Finite Group Databasebeta

← #32985314

Subgroups — #32985314

Order 18688.

510 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988935C211yesmaximal, characteristic
2#22610858C211yesmaximal, characteristic
3#22609800C211yesmaximal, characteristic, complemented
4C2×#17988339C211yesmaximal, characteristic, complemented
5C2×#17988336C211yesmaximal, characteristic, complemented
6C2×#17988330C211yesmaximal, characteristic
7C73×#1987C211yesmaximal, characteristic, complemented
8C2×#14556588C2211yescharacteristic
9#17988935—22no—
10#17988935C414yescomplemented
11C2×C4⋊#10596831C2211yescharacteristic
12C2×C4⋊#10596831C2211yescharacteristic
13C2×#14556511—21no—
14C2×#14556511—21no—
15C2×#14556511—22no—
16C2×#14556502—21no—
17C2×C4×#10596831C2211yescharacteristic
18#17988342—21no—
19#17988339—22no—
20#17988336—22no—
21#17988330—22no—
22C2×C4⋊Q8×C73C2211yescharacteristic
23C2×C4⋊C8×C73C2211yescharacteristic
24C4×C4⋊C4×C73C2211yescharacteristic
25#340×C73—22no—
26C22×D292—21no—
27C22×D292—21no—
28C22×D292—21no—
29C22×#10596831—21no—
30C22×#10596831—21no—
31C22×#10596831D411yescharacteristic
32C22×#10596831D411yescharacteristic
33#14556588—21no—
34#14556588—21no—
35#14556588—21no—
36#14556588C2×C412yes—
37C4⋊#10596831—21no—
38C4⋊#10596831—21no—
39C4⋊#10596831—21no—
40C4⋊#10596831D412yes—
41#14556511—24no—
42#14556511—24no—
43#14556511—44no—
44#14556502—42no—
45C4×#10596831C2×C412yes—
46C4×#10596831D412yes—
47C22×Q8×C73—21no—
48C22×C8×C73—21no—
49C4⋊Q8×C73—21no—
50C4⋊Q8×C73—21no—
51C4⋊Q8×C73—21no—
52C4⋊Q8×C73C2×C412yes—
53C2×C4⋊C4×C73—21no—
54C2×C4⋊C4×C73—21no—
55C2×C4⋊C4×C73D412yes—
56C2×C42×C73—21no—
57C2×C42×C73C2311yescharacteristic
58C4⋊C8×C73—21no—
59C4⋊C8×C73—21no—
60#95×C73—21no—
61C23×D73—41no—
62C2×D292—22no—
63C2×D292—22no—
64C2×D292—22no—
65C2×D292—41no—
66C2×D292—41no—
67C2×D292—42no—
68C2×D292—41no—
69C2×D292—42no—
70C2×D292—41no—
71C2×D292—41no—
72C2×D292—41no—
73C2×D292—41no—
74C2×D292—82no—
75C2×#10596831—21no—
76C2×#10596831—21no—
77C2×#10596831—21no—
78C2×#10596831—21no—
79C2×#10596831—22no—
80C2×#10596831—41no—
81C2×#10596831—41no—
82C2×#10596831—41no—
83C2×#10596831—42no—
84C2×#10596831C22⋊C412yes—
85C2×#10596831C22⋊C412yes—
86C2×Q8×C73—22no—
87C2×Q8×C73—41no—
88C2×Q8×C73—82no—
89C22×C4×C73—22no—
90C22×C4×C73—41no—
91C22×C4×C73C2×D411yescharacteristic
92C22×C4×C73C2×D411yescharacteristic
93C22×C4×C73#4111yescharacteristic
94C2×C8×C73—41no—
95C2×C8×C73—42no—
96C4⋊C4×C73—22no—
97C4⋊C4×C73—22no—
98C4⋊C4×C73—22no—
99C4⋊C4×C73—22no—
100C4⋊C4×C73—41no—
101C4⋊C4×C73—41no—
102C42×C73—41no—
103C42×C73—41no—
104C42×C73C22×C411yescharacteristic
105C42×C73C2×D411yescharacteristic
106C42×C73C2×D411yescharacteristic
107C42×C73#4111yescharacteristic
108C23×C73#12111yescharacteristic
109C22×D73—42no—
110C22×D73—81no—
111C22×D73—81no—
112C22×D73—82no—
113C22×D73—82no—
114Q8×C73—82no—
115Q8×C73—82no—
116C2×C4×C73—22no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—82no—
122C2×C4×C73C2×C22⋊C411yescharacteristic
123C2×C4×C73C4×D411yescharacteristic
124C2×C4×C73C4×D411yescharacteristic
125C2×C4×C73#12111yescharacteristic
126C2×C4×C73#12411yescharacteristic
127C2×C4×C73#12711yescharacteristic
128C2×C4×C73#13511yescharacteristic
129C2×C4×C73#13511yescharacteristic
130C2×C4×C73AGL(1,ℤ/8)12yes—
131C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C73#13712yes—
133C2×C4×C73#13711yescharacteristic
134D292—42no—
135D292—42no—
136D292—44no—
137D292—82no—
138D292—82no—
139D292—81no—
140D292—81no—
141D292—81no—
142D292—81no—
143C8×C73—81no—
144#10596831—22no—
145#10596831—22no—
146#10596831—42no—
147#10596831—81no—
148C22×C73#39011yescharacteristic
149C22×C73#46311yescharacteristic
150C22×C73#46411yescharacteristic
151C22×C73#46812yes—
152C22×C73#47212yes—
153C2×D73—82no—
154C2×D73—82no—
155C2×D73—82no—
156C2×D73—162no—
157C2×D73—162no—
158C4×C73—21no—
159C4×C73—21no—
160C4×C73—82no—
161C4×C73—82no—
162C4×C73#41611yescharacteristic
163C4×C73#41711yescharacteristic
164C4×C73#44012yes—
165C4×C73#46311yescharacteristic
166C4×C73#46411yescharacteristic
167C4×C73#49611yescharacteristic
168C4×C73#49711yescharacteristic
169#9747—731nomaximal, Sylow
170C2×C73#169712yes—
171C2×C73#169812yes—
172C2×C73#197311yescharacteristic
173C2×C73#197811yescharacteristic
174C2×C73#198311yescharacteristic
175D73—162no—
176C2×#486—731no—
177#1987D7311yescharacteristic, complemented
178#1982—731no—
179#1770—731no—
180C2×#335—731no—
181C2×#333—731no—
182C2×#331—731no—
183C73#974711yescharacteristic, complemented, Sylow
184C2×C4⋊Q8C2×D7311yescharacteristic
185C2×#127—731no—
186#486—734no—
187#486—1462no—
188C2×C4⋊C8—731no—
189C2×C4⋊C8—731no—
190C2×C4⋊C8C2×D7311yescharacteristic
191C2×#102—1462no—
192C2×#102—1461no—
193C2×#102—1461no—
194C2×#97—1461no—
195C2×C4×C8—731no—
196C4×C4⋊C4C2×D7311yescharacteristic
197#340—22no—
198#336—1461no—
199#335—1462no—
200#333—1462no—
201#331—1462no—
202C22×Q8—21no—
203C22×D4—1461no—
204C22×D4—1461no—
205C22×D4—1461no—
206C22×C8—21no—
207C22×C8—731no—
208C22×C8—731no—
209C22×C8—1461no—
210C22×C8—1461no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8C4×D7312yes—
215#127—732no—
216#127—1461no—
217#127—1461no—
218#127—1461no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22⋊D7312yes—
222C2×C42—21no—
223C2×C42C22×D7311yescharacteristic
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—732no—
227C4⋊C8—1461no—
228C4⋊C8—1461no—
229C4⋊C8—1461no—
230#102—2924no—
231#102—1464no—
232#102—1464no—
233#97—2922no—
234C4×C8—732no—
235C4×C8—732no—
236#95—21no—
237C24—2921no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—82no—
241C2×D4—2921no—
242C2×D4—2921no—
243C2×D4—2921no—
244C2×D4—2921no—
245C2×D4—2921no—
246C2×D4—2922no—
247C2×D4—2921no—
248C2×D4—2922no—
249C2×D4—2921no—
250C2×D4—5842no—
251C2×D4—1462no—
252C2×D4—1462no—
253C2×D4—1462no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4D4×D7311yescharacteristic
257C22×C4C2×C22⋊D7311yescharacteristic
258C22×C4#1192402711yescharacteristic
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—2921no—
262C2×C8—2921no—
263C2×C8—2921no—
264C2×C8—2922no—
265C2×C8—732no—
266C2×C8—732no—
267C2×C8—1462no—
268C2×C8—1461no—
269C2×C8—1461no—
270C2×C8—1461no—
271C2×C8—1461no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C42—41no—
279C42—41no—
280C42C2×C4×D7311yescharacteristic
281C42D4×D7311yescharacteristic
282C42D4×D7311yescharacteristic
283C42#1192402811yescharacteristic
284C23—2922no—
285C23—5841no—
286C23—5841no—
287C23—5842no—
288C23—5842no—
289C23#1455656311yescharacteristic
290Q8—82no—
291Q8—82no—
292D4—2924no—
293D4—2922no—
294D4—2922no—
295D4—5842no—
296D4—5842no—
297D4—5841no—
298D4—5841no—
299D4—5841no—
300D4—5841no—
301C2×C4—22no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4C22⋊C4×D7311yescharacteristic
308C2×C4#1455655111yescharacteristic
309C2×C4C4×C22⋊D7311yescharacteristic
310C2×C4#1455656411yescharacteristic
311C2×C4#1455657711yescharacteristic
312C2×C4#1455658711yescharacteristic
313C2×C4#1455661211yescharacteristic
314C2×C4#1455661411yescharacteristic
315C2×C4#1455661612yes—
316C2×C4#1455661811yescharacteristic
317C2×C4#1455662212yes—
318C2×C4#1455662311yescharacteristic
319C8—81no—
320C8—2922no—
321C8—5841no—
322C8—1462no—
323C8—1462no—
324C22—5842no—
325C22—5842no—
326C22—5842no—
327C22—11682no—
328C22—11682no—
329C22#1798848711yescharacteristic
330C22#1798880411yescharacteristic
331C22#1798881111yescharacteristic
332C22#1798883512yes—
333C22#1798885512yes—
334C4—21no—
335C4—21no—
336C4—82no—
337C4—82no—
338C4#1798858211yescharacteristic
339C4#1798858711yescharacteristic
340C4#1798868512yes—
341C4#1798880511yescharacteristic
342C4#1798881211yescharacteristic
343C4#1798897711yescharacteristic
344C4#1798897911yescharacteristic
345C2—11682no—
346C2#2260946612yes—
347C2#2260947312yes—
348C2#2261081511yescharacteristic
349C2#2261084011yescharacteristic
350C2#2261086011yescharacteristic
subgroupG/H
centerC23#14556563
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1987C2
socleC23×C73#121
prime radicalC2×#14556588C22
cosocleC2×C42×C73C23
hypercenter#1987D73
hypocenterC73#9747
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
3. #22609800C2semidirect
4. C2×#17988339C2semidirect
5. C2×#17988336C2semidirect
7. C73×#1987C2semidirect
10. #17988935C4semidirect
177. #1987D73semidirect
183. C73#9747semidirect

Maximal subgroups

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

subgroupindex
C2×#179889352
#226108582
#226098002
C2×#179883392
C2×#179883362
C2×#179883302
C73×#19872
#974773