Finite Group Databasebeta

← #34030244

Subgroups — #34030244

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626900C211yesmaximal, characteristic, complemented
2#22626898C211yesmaximal, characteristic, complemented
3#22626322C214yesmaximal, complemented
4#22626319C214yesmaximal, complemented
5#22626274C211yesmaximal, characteristic, complemented
6C2×#17989486C211yesmaximal, characteristic, complemented
7C2×#17989485C211yesmaximal, characteristic, complemented
8#22626140C211yesmaximal, characteristic, complemented
9#22618830C211yesmaximal, characteristic
10#17989646—22no—
11#17989646C2214yescomplemented
12#17989645—22no—
13#17989645C2214yescomplemented
14#17989643—22no—
15C2×#14556665—22no—
16C2×#14556665C2211yescharacteristic
17C2×#14556665C2211yescharacteristic
18C2×#14556665C2211yescharacteristic
19C22×#11924031C2211yescharacteristic
20#17989510C2214yescomplemented
21#17989486C2212yescomplemented
22#17989486C2212yescomplemented
23#17989485C2212yescomplemented
24#17989485C2212yescomplemented
25#17989475C2212yescomplemented
26C4×#11924031C2212yescomplemented
27#17989385C411yescharacteristic
28C2×#14556599C2211yescharacteristic
29C2×#14556598C2211yescharacteristic
30#17988706C414yes—
31#17988701C2214yes—
32#17988635C411yescharacteristic
33#17988605C411yescharacteristic
34#17988543C2211yescharacteristic
35#421×C73C411yescharacteristic
36#14556666—24no—
37#14556665—24no—
38#14556665—21no—
39#14556665—21no—
40#14556665D412yes—
41#14556665D412yes—
42#14556665C2312yes—
43#14556665C2312yes—
44#14556664—24no—
45C2×#11924031—24no—
46C2×#11924031—21no—
47C2×#11924031—21no—
48C2×#11924031—21no—
49C2×#11924031—21no—
50C2×#11924031C2311yescharacteristic
51C2×#11924031C2311yescharacteristic
52C2×#11924031C2312yes—
53C2×#11924028C2×C411yescharacteristic
54#14556622C812yes—
55#14556621C812yes—
56#14556617C812yes—
57#14556616C812yes—
58#14556611C2×C414yes—
59#14556599—21no—
60#14556599—21no—
61#14556599C2×C412yes—
62#14556599C2312yes—
63#14556598—21no—
64#14556598—21no—
65#14556598C2×C412yes—
66#14556598C2312yes—
67C2×#11924012—21no—
68C2×#11924012C2×C411yescharacteristic
69C2×#11924012C2×C411yescharacteristic
70C2×#11924011—21no—
71C22×#10596831—21no—
72C22×#10596831C2311yescharacteristic
73C2×C8×D73—21no—
74#14556526C2×C414yes—
75#14556519C2×C414yes—
76#14556502C2×C412yes—
77C4×#10596831C2312yes—
78C2×#41×C73C2×C411yescharacteristic
79C2×M16×C73C2×C411yescharacteristic
80#117×C73C2×C411yescharacteristic
81#104×C73C2×C414yes—
82#11924031—22no—
83#11924031—22no—
84#11924031—21no—
85#11924031—21no—
86#11924031C2×D414yes—
87#11924028—21no—
88#11924028—21no—
89#11924028C2×C812yes—
90#11924028C22×C412yes—
91C2×Dic146C2×C811yescharacteristic
92C2×D292C2×C811yescharacteristic
93C2×C22⋊D73—21no—
94C2×C4×D73—21no—
95#11924018C2×C814yes—
96#11924016C2×C814yes—
97#11924015C2×C814yes—
98#11924013C2×C814yes—
99#11924012—21no—
100#11924012—22no—
101#11924012—22no—
102#11924012C2×C812yes—
103#11924012C2×C812yes—
104#11924012C22×C412yes—
105#11924011—22no—
106#11924011—22no—
107C2×#10596831—21no—
108C2×#10596831—22no—
109C2×#10596831—22no—
110C2×#10596831C22×C411yescharacteristic
111C2×#10596831C22×C412yes—
112C2×#10596831C2×D411yescharacteristic
113C2×#10596831C2×D411yescharacteristic
114C2×#10596831#4111yescharacteristic
115C2×#10596831#4111yescharacteristic
116C2×#10596831C2411yescharacteristic
117C8×D73—22no—
118C8×D73—22no—
119#41×C73—21no—
120#41×C73—21no—
121#41×C73C2×C812yes—
122#41×C73C22×C412yes—
123C2×Q8×C73C2×C811yescharacteristic
124C2×D4×C73—21no—
125C2×D4×C73C2×C811yescharacteristic
126C22×C4×C73—21no—
127C22×C4×C73C22×C411yescharacteristic
128M16×C73—21no—
129M16×C73C22×C412yes—
130C2×C8×C73—21no—
131C4⋊C4×C73—21no—
132C22⋊C4×C73—21no—
133C42×C73C22×C412yes—
134C23×C73—21no—
135C23×C73#13111yescharacteristic
136C22×D73—21no—
137Q8×C73—21no—
138Q8×C73C22×C812yes—
139D4×C73—22no—
140D4×C73—22no—
141D4×C73—21no—
142D4×C73C22×C812yes—
143C2×C4×C73—21no—
144C2×C4×C73—22no—
145C2×C4×C73—22no—
146C2×C4×C73—22no—
147C2×C4×C73C4×D411yescharacteristic
148C2×C4×C73C4×D411yescharacteristic
149C2×C4×C73C22×C811yescharacteristic
150C2×C4×C73C22×C811yescharacteristic
151C2×C4×C73#13111yescharacteristic
152C2×C4×C73C23×C411yescharacteristic
153C22⋊D73—22no—
154C22⋊D73—22no—
155C2×Dic73—21no—
156D292—21no—
157D292C22×C812yes—
158C4×D73—22no—
159C4×D73—22no—
160Dic146—21no—
161Dic146C22×C812yes—
162C8×C73—22no—
163#10596831—22no—
164#10596831C4×D411yescharacteristic
165#10596831C4×D411yescharacteristic
166#10596831C22×C814yes—
167#10596831C22×D411yescharacteristic
168#10596831C2×#4111yescharacteristic
169C22×C73—22no—
170C22×C73—41no—
171C22×C73—81no—
172C22×C73C8×D411yescharacteristic
173C22×C73C8×D411yescharacteristic
174C22×C73C2×#13111yescharacteristic
175C2×D73—22no—
176C2×D73—81no—
177C4×C73—22no—
178C4×C73—42no—
179C4×C73C8×D411yescharacteristic
180C4×C73C8×D411yescharacteristic
181C4×C73C2×C4×D411yescharacteristic
182C4×C73C23×C811yescharacteristic
183Dic73—22no—
184#33442—731nomaximal, Sylow
185C2×C73—21no—
186C2×C73—21no—
187C2×C73—21no—
188C2×C73—42no—
189C2×C73C2×C8×D411yescharacteristic
190D73—42no—
191#3407—731no—
192#3407—731no—
193#2955—731no—
194#2171—734no—
195#2170—734no—
196#2153—731no—
197C2×#350—731no—
198C2×#350—731no—
199#2108—731no—
200C73#3344211yescharacteristic, complemented, Sylow
201#575—731no—
202C2×#131—731no—
203C2×#131—731no—
204#504—734no—
205#504—734no—
206#504—1462no—
207#504—1462no—
208C2×M32—731no—
209C2×M32—731no—
210C2×M32—731no—
211C2×M32—1462no—
212C2×M32—1462no—
213C22×C16—731no—
214#444—734no—
215#443—734no—
216#430—731no—
217#421—731no—
218#421#6440811yescharacteristic
219#405—731no—
220#364—734no—
221#350—732no—
222#350—732no—
223#350—732no—
224#350—732no—
225#346—732no—
226C4×C16—732no—
227C2×#41—731no—
228C2×#41C2×#6440811yescharacteristic
229#137—732no—
230#137—732no—
231AGL(1,ℤ/8)—732no—
232AGL(1,ℤ/8)—732no—
233#135—734no—
234#131—732no—
235#131—732no—
236#131—732no—
237#131—732no—
238#131—1461no—
239#131—1461no—
240#131—1461no—
241#131—1461no—
242C2×M16—731no—
243C2×M16—731no—
244C2×M16—1461no—
245C2×M16—1461no—
246C2×M16C2×#6440811yescharacteristic
247C22×C8—731no—
248C22×C8—1461no—
249C22×C8—1461no—
250#117C2×#6440811yescharacteristic
251M32—732no—
252M32—732no—
253M32—732no—
254M32—732no—
255M32—1464no—
256M32—1464no—
257M32—1461no—
258M32—1461no—
259C2×C16—732no—
260C2×C16—731no—
261C2×C16—731no—
262C2×C16—1464no—
263C2×C16—1464no—
264C2×C16—1461no—
265C2×C16—1461no—
266C2×C16—1461no—
267C2×C16—1461no—
268#108—734no—
269#104—734no—
270#104C2×#6440814yes—
271#97—732no—
272C4×C8—732no—
273#41—21no—
274#41—21no—
275#41—732no—
276#41—732no—
277#41—1461no—
278#41—1461no—
279#41#1192403612yes—
280#41C22×#6440812yes—
281C2×Q8—731no—
282C2×Q8#1192403611yescharacteristic
283C2×D4—21no—
284C2×D4—731no—
285C2×D4—1461no—
286C2×D4#1192403611yescharacteristic
287C22×C4—21no—
288C22×C4—1461no—
289C22×C4C22×#6440811yescharacteristic
290Q16—734no—
291SD16—734no—
292SD16—734no—
293D8—734no—
294M16—21no—
295M16—732no—
296M16—732no—
297M16—732no—
298M16—1462no—
299M16—1462no—
300M16—1462no—
301M16—1462no—
302M16—1461no—
303M16C22×#6440812yes—
304C2×C8—21no—
305C2×C8—732no—
306C2×C8—731no—
307C2×C8—731no—
308C2×C8—731no—
309C2×C8—731no—
310C2×C8—731no—
311C2×C8—731no—
312C2×C8—1462no—
313C2×C8—1462no—
314C2×C8—1462no—
315C2×C8—1462no—
316C2×C8—1461no—
317C4⋊C4—21no—
318C22⋊C4—21no—
319C42C22×#6440812yes—
320C16—734no—
321C16—1461no—
322C16—1461no—
323C16—1462no—
324C16—1462no—
325C23—21no—
326C23—1461no—
327C23#1455667811yescharacteristic
328Q8—21no—
329Q8—732no—
330Q8—1461no—
331Q8C2×#1192403612yes—
332D4—22no—
333D4—22no—
334D4—21no—
335D4—732no—
336D4—1462no—
337D4—1462no—
338D4—1461no—
339D4C2×#1192403612yes—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—1461no—
345C2×C4—1462no—
346C2×C4—1462no—
347C2×C4D4×#6440811yescharacteristic
348C2×C4D4×#6440811yescharacteristic
349C2×C4C2×#1192403611yescharacteristic
350C2×C4C2×#1192403611yescharacteristic
351C2×C4#1455667811yescharacteristic
352C2×C4C23×#6440811yescharacteristic
353C8—22no—
354C8—731no—
355C8—731no—
356C8—731no—
357C8—731no—
358C8—734no—
359C8—1462no—
360C22—22no—
361C22—41no—
362C22—81no—
363C22—5841no—
364C22—1462no—
365C22#1798961411yescharacteristic
366C22#1798961411yescharacteristic
367C22C2×#1455667811yescharacteristic
368C4—22no—
369C4—42no—
370C4—1462no—
371C4#1798961411yescharacteristic
372C4#1798961411yescharacteristic
373C4C2×D4×#6440811yescharacteristic
374C4C22×#1192403611yescharacteristic
375C2—21no—
376C2—21no—
377C2—21no—
378C2—42no—
379C2—2922no—
380C2C2×#1798961411yescharacteristic
subgroupG/H
centerC4C2×D4×#64408
commutatorC4×C73C23×C8
FrattiniC2×C4C23×#64408
Fitting#421×C73C4
socleC2×C73C2×C8×D4
prime radical#17989385C4
cosocleC2×#10596831C24
hypercenter#421#64408
hypocenterC73#33442
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #22626900C2semidirect
2. #22626898C2semidirect
3. #22626322C2semidirect
4. #22626319C2semidirect
5. #22626274C2semidirect
6. C2×#17989486C2semidirect
7. C2×#17989485C2semidirect
8. #22626140C2semidirect
11. #17989646C22semidirect
13. #17989645C22semidirect
20. #17989510C22semidirect
21. #17989486C22semidirect
22. #17989486C22semidirect
23. #17989485C22semidirect
24. #17989485C22semidirect
25. #17989475C22semidirect
26. C4×#11924031C22semidirect
200. C73#33442semidirect

Maximal subgroups

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

subgroupindex
#226269002
#226268982
#226263222
#226263192
#226262742
C2×#179894862
C2×#179894852
#226261402
#226188302
#3344273