Finite Group Databasebeta

← #33239435

Subgroups — #33239435

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
1C2×#17988709C211yesmaximal, characteristic, complemented
2#22618732C211yesmaximal, characteristic, complemented
3#22611830C214yesmaximal, complemented
4#22611823C214yesmaximal, complemented
5#22611704C211yesmaximal, characteristic, complemented
6#22611680C211yesmaximal, characteristic, complemented
7#22611667C211yesmaximal, characteristic, complemented
8C2×#17988364C211yesmaximal, characteristic, complemented
9C73×#3414C211yesmaximal, characteristic
10C2×#14556598—21no—
11#41×Dic73—21no—
12C2×#14556531C2211yescharacteristic
13#17988709—22no—
14#17988709C2214yes—
15#17988699C2212yescomplemented
16#17988694C2212yescomplemented
17#17988688—22no—
18C2×#14556527C2211yescharacteristic
19#17988625C2211yescharacteristic
20C2×#14556518—21no—
21#17988583—21no—
22C2×#14556501C2211yescharacteristic
23#17988410C2214yescomplemented
24#17988403C2212yescomplemented
25#17988400C2212yescomplemented
26#17988397C2214yes—
27#17988364C2214yescomplemented
28C2×#135×C73C2211yescharacteristic
29C2×#135×C73C2211yescharacteristic
30#510×C73C412yescomplemented
31#510×C73C2212yes—
32#509×C73C412yescomplemented
33#509×C73C2212yes—
34#508×C73C414yescomplemented
35C2×M32×C73C2211yescharacteristic
36#14556598—24no—
37#14556598—42no—
38C2×#11924012—21no—
39C2×#11924012—41no—
40C22×#10596831—41no—
41C22×#10596831D411yescharacteristic
42Q8×Dic73—22no—
43D4×Dic73—22no—
44D4×Dic73—42no—
45#14556550—21no—
46#14556550—41no—
47C2×C4×Dic73—41no—
48C2×C4×Dic73D411yescharacteristic
49#14556531—21no—
50#14556531C2312yes—
51C2×#11924003—21no—
52#14556527—21no—
53#14556527C2312yes—
54#14556525—21no—
55#14556523C2312yes—
56#14556518—24no—
57#14556515—22no—
58#14556512—22no—
59#14556501D412yes—
60#14556501C2312yes—
61C2×#41×C73—21no—
62C2×#41×C73—21no—
63#135×C73—21no—
64#135×C73—21no—
65#135×C73—21no—
66#135×C73—21no—
67#135×C73C2×C412yes—
68#135×C73C2×C412yes—
69#135×C73C2×C412yescomplemented
70#135×C73C2312yes—
71C2×Q16×C73C2×C411yescharacteristic, complemented
72C2×Q16×C73C2311yescharacteristic
73C2×SD16×C73—21no—
74C2×SD16×C73—21no—
75C2×D8×C73C2×C411yescharacteristic, complemented
76C2×D8×C73C2311yescharacteristic
77C22×C8×C73C2311yescharacteristic
78Q32×C73C2×C414yescomplemented
79SD32×C73C2×C414yescomplemented
80SD32×C73C2×C414yescomplemented
81D16×C73C2×C414yescomplemented
82M32×C73C2×C412yescomplemented
83M32×C73C2312yes—
84C2×C16×C73C2×C412yescomplemented
85C22×Dic73—21no—
86C22×Dic73—42no—
87#11924012—22no—
88#11924012—44no—
89#11924012—41no—
90C2×#10596831—21no—
91C2×#10596831—21no—
92C2×#10596831—44no—
93C2×#10596831—41no—
94C2×#10596831C2×D412yes—
95C4⋊Dic73—22no—
96C4⋊Dic73—42no—
97C22⋊Dic73—42no—
98C22⋊Dic73—41no—
99C4×Dic73—21no—
100C4×Dic73—42no—
101C4×Dic73—41no—
102C4×Dic73C2×D412yes—
103#11924003—22no—
104#41×C73—22no—
105#41×C73—22no—
106#41×C73—22no—
107#41×C73—22no—
108#41×C73—41no—
109#41×C73—41no—
110#41×C73—41no—
111#41×C73—41no—
112C2×Q8×C73—21no—
113C2×Q8×C73—21no—
114C2×D4×C73—21no—
115C2×D4×C73—21no—
116C2×D4×C73—41no—
117C2×D4×C73—41no—
118C22×C4×C73—41no—
119C22×C4×C73—41no—
120C22×C4×C73C2×D411yescharacteristic
121Q16×C73—21no—
122Q16×C73—21no—
123Q16×C73C22×C412yes—
124Q16×C73C22×C412yes—
125SD16×C73—22no—
126SD16×C73—22no—
127SD16×C73—22no—
128SD16×C73—22no—
129D8×C73—21no—
130D8×C73—21no—
131D8×C73C22×C412yes—
132D8×C73C22×C412yes—
133C2×C8×C73C22×C411yescharacteristic
134C2×C8×C73C22×C411yescharacteristic
135C2×C8×C73C2×D411yescharacteristic
136C2×C8×C73#4111yescharacteristic
137C2×C8×C73#4111yescharacteristic
138C2×C8×C73C2411yescharacteristic
139C16×C73C22×C414yes—
140C23×C73—41no—
141C23×C73—41no—
142C23×C73AGL(1,ℤ/8)11yescharacteristic
143Q8×C73—22no—
144Q8×C73—22no—
145Q8×C73—41no—
146Q8×C73—41no—
147D4×C73—22no—
148D4×C73—22no—
149D4×C73—42no—
150D4×C73—42no—
151D4×C73—42no—
152D4×C73—42no—
153D4×C73—41no—
154D4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73—42no—
158C2×C4×C73—42no—
159C2×C4×C73—42no—
160C2×C4×C73—42no—
161C2×C4×C73C4×D411yescharacteristic
162C2×C4×C73C4×D411yescharacteristic
163C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C73C22×D411yescharacteristic
167C2×Dic73—22no—
168C2×Dic73—42no—
169C2×Dic73—81no—
170C2×Dic73—84no—
171C8×C73C4×D411yescharacteristic
172C8×C73C4×D411yescharacteristic
173C8×C73C23×C411yescharacteristic
174C8×C73C2×#4111yescharacteristic
175#10596831—22no—
176#10596831—42no—
177C22×C73—41no—
178C22×C73—42no—
179C22×C73—42no—
180C22×C73—82no—
181C22×C73—82no—
182C22×C73#44211yescharacteristic
183C22×C73#44211yescharacteristic
184C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
185C4×C73—42no—
186C4×C73—42no—
187C4×C73#44211yescharacteristic
188C4×C73#44211yescharacteristic
189C4×C73C2×C4×D411yescharacteristic
190C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
191Dic73—42no—
192Dic73—82no—
193#33468—731nomaximal, Sylow
194C2×C73—21no—
195C2×C73—21no—
196C2×C73—21no—
197C2×C73—82no—
198C2×C73—82no—
199C2×C73C2×#44211yescharacteristic
200#3414D7311yescharacteristic
201C2×#444—731no—
202#2943—731no—
203AGL(1,ℤ/16)—734no—
204#2180—734no—
205#2155—731no—
206#2148—731no—
207#2145—731no—
208C2×#347—731no—
209C73#3346811yescharacteristic, complemented, Sylow
210C2×#135C2×D7311yescharacteristic
211C2×#135C2×D7311yescharacteristic
212C2×#131—1461no—
213C4×#41—1461no—
214#510Dic7312yescomplemented
215#510C2×D7312yes—
216#509Dic7312yescomplemented
217#509C2×D7312yes—
218#508Dic7314yescomplemented
219C2×M32—731no—
220C2×M32C2×D7311yescharacteristic
221#444—734no—
222#444—1462no—
223#442—732no—
224#441—732no—
225#440—1462no—
226C2×#108—731no—
227#427—731no—
228C2×#104—1461no—
229#416—1461no—
230C2×#97—731no—
231#365—734no—
232#362—732no—
233#361—732no—
234#360—734no—
235#347—734no—
236C2×#41—21no—
237C2×#41—21no—
238#135—21no—
239#135—21no—
240#135—21no—
241#135—21no—
242#135C2×Dic7312yes—
243#135C2×Dic7312yes—
244#135C2×Dic7312yescomplemented
245#135C22×D7312yes—
246C2×Q16C2×Dic7311yescharacteristic, complemented
247C2×Q16C22×D7311yescharacteristic
248C2×SD16—21no—
249C2×SD16—21no—
250C2×D8C2×Dic7311yescharacteristic, complemented
251C2×D8C22×D7311yescharacteristic
252#131—2922no—
253#131—1464no—
254C2×M16—2921no—
255C2×M16—1461no—
256C22×C8—2921no—
257C22×C8—731no—
258C22×C8C22×D7311yescharacteristic
259C4×Q8—1462no—
260C4×D4—2922no—
261C4×D4—1462no—
262#117—2921no—
263#117—1461no—
264C2×C42—2921no—
265C2×C42—731no—
266Q32C2×Dic7314yescomplemented
267SD32C2×Dic7314yescomplemented
268SD32C2×Dic7314yescomplemented
269D16C2×Dic7314yescomplemented
270M32—732no—
271M32—1461no—
272M32C2×Dic7312yescomplemented
273M32C22×D7312yes—
274C2×C16—1461no—
275C2×C16C2×Dic7312yescomplemented
276#108—732no—
277#108—1461no—
278#107—1461no—
279#106—732no—
280#104—1464no—
281#103—1462no—
282#102—1462no—
283#97—732no—
284#97—732no—
285#41—22no—
286#41—22no—
287#41—22no—
288#41—22no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—2922no—
302C22×C4—1461no—
303C22×C4D4×D7311yescharacteristic
304Q16—21no—
305Q16—21no—
306Q16C22×Dic7312yes—
307Q16C22×Dic7312yes—
308SD16—22no—
309SD16—22no—
310SD16—22no—
311SD16—22no—
312D8—21no—
313D8—21no—
314D8C22×Dic7312yes—
315D8C22×Dic7312yes—
316M16—2921no—
317M16—2924no—
318M16—1462no—
319C2×C8—2921no—
320C2×C8—2924no—
321C2×C8—732no—
322C2×C8—1461no—
323C2×C8—1461no—
324C2×C8C22×Dic7311yescharacteristic
325C2×C8C22×Dic7311yescharacteristic
326C2×C8D4×D7311yescharacteristic
327C2×C8#1192402711yescharacteristic
328C2×C8#1192402711yescharacteristic
329C2×C8C23×D7311yescharacteristic
330C4⋊C4—2922no—
331C4⋊C4—1462no—
332C22⋊C4—2921no—
333C22⋊C4—2922no—
334C42—2922no—
335C42—2921no—
336C42—732no—
337C42—1461no—
338C16—1462no—
339C16C22×Dic7314yes—
340C23—41no—
341C23—41no—
342C23#1455661511yescharacteristic
343Q8—22no—
344Q8—22no—
345Q8—41no—
346Q8—41no—
347D4—22no—
348D4—22no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—42no—
353D4—41no—
354D4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—2922no—
362C2×C4—5841no—
363C2×C4—5844no—
364C2×C4—1462no—
365C2×C4D4×Dic7311yescharacteristic
366C2×C4D4×Dic7311yescharacteristic
367C2×C4#1455661511yescharacteristic
368C2×C4#1455661511yescharacteristic
369C2×C4#1455661511yescharacteristic
370C2×C4C2×D4×D7311yescharacteristic
371C8—2922no—
372C8—1462no—
373C8D4×Dic7311yescharacteristic
374C8D4×Dic7311yescharacteristic
375C8C23×Dic7311yescharacteristic
376C8C2×#1192402711yescharacteristic
377C22—41no—
378C22—42no—
379C22—42no—
380C22—82no—
381C22—82no—
382C22#1798869911yescharacteristic
383C22#1798869911yescharacteristic
384C22C2×#1455661511yescharacteristic
385C4—42no—
386C4—42no—
387C4—2922no—
388C4—5842no—
389C4#1798869911yescharacteristic
390C4#1798869911yescharacteristic
391C4C2×D4×Dic7311yescharacteristic
392C4C2×#1455661511yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—82no—
398C2C2×#1798869911yescharacteristic
subgroupG/H
centerC4#17988699
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
Fitting, prime radicalC73×#3414C2
socleC2×C73C2×#442
cosocleC2×C8×C73C24
hypercenter#3414D73
hypocenterC73#33468
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. C2×#17988709C2semidirect
2. #22618732C2semidirect
3. #22611830C2semidirect
4. #22611823C2semidirect
5. #22611704C2semidirect
6. #22611680C2semidirect
7. #22611667C2semidirect
8. C2×#17988364C2semidirect
15. #17988699C22semidirect
16. #17988694C22semidirect
23. #17988410C22semidirect
24. #17988403C22semidirect
25. #17988400C22semidirect
27. #17988364C22semidirect
30. #510×C73C4semidirect
32. #509×C73C4semidirect
34. #508×C73C4semidirect
69. #135×C73C2×C4semidirect
71. C2×Q16×C73C2×C4semidirect
75. C2×D8×C73C2×C4semidirect
78. Q32×C73C2×C4semidirect
79. SD32×C73C2×C4semidirect
80. SD32×C73C2×C4semidirect
81. D16×C73C2×C4semidirect
82. M32×C73C2×C4semidirect
84. C2×C16×C73C2×C4semidirect
209. C73#33468semidirect
214. #510Dic73semidirect
216. #509Dic73semidirect
218. #508Dic73semidirect
244. #135C2×Dic73semidirect
246. C2×Q16C2×Dic73semidirect
250. C2×D8C2×Dic73semidirect
266. Q32C2×Dic73semidirect
267. SD32C2×Dic73semidirect
268. SD32C2×Dic73semidirect
269. D16C2×Dic73semidirect
272. M32C2×Dic73semidirect
275. C2×C16C2×Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179887092
#226187322
#226118302
#226118232
#226117042
#226116802
#226116672
C2×#179883642
C73×#34142
#3346873