Finite Group Databasebeta

← #34441730

Subgroups — #34441730

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#23776536C211yesmaximal, characteristic
2#23772318C211yesmaximal, characteristic, complemented
3#23772285C211yesmaximal, characteristic, complemented
4#23765300C214yesmaximal, complemented
5#23765293C214yesmaximal, complemented
6C2×#19098386C211yesmaximal, characteristic, complemented
7#23765150C211yesmaximal, characteristic, complemented
8C2×#19098374C211yesmaximal, characteristic, complemented
9C79×#2110C211yesmaximal, characteristic, complemented
10C2×#14579183C2211yescharacteristic
11C2×#14579183C2211yescharacteristic
12#19099329—21no—
13#19099043—21no—
14#19099006C412yescomplemented
15#19099006C2212yes—
16#19099001C412yescomplemented
17#19099001C2212yes—
18#19098995C414yescomplemented
19#19098686C2214yes—
20#19098681—22no—
21#19098676C2212yescomplemented
22#19098671C2212yescomplemented
23C4×#11929613—22no—
24C2×#14579097C2211yescharacteristic
25C2×#14579093C2211yescharacteristic
26#19098581—21no—
27#19098565—21no—
28#19098386C2214yescomplemented
29#19098380C2212yescomplemented
30#19098377C2212yescomplemented
31#19098374C2214yes—
32C2×M32×C79C2211yescharacteristic
33C2×M32×C79C2211yescharacteristic
34#405×C79C2211yescharacteristic
35#347×C79C2214yescomplemented
36C2×#11929624—21no—
37C2×#11929624—21no—
38#14579183—21no—
39#14579183—21no—
40#14579183—21no—
41#14579183—21no—
42#14579183C2×C412yes—
43#14579183C2×C412yes—
44#14579183C2×C412yescomplemented
45#14579183C2312yes—
46C2×Dic316C2×C411yescharacteristic, complemented
47C2×Dic316C2311yescharacteristic
48C2×#11929613—21no—
49C2×#11929613—21no—
50C2×D632C2×C411yescharacteristic, complemented
51C2×D632C2311yescharacteristic
52#14579169—42no—
53#14579167—24no—
54C2×#11929608—21no—
55M16×D79—42no—
56C4×Dic158—22no—
57C4×D316—22no—
58(C22⋊D79)⋊C4—42no—
59#14579119—41no—
60C2×C4⋊Dic79—21no—
61C4⋊C4×D79—41no—
62Dic632C2×C414yescomplemented
63#14579106C2×C414yescomplemented
64#14579106C2×C414yescomplemented
65D1264C2×C414yescomplemented
66#14579097—21no—
67#14579097C2312yes—
68#14579093—21no—
69#14579093C2312yes—
70#14579089—24no—
71#14579086—22no—
72#14579083—22no—
73C2×M16×C79D411yescharacteristic
74C22×C8×C79C2311yescharacteristic
75#117×C79D411yescharacteristic
76M32×C79—21no—
77M32×C79C2×C412yescomplemented
78M32×C79C2312yes—
79M32×C79C2312yes—
80C2×C16×C79—21no—
81C2×C16×C79C2×C412yescomplemented
82#97×C79C2312yes—
83C4×C8×C79D412yes—
84#11929624—22no—
85#11929624—22no—
86#11929624—22no—
87#11929624—22no—
88#11929624—41no—
89#11929624—41no—
90#11929624—41no—
91#11929624—41no—
92C2×Dic158—21no—
93C2×Dic158—21no—
94C2×D316—21no—
95C2×D316—21no—
96C2×C22⋊D79—41no—
97C2×C22⋊D79—41no—
98C22×Dic79—41no—
99C2×C4×D79—42no—
100C2×C4×D79—41no—
101C2×C4×D79—41no—
102Dic316—21no—
103Dic316—21no—
104Dic316C22×C412yes—
105Dic316C22×C412yes—
106#11929613—22no—
107#11929613—22no—
108#11929613—22no—
109#11929613—22no—
110D632—21no—
111D632—21no—
112D632C22×C412yes—
113D632C22×C412yes—
114#11929608—22no—
115#11929608—41no—
116#11929607—44no—
117C2×C79⋊C8—41no—
118C8×D79—44no—
119C4⋊Dic79—22no—
120C4⋊Dic79—41no—
121Dic79⋊C4—42no—
122(C2×D79)⋊C4—42no—
123C4×Dic79—42no—
124C22×C4×C79C2×D411yescharacteristic
125M16×C79—21no—
126M16×C79—21no—
127C2×C8×C79C22×C411yescharacteristic
128C2×C8×C79C22×C411yescharacteristic
129C2×C8×C79C2×D412yes—
130C2×C8×C79C2×D411yescharacteristic
131C2×C8×C79#4111yescharacteristic
132C2×C8×C79#4111yescharacteristic
133C2×C8×C79C2411yescharacteristic
134C4⋊C4×C79—21no—
135C22⋊C4×C79—21no—
136C42×C79C2×D412yes—
137C16×C79—22no—
138C16×C79C22×C414yes—
139C23×C79AGL(1,ℤ/8)11yescharacteristic
140C22×D79—41no—
141C22×D79—41no—
142C2×C4×C79—22no—
143C2×C4×C79C4×D411yescharacteristic
144C2×C4×C79C4×D411yescharacteristic
145C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C79C22×D411yescharacteristic
149C22⋊D79—42no—
150C22⋊D79—42no—
151C22⋊D79—42no—
152C22⋊D79—42no—
153C2×Dic79—41no—
154C2×Dic79—41no—
155C2×Dic79—42no—
156C2×Dic79—161no—
157D316—22no—
158D316—22no—
159D316—41no—
160D316—41no—
161C4×D79—42no—
162C4×D79—42no—
163C4×D79—42no—
164C4×D79—42no—
165C4×D79—162no—
166Dic158—22no—
167Dic158—22no—
168Dic158—41no—
169Dic158—41no—
170C8×C79—22no—
171C8×C79C4×D411yescharacteristic
172C8×C79C4×D411yescharacteristic
173C8×C79C23×C411yescharacteristic
174C8×C79C2×#4111yescharacteristic
175C79⋊C8—42no—
176C22×C79—41no—
177C22×C79#44211yescharacteristic
178C22×C79#44211yescharacteristic
179C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
180C2×D79—42no—
181C2×D79—42no—
182C2×D79—81no—
183C2×D79—81no—
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×C79C2×#44211yescharacteristic
198D79—82no—
199D79—82no—
200#3414—791no—
201#2957—791no—
202#2952—791no—
203AGL(1,ℤ/16)—794no—
204#2180—794no—
205C2×#365—791no—
206#2148—791no—
207C2×#360—791no—
208#2110D7911yescharacteristic, complemented
209C79#3346911yescharacteristic, complemented, Sylow
210C2×#135—791no—
211C2×#135—791no—
212#568—1581no—
213#520—1581no—
214#510—792no—
215#510—792no—
216#509—792no—
217#509—792no—
218#508—794no—
219C2×M32C2×D7911yescharacteristic
220C2×M32C2×D7911yescharacteristic
221#444—794no—
222#443—1582no—
223#442—792no—
224#441—792no—
225C4×SD16—1582no—
226C2×#108—791no—
227C2×#106—791no—
228#421—1581no—
229#417—1581no—
230#405C2×D7911yescharacteristic
231#365—794no—
232#362—792no—
233#361—792no—
234#360—794no—
235#347C2×D7914yescomplemented
236C2×#41—1581no—
237C2×#41—1581no—
238#135—792no—
239#135—792no—
240#135—792no—
241#135—792no—
242#135—1581no—
243#135—1581no—
244#135—1581no—
245#135—1581no—
246C2×Q16—791no—
247C2×Q16—791no—
248C2×SD16—1581no—
249C2×SD16—1581no—
250C2×D8—791no—
251C2×D8—791no—
252#131—3162no—
253#131—1584no—
254C2×M16—3162no—
255C2×M16—1581no—
256C2×M16D31611yescharacteristic
257C22×C8C22×D7911yescharacteristic
258C4×Q8—1582no—
259C4×D4—3162no—
260C4×D4—1582no—
261#117—3161no—
262#117D31611yescharacteristic
263C2×C4⋊C4—3161no—
264C2×C4⋊C4—1581no—
265Q32—794no—
266SD32—794no—
267SD32—794no—
268D16—794no—
269M32—21no—
270M32C4×D7912yescomplemented
271M32C22×D7912yes—
272M32C22×D7912yes—
273C2×C16—21no—
274C2×C16C4×D7912yescomplemented
275#108—792no—
276#108—1581no—
277#106—792no—
278#106—1581no—
279#104—1584no—
280#103—1582no—
281#102—1582no—
282#97C22×D7912yes—
283C4×C8D31612yes—
284#41—3161no—
285#41—3161no—
286#41—3161no—
287#41—3161no—
288#41—1582no—
289#41—1582no—
290#41—1582no—
291#41—1582no—
292C2×Q8—1581no—
293C2×Q8—1581no—
294C2×D4—3161no—
295C2×D4—3161no—
296C2×D4—1581no—
297C2×D4—1581no—
298C22×C4—3162no—
299C22×C4—3161no—
300C22×C4—3161no—
301C22×C4—3161no—
302C22×C4C2×D31611yescharacteristic
303Q16—792no—
304Q16—792no—
305Q16—1581no—
306Q16—1581no—
307SD16—1582no—
308SD16—1582no—
309SD16—1582no—
310SD16—1582no—
311D8—792no—
312D8—792no—
313D8—1581no—
314D8—1581no—
315M16—21no—
316M16—21no—
317M16—3164no—
318M16—3161no—
319M16—1582no—
320C2×C8—3164no—
321C2×C8—3161no—
322C2×C8C2×C4×D7911yescharacteristic
323C2×C8C2×C4×D7911yescharacteristic
324C2×C8C2×D31612yes—
325C2×C8C2×D31611yescharacteristic
326C2×C8#1192962411yescharacteristic
327C2×C8#1192962411yescharacteristic
328C2×C8C23×D7911yescharacteristic
329C4⋊C4—21no—
330C4⋊C4—3162no—
331C4⋊C4—3161no—
332C4⋊C4—1582no—
333C22⋊C4—21no—
334C22⋊C4—3162no—
335C42—3162no—
336C42C2×D31612yes—
337C16—22no—
338C16C2×C4×D7914yes—
339C23—3161no—
340C23—3161no—
341C23#1457918911yescharacteristic
342Q8—3161no—
343Q8—3161no—
344Q8—1582no—
345Q8—1582no—
346D4—3162no—
347D4—3162no—
348D4—3162no—
349D4—3162no—
350D4—3161no—
351D4—3161no—
352D4—1582no—
353D4—1582no—
354C2×C4—22no—
355C2×C4—3161no—
356C2×C4—3161no—
357C2×C4—3162no—
358C2×C4—3162no—
359C2×C4—3162no—
360C2×C4—3162no—
361C2×C4—3162no—
362C2×C4—12642no—
363C2×C4—12641no—
364C2×C4C4×D31611yescharacteristic
365C2×C4C4×D31611yescharacteristic
366C2×C4#1457918911yescharacteristic
367C2×C4#1457918911yescharacteristic
368C2×C4#1457918911yescharacteristic
369C2×C4C22×D31611yescharacteristic
370C8—22no—
371C8—3162no—
372C8C4×D31611yescharacteristic
373C8C4×D31611yescharacteristic
374C8C22×C4×D7911yescharacteristic
375C8C2×#1192962411yescharacteristic
376C22—41no—
377C22—3162no—
378C22—3162no—
379C22—6321no—
380C22—6321no—
381C22—12641no—
382C22#1909867611yescharacteristic
383C22#1909867611yescharacteristic
384C22C2×#1457918911yescharacteristic
385C4—42no—
386C4—3162no—
387C4—3162no—
388C4—6322no—
389C4#1909867611yescharacteristic
390C4#1909867611yescharacteristic
391C4C2×C4×D31611yescharacteristic
392C4C2×#1457918911yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—6322no—
397C2—6322no—
398C2C2×#1909867611yescharacteristic
subgroupG/H
centerC2C2×#19098676
commutatorC8×C79C23×C4
FrattiniC2×C8C23×D79
FittingC79×#2110C2
socleC2×C79C2×#442
prime radical#23776536C2
cosocleC2×C8×C79C24
hypercenter#2110D79
hypocenterC79#33469
LCSG ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. #23772318C2semidirect
3. #23772285C2semidirect
4. #23765300C2semidirect
5. #23765293C2semidirect
6. C2×#19098386C2semidirect
7. #23765150C2semidirect
8. C2×#19098374C2semidirect
9. C79×#2110C2semidirect
14. #19099006C4semidirect
16. #19099001C4semidirect
18. #19098995C4semidirect
21. #19098676C22semidirect
22. #19098671C22semidirect
28. #19098386C22semidirect
29. #19098380C22semidirect
30. #19098377C22semidirect
35. #347×C79C22semidirect
44. #14579183C2×C4semidirect
46. C2×Dic316C2×C4semidirect
50. C2×D632C2×C4semidirect
62. Dic632C2×C4semidirect
63. #14579106C2×C4semidirect
64. #14579106C2×C4semidirect
65. D1264C2×C4semidirect
77. M32×C79C2×C4semidirect
81. C2×C16×C79C2×C4semidirect
208. #2110D79semidirect
209. C79#33469semidirect
235. #347C2×D79semidirect
270. M32C4×D79semidirect
274. C2×C16C4×D79semidirect

Maximal subgroups

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

subgroupindex
#237765362
#237723182
#237722852
#237653002
#237652932
C2×#190983862
#237651502
C2×#190983742
C79×#21102
#3346979