Finite Group Databasebeta

← #34013074

Subgroups — #34013074

Order 18688.

544 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 90 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#22625792C212yesmaximal
3#22625026C212yesmaximal
4C4×#14556580C211yesmaximal, characteristic
5C2×#14556650C2214yes—
6#17989529—22no—
7#17989493—24no—
8#17989465—22no—
9C2×#14556580C2211yescharacteristic
10C2×C4×D292C2211yescharacteristic
11#17988494C411yescharacteristic, complemented
12#17988488C2211yescharacteristic
13#17988477—22no—
14#17988452C411yescharacteristic, complemented
15C4×#11924005—22no—
16C42×Dic73C411yescharacteristic, complemented
17C2×C4×Q8×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—
28C4×D292—41no—
29C4×D292—41no—
30C2×C4⋊Dic73C2×C411yescharacteristic
31C2×#11924005—22no—
32C2×#11924005—22no—
33C2×#11924005D412yes—
34C2×C4×Dic73—21no—
35C2×C4×Dic73—21no—
36C2×C4×Dic73C2×C411yescharacteristic
37C2×C4×Dic73C2×C411yescharacteristic
38#14556498—22no—
39#14556498—22no—
40C22×Q8×C73C2×C411yescharacteristic
41C4×Q8×C73—41no—
42C4×Q8×C73—41no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73C2×C411yescharacteristic
45C2×C42×C73—21no—
46C2×C42×C73C2×C411yescharacteristic
47C22×#64408—44no—
48C4⋊#64408—84no—
49C22⋊#64408—44no—
50C22⋊#64408—44no—
51C23×D73C2×D412yes—
52C2×D292—41no—
53C2×D292—41no—
54C2×D292C2×D411yescharacteristic
55C2×D292C2×Q811yescharacteristic
56C2×D292#4112yes—
57C22×Dic73—21no—
58C22×Dic73—21no—
59C22×Dic73C22⋊C412yes—
60C2×C4×D73—22no—
61C2×C4×D73—22no—
62C2×C4×D73—44no—
63C4⋊Dic73—41no—
64#11924005—24no—
65#11924005—24no—
66#11924005—42no—
67#11924005—44no—
68C4×Dic73—21no—
69C4×Dic73—42no—
70C4×Dic73—42no—
71C4×Dic73—41no—
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⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22×C411yescharacteristic
82C4⋊C4×C73—41no—
83C4⋊C4×C73—42no—
84C42×C73—21no—
85C42×C73—21no—
86C42×C73—41no—
87C42×C73—41no—
88C23×C73C2×C22⋊C411yescharacteristic
89C22×D73—22no—
90C22×D73—22no—
91C22×D73—42no—
92C22×D73—42no—
93C22×D73#12112yes—
94C22×D73C22⋊Q812yes—
95C2×#64408—84no—
96C2×#64408—84no—
97C2×#64408—84no—
98Q8×C73—42no—
99Q8×C73—42no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—22no—
103C2×C4×C73—22no—
104C2×C4×C73—21no—
105C2×C4×C73—41no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—42no—
109C2×C4×C73—41no—
110C2×C4×C73#9911yescharacteristic
111C2×C4×C73#9911yescharacteristic
112C2×C4×C73C2×C4⋊C411yescharacteristic
113C2×C4×C73#11711yescharacteristic
114C2×Dic73—22no—
115C2×Dic73—22no—
116C2×Dic73—22no—
117C2×Dic73—41no—
118C2×Dic73—41no—
119C2×Dic73—41no—
120C2×Dic73—41no—
121C2×Dic73—42no—
122D292—41no—
123D292—41no—
124D292—42no—
125C4×D73—44no—
126C4×D73—82no—
127C22×C73—21no—
128C22×C73—21no—
129C22×C73#38011yescharacteristic
130C22×C73C2×#9911yescharacteristic
131C22×C73#41011yescharacteristic
132C2×D73—22no—
133C2×D73—22no—
134C2×D73—42no—
135C2×D73—42no—
136C2×D73—42no—
137C2×D73—42no—
138C2×D73—82no—
139#64408—164no—
140C4×C73—21no—
141C4×C73—21no—
142C4×C73—21no—
143C4×C73—42no—
144C4×C73—41no—
145C4×C73—81no—
146C4×C73#35412yescomplemented
147Dic73—42no—
148Dic73—42no—
149Dic73—42no—
150#12798—731nomaximal, Sylow
151C2×C73—21no—
152C2×C73—21no—
153C2×C73#179911yescharacteristic
154C2×C73C2×#35411yescharacteristic
155C2×C73#212611yescharacteristic
156D73—42no—
157D73—42no—
158C4×#124—731no—
159C2×#354—732no—
160#1799—732no—
161#1393—732no—
162C73#1279811yescharacteristic, complemented, Sylow
163C2×#124—731no—
164C2×C4×Q8#6440811yescharacteristic, complemented
165C2×C4×D4—731no—
166C2×#99—734no—
167#391—731no—
168#390—731no—
169#388—1462no—
170#382—731no—
171#380—1462no—
172C4×C22⋊C4—1462no—
173C43—731no—
174#354—1464no—
175#342—1462no—
176C22×Q8C2×#6440811yescharacteristic
177C22×D4—731no—
178C23×C4—732no—
179#124—732no—
180#124—732no—
181#124—1462no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×D4—2921no—
185C4×D4—2921no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—2922no—
188C2×C4⋊C4—731no—
189C2×C4⋊C4C2×#6440811yescharacteristic
190C2×C22⋊C4—732no—
191C2×C22⋊C4—1462no—
192C2×C22⋊C4—1464no—
193C2×C22⋊C4—1462no—
194C2×C42—21no—
195C2×C42—731no—
196C2×C42—731no—
197C2×C42—1461no—
198C2×C42—1461no—
199C2×C42C2×#6440811yescharacteristic
200#99—1464no—
201#99—1464no—
202#95—2922no—
203#95—1462no—
204#95—1462no—
205C24—732no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8C4⋊#6440812yescomplemented
210C2×D4—2921no—
211C2×D4—2921no—
212C2×D4—731no—
213C2×D4—731no—
214C2×D4—732no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—2924no—
218C22×C4—2924no—
219C22×C4—732no—
220C22×C4—1461no—
221C22×C4—1461no—
222C22×C4—1462no—
223C22×C4—1462no—
224C22×C4C22⋊#6440811yescharacteristic
225C22×C4C22⋊#6440811yescharacteristic
226C22×C4C22×#6440811yescharacteristic
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—2921no—
230C4⋊C4—5844no—
231C22⋊C4—2922no—
232C22⋊C4—2924no—
233C22⋊C4—2924no—
234C22⋊C4—2924no—
235C22⋊C4—1464no—
236C22⋊C4—1464no—
237C42—21no—
238C42—21no—
239C42—41no—
240C42—41no—
241C42—2921no—
242C42—2922no—
243C42—2922no—
244C42—732no—
245C42—1461no—
246C23—2922no—
247C23—2922no—
248C23—732no—
249C23—732no—
250C23—1462no—
251C23—1462no—
252C23C2×C22⋊#6440811yescharacteristic
253Q8—42no—
254Q8—42no—
255D4—2921no—
256D4—2921no—
257D4—2922no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—2921no—
269C2×C4—2921no—
270C2×C4—2921no—
271C2×C4—2921no—
272C2×C4—2924no—
273C2×C4—2922no—
274C2×C4—5842no—
275C2×C4—5844no—
276C2×C4—5844no—
277C2×C4—5844no—
278C2×C4—1462no—
279C2×C4—1462no—
280C2×C4—1462no—
281C2×C4#1455664911yescharacteristic
282C2×C4#1455664911yescharacteristic
283C2×C4C2×C4⋊#6440811yescharacteristic
284C2×C4#1455667011yescharacteristic
285C22—21no—
286C22—21no—
287C22—2922no—
288C22—2922no—
289C22—2922no—
290C22—2922no—
291C22—5842no—
292C22—1462no—
293C22—1462no—
294C22#1798952911yescharacteristic
295C22C2×#1455664911yescharacteristic
296C22#1798956611yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—41no—
301C4—42no—
302C4—81no—
303C4—2922no—
304C4—2922no—
305C4—2922no—
306C4—11684no—
307C4#1798949312yescomplemented
308C2—21no—
309C2—21no—
310C2—2922no—
311C2—2922no—
312C2#2262579111yescharacteristic
313C2C2×#1798949311yescharacteristic
314C2#2262620211yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4×Q8×C73C4
socleC22×C73C2×#99
prime radical, cosocleC22×D292C23
hypercenterC2×C4×Q8#64408
hypocenterC73#12798
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
11. #17988494C4semidirect
14. #17988452C4semidirect
16. C42×Dic73C4semidirect
17. C2×C4×Q8×C73C4semidirect
72. C4×Dic73C4⋊C4semidirect
76. C2×Q8×C73C4⋊C4semidirect
146. C4×C73#354semidirect
162. C73#12798semidirect
164. C2×C4×Q8#64408semidirect
209. C2×Q8C4⋊#64408semidirect
307. C4#17989493semidirect

Maximal subgroups

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

subgroupindex
C2×#179894932
#226257922
#226250262
C4×#145565802
#1279873