4⋊#64408—84no—
47C22⋊#64408—44no—
48C22⋊#64408—44no—
49C23×D73C2×D412yes—
50C2×D292—41no—
51C2×D292—41no—
52C2×D292C2×D411yescharacteristic
53C2×D292C2×Q811yescharacteristic
54C2×D292#4112yes—
55C22×Dic73—21no—
56C22×Dic73—21no—
57C22×Dic73C22⋊C411yescharacteristic
58C22×Dic73C22⋊C411yescharacteristic
59C2×C4×D73—22no—
60C2×C4×D73—22no—
61C2×C4×D73—82no—
62C4⋊Dic73—42no—
63C4⋊Dic73—41no—
64#11924007—82no—
65#11924005—24no—
66#11924005—24no—
67#11924005—42no—
68#11924005—82no—
69C4×Dic73—21no—
70C4×Dic73—22no—
71C4×Dic73—42no—
72C4×Dic73C4⋊C412yescomplemented
73C2×Q8×C73—21no—
74C2×Q8×C73—41no—
75C2×Q8×C73—41no—
76C2×Q8×C73C4⋊C412yescomplemented
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73C22⋊C412yes—
80C22×C4×C73C22×C411yescharacteristic
81C4⋊C4×C73—41no—
82C42×C73—41no—
83C23×C73C2×C22⋊C411yescharacteristic
84C22×D73—22no—
85C22×D73—22no—
86C22×D73—42no—
87C22×D73—42no—
88C22×D73#12112yes—
89C22×D73C22⋊Q812yes—
90C2×#64408—84no—
91C2×#64408—84no—
92C2×#64408—84no—
93Q8×C73—42no—
94Q8×C73—42no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—22no—
98C2×C4×C73—22no—
99C2×C4×C73—41no—
100C2×C4×C73—41no—
101C2×C4×C73—42no—
102C2×C4×C73—41no—
103C2×C4×C73—41no—
104C2×C4×C73—41no—
105C2×C4×C73C2×C4⋊C411yescharacteristic
106C2×C4×C73#11711yescharacteristic
107C2×Dic73—22no—
108C2×Dic73—21no—
109C2×Dic73—41no—
110C2×Dic73—41no—
111C2×Dic73—41no—
112C2×Dic73—41no—
113C2×Dic73—42no—
114C2×Dic73—41no—
115C2×Dic73#9912yes—
116D292—41no—
117D292—41no—
118D292—42no—
119C4×D73—44no—
120C4×D73—82no—
121C22×C73—21no—
122C22×C73—21no—
123C22×C73#38011yescharacteristic
124C22×C73C2×#9911yescharacteristic
125C22×C73#41011yescharacteristic
126C2×D73—22no—
127C2×D73—22no—
128C2×D73—42no—
129C2×D73—42no—
130C2×D73—42no—
131C2×D73—42no—
132C2×D73—82no—
133#64408—164no—
134C4×C73—21no—
135C4×C73—21no—
136C4×C73—42no—
137C4×C73—42no—
138C4×C73—81no—
139Dic73—22no—
140Dic73—42no—
141Dic73—41no—
142Dic73—81no—
143#12800—731nomaximal, Sylow
144C2×C73—21no—
145C2×C73—21no—
146C2×C73#179911yescharacteristic
147C2×C73#212711yescharacteristic
148C2×C73#212911yescharacteristic
149D73—42no—
150D73—42no—
151#2326—731no—
152C2×#354—732no—
153#1799—732no—
154#1393—732no—
155C73#1280011yescharacteristic, complemented, Sylow
156C2×#124—731no—
157C2×C4×Q8—731no—
158C2×C4×D4—731no—
159C2×#99—734no—
160#391#6440811yescharacteristic, complemented
161#390—731no—
162#388—1462no—
163#385—1462no—
164#384—731no—
165#380—1462no—
166#376—731no—
167#354—1464no—
168#342—1462no—
169C22×Q8C2×#6440811yescharacteristic
170C22×D4—731no—
171C23×C4—732no—
172#124—732no—
173#124—732no—
174#124—1462no—
175C4×Q8—2922no—
176C4×D4—2922no—
177C2×C4⋊C4—2922no—
178C2×C4⋊C4—731no—
179C2×C4⋊C4—1461no—
180C2×C4⋊C4—1462no—
181C2×C4⋊C4C2×#6440811yescharacteristic
182C2×C22⋊C4—732no—
183C2×C22⋊C4—1464no—
184C2×C22⋊C4—1462no—
185C2×C22⋊C4—1462no—
186C2×C42—731no—
187C2×C42—731no—
188C2×C42—1461no—
189C2×C42C2×#6440811yescharacteristic
190#99—1464no—
191#99—1464no—
192#95—22no—
193#95—2922no—
194#95—1461no—
195#95—1461no—
196C24—732no—
197C2×Q8—21no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8C4⋊#6440812yescomplemented
201C2×D4—2921no—
202C2×D4—2921no—
203C2×D4—731no—
204C2×D4—731no—
205C2×D4—732no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—2924no—
209C22×C4—5842no—
210C22×C4—731no—
211C22×C4—731no—
212C22×C4—1462no—
213C22×C4—1462no—
214C22×C4—1461no—
215C22×C4—1461no—
216C22×C4C22⋊#6440812yes—
217C22×C4C22×#6440811yescharacteristic
218C4⋊C4—41no—
219C4⋊C4—2921no—
220C4⋊C4—2922no—
221C4⋊C4—5842no—
222C4⋊C4—5844no—
223C22⋊C4—2922no—
224C22⋊C4—2924no—
225C22⋊C4—2924no—
226C22⋊C4—5842no—
227C22⋊C4—1464no—
228C22⋊C4—1464no—
229C42—41no—
230C42—2922no—
231C42—732no—
232C42—1462no—
233C42—1461no—
234C23—2922no—
235C23—2922no—
236C23—732no—
237C23—732no—
238C23—1462no—
239C23—1462no—
240C23C2×C22⋊#6440811yescharacteristic
241Q8—42no—
242Q8—42no—
243D4—2921no—
244D4—2921no—
245D4—2922no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—2924no—
257C2×C4—2921no—
258C2×C4—2921no—
259C2×C4—2921no—
260C2×C4—2922no—
261C2×C4—2921no—
262C2×C4—2921no—
263C2×C4—5842no—
264C2×C4—5844no—
265C2×C4—5844no—
266C2×C4—5844no—
267C2×C4—732no—
268C2×C4—1462no—
269C2×C4—1461no—
270C2×C4C2×C4⋊#6440811yescharacteristic
271C2×C4#1455667011yescharacteristic
272C22—21no—
273C22—21no—
274C22—2922no—
275C22—2922no—
276C22—2922no—
277C22—2922no—
278C22—5842no—
279C22—1462no—
280C22—1462no—
281C22#1798952911yescharacteristic
282C22D73⋊#9911yescharacteristic
283C22#1798956711yescharacteristic
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—81no—
289C4—2922no—
290C4—2921no—
291C4—5841no—
292C4—11684no—
293C4—1462no—
294C2—21no—
295C2—21no—
296C2—2922no—
297C2—2922no—
298C2#2262579311yescharacteristic
299C2#2262620911yescharacteristic
300C2#2262621411yescharacteristic
subgroupG/H
centerC22D73⋊#99
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#391×C73C4
socleC22×C73C2×#99
prime radical, cosocleC22×D292C23
hypercenter#391#64408
hypocenterC73#12800
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #391

Complements

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

NH ≅ G/Ntype
10. C2×Q8×Dic73C4semidirect
15. #17988456C4semidirect
16. #17988433C4semidirect
17. #391×C73C4semidirect
72. C4×Dic73C4⋊C4semidirect
76. C2×Q8×C73C4⋊C4semidirect
155. C73#12800semidirect
160. #391#64408semidirect
200. C2×Q8C4⋊#64408semidirect

Maximal subgroups

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

subgroupindex
C2×#179894932
#226257942
#226250252
#226126452
#1280073