Finite Group Databasebeta

← #33005587

Subgroups — #33005587

Order 18688.

446 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556536C211yesmaximal, characteristic
2C2×#17988405C211yesmaximal, characteristic, complemented
3C2×#17988390C211yesmaximal, characteristic, complemented
4C2×#17988387C211yesmaximal, characteristic
5#22610795C211yesmaximal, characteristic
6#22610790C211yesmaximal, characteristic, complemented
7C73×#1380C211yesmaximal, characteristic, complemented
8C22×Dic292C2211yescharacteristic
9C22×D584C2211yescharacteristic
10C2×#14556536—22no—
11C2×#14556536—22no—
12C2×#14556536—22no—
13C22×#11924003C2211yescharacteristic
14C2×#14556514—21no—
15C2×#14556511—21no—
16C2×C8×Dic73C2211yescharacteristic
17#17988492—21no—
18#17988488—21no—
19#17988405—22no—
20#17988390—22no—
21#17988387—22no—
22C22×C16×C73C2211yescharacteristic
23C2×#107×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25C22×Dic146—21no—
26C22×D292—21no—
27C2×Dic292—21no—
28C2×Dic292—21no—
29C2×Dic292—21no—
30C2×Dic292—21no—
31C2×Dic292—21no—
32C2×Dic292D412yes—
33C2×D584—21no—
34C2×D584—21no—
35C2×D584—21no—
36C2×D584—21no—
37C2×D584—21no—
38C2×D584D412yes—
39C22×#10596831D411yescharacteristic
40C2×#11924005—41no—
41C2×#11924005—41no—
42C2×C4×Dic73D411yescharacteristic
43#14556536—28no—
44C2×#11924003—21no—
45C2×#11924003—21no—
46C2×#11924003—21no—
47#14556514—42no—
48#14556511—42no—
49C8×Dic73—21no—
50C8×Dic73—21no—
51#14556498—41no—
52#14556498—41no—
53C22×C8×C73C2311yescharacteristic
54C2×C4⋊C4×C73—21no—
55C2×C4⋊C4×C73—21no—
56C2×C16×C73—21no—
57C2×C16×C73—21no—
58C2×C16×C73D412yes—
59#107×C73—21no—
60#107×C73—21no—
61#107×C73—21no—
62#107×C73—21no—
63C23×D73—41no—
64C2×Dic146—22no—
65C2×Dic146—41no—
66C2×Dic146—41no—
67C2×Dic146—41no—
68C2×Dic146—41no—
69C2×Dic146—41no—
70C2×D292—22no—
71C2×D292—41no—
72C2×D292—41no—
73C2×D292—41no—
74C2×D292—41no—
75C2×D292—41no—
76C22×Dic73—41no—
77C22×Dic73D811yescharacteristic
78C22×Dic73D811yescharacteristic
79Dic292—22no—
80Dic292—22no—
81Dic292—41no—
82Dic292—41no—
83D584—22no—
84D584—22no—
85D584—41no—
86D584—41no—
87C2×#10596831—21no—
88C2×#10596831—21no—
89C2×#10596831—21no—
90#11924005—82no—
91#11924005—82no—
92C4×Dic73—21no—
93C4×Dic73—21no—
94#11924003—41no—
95C22×C4×C73—41no—
96C22×C4×C73—41no—
97C22×C4×C73C2×D411yescharacteristic
98C2×C8×C73C2×D411yescharacteristic
99C2×C8×C73C2×D411yescharacteristic
100C2×C8×C73C2×D411yescharacteristic
101C2×C8×C73#4111yescharacteristic
102C2×C8×C73#4111yescharacteristic
103C2×C8×C73#4111yescharacteristic
104C4⋊C4×C73—41no—
105C4⋊C4×C73—41no—
106C4⋊C4×C73—41no—
107C4⋊C4×C73—41no—
108C16×C73—22no—
109C23×C73C2×D811yescharacteristic
110C22×D73—42no—
111C22×D73—81no—
112C22×D73—81no—
113C22×D73—81no—
114C22×D73—81no—
115C22×D73—81no—
116C22×D73—81no—
117C2×C4×C73—81no—
118C2×C4×C73—81no—
119C2×C4×C73—81no—
120C2×C4×C73—81no—
121C2×C4×C73—81no—
122C2×C4×C73—81no—
123C2×C4×C73#12111yescharacteristic
124C2×C4×C73#12111yescharacteristic
125C2×C4×C73#12111yescharacteristic
126C2×C4×C73#13511yescharacteristic
127C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
129C2×Dic73—21no—
130C2×Dic73—21no—
131C2×Dic73—21no—
132C2×Dic73—21no—
133C2×Dic73—21no—
134C2×Dic73—42no—
135C2×Dic73—81no—
136C2×Dic73—81no—
137C2×Dic73SD3212yescomplemented
138D292—42no—
139D292—42no—
140D292—81no—
141D292—81no—
142Dic146—42no—
143Dic146—42no—
144Dic146—81no—
145Dic146—81no—
146C8×C73#12011yescharacteristic
147C8×C73#12311yescharacteristic
148C8×C73#12311yescharacteristic
149C8×C73#12411yescharacteristic
150#10596831—41no—
151C22×C73#45911yescharacteristic
152C22×C73#45911yescharacteristic
153C22×C73#46611yescharacteristic
154C22×C73C2×SD3211yescharacteristic
155C22×C73#50811yescharacteristic
156C22×C73#50911yescharacteristic
157C22×C73#51011yescharacteristic
158C2×D73—82no—
159C2×D73—82no—
160C2×D73—82no—
161C2×D73—161no—
162C2×D73—161no—
163C2×D73—161no—
164C2×D73—161no—
165C4×C73—161no—
166C4×C73—161no—
167C4×C73#39411yescharacteristic
168C4×C73#46311yescharacteristic
169C4×C73#46311yescharacteristic
170C4×C73#46911yescharacteristic
171Dic73—22no—
172Dic73—41no—
173Dic73—82no—
174#13449—731nomaximal, Sylow
175C2×C73#205411yescharacteristic
176C2×C73#220711yescharacteristic
177C2×C73#220811yescharacteristic
178C2×C73#221011yescharacteristic
179C2×C73#221111yescharacteristic
180C2×C73#221711yescharacteristic
181C2×C73#222011yescharacteristic
182D73—162no—
183C22×SD32—731no—
184C2×C4⋊C16—731no—
185C2×#358—731no—
186C2×#357—731no—
187#1970—731no—
188#1969—731no—
189#1380D7311yescharacteristic, complemented
190C73#1344911yescharacteristic, complemented, Sylow
191C22×Q16—731no—
192C22×D8—731no—
193C2×SD32—1462no—
194C2×SD32—1462no—
195C2×SD32—1462no—
196C22×C16—731no—
197C22×C16C2×D7311yescharacteristic
198C2×#107C2×D7311yescharacteristic
199C2×#107C2×D7311yescharacteristic
200C2×#103—1461no—
201C2×#102—1461no—
202C2×C4×C8—731no—
203#391—1461no—
204#390—1461no—
205C4⋊C16—1462no—
206#358—1462no—
207#357—1462no—
208C22×Q8—1461no—
209C22×D4—1461no—
210C2×Q16—732no—
211C2×Q16—1461no—
212C2×Q16—1461no—
213C2×Q16—1461no—
214C2×Q16—1461no—
215C2×Q16—1461no—
216C2×D8—732no—
217C2×D8—1461no—
218C2×D8—1461no—
219C2×D8—1461no—
220C2×D8—1461no—
221C2×D8—1461no—
222C22×C8—731no—
223C22×C8C22×D7311yescharacteristic
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C22⋊C4—2921no—
227C2×C22⋊C4—2921no—
228C2×C42—731no—
229SD32—1468no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—1461no—
233C2×C16—1461no—
234C2×C16—1461no—
235C2×C16D29212yes—
236#107—21no—
237#107—21no—
238#107—21no—
239#107—21no—
240#103—2922no—
241#102—2922no—
242C4×C8—1461no—
243C4×C8—1461no—
244#95—2921no—
245#95—2921no—
246C24—2921no—
247C2×Q8—2921no—
248C2×Q8—2921no—
249C2×Q8—2921no—
250C2×Q8—2921no—
251C2×Q8—2921no—
252C2×Q8—1462no—
253C2×D4—2921no—
254C2×D4—2921no—
255C2×D4—2921no—
256C2×D4—2921no—
257C2×D4—2921no—
258C2×D4—1462no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—2921no—
262C22×C4—731no—
263C22×C4—731no—
264C22×C4D4×D7311yescharacteristic
265Q16—2921no—
266Q16—2921no—
267Q16—1462no—
268Q16—1462no—
269D8—2921no—
270D8—2921no—
271D8—1462no—
272D8—1462no—
273C2×C8—1461no—
274C2×C8—1461no—
275C2×C8—1461no—
276C2×C8D4×D7311yescharacteristic
277C2×C8D4×D7311yescharacteristic
278C2×C8C2×D29211yescharacteristic
279C2×C8#1192402811yescharacteristic
280C2×C8#1192402811yescharacteristic
281C2×C8#1192402911yescharacteristic
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C22⋊C4—5842no—
287C22⋊C4—5842no—
288C42—1461no—
289C42—1461no—
290C16—22no—
291C16—2921no—
292C23—2922no—
293C23—5841no—
294C23—5841no—
295C23—5841no—
296C23—5841no—
297C23—5841no—
298C23—5841no—
299C23D8×D7311yescharacteristic
300Q8—2922no—
301Q8—2922no—
302Q8—5841no—
303Q8—5841no—
304D4—2922no—
305D4—2922no—
306D4—5841no—
307D4—5841no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—2922no—
315C2×C4—5841no—
316C2×C4—5841no—
317C2×C4—732no—
318C2×C4—1461no—
319C2×C4—1461no—
320C2×C4—1461no—
321C2×C4—1461no—
322C2×C4—1461no—
323C2×C4#1455656411yescharacteristic
324C2×C4#1455656411yescharacteristic
325C2×C4#1455656611yescharacteristic
326C2×C4#1455661411yescharacteristic
327C2×C4#1455661811yescharacteristic
328C2×C4#1455661811yescharacteristic
329C8—2921no—
330C8#1455656111yescharacteristic
331C8#1455657611yescharacteristic
332C8#1455657611yescharacteristic
333C8#1455657911yescharacteristic
334C22—5842no—
335C22—5842no—
336C22—5842no—
337C22—11681no—
338C22—11681no—
339C22—11681no—
340C22—11681no—
341C22#1798878111yescharacteristic
342C22#1798878111yescharacteristic
343C22#1798882411yescharacteristic
344C22SD32×D7311yescharacteristic
345C22#1798901811yescharacteristic
346C22#1798902111yescharacteristic
347C22#1798902911yescharacteristic
348C4—161no—
349C4—161no—
350C4—2921no—
351C4—5842no—
352C4—1462no—
353C4#1798850511yescharacteristic
354C4#1798880511yescharacteristic
355C4#1798880511yescharacteristic
356C4#1798884111yescharacteristic
357C2—11682no—
358C2#2261124311yescharacteristic
359C2#2261197011yescharacteristic
360C2#2261197511yescharacteristic
361C2#2261198511yescharacteristic
362C2#2261199611yescharacteristic
363C2#2261202811yescharacteristic
364C2#2261204511yescharacteristic
subgroupG/H
centerC23D8×D73
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC73×#1380C2
socleC23×C73C2×D8
prime radicalC22×D584C22
hypercenter#1380D73
hypocenterC73#13449
LCSG ▷ C22×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#17988405C2semidirect
3. C2×#17988390C2semidirect
6. #22610790C2semidirect
7. C73×#1380C2semidirect
137. C2×Dic73SD32semidirect
189. #1380D73semidirect
190. C73#13449semidirect

Maximal subgroups

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

subgroupindex
C22×#145565362
C2×#179884052
C2×#179883902
C2×#179883872
#226107952
#226107902
C73×#13802
#1344973