Finite Group Databasebeta

← #34441731

Subgroups — #34441731

Order 20224.

646 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
2#23772316C211yesmaximal, characteristic, complemented
3#23772284C211yesmaximal, characteristic, complemented
4#23765301C214yesmaximal, complemented
5#23765294C214yesmaximal, complemented
6C2×#19098387C211yesmaximal, characteristic, complemented
7C2×#19098374C211yesmaximal, characteristic, complemented
8#23765004C211yesmaximal, characteristic, complemented
9C79×#2148C211yesmaximal, characteristic, complemented
10C2×#14579183C2211yescharacteristic
11#19099329—21no—
12#19099046—21no—
13#19099004C2212yes—
14#19099003C412yescomplemented
15#19098999C2212yes—
16#19098998C412yescomplemented
17#19098993C414yescomplemented
18C2×#14579101C2211yescharacteristic
19#19098684C2214yes—
20#19098679—22no—
21#19098675C2212yescomplemented
22#19098670C2212yescomplemented
23SD16×Dic79—22no—
24C2×#14579093C2211yescharacteristic
25#19098579—21no—
26#19098564—21no—
27#19098518C2211yescharacteristic
28#19098387C2214yescomplemented
29#19098374C2214yes—
30#19098340C2214yescomplemented
31C2×#135×C79C2211yescharacteristic
32C2×M32×C79C2211yescharacteristic
33C2×#108×C79C2211yescharacteristic
34#362×C79C2212yescomplemented
35#361×C79C2212yescomplemented
36C2×#11929624—21no—
37#14579183—21no—
38#14579183—21no—
39#14579183C2×C412yescomplemented
40#14579183C2312yes—
41C2×Dic316C2×C411yescharacteristic, complemented
42C2×#11929613—21no—
43C2×D632C2×C411yescharacteristic, complemented
44#14579169—24no—
45#14579167—42no—
46C2×#11929608D411yescharacteristic
47M16×D79—42no—
48Q8×Dic79—22no—
49D4×Dic79—22no—
50D4×Dic79—42no—
51#14579120—41no—
52#14579120D411yescharacteristic
53C2×C4⋊Dic79—21no—
54C2×C4⋊Dic79—41no—
55#14579107C2×C414yescomplemented
56#14579105C2×C414yescomplemented
57#14579104C2×C414yescomplemented
58#14579102C2×C414yescomplemented
59#14579101C2×C412yescomplemented
60#14579101C2312yes—
61C2×C79⋊C16C2×C412yescomplemented
62#14579093—21no—
63#14579093C2312yes—
64#14579087—24no—
65#14579085—22no—
66#14579082—22no—
67#14579071C2312yes—
68C8×Dic79D412yes—
69C2×#41×C79—21no—
70#135×C79—21no—
71#135×C79—21no—
72#135×C79C2×C412yes—
73#135×C79C2×C412yes—
74C2×Q16×C79C2311yescharacteristic
75C2×SD16×C79—21no—
76C2×D8×C79C2311yescharacteristic
77C2×M16×C79—21no—
78C22×C8×C79C2311yescharacteristic
79M32×C79—21no—
80M32×C79C2312yes—
81C2×C16×C79—21no—
82#108×C79—21no—
83#108×C79C2312yes—
84#11929624—22no—
85#11929624—22no—
86#11929624—41no—
87#11929624—41no—
88C2×Dic158—21no—
89C2×D316—21no—
90C2×C22⋊D79—41no—
91C22×Dic79—42no—
92C22×Dic79—41no—
93C2×C4×D79—41no—
94Dic316—21no—
95Dic316C22×C412yes—
96#11929613—22no—
97#11929613—22no—
98D632—21no—
99D632C22×C412yes—
100#11929608—21no—
101#11929608—21no—
102#11929607—44no—
103C2×C79⋊C8C2×D412yes—
104C8×D79—44no—
105C4⋊Dic79—21no—
106C4⋊Dic79—22no—
107C4⋊Dic79—42no—
108C4⋊Dic79—41no—
109C22⋊Dic79—21no—
110C22⋊Dic79—42no—
111C4×Dic79—42no—
112C4×Dic79C2×D412yes—
113C79⋊C16C22×C414yes—
114#41×C79—22no—
115#41×C79—22no—
116#41×C79—41no—
117#41×C79—41no—
118C2×Q8×C79—21no—
119C2×D4×C79—21no—
120C2×D4×C79—41no—
121C22×C4×C79—41no—
122C22×C4×C79C2×D411yescharacteristic
123Q16×C79—21no—
124Q16×C79C22×C412yes—
125SD16×C79—22no—
126SD16×C79—22no—
127D8×C79—21no—
128D8×C79C22×C412yes—
129M16×C79—22no—
130M16×C79—41no—
131C2×C8×C79—41no—
132C2×C8×C79C22×C411yescharacteristic
133C2×C8×C79C22×C411yescharacteristic
134C2×C8×C79C2×D411yescharacteristic
135C2×C8×C79#4111yescharacteristic
136C2×C8×C79#4111yescharacteristic
137C2×C8×C79C2411yescharacteristic
138C16×C79—22no—
139C23×C79—41no—
140C23×C79AGL(1,ℤ/8)11yescharacteristic
141C22×D79—41no—
142Q8×C79—22no—
143Q8×C79—41no—
144D4×C79—22no—
145D4×C79—42no—
146D4×C79—42no—
147D4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79—42no—
150C2×C4×C79—42no—
151C2×C4×C79C4×D411yescharacteristic
152C2×C4×C79C4×D411yescharacteristic
153C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C79C22×D411yescharacteristic
157C22⋊D79—42no—
158C22⋊D79—42no—
159C2×Dic79—22no—
160C2×Dic79—41no—
161C2×Dic79—42no—
162C2×Dic79—162no—
163C2×Dic79—161no—
164D316—22no—
165D316—41no—
166C4×D79—42no—
167C4×D79—42no—
168Dic158—22no—
169Dic158—41no—
170C8×C79—42no—
171C8×C79C4×D411yescharacteristic
172C8×C79C4×D411yescharacteristic
173C8×C79C23×C411yescharacteristic
174C8×C79C2×#4111yescharacteristic
175C79⋊C8—22no—
176C22×C79—41no—
177C22×C79—42no—
178C22×C79—81no—
179C22×C79—81no—
180C22×C79#44211yescharacteristic
181C22×C79#44211yescharacteristic
182C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
183C2×D79—42no—
184C2×D79—161no—
185C4×C79—42no—
186C4×C79#44211yescharacteristic
187C4×C79#44211yescharacteristic
188C4×C79C2×C4×D411yescharacteristic
189C4×C79C2×AGL(1,ℤ/8)11yescharacteristic
190Dic79—42no—
191Dic79—42no—
192Dic79—82no—
193#33469—791nomaximal, Sylow
194C2×C79—21no—
195C2×C79—21no—
196C2×C79—21no—
197C2×C79—82no—
198C2×C79C2×#44211yescharacteristic
199D79—82no—
200#3414—791no—
201#2957—791no—
202#2952—791no—
203AGL(1,ℤ/16)—794no—
204#2180—794no—
205C2×#365—791no—
206#2148D7911yescharacteristic, complemented
207C2×#360—791no—
208#2110—791no—
209C79#3346911yescharacteristic, complemented, Sylow
210C2×#135—791no—
211C2×#135C2×D7911yescharacteristic
212#568—1581no—
213#520—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#443—1582no—
223#442—792no—
224#441—792no—
225C4×SD16—1582no—
226C2×#108C2×D7911yescharacteristic
227C2×#106—791no—
228#421—1581no—
229#417—1581no—
230#405—791no—
231#365—794no—
232#362C2×D7912yescomplemented
233#361C2×D7912yescomplemented
234#360—794no—
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×D7912yes—
245#135C4×D7912yes—
246C2×Q16—791no—
247C2×Q16C22×D7911yescharacteristic
248C2×SD16—21no—
249C2×SD16—1581no—
250C2×D8—791no—
251C2×D8C22×D7911yescharacteristic
252#131—3162no—
253#131—1584no—
254C2×M16—21no—
255C2×M16—3162no—
256C2×M16—791no—
257C22×C8C22×D7911yescharacteristic
258C4×Q8—1582no—
259C4×D4—3162no—
260C4×D4—1582no—
261#117—3161no—
262#117—791no—
263C2×C4⋊C4—3161no—
264C2×C4⋊C4—1581no—
265Q32—794no—
266SD32—794no—
267SD32—794no—
268D16—794no—
269M32—21no—
270M32—792no—
271M32—792no—
272M32C22×D7912yes—
273C2×C16—21no—
274C2×C16—792no—
275#108—21no—
276#108C22×D7912yes—
277#106—792no—
278#106—1581no—
279#104—1584no—
280#103—1582no—
281#102—1582no—
282#97—792no—
283C4×C8—792no—
284#41—22no—
285#41—22no—
286#41—41no—
287#41—41no—
288#41—3161no—
289#41—3161no—
290#41—1582no—
291#41—1582no—
292C2×Q8—21no—
293C2×Q8—1581no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—3161no—
297C2×D4—1581no—
298C22×C4—41no—
299C22×C4—3161no—
300C22×C4—3161no—
301C22×C4—3162no—
302C22×C4D4×D7911yescharacteristic
303Q16—21no—
304Q16—792no—
305Q16—1581no—
306Q16C2×C4×D7912yes—
307SD16—22no—
308SD16—22no—
309SD16—1582no—
310SD16—1582no—
311D8—21no—
312D8—792no—
313D8—1581no—
314D8C2×C4×D7912yes—
315M16—22no—
316M16—41no—
317M16—3164no—
318M16—1581no—
319M16—1581no—
320C2×C8—41no—
321C2×C8—3164no—
322C2×C8—792no—
323C2×C8C2×C4×D7911yescharacteristic
324C2×C8C2×C4×D7911yescharacteristic
325C2×C8D4×D7911yescharacteristic
326C2×C8#1192962311yescharacteristic
327C2×C8#1192962311yescharacteristic
328C2×C8C23×D7911yescharacteristic
329C4⋊C4—3162no—
330C4⋊C4—3161no—
331C4⋊C4—1582no—
332C4⋊C4—1581no—
333C22⋊C4—3162no—
334C22⋊C4—1581no—
335C42—3162no—
336C42—792no—
337C16—22no—
338C16—794no—
339C23—41no—
340C23—3161no—
341C23#1457918811yescharacteristic
342Q8—22no—
343Q8—41no—
344Q8—3161no—
345Q8—1582no—
346D4—22no—
347D4—42no—
348D4—42no—
349D4—41no—
350D4—3162no—
351D4—3162no—
352D4—3161no—
353D4—1582no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—3161no—
358C2×C4—3162no—
359C2×C4—3162no—
360C2×C4—3162no—
361C2×C4—1582no—
362C2×C4—12642no—
363C2×C4—12641no—
364C2×C4(C22⋊D79)⋊C411yescharacteristic
365C2×C4(C22⋊D79)⋊C411yescharacteristic
366C2×C4#1457918511yescharacteristic
367C2×C4#1457918511yescharacteristic
368C2×C4#1457918811yescharacteristic
369C2×C4C2×D4×D7911yescharacteristic
370C8—42no—
371C8—1582no—
372C8(C22⋊D79)⋊C411yescharacteristic
373C8(C22⋊D79)⋊C411yescharacteristic
374C8C22×C4×D7911yescharacteristic
375C8C2×#1192962311yescharacteristic
376C22—41no—
377C22—42no—
378C22—81no—
379C22—81no—
380C22—3162no—
381C22—12641no—
382C22#1909867211yescharacteristic
383C22#1909867211yescharacteristic
384C22C2×#1457918811yescharacteristic
385C4—42no—
386C4—3162no—
387C4—3162no—
388C4—6322no—
389C4#1909867211yescharacteristic
390C4#1909867211yescharacteristic
391C4C2×(C22⋊D79)⋊C411yescharacteristic
392C4C2×#1457918511yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—6322no—
398C2C2×#1909867211yescharacteristic
subgroupG/H
centerC2C2×#19098672
commutatorC8×C79C23×C4
FrattiniC2×C8C23×D79
FittingC79×#2148C2
socleC2×C79C2×#442
prime radical#23776534C2
cosocleC2×C8×C79C24
hypercenter#2148D79
hypocenterC79#33469
LCSG ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. #23772316C2semidirect
3. #23772284C2semidirect
4. #23765301C2semidirect
5. #23765294C2semidirect
6. C2×#19098387C2semidirect
7. C2×#19098374C2semidirect
8. #23765004C2semidirect
9. C79×#2148C2semidirect
14. #19099003C4semidirect
16. #19098998C4semidirect
17. #19098993C4semidirect
21. #19098675C22semidirect
22. #19098670C22semidirect
28. #19098387C22semidirect
30. #19098340C22semidirect
34. #362×C79C22semidirect
35. #361×C79C22semidirect
39. #14579183C2×C4semidirect
41. C2×Dic316C2×C4semidirect
43. C2×D632C2×C4semidirect
55. #14579107C2×C4semidirect
56. #14579105C2×C4semidirect
57. #14579104C2×C4semidirect
58. #14579102C2×C4semidirect
59. #14579101C2×C4semidirect
61. C2×C79⋊C16C2×C4semidirect
206. #2148D79semidirect
209. C79#33469semidirect
232. #362C2×D79semidirect
233. #361C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237765342
#237723162
#237722842
#237653012
#237652942
C2×#190983872
C2×#190983742
#237650042
C79×#21482
#3346979