Finite Group Databasebeta

← #33239373

Subgroups — #33239373

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989017C211yesmaximal, characteristic
2#22618807C211yesmaximal, characteristic, complemented
3#22618805C211yesmaximal, characteristic, complemented
4#22611804C212yesmaximal, complemented
5#22611799C212yesmaximal, complemented
6#22611786C212yesmaximal, complemented
7#22611780C212yesmaximal, complemented
8C2×#17988414C211yesmaximal, characteristic, complemented
9#22611644C211yesmaximal, characteristic, complemented
10#22611530C211yesmaximal, characteristic, complemented
11C73×#2139C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13#17989070—21no—
14#17989068—21no—
15#17989017C418yescomplemented
16C2×#14556537C2211yescharacteristic
17C2×#14556535C2211yescharacteristic
18C2×#14556534C2211yescharacteristic
19C2×#14556532C2211yescharacteristic
20C22×#11924003C2211yescharacteristic
21#17988688—22no—
22#17988686—22no—
23Q16×Dic73C2212yes—
24#17988681C2212yes—
25D8×Dic73C2212yes—
26#17988669C2212yes—
27C2×#14556525C2211yescharacteristic
28#17988588—21no—
29#17988586—21no—
30#17988542C2211yescharacteristic
31#17988414C2212yes—
32#17988414C2212yes—
33#17988392C2212yes—
34#17988389C2212yes—
35#17988362C2212yes—
36C16×Dic73C2212yes—
37C2×#135×C73C2211yescharacteristic
38C2×M32×C73C2211yescharacteristic
39C2×#107×C73C2211yescharacteristic
40#358×C73C2212yes—
41#357×C73C2212yes—
42C2×#11924028—21no—
43#14556613—22no—
44#14556613C2×C414yescomplemented
45C2×Dic292C2311yescharacteristic
46C2×#11924017—21no—
47C2×D584C2311yescharacteristic
48C2×#11924012D411yescharacteristic
49Q8×Dic73—22no—
50#14556556—22no—
51D4×Dic73—22no—
52D4×Dic73—42no—
53#14556553—22no—
54C4×C22⋊D73—42no—
55#14556550—41no—
56#14556550D411yescharacteristic
57#14556549—41no—
58C2×C4⋊Dic73—21no—
59C2×C4⋊Dic73—41no—
60C4⋊C4×D73—41no—
61#14556537C2×C414yescomplemented
62#14556535C2×C414yescomplemented
63#14556534C2×C414yescomplemented
64#14556532C2×C414yescomplemented
65C2×#11924003C2×C414yescomplemented
66C2×#11924003C2311yescharacteristic
67C2×#11924003C2311yescharacteristic
68#14556525—21no—
69#14556525C2312yes—
70#14556515—22no—
71#14556514—22no—
72#14556512—22no—
73#14556511—22no—
74#14556501D412yes—
75C8×Dic73C2312yes—
76C2×#41×C73—21no—
77#135×C73—22no—
78#135×C73C2×C414yescomplemented
79C2×Q16×C73C2311yescharacteristic
80C2×SD16×C73—21no—
81C2×D8×C73C2311yescharacteristic
82C22×C8×C73C2311yescharacteristic
83C2×C4⋊C4×C73—21no—
84M32×C73—22no—
85C2×C16×C73C2312yes—
86#107×C73—21no—
87#107×C73C2312yes—
88#11924028—24no—
89#11924028—42no—
90C2×Dic146—21no—
91C2×D292—21no—
92C2×C22⋊D73—41no—
93C22×Dic73—41no—
94C22×Dic73—42no—
95C2×C4×D73—41no—
96C2×C4×D73—42no—
97Dic292—21no—
98Dic292C22×C412yes—
99#11924017—24no—
100D584—21no—
101D584C22×C412yes—
102#11924012—22no—
103C2×#10596831C2×D412yes—
104C4⋊Dic73—22no—
105C4⋊Dic73—41no—
106C4⋊Dic73—42no—
107C4⋊Dic73D811yescharacteristic
108C4⋊Dic73D811yescharacteristic
109#11924007—42no—
110C22⋊Dic73—42no—
111C22⋊Dic73D812yes—
112#11924005—42no—
113C4×Dic73—42no—
114C4×Dic73—42no—
115C4×Dic73C2×D412yes—
116#11924003C22×C412yes—
117#11924003C22×C412yes—
118#41×C73—24no—
119#41×C73—42no—
120C2×Q8×C73—21no—
121C2×D4×C73—21no—
122C2×D4×C73—41no—
123C22×C4×C73—41no—
124C22×C4×C73—41no—
125C22×C4×C73C2×D411yescharacteristic
126Q16×C73—21no—
127Q16×C73C22×C412yes—
128SD16×C73—24no—
129D8×C73—21no—
130D8×C73C22×C412yes—
131C2×C8×C73C22×C412yes—
132C2×C8×C73C2×D411yescharacteristic
133C2×C8×C73#4112yes—
134C2×C8×C73C2411yescharacteristic
135C4⋊C4×C73—22no—
136C4⋊C4×C73—41no—
137C16×C73—22no—
138C23×C73—41no—
139C23×C73C2×D811yescharacteristic
140C22×D73—41no—
141Q8×C73—22no—
142Q8×C73—41no—
143D4×C73—22no—
144D4×C73—44no—
145D4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—44no—
148C2×C4×C73—42no—
149C2×C4×C73—161no—
150C2×C4×C73C4×D412yes—
151C2×C4×C73C2×D811yescharacteristic
152C2×C4×C73#13512yes—
153C2×C4×C73C22×D411yescharacteristic
154C22⋊D73—44no—
155C2×Dic73—41no—
156C2×Dic73—42no—
157C2×Dic73—162no—
158C2×Dic73—161no—
159C2×Dic73C2×D812yes—
160C2×Dic73C2×D812yes—
161D292—22no—
162D292—41no—
163C4×D73—44no—
164C4×D73—162no—
165Dic146—22no—
166Dic146—41no—
167C8×C73C4×D412yes—
168C8×C73C23×C411yescharacteristic
169C8×C73C2×#4111yescharacteristic
170#10596831—22no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—42no—
174C22×C73—81no—
175C22×C73—81no—
176C22×C73C4×D812yes—
177C22×C73C22×D811yescharacteristic
178C2×D73—42no—
179C2×D73—81no—
180C2×D73—81no—
181C4×C73—42no—
182C4×C73—82no—
183C4×C73C4×D812yes—
184C4×C73C2×C4×D411yescharacteristic
185C4×C73C2×#13511yescharacteristic
186Dic73—22no—
187Dic73—22no—
188Dic73—42no—
189Dic73—82no—
190#33460—731nomaximal, Sylow
191C2×C73—22no—
192C2×C73—82no—
193C2×C73C2×C4×D811yescharacteristic
194C2×C73#341611yescharacteristic
195C2×C73#341611yescharacteristic
196D73—82no—
197C2×#508—731no—
198#2951—731no—
199#2951—731no—
200#2177—732no—
201#2176—732no—
202C4×SD32—732no—
203C4×SD32—732no—
204C2×#367—731no—
205#2139—731no—
206#2139D7311yescharacteristic, complemented
207#2108—731no—
208C73#3346011yescharacteristic, complemented, Sylow
209C2×#135—731no—
210C2×#135C2×D7311yescharacteristic
211#520—1461no—
212#520—1461no—
213#508—738no—
214C2×Q32—731no—
215C2×SD32—731no—
216C2×SD32—731no—
217C2×D16—731no—
218C2×M32C2×D7311yescharacteristic
219C22×C16—731no—
220#440—1462no—
221#440—1462no—
222C4×Q16—732no—
223C4×Q16—732no—
224C4×D8—732no—
225C4×D8—732no—
226C2×#107—731no—
227C2×#107C2×D7311yescharacteristic
228#417—1461no—
229#417—1461no—
230#405—731no—
231#367—732no—
232#367—732no—
233#358—732no—
234#358C2×D7312yes—
235#357—732no—
236#357C2×D7312yes—
237#346—732no—
238C4×C16—732no—
239C2×#41—21no—
240C2×#41—1461no—
241#135—22no—
242#135—734no—
243#135—1462no—
244#135C4×D7314yescomplemented
245C2×Q16—731no—
246C2×Q16C22×D7311yescharacteristic
247C2×SD16—21no—
248C2×SD16—1461no—
249C2×D8—731no—
250C2×D8C22×D7311yescharacteristic
251C2×M16—731no—
252C22×C8C22×D7311yescharacteristic
253C4×Q8—1462no—
254C4×Q8—1462no—
255C4×D4—2922no—
256C4×D4—2922no—
257C4×D4—1462no—
258C4×D4—1462no—
259#117—2921no—
260#117—2921no—
261#117—731no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—2921no—
264C2×C4⋊C4—2921no—
265C2×C4⋊C4—1461no—
266Q32—734no—
267SD32—734no—
268SD32—734no—
269D16—734no—
270M32—22no—
271C2×C16—734no—
272C2×C16—731no—
273C2×C16—731no—
274C2×C16C22×D7312yes—
275#107—21no—
276#107—732no—
277#107—1461no—
278#107C22×D7312yes—
279#103—1462no—
280#103—1462no—
281#102—1462no—
282#102—1462no—
283#97—732no—
284C4×C8—732no—
285#41—24no—
286#41—42no—
287#41—2922no—
288#41—1464no—
289C2×Q8—21no—
290C2×Q8—1461no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—2921no—
294C2×D4—1461no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—2922no—
298C22×C4—2921no—
299C22×C4—2921no—
300C22×C4—2922no—
301C22×C4D4×D7311yescharacteristic
302Q16—21no—
303Q16—732no—
304Q16—1461no—
305Q16C2×C4×D7312yes—
306SD16—24no—
307SD16—1464no—
308D8—21no—
309D8—732no—
310D8—1461no—
311D8C2×C4×D7312yes—
312M16—1462no—
313C2×C8—732no—
314C2×C8C2×C4×D7312yes—
315C2×C8D4×D7311yescharacteristic
316C2×C8#1192402712yes—
317C2×C8C23×D7311yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—2922no—
321C4⋊C4—2922no—
322C4⋊C4—2921no—
323C4⋊C4—731no—
324C4⋊C4—731no—
325C4⋊C4—1462no—
326C22⋊C4—2922no—
327C22⋊C4—2922no—
328C22⋊C4—732no—
329C42—2922no—
330C42—2922no—
331C42—732no—
332C16—22no—
333C16—732no—
334C16—732no—
335C23—41no—
336C23—2921no—
337C23D8×D7311yescharacteristic
338Q8—22no—
339Q8—41no—
340Q8—2921no—
341Q8—1462no—
342D4—22no—
343D4—44no—
344D4—41no—
345D4—2924no—
346D4—2921no—
347D4—1462no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—161no—
352C2×C4—2921no—
353C2×C4—2924no—
354C2×C4—2922no—
355C2×C4—732no—
356C2×C4—732no—
357C2×C4—11682no—
358C2×C4—11681no—
359C2×C4—11682no—
360C2×C4#1455655112yes—
361C2×C4D8×D7311yescharacteristic
362C2×C4#1455661212yes—
363C2×C4C2×D4×D7311yescharacteristic
364C8—1462no—
365C8#1455655112yes—
366C8C22×C4×D7311yescharacteristic
367C8C2×#1192402711yescharacteristic
368C22—21no—
369C22—21no—
370C22—42no—
371C22—81no—
372C22—81no—
373C22—2922no—
374C22—5841no—
375C22—5841no—
376C22#1798866712yes—
377C22C2×D8×D7311yescharacteristic
378C4—42no—
379C4—82no—
380C4—2922no—
381C4—5842no—
382C4—1462no—
383C4—1462no—
384C4#1798866712yes—
385C4C2×#1455655111yescharacteristic
386C4C2×#1455661211yescharacteristic
387C2—22no—
388C2—82no—
389C2—5842no—
390C2C2×#1798866711yescharacteristic
391C2#2262307611yescharacteristic
392C2#2262307711yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2139C2
socleC22×C73C22×D8
prime radicalC2×#17989017C2
cosocleC2×C8×C73C24
hypercenter#2139D73
hypocenterC73#33460
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
2. #22618807C2semidirect
3. #22618805C2semidirect
4. #22611804C2semidirect
5. #22611799C2semidirect
6. #22611786C2semidirect
7. #22611780C2semidirect
8. C2×#17988414C2semidirect
9. #22611644C2semidirect
10. #22611530C2semidirect
11. C73×#2139C2semidirect
15. #17989017C4semidirect
44. #14556613C2×C4semidirect
61. #14556537C2×C4semidirect
62. #14556535C2×C4semidirect
63. #14556534C2×C4semidirect
64. #14556532C2×C4semidirect
65. C2×#11924003C2×C4semidirect
78. #135×C73C2×C4semidirect
206. #2139D73semidirect
208. C73#33460semidirect
244. #135C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890172
#226188072
#226188052
#226118042
#226117992
#226117862
#226117802
C2×#179884142
#226116442
#226115302
C73×#21392
#3346073