Finite Group Databasebeta

← #33239433

Subgroups — #33239433

Order 18688.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623067C211yesmaximal, characteristic
2C2×#17988708C211yesmaximal, characteristic, complemented
3#22618731C211yesmaximal, characteristic, complemented
4#22611828C214yesmaximal, complemented
5#22611821C214yesmaximal, complemented
6#22611678C211yesmaximal, characteristic, complemented
7#22611666C211yesmaximal, characteristic, complemented
8C2×#17988364C211yesmaximal, characteristic, complemented
9C73×#2155C211yesmaximal, characteristic, complemented
10C2×#14556613C2211yescharacteristic
11C2×#14556613C2211yescharacteristic
12C2×#14556599—21no—
13#17989055—21no—
14#17989030C412yescomplemented
15#17989030C2212yes—
16#17989025C412yescomplemented
17#17989025C2212yes—
18#17989019C414yescomplemented
19C2×#14556531C2211yescharacteristic
20#17988708—22no—
21#17988708C2214yes—
22#17988698C2212yescomplemented
23#17988693C2212yescomplemented
24#17988686—22no—
25C2×#14556517—21no—
26#17988581—21no—
27C2×#14556501C2211yescharacteristic
28#17988402C2212yescomplemented
29#17988399C2212yescomplemented
30#17988396C2214yes—
31#17988364C2214yescomplemented
32C2×M32×C73C2211yescharacteristic
33C2×#108×C73C2211yescharacteristic
34#427×C73C2211yescharacteristic
35#365×C73C2214yescomplemented
36C2×#11924028—21no—
37C2×#11924028—21no—
38#14556613—21no—
39#14556613—21no—
40#14556613—21no—
41#14556613—21no—
42#14556613C2×C412yes—
43#14556613C2×C412yes—
44#14556613C2×C412yescomplemented
45#14556613C2312yes—
46C2×Dic292C2×C411yescharacteristic, complemented
47C2×Dic292C2311yescharacteristic
48C2×#11924017—21no—
49C2×#11924017—21no—
50C2×D584C2×C411yescharacteristic, complemented
51C2×D584C2311yescharacteristic
52#14556599—24no—
53#14556599—42no—
54C2×#11924011—41no—
55C22×#10596831D411yescharacteristic
56C2×C8×D73—41no—
57#14556556—22no—
58#14556553—22no—
59#14556551—42no—
60#14556548—41no—
61C2×C4×Dic73D411yescharacteristic
62C42×D73—41no—
63Dic584C2×C414yescomplemented
64#14556536C2×C414yescomplemented
65#14556536C2×C414yescomplemented
66D1168C2×C414yescomplemented
67#14556531—21no—
68#14556531C2312yes—
69C2×#11924003—21no—
70#14556517—24no—
71#14556514—22no—
72#14556511—22no—
73#14556501D412yes—
74#14556501C2312yes—
75C2×M16×C73—21no—
76C22×C8×C73C2311yescharacteristic
77#117×C73—21no—
78M32×C73C2×C412yescomplemented
79M32×C73C2312yes—
80C2×C16×C73C2×C412yescomplemented
81#108×C73—21no—
82#108×C73C2312yes—
83#107×C73—21no—
84#106×C73C2312yes—
85#11924028—22no—
86#11924028—22no—
87#11924028—22no—
88#11924028—22no—
89#11924028—41no—
90#11924028—41no—
91#11924028—41no—
92#11924028—41no—
93C2×Dic146—21no—
94C2×Dic146—21no—
95C2×D292—21no—
96C2×D292—21no—
97C2×C22⋊D73—41no—
98C2×C22⋊D73—41no—
99C22×Dic73—21no—
100C2×C4×D73—41no—
101C2×C4×D73—41no—
102C2×C4×D73—42no—
103Dic292—21no—
104Dic292—21no—
105Dic292C22×C412yes—
106Dic292C22×C412yes—
107#11924017—22no—
108#11924017—22no—
109#11924017—22no—
110#11924017—22no—
111D584—21no—
112D584—21no—
113D584C22×C412yes—
114D584C22×C412yes—
115#11924011—44no—
116C2×#10596831—21no—
117C2×#10596831—21no—
118C2×#10596831C2×D412yes—
119C8×D73—44no—
120#11924007—42no—
121#11924005—42no—
122C4×Dic73—21no—
123C4×Dic73—42no—
124C4×Dic73C2×D412yes—
125#11924003—22no—
126C22×C4×C73C2×D411yescharacteristic
127M16×C73—22no—
128M16×C73—41no—
129C2×C8×C73—41no—
130C2×C8×C73C22×C411yescharacteristic
131C2×C8×C73C22×C411yescharacteristic
132C2×C8×C73C2×D411yescharacteristic
133C2×C8×C73#4111yescharacteristic
134C2×C8×C73#4111yescharacteristic
135C2×C8×C73C2411yescharacteristic
136C4⋊C4×C73—22no—
137C22⋊C4×C73—41no—
138C42×C73—41no—
139C16×C73C22×C414yes—
140C23×C73AGL(1,ℤ/8)11yescharacteristic
141C22×D73—41no—
142C22×D73—41no—
143C2×C4×C73—42no—
144C2×C4×C73C4×D411yescharacteristic
145C2×C4×C73C4×D411yescharacteristic
146C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C73C22×D411yescharacteristic
150C22⋊D73—42no—
151C22⋊D73—42no—
152C22⋊D73—42no—
153C22⋊D73—42no—
154C2×Dic73—22no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—81no—
158D292—22no—
159D292—22no—
160D292—41no—
161D292—41no—
162C4×D73—42no—
163C4×D73—42no—
164C4×D73—42no—
165C4×D73—42no—
166C4×D73—84no—
167Dic146—22no—
168Dic146—22no—
169Dic146—41no—
170Dic146—41no—
171C8×C73—42no—
172C8×C73C4×D411yescharacteristic
173C8×C73C4×D411yescharacteristic
174C8×C73C23×C411yescharacteristic
175C8×C73C2×#4111yescharacteristic
176#10596831—22no—
177C22×C73—41no—
178C22×C73#44211yescharacteristic
179C22×C73#44211yescharacteristic
180C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
181C2×D73—42no—
182C2×D73—42no—
183C2×D73—82no—
184C2×D73—82no—
185C4×C73—82no—
186C4×C73#44211yescharacteristic
187C4×C73#44211yescharacteristic
188C4×C73C2×C4×D411yescharacteristic
189C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
190Dic73—42no—
191Dic73—42no—
192Dic73—42no—
193#33468—731nomaximal, Sylow
194C2×C73—21no—
195C2×C73—21no—
196C2×C73—21no—
197C2×C73C2×#44211yescharacteristic
198D73—82no—
199D73—82no—
200#3414—731no—
201C2×#444—731no—
202#2943—731no—
203AGL(1,ℤ/16)—734no—
204#2180—734no—
205#2155D7311yescharacteristic, complemented
206#2148—731no—
207#2145—731no—
208C2×#347—731no—
209C73#3346811yescharacteristic, complemented, Sylow
210C2×#135—731no—
211C2×#135—731no—
212C2×#131—1461no—
213C4×#41—1461no—
214#510—732no—
215#510—732no—
216#509—732no—
217#509—732no—
218#508—734no—
219C2×M32—731no—
220C2×M32C2×D7311yescharacteristic
221#444—734no—
222#444—1462no—
223#442—732no—
224#441—732no—
225#440—1462no—
226C2×#108C2×D7311yescharacteristic
227#427C2×D7311yescharacteristic
228C2×#104—1461no—
229#416—1461no—
230C2×#97—731no—
231#365C2×D7314yescomplemented
232#362—732no—
233#361—732no—
234#360—734no—
235#347—734no—
236C2×#41—1461no—
237C2×#41—1461no—
238#135—732no—
239#135—732no—
240#135—732no—
241#135—732no—
242#135—1461no—
243#135—1461no—
244#135—1461no—
245#135—1461no—
246C2×Q16—731no—
247C2×Q16—731no—
248C2×SD16—1461no—
249C2×SD16—1461no—
250C2×D8—731no—
251C2×D8—731no—
252#131—2922no—
253#131—1464no—
254C2×M16—21no—
255C2×M16—2921no—
256C22×C8—2921no—
257C22×C8—731no—
258C22×C8C22×D7311yescharacteristic
259C4×Q8—1462no—
260C4×D4—2922no—
261C4×D4—1462no—
262#117—21no—
263#117—2921no—
264C2×C42—2921no—
265C2×C42—731no—
266Q32—734no—
267SD32—734no—
268SD32—734no—
269D16—734no—
270M32—732no—
271M32—1461no—
272M32C4×D7312yescomplemented
273M32C22×D7312yes—
274C2×C16—1461no—
275C2×C16C4×D7312yescomplemented
276#108—21no—
277#108C22×D7312yes—
278#107—21no—
279#106C22×D7312yes—
280#104—1464no—
281#103—1462no—
282#102—1462no—
283#97—732no—
284#97—732no—
285#41—2921no—
286#41—2921no—
287#41—2921no—
288#41—2921no—
289#41—1462no—
290#41—1462no—
291#41—1462no—
292#41—1462no—
293C2×Q8—1461no—
294C2×Q8—1461no—
295C2×D4—2921no—
296C2×D4—2921no—
297C2×D4—1461no—
298C2×D4—1461no—
299C22×C4—2922no—
300C22×C4—2921no—
301C22×C4—2921no—
302C22×C4—1461no—
303C22×C4D4×D7311yescharacteristic
304Q16—732no—
305Q16—732no—
306Q16—1461no—
307Q16—1461no—
308SD16—1462no—
309SD16—1462no—
310SD16—1462no—
311SD16—1462no—
312D8—732no—
313D8—732no—
314D8—1461no—
315D8—1461no—
316M16—22no—
317M16—41no—
318M16—2924no—
319C2×C8—41no—
320C2×C8—2924no—
321C2×C8—732no—
322C2×C8—1461no—
323C2×C8—1461no—
324C2×C8C2×C4×D7311yescharacteristic
325C2×C8C2×C4×D7311yescharacteristic
326C2×C8D4×D7311yescharacteristic
327C2×C8#1192402911yescharacteristic
328C2×C8#1192402911yescharacteristic
329C2×C8C23×D7311yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—2922no—
332C22⋊C4—41no—
333C22⋊C4—2922no—
334C42—41no—
335C42—2922no—
336C42—732no—
337C42—1461no—
338C16—1462no—
339C16C2×C4×D7314yes—
340C23—2921no—
341C23—2921no—
342C23#1455661811yescharacteristic
343Q8—2921no—
344Q8—2921no—
345Q8—1462no—
346Q8—1462no—
347D4—2922no—
348D4—2922no—
349D4—2922no—
350D4—2922no—
351D4—2921no—
352D4—2921no—
353D4—1462no—
354D4—1462no—
355C2×C4—42no—
356C2×C4—2921no—
357C2×C4—2921no—
358C2×C4—2922no—
359C2×C4—2922no—
360C2×C4—2922no—
361C2×C4—2922no—
362C2×C4—5844no—
363C2×C4—5841no—
364C2×C4—1462no—
365C2×C4#1455655311yescharacteristic
366C2×C4#1455655311yescharacteristic
367C2×C4#1455661811yescharacteristic
368C2×C4#1455661811yescharacteristic
369C2×C4#1455661811yescharacteristic
370C2×C4C2×D4×D7311yescharacteristic
371C8—42no—
372C8—1462no—
373C8#1455655311yescharacteristic
374C8#1455655311yescharacteristic
375C8C22×C4×D7311yescharacteristic
376C8C2×#1192402911yescharacteristic
377C22—41no—
378C22—2922no—
379C22—2922no—
380C22—5842no—
381C22—5842no—
382C22#1798869811yescharacteristic
383C22#1798869811yescharacteristic
384C22C2×#1455661811yescharacteristic
385C4—82no—
386C4—2922no—
387C4—2922no—
388C4—2922no—
389C4#1798869811yescharacteristic
390C4#1798869811yescharacteristic
391C4C2×#1455655311yescharacteristic
392C4C2×#1455661811yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—5842no—
397C2—5842no—
398C2C2×#1798869811yescharacteristic
subgroupG/H
centerC4#17988698
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2155C2
socleC2×C73C2×#442
prime radical#22623067C2
cosocleC2×C8×C73C24
hypercenter#2155D73
hypocenterC73#33468
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2155

Complements

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

NH ≅ G/Ntype
2. C2×#17988708C2semidirect
3. #22618731C2semidirect
4. #22611828C2semidirect
5. #22611821C2semidirect
6. #22611678C2semidirect
7. #22611666C2semidirect
8. C2×#17988364C2semidirect
9. C73×#2155C2semidirect
14. #17989030C4semidirect
16. #17989025C4semidirect
18. #17989019C4semidirect
22. #17988698C22semidirect
23. #17988693C22semidirect
28. #17988402C22semidirect
29. #17988399C22semidirect
31. #17988364C22semidirect
35. #365×C73C22semidirect
44. #14556613C2×C4semidirect
46. C2×Dic292C2×C4semidirect
50. C2×D584C2×C4semidirect
63. Dic584C2×C4semidirect
64. #14556536C2×C4semidirect
65. #14556536C2×C4semidirect
66. D1168C2×C4semidirect
78. M32×C73C2×C4semidirect
80. C2×C16×C73C2×C4semidirect
205. #2155D73semidirect
209. C73#33468semidirect
231. #365C2×D73semidirect
272. M32C4×D73semidirect
275. C2×C16C4×D73semidirect

Maximal subgroups

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

subgroupindex
#226230672
C2×#179887082
#226187312
#226118282
#226118212
#226116782
#226116662
C2×#179883642
C73×#21552
#3346873