Finite Group Databasebeta

← #34441715

Subgroups — #34441715

Order 20224.

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#23776534C211yesmaximal, characteristic
2C2×#19098685C211yesmaximal, characteristic, complemented
3#23772200C211yesmaximal, characteristic, complemented
4#23765295C214yesmaximal, complemented
5#23765288C214yesmaximal, complemented
6#23765172C211yesmaximal, characteristic, complemented
7#23765150C211yesmaximal, characteristic, complemented
8C2×#19098340C211yesmaximal, characteristic, complemented
9C79×#2145C211yesmaximal, characteristic, complemented
10C2×#14579183C2211yescharacteristic
11C2×#14579168—21no—
12#19099031—21no—
13#19099004C412yescomplemented
14#19099003C2212yes—
15#19098999C412yescomplemented
16#19098998C2212yes—
17#19098993C414yescomplemented
18C2×#14579101C2211yescharacteristic
19#19098685—22no—
20#19098685C2214yes—
21#19098674C2212yescomplemented
22#19098669C2212yescomplemented
23#19098662—22no—
24C2×#14579097C2211yescharacteristic
25C2×#14579088—21no—
26#19098557—21no—
27C2×#14579071C2211yescharacteristic
28#19098385C2214yescomplemented
29#19098380C2212yescomplemented
30#19098377C2212yescomplemented
31#19098340C2214yescomplemented
32C2×#135×C79C2211yescharacteristic
33C2×M32×C79C2211yescharacteristic
34#427×C79C2211yescharacteristic
35#360×C79C2214yes—
36C2×#11929624—21no—
37#14579183—21no—
38#14579183—21no—
39#14579183C2×C412yes—
40#14579183C2×C412yes—
41C2×Dic316C2311yescharacteristic
42C2×#11929613—21no—
43C2×D632C2311yescharacteristic
44#14579168—24no—
45#14579168—42no—
46C2×#11929608—21no—
47C2×#11929608—41no—
48C22×C79⋊C8—41no—
49C22×C79⋊C8D411yescharacteristic
50#14579126—22no—
51#14579123—22no—
52(C22⋊D79)⋊C4—42no—
53#14579118—41no—
54C2×C4×Dic79D411yescharacteristic
55C42×D79—41no—
56#14579107C2×C414yescomplemented
57#14579105C2×C414yescomplemented
58#14579104C2×C414yescomplemented
59#14579102C2×C414yescomplemented
60#14579101C2×C412yescomplemented
61#14579101C2312yes—
62C2×C79⋊C16C2×C412yescomplemented
63#14579097—21no—
64#14579097C2312yes—
65#14579088—24no—
66#14579084—22no—
67#14579081—22no—
68#14579071D412yes—
69#14579071C2312yes—
70C2×#41×C79—21no—
71#135×C79—21no—
72#135×C79—21no—
73#135×C79C2×C412yescomplemented
74#135×C79C2312yes—
75C2×Q16×C79C2×C411yescharacteristic, complemented
76C2×SD16×C79—21no—
77C2×D8×C79C2×C411yescharacteristic, complemented
78C22×C8×C79C2311yescharacteristic
79#117×C79—21no—
80M32×C79—21no—
81M32×C79C2312yes—
82C2×C16×C79—21no—
83#107×C79—21no—
84#106×C79C2312yes—
85#11929624—22no—
86#11929624—22no—
87#11929624—41no—
88#11929624—41no—
89C2×Dic158—21no—
90C2×D316—21no—
91C2×C22⋊D79—41no—
92C22×Dic79—21no—
93C2×C4×D79—42no—
94C2×C4×D79—41no—
95Dic316—21no—
96Dic316C22×C412yes—
97#11929613—22no—
98#11929613—22no—
99D632—21no—
100D632C22×C412yes—
101#11929608—22no—
102#11929608—44no—
103#11929608—41no—
104C2×C79⋊C8—21no—
105C2×C79⋊C8—21no—
106C2×C79⋊C8—44no—
107C2×C79⋊C8—41no—
108C2×C79⋊C8C2×D412yes—
109Dic79⋊C4—42no—
110(C2×D79)⋊C4—42no—
111C4×Dic79—21no—
112C4×Dic79—42no—
113C4×Dic79C2×D412yes—
114C79⋊C16C22×C414yes—
115#41×C79—22no—
116#41×C79—22no—
117#41×C79—41no—
118#41×C79—41no—
119C2×Q8×C79—21no—
120C2×D4×C79—21no—
121C2×D4×C79—41no—
122C22×C4×C79—41no—
123C22×C4×C79C2×D411yescharacteristic
124Q16×C79—21no—
125Q16×C79C22×C412yes—
126SD16×C79—22no—
127SD16×C79—22no—
128D8×C79—21no—
129D8×C79C22×C412yes—
130C2×C8×C79C22×C411yescharacteristic
131C2×C8×C79C22×C411yescharacteristic
132C2×C8×C79C2×D411yescharacteristic
133C2×C8×C79#4111yescharacteristic
134C2×C8×C79#4111yescharacteristic
135C2×C8×C79C2411yescharacteristic
136C4⋊C4×C79—22no—
137C22⋊C4×C79—41no—
138C42×C79—41no—
139C16×C79—22no—
140C23×C79—41no—
141C23×C79AGL(1,ℤ/8)11yescharacteristic
142C22×D79—41no—
143Q8×C79—22no—
144Q8×C79—41no—
145D4×C79—22no—
146D4×C79—42no—
147D4×C79—42no—
148D4×C79—41no—
149C2×C4×C79—41no—
150C2×C4×C79—42no—
151C2×C4×C79—42no—
152C2×C4×C79—42no—
153C2×C4×C79C4×D411yescharacteristic
154C2×C4×C79C4×D411yescharacteristic
155C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C79C22×D411yescharacteristic
159C22⋊D79—42no—
160C22⋊D79—42no—
161C2×Dic79—22no—
162C2×Dic79—41no—
163C2×Dic79—81no—
164D316—22no—
165D316—41no—
166C4×D79—42no—
167C4×D79—42no—
168C4×D79—84no—
169Dic158—22no—
170Dic158—41no—
171C8×C79C4×D411yescharacteristic
172C8×C79C4×D411yescharacteristic
173C8×C79C23×C411yescharacteristic
174C8×C79C2×#4111yescharacteristic
175C79⋊C8—22no—
176C79⋊C8—42no—
177C22×C79—41no—
178C22×C79—42no—
179C22×C79—82no—
180C22×C79#44211yescharacteristic
181C22×C79#44211yescharacteristic
182C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
183C2×D79—42no—
184C2×D79—82no—
185C4×C79—42no—
186C4×C79—82no—
187C4×C79#44211yescharacteristic
188C4×C79#44211yescharacteristic
189C4×C79C2×C4×D411yescharacteristic
190C4×C79C2×AGL(1,ℤ/8)11yescharacteristic
191Dic79—42no—
192Dic79—42no—
193#33468—791nomaximal, Sylow
194C2×C79—21no—
195C2×C79—21no—
196C2×C79—21no—
197C2×C79—82no—
198C2×C79C2×#44211yescharacteristic
199D79—82no—
200#3414—791no—
201C2×#444—791no—
202#2943—791no—
203AGL(1,ℤ/16)—794no—
204#2180—794no—
205#2155—791no—
206#2148—791no—
207#2145D7911yescharacteristic, complemented
208C2×#347—791no—
209C79#3346811yescharacteristic, complemented, Sylow
210C2×#135—791no—
211C2×#135C2×D7911yescharacteristic
212C2×#131—1581no—
213C4×#41—1581no—
214#510—792no—
215#510—792no—
216#509—792no—
217#509—792no—
218#508—794no—
219C2×M32—791no—
220C2×M32C2×D7911yescharacteristic
221#444—794no—
222#444—1582no—
223#442—792no—
224#441—792no—
225#440—1582no—
226C2×#108—791no—
227#427C2×D7911yescharacteristic
228C2×#104—1581no—
229#416—1581no—
230C2×#97—791no—
231#365—794no—
232#362—792no—
233#361—792no—
234#360C2×D7914yes—
235#347—794no—
236C2×#41—21no—
237C2×#41—1581no—
238#135—21no—
239#135—21no—
240#135—792no—
241#135—792no—
242#135—1581no—
243#135—1581no—
244#135C4×D7912yescomplemented
245#135C22×D7912yes—
246C2×Q16—791no—
247C2×Q16C4×D7911yescharacteristic, complemented
248C2×SD16—21no—
249C2×SD16—1581no—
250C2×D8—791no—
251C2×D8C4×D7911yescharacteristic, complemented
252#131—3162no—
253#131—1584no—
254C2×M16—3161no—
255C2×M16—1581no—
256C22×C8—3161no—
257C22×C8—791no—
258C22×C8C22×D7911yescharacteristic
259C4×Q8—1582no—
260C4×D4—3162no—
261C4×D4—1582no—
262#117—21no—
263#117—3161no—
264C2×C42—3161no—
265C2×C42—791no—
266Q32—794no—
267SD32—794no—
268SD32—794no—
269D16—794no—
270M32—21no—
271M32—792no—
272M32—792no—
273M32C22×D7912yes—
274C2×C16—21no—
275C2×C16—792no—
276#108—792no—
277#108—1581no—
278#107—21no—
279#106C22×D7912yes—
280#104—1584no—
281#103—1582no—
282#102—1582no—
283#97—792no—
284#97—792no—
285#41—22no—
286#41—22no—
287#41—41no—
288#41—41no—
289#41—3161no—
290#41—3161no—
291#41—1582no—
292#41—1582no—
293C2×Q8—21no—
294C2×Q8—1581no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—3161no—
298C2×D4—1581no—
299C22×C4—41no—
300C22×C4—3162no—
301C22×C4—3161no—
302C22×C4—1581no—
303C22×C4D4×D7911yescharacteristic
304Q16—21no—
305Q16—792no—
306Q16—1581no—
307Q16C2×C4×D7912yes—
308SD16—22no—
309SD16—22no—
310SD16—1582no—
311SD16—1582no—
312D8—21no—
313D8—792no—
314D8—1581no—
315D8C2×C4×D7912yes—
316M16—3161no—
317M16—3164no—
318M16—1582no—
319C2×C8—3161no—
320C2×C8—3164no—
321C2×C8—792no—
322C2×C8—1581no—
323C2×C8—1581no—
324C2×C8C2×C4×D7911yescharacteristic
325C2×C8C2×C4×D7911yescharacteristic
326C2×C8D4×D7911yescharacteristic
327C2×C8#1192962311yescharacteristic
328C2×C8#1192962311yescharacteristic
329C2×C8C23×D7911yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—3162no—
332C22⋊C4—41no—
333C22⋊C4—3162no—
334C42—41no—
335C42—3162no—
336C42—792no—
337C42—1581no—
338C16—22no—
339C16—794no—
340C23—41no—
341C23—3161no—
342C23#1457918511yescharacteristic
343Q8—22no—
344Q8—41no—
345Q8—3161no—
346Q8—1582no—
347D4—22no—
348D4—42no—
349D4—42no—
350D4—41no—
351D4—3162no—
352D4—3162no—
353D4—3161no—
354D4—1582no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—3161no—
360C2×C4—3162no—
361C2×C4—3162no—
362C2×C4—6324no—
363C2×C4—6321no—
364C2×C4—1582no—
365C2×C4(C22⋊D79)⋊C411yescharacteristic
366C2×C4(C22⋊D79)⋊C411yescharacteristic
367C2×C4#1457918511yescharacteristic
368C2×C4#1457918811yescharacteristic
369C2×C4#1457918811yescharacteristic
370C2×C4C2×D4×D7911yescharacteristic
371C8—3162no—
372C8—1582no—
373C8(C22⋊D79)⋊C411yescharacteristic
374C8(C22⋊D79)⋊C411yescharacteristic
375C8C22×C4×D7911yescharacteristic
376C8C2×#1192962311yescharacteristic
377C22—41no—
378C22—42no—
379C22—82no—
380C22—3162no—
381C22—6322no—
382C22#1909867211yescharacteristic
383C22#1909867211yescharacteristic
384C22C2×#1457918511yescharacteristic
385C4—42no—
386C4—82no—
387C4—3162no—
388C4—3162no—
389C4#1909867211yescharacteristic
390C4#1909867211yescharacteristic
391C4C2×(C22⋊D79)⋊C411yescharacteristic
392C4C2×#1457918811yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—6322no—
398C2C2×#1909867211yescharacteristic
subgroupG/H
centerC4#19098672
commutatorC8×C79C23×C4
FrattiniC2×C8C23×D79
FittingC79×#2145C2
socleC2×C79C2×#442
prime radical#23776534C2
cosocleC2×C8×C79C24
hypercenter#2145D79
hypocenterC79#33468
LCSG ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2145

Complements

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

NH ≅ G/Ntype
2. C2×#19098685C2semidirect
3. #23772200C2semidirect
4. #23765295C2semidirect
5. #23765288C2semidirect
6. #23765172C2semidirect
7. #23765150C2semidirect
8. C2×#19098340C2semidirect
9. C79×#2145C2semidirect
13. #19099004C4semidirect
15. #19098999C4semidirect
17. #19098993C4semidirect
21. #19098674C22semidirect
22. #19098669C22semidirect
28. #19098385C22semidirect
29. #19098380C22semidirect
30. #19098377C22semidirect
31. #19098340C22semidirect
56. #14579107C2×C4semidirect
57. #14579105C2×C4semidirect
58. #14579104C2×C4semidirect
59. #14579102C2×C4semidirect
60. #14579101C2×C4semidirect
62. C2×C79⋊C16C2×C4semidirect
73. #135×C79C2×C4semidirect
75. C2×Q16×C79C2×C4semidirect
77. C2×D8×C79C2×C4semidirect
207. #2145D79semidirect
209. C79#33468semidirect
244. #135C4×D79semidirect
247. C2×Q16C4×D79semidirect
251. C2×D8C4×D79semidirect

Maximal subgroups

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

subgroupindex
#237765342
C2×#190986852
#237722002
#237652952
#237652882
#237651722
#237651502
C2×#190983402
C79×#21452
#3346879