Finite Group Databasebeta

← #34020361

Subgroups — #34020361

Order 18688.

990 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989609C211yesmaximal, characteristic
2C2×C4×#11924036C211yesmaximal, characteristic
3#22625727C212yesmaximal, complemented
4C4⋊C4×#10596833C212yesmaximal, complemented
5#22625669C212yesmaximal
6#22625665C212yesmaximal
7#22625650C212yesmaximal, complemented
8#22625588C212yesmaximal, complemented
9#22612324C211yesmaximal, characteristic
10C22×#11924036C2211yescharacteristic
11C22×#11924036C2211yescharacteristic
12#17989609C418yescomplemented
13C2×C4⋊#10596833C2212yes—
14C2×#14556648C2212yes—
15C2×#14556646C2214yes—
16C2×#14556646C2212yes—
17C4×#11924036C418yescomplemented
18C2×C4×#10596833C2214yes—
19C2×C4×#10596833C2211yescharacteristic
20C2×C4×#10596833C2211yescharacteristic
21C2×C4×#10596833C2211yescharacteristic
22C2×C4×#10596833C2211yescharacteristic
23#17989453C2214yes—
24#17989453C2214yes—
25C2×C42×D73C2211yescharacteristic
26#17988454C412yes—
27#17988453C411yescharacteristic
28#17988449C412yes—
29#17988444C412yes—
30C4⋊C4×Dic73C2212yes—
31#17988434C2212yes—
32#17988432C2212yes—
33#382×C73C411yescharacteristic
34C2×#11924036—24no—
35C2×#11924036—24no—
36C2×#11924036C2×C414yes—
37C2×#11924036C2×C414yes—
38C22×#10596833C2311yescharacteristic
39C22×#10596833C2311yescharacteristic
40C22×#10596833C2314yes—
41C22×#10596833C2311yescharacteristic
42C22×#10596833C2311yescharacteristic
43C4⋊#10596833C2×C418yes—
44#14556648C2×C418yes—
45#14556646—28no—
46#14556646C2×C418yes—
47C4×#10596833—28no—
48C4×#10596833C2×C414yes—
49C4×#10596833C2×C414yes—
50C4×#10596833C2×C414yes—
51C4×#10596833C2×C414yes—
52C22×C4×D73C2×C411yescharacteristic
53C22×C4×D73C2×C411yescharacteristic
54C22×C4×D73C2311yescharacteristic
55C2×C4⋊Dic73C2×C412yes—
56C2×#11924005C2×C414yes—
57C2×C4×Dic73C2×C411yescharacteristic
58C2×C4×Dic73C2314yes—
59C2×C4×Dic73C2311yescharacteristic
60C2×C4×Dic73C2311yescharacteristic
61C42×D73C2×C414yes—
62C42×D73C2×C414yes—
63#14556498C2×C414yes—
64#14556498C2×C412yes—
65#14556498C2×C412yes—
66#14556498C2×C414yes—
67C2×C4⋊C4×C73C2×C412yes—
68C2×C42×C73C2×C411yescharacteristic
69#95×C73C2×C412yes—
70#95×C73C2×C412yes—
71#11924036—28no—
72#11924036—28no—
73C2×#10596833—22no—
74C2×#10596833—22no—
75C2×#10596833—24no—
76C2×#10596833—24no—
77C2×#10596833C22×C412yes—
78C2×#10596833C22×C412yes—
79C2×#10596833C22×C418yes—
80C2×#10596833C22×C412yes—
81C2×#10596833C22×C412yes—
82C2×#10596833#4114yes—
83C2×#10596833#4114yes—
84C23×D73C22×C411yescharacteristic
85C22×Dic73C22×C414yes—
86C22×Dic73C22×C411yescharacteristic
87C22×Dic73C22×C411yescharacteristic
88C22×Dic73C2411yescharacteristic
89C2×C4×D73—24no—
90C2×C4×D73—22no—
91C2×C4×D73—22no—
92C2×C4×D73—22no—
93C2×C4×D73—22no—
94C2×C4×D73C4214yes—
95C2×C4×D73C4214yes—
96C2×C4×D73C22×C412yes—
97C2×C4×D73C22×C412yes—
98C4⋊Dic73—24no—
99#11924005—28no—
100C4×Dic73—24no—
101C4×Dic73—24no—
102C4×Dic73C4214yes—
103C4×Dic73C22×C412yes—
104C4×Dic73C22×C412yes—
105C4×Dic73C22×C412yes—
106C4×Dic73C22×C412yes—
107C22×C4×C73C22×C414yes—
108C22×C4×C73C22×C411yescharacteristic
109C22×C4×C73C22×C411yescharacteristic
110C22×C4×C73C22×C411yescharacteristic
111C4⋊C4×C73—24no—
112C42×C73C4214yes—
113C23×C73C23×C411yescharacteristic
114C22×D73—22no—
115C22×D73—21no—
116C22×D73—21no—
117C22×D73C2×C4212yes—
118C22×D73#13114yes—
119C2×C4×C73—22no—
120C2×C4×C73—22no—
121C2×C4×C73—24no—
122C2×C4×C73—24no—
123C2×C4×C73—24no—
124C2×C4×C73C2×C4212yes—
125C2×C4×C73C2×C4212yes—
126C2×C4×C73C2×C4212yes—
127C2×C4×C73#13112yes—
128C2×C4×C73#13112yes—
129C2×Dic73—21no—
130C2×Dic73—22no—
131C2×Dic73—21no—
132C2×Dic73—24no—
133C2×Dic73—24no—
134C2×Dic73—24no—
135C2×Dic73C2×C4212yes—
136C2×Dic73C2×C4212yes—
137C2×Dic73C23×C411yescharacteristic
138C2×Dic73C23×C411yescharacteristic
139C2×Dic73C2×#4112yes—
140C2×Dic73C2×#4112yes—
141C4×D73—28no—
142C4×D73—28no—
143C4×D73—22no—
144C4×D73—22no—
145C4×D73—24no—
146#10596833—22no—
147#10596833—22no—
148#10596833—28no—
149#10596833#11714yes—
150#10596833#11714yes—
151C22×C73C22×C4211yescharacteristic
152C22×C73C4×#4112yes—
153C22×C73C2×#13111yescharacteristic
154C22×C73C2×#13112yes—
155C22×C73C2×#13111yescharacteristic
156C2×D73—22no—
157C2×D73—24no—
158C2×D73—24no—
159C2×D73—22no—
160C2×D73#40514yes—
161C4×C73—22no—
162C4×C73—22no—
163C4×C73—44no—
164C4×C73#40514yes—
165Dic73—22no—
166Dic73—22no—
167Dic73—44no—
168Dic73C2×#11711yescharacteristic
169Dic73C2×#11711yescharacteristic
170Dic73C4×#4112yes—
171#20476—731nomaximal, Sylow
172C2×C73C4×#11711yescharacteristic
173C2×C73C2×#40511yescharacteristic
174C2×C73C2×#40511yescharacteristic
175C2×C73#296612yes—
176C2×C73#296612yes—
177D73—24no—
178C2×#432—731no—
179C22×C4×C8—731no—
180#2283—731no—
181#1770—732no—
182C8×C4⋊C4—732no—
183#1752—732no—
184C8×C22⋊C4—732no—
185#1746—732no—
186#1726—732no—
187C73#2047611yescharacteristic, complemented, Sylow
188C23×C8—731no—
189C23×C8—731no—
190C22×C42—731no—
191#432—738no—
192C2×C4⋊C8—732no—
193C2×C22⋊C8—732no—
194C2×#97—732no—
195C2×#97—734no—
196C2×C4×C8—731no—
197C2×C4×C8—731no—
198C2×C4×C8—731no—
199C2×C4×C8—731no—
200C2×C4×C8—738no—
201C2×C4×C8—734no—
202#383—732no—
203#382—731no—
204#382#6440811yescharacteristic
205#381—732no—
206#380—732no—
207C4×C4⋊C4—732no—
208C4×C22⋊C4—732no—
209#376—732no—
210#336—734no—
211#336—734no—
212C23×C4—731no—
213C23×C4—731no—
214C23×C4—731no—
215C22×C8—731no—
216C22×C8—731no—
217C22×C8—731no—
218C22×C8—731no—
219C22×C8—734no—
220C22×C8—734no—
221C22×C8—734no—
222C22×C8—1464no—
223C22×C8—1464no—
224C2×C4⋊C4—732no—
225C2×C4⋊C4C2×#6440812yes—
226C2×C22⋊C4—734no—
227C2×C42—731no—
228C2×C42—731no—
229C2×C42—731no—
230C2×C42—734no—
231C2×C42—734no—
232C2×C42—734no—
233C2×C42C2×#6440811yescharacteristic
234C4⋊C8—738no—
235C22⋊C8—738no—
236#97—738no—
237#97—1468no—
238C4×C8—734no—
239C4×C8—734no—
240C4×C8—734no—
241C4×C8—734no—
242C4×C8—1468no—
243#95—734no—
244#95—732no—
245#95—732no—
246#95—734no—
247#95C2×#6440812yes—
248#95C2×#6440812yes—
249C24—731no—
250C22×C4—731no—
251C22×C4—731no—
252C22×C4—731no—
253C22×C4—734no—
254C22×C4—734no—
255C22×C4—732no—
256C22×C4—732no—
257C22×C4—734no—
258C22×C4—1462no—
259C22×C4—1462no—
260C22×C4—1464no—
261C22×C4—1462no—
262C22×C4—1462no—
263C22×C4C22×#6440814yes—
264C22×C4C22×#6440811yescharacteristic
265C22×C4C22×#6440811yescharacteristic
266C22×C4C22×#6440811yescharacteristic
267C2×C8—732no—
268C2×C8—732no—
269C2×C8—734no—
270C2×C8—734no—
271C2×C8—732no—
272C2×C8—732no—
273C2×C8—738no—
274C2×C8—1468no—
275C2×C8—1468no—
276C2×C8—1464no—
277C2×C8—1464no—
278C2×C8—1462no—
279C2×C8—1462no—
280C4⋊C4—24no—
281C4⋊C4—1464no—
282C22⋊C4—1468no—
283C42—734no—
284C42—732no—
285C42—732no—
286C42—732no—
287C42—732no—
288C42—1464no—
289C42—1464no—
290C42C4×#6440814yes—
291C23—732no—
292C23—734no—
293C23—1462no—
294C23—1461no—
295C23—1461no—
296C23C23×#6440811yescharacteristic
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—731no—
303C2×C4—731no—
304C2×C4—732no—
305C2×C4—732no—
306C2×C4—732no—
307C2×C4—732no—
308C2×C4—1462no—
309C2×C4—1462no—
310C2×C4—1464no—
311C2×C4—1468no—
312C2×C4—1464no—
313C2×C4—1464no—
314C2×C4—1464no—
315C2×C4—1461no—
316C2×C4—1462no—
317C2×C4—1461no—
318C2×C4—1468no—
319C2×C4C2×C4×#6440812yes—
320C2×C4C2×C4×#6440812yes—
321C2×C4C2×C4×#6440812yes—
322C2×C4#1455667812yes—
323C2×C4#1455667912yes—
324C8—734no—
325C8—734no—
326C8—1462no—
327C8—1462no—
328C8—1468no—
329C22—734no—
330C22—1462no—
331C22—1464no—
332C22—1462no—
333C22—1464no—
334C22C22×C4×#6440811yescharacteristic
335C22#41×#6440812yes—
336C22C2×#1455667811yescharacteristic
337C22D73⋊#13112yes—
338C22C2×#1455667911yescharacteristic
339C4—22no—
340C4—22no—
341C4—44no—
342C4—2924no—
343C4—731no—
344C4—731no—
345C4—732no—
346C4—1462no—
347C4—1462no—
348C4#1798955214yes—
349C2—1464no—
350C2#117×#6440811yescharacteristic
351C2#2262649111yescharacteristic
352C2C2×#1798955211yescharacteristic
353C2#2262674512yes—
354C2#2262674812yes—
subgroupG/H
center, FrattiniC23C23×#64408
commutatorC22×C73C22×C42
Fitting#382×C73C4
socleC23×C73C23×C4
prime radicalC23×D73C22×C4
cosocleC22×Dic73C24
hypercenter#382#64408
hypocenterC73#20476
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
3. #22625727C2semidirect
4. C4⋊C4×#10596833C2semidirect
7. #22625650C2semidirect
8. #22625588C2semidirect
12. #17989609C4semidirect
17. C4×#11924036C4semidirect
187. C73#20476semidirect

Maximal subgroups

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

subgroupindex
C2×#179896092
C2×C4×#119240362
#226257272
C4⋊C4×#105968332
#226256692
#226256652
#226256502
#226255882
#226123242
#2047673