Finite Group Databasebeta

← #34441611

Subgroups — #34441611

Order 20224.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098999C211yesmaximal, characteristic
2#23772277C211yesmaximal, characteristic, complemented
3#23765274C212yesmaximal, complemented
4#23765264C212yesmaximal, complemented
5C4×#14579105C212yesmaximal, complemented
6C4×#14579102C212yesmaximal, complemented
7#23765189C211yesmaximal, characteristic, complemented
8#23765105C211yesmaximal, characteristic, complemented
9C2×#19098364C211yesmaximal, characteristic, complemented
10#23764997C211yesmaximal, characteristic, complemented
11C79×#2948C211yesmaximal, characteristic, complemented
12C2×#14579183C2211yescharacteristic
13#19099043—21no—
14#19098999C418yescomplemented
15C2×#14579105C2212yes—
16C2×#14579102C2212yes—
17C2×#14579101C2211yescharacteristic
18C22×C79⋊C16C2211yescharacteristic
19#19098665—22no—
20C4×Dic316C2212yescomplemented
21C4×D632C2212yescomplemented
22C2×#14579095C2211yescharacteristic
23#19098565—21no—
24#19098390C2212yescomplemented
25#19098388C2212yescomplemented
26#19098366C2211yescharacteristic, complemented
27#19098366C2211yescharacteristic, complemented
28#19098364C2212yes—
29#19098364C2212yes—
30#19098363C2211yescharacteristic, complemented
31#19098363C2211yescharacteristic, complemented
32#19098339C2212yescomplemented
33C4×C79⋊C16C2212yescomplemented
34C22×D8×C79C2211yescharacteristic
35#518×C79—21no—
36#442×C79—22no—
37C4×D8×C79C2212yes—
38C4×D8×C79C2212yes—
39#427×C79C2211yescharacteristic
40#418×C79—21no—
41#405×C79C2211yescharacteristic
42C2×#11929624—21no—
43#14579183—22no—
44#14579183C2×C414yescomplemented
45C2×Dic316C2311yescharacteristic
46C2×#11929613—21no—
47C2×D632C2311yescharacteristic
48C4×Dic158—22no—
49C4×D316—22no—
50(C22⋊D79)⋊C4—42no—
51#14579119—41no—
52C2×C4⋊Dic79—21no—
53C4⋊C4×D79—41no—
54#14579105C2×C418yescomplemented
55#14579102C2×C418yescomplemented
56#14579101C2×C414yescomplemented
57C2×C79⋊C16—22no—
58C2×C79⋊C16C2311yescharacteristic
59C2×C79⋊C16C2311yescharacteristic
60C2×C79⋊C16C2312yes—
61#14579095—21no—
62#14579095C2312yes—
63#14579086—22no—
64#14579083—22no—
65C22×D4×C79—21no—
66C2×D8×C79—21no—
67C2×D8×C79—22no—
68C2×D8×C79C2×C414yescomplemented
69C2×D8×C79C2312yes—
70C2×M16×C79D411yescharacteristic
71C22×C8×C79C2311yescharacteristic
72C4×D4×C79—22no—
73C4×D4×C79—22no—
74C4×D4×C79—41no—
75C4×D4×C79—41no—
76#117×C79—21no—
77#117×C79D411yescharacteristic
78C2×C22⋊C4×C79—42no—
79#107×C79C2311yescharacteristic
80#107×C79C2311yescharacteristic
81#106×C79—21no—
82#102×C79—22no—
83#102×C79—22no—
84#97×C79D412yes—
85C4×C8×C79C2312yes—
86#11929624—24no—
87#11929624—42no—
88C2×Dic158—21no—
89C2×D316—21no—
90C2×C22⋊D79—41no—
91C22×Dic79—41no—
92C2×C4×D79—41no—
93C2×C4×D79—42no—
94Dic316—21no—
95Dic316C22×C412yes—
96#11929613—24no—
97D632—21no—
98D632C22×C412yes—
99C4⋊Dic79—22no—
100C4⋊Dic79—41no—
101Dic79⋊C4—42no—
102(C2×D79)⋊C4—42no—
103C4×Dic79—42no—
104C79⋊C16—21no—
105C79⋊C16—21no—
106C79⋊C16C22×C414yes—
107C24×C79—41no—
108C2×D4×C79—24no—
109C2×D4×C79—22no—
110C2×D4×C79—41no—
111C2×D4×C79—42no—
112C22×C4×C79—42no—
113C22×C4×C79—42no—
114C22×C4×C79C2×D411yescharacteristic
115D8×C79—24no—
116D8×C79—22no—
117D8×C79C22×C414yes—
118M16×C79—22no—
119C2×C8×C79C22×C412yes—
120C2×C8×C79C2×D412yes—
121C2×C8×C79C2×D411yescharacteristic
122C2×C8×C79#4112yes—
123C2×C8×C79C2411yescharacteristic
124C4⋊C4×C79—21no—
125C4⋊C4×C79—21no—
126C4⋊C4×C79D811yescharacteristic
127C4⋊C4×C79D811yescharacteristic
128C22⋊C4×C79—41no—
129C22⋊C4×C79—42no—
130C22⋊C4×C79—42no—
131C22⋊C4×C79D812yes—
132C42×C79—41no—
133C42×C79C2×D412yes—
134C23×C79—42no—
135C23×C79—44no—
136C23×C79—161no—
137C23×C79—161no—
138C23×C79C2×D811yescharacteristic
139C22×D79—41no—
140D4×C79—24no—
141D4×C79—42no—
142D4×C79—44no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—162no—
146C2×C4×C79—162no—
147C2×C4×C79C4×D412yes—
148C2×C4×C79C2×D812yes—
149C2×C4×C79C2×D812yes—
150C2×C4×C79C2×D811yescharacteristic
151C2×C4×C79#13512yes—
152C2×C4×C79C22×D411yescharacteristic
153C22⋊D79—44no—
154C2×Dic79—41no—
155C2×Dic79—42no—
156C2×Dic79—161no—
157D316—22no—
158D316—41no—
159C4×D79—44no—
160C4×D79—162no—
161Dic158—22no—
162Dic158—41no—
163C8×C79—22no—
164C8×C79C4×D412yes—
165C8×C79C23×C411yescharacteristic
166C8×C79C2×#4111yescharacteristic
167C22×C79—21no—
168C22×C79—21no—
169C22×C79—44no—
170C22×C79—82no—
171C22×C79—82no—
172C22×C79—164no—
173C22×C79C4×D812yes—
174C22×C79C22×D811yescharacteristic
175C2×D79—42no—
176C2×D79—81no—
177C2×D79—81no—
178C4×C79—22no—
179C4×C79—22no—
180C4×C79—81no—
181C4×C79—81no—
182C4×C79C4×D812yes—
183C4×C79C2×C4×D411yescharacteristic
184C4×C79C2×#13511yescharacteristic
185Dic79—42no—
186Dic79—82no—
187#33454—791nomaximal, Sylow
188C2×C79—22no—
189C2×C79—84no—
190C2×C79C2×C4×D811yescharacteristic
191C2×C79#341511yescharacteristic
192C2×C79#341611yescharacteristic
193D79—82no—
194C2×#509—791no—
195#2951—791no—
196#2948D7911yescharacteristic, complemented
197#2177—792no—
198#2175—792no—
199C4×SD32—792no—
200C4×D16—792no—
201#2159—791no—
202#2138—791no—
203C2×#357—791no—
204#2108—791no—
205C79#3345411yescharacteristic, complemented, Sylow
206C2×#135—791no—
207C22×D8C2×D7911yescharacteristic
208#520—1581no—
209#518—21no—
210#509—798no—
211C2×SD32—792no—
212C2×D16—792no—
213C2×M32—791no—
214C22×C16—791no—
215#442—22no—
216#440—1582no—
217C4×Q16—792no—
218C4×D8—792no—
219C4×D8C2×D7912yes—
220C4×D8C2×D7912yes—
221#427C2×D7911yescharacteristic
222C2×#107—791no—
223#418—21no—
224#417—1581no—
225#405C2×D7911yescharacteristic
226#367—792no—
227#366—792no—
228#358—791no—
229#358—791no—
230#357—792no—
231#357—792no—
232#357—791no—
233#357—791no—
234#346—792no—
235C4×C16—792no—
236C2×#41—1581no—
237C22×D4—21no—
238#135—794no—
239#135—1582no—
240C2×Q16—791no—
241C2×SD16—1581no—
242C2×D8—21no—
243C2×D8—22no—
244C2×D8—791no—
245C2×D8C4×D7914yescomplemented
246C2×D8C22×D7912yes—
247C2×M16C22⋊D7911yescharacteristic
248C22×C8C22×D7911yescharacteristic
249C4×Q8—1582no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—41no—
253C4×D4—41no—
254C4×D4—3162no—
255C4×D4—1582no—
256#117—21no—
257#117—3161no—
258#117C22⋊D7911yescharacteristic
259C2×C4⋊C4—3161no—
260C2×C4⋊C4—1581no—
261C2×C22⋊C4—42no—
262SD32—798no—
263D16—798no—
264M32—794no—
265C2×C16—792no—
266C2×C16—791no—
267C2×C16—791no—
268C2×C16—1582no—
269#107—792no—
270#107—1581no—
271#107C22×D7911yescharacteristic
272#107C22×D7911yescharacteristic
273#106—21no—
274#103—1582no—
275#102—22no—
276#102—22no—
277#102—1582no—
278#97C22⋊D7912yes—
279C4×C8C22×D7912yes—
280C24—41no—
281#41—3162no—
282#41—1584no—
283C2×Q8—1581no—
284C2×D4—24no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—3161no—
289C2×D4—1581no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—3161no—
293C22×C4—3161no—
294C22×C4—3162no—
295C22×C4C2×C22⋊D7911yescharacteristic
296Q16—792no—
297Q16—1581no—
298SD16—1584no—
299D8—22no—
300D8—24no—
301D8—792no—
302D8—1581no—
303D8C2×C4×D7914yes—
304M16—22no—
305C2×C8C2×C4×D7912yes—
306C2×C8C2×C22⋊D7912yes—
307C2×C8C2×C22⋊D7911yescharacteristic
308C2×C8#1192962412yes—
309C2×C8C23×D7911yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—3162no—
313C4⋊C4—3161no—
314C4⋊C4—1582no—
315C4⋊C4#1192960911yescharacteristic
316C4⋊C4#1192960911yescharacteristic
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—3162no—
321C22⋊C4#1192960912yes—
322C42—41no—
323C42—3162no—
324C42C2×C22⋊D7912yes—
325C16—794no—
326C16—1581no—
327C16—1581no—
328C23—42no—
329C23—44no—
330C23—161no—
331C23—161no—
332C23—3161no—
333C23C2×#1192960911yescharacteristic
334Q8—3161no—
335Q8—1582no—
336D4—24no—
337D4—44no—
338D4—42no—
339D4—3164no—
340D4—3161no—
341D4—1582no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—162no—
345C2×C4—162no—
346C2×C4—3161no—
347C2×C4—3164no—
348C2×C4—3162no—
349C2×C4—12642no—
350C2×C4—12641no—
351C2×C4C4×C22⋊D7912yes—
352C2×C4C2×#1192960912yes—
353C2×C4C2×#1192960912yes—
354C2×C4C2×#1192960911yescharacteristic
355C2×C4#1457918112yes—
356C2×C4C22×C22⋊D7911yescharacteristic
357C8—22no—
358C8C4×C22⋊D7912yes—
359C8C22×C4×D7911yescharacteristic
360C8C2×#1192962411yescharacteristic
361C22—21no—
362C22—21no—
363C22—44no—
364C22—82no—
365C22—82no—
366C22—164no—
367C22—3162no—
368C22—6321no—
369C22—6321no—
370C22C4×#1192960912yes—
371C22C22×#1192960911yescharacteristic
372C4—22no—
373C4—22no—
374C4—81no—
375C4—81no—
376C4—3162no—
377C4—6322no—
378C4C4×#1192960912yes—
379C4C2×C4×C22⋊D7911yescharacteristic
380C4C2×#1457918111yescharacteristic
381C2—22no—
382C2—84no—
383C2—6322no—
384C2C2×C4×#1192960911yescharacteristic
385C2#2377653911yescharacteristic
386C2#2377654711yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutatorC8×C79C23×C4
FrattiniC2×C8C23×D79
FittingC79×#2948C2
socleC22×C79C22×D8
prime radicalC2×#19098999C2
cosocleC2×C8×C79C24
hypercenter#2948D79
hypocenterC79#33454
LCSG ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2948

Complements

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

NH ≅ G/Ntype
2. #23772277C2semidirect
3. #23765274C2semidirect
4. #23765264C2semidirect
5. C4×#14579105C2semidirect
6. C4×#14579102C2semidirect
7. #23765189C2semidirect
8. #23765105C2semidirect
9. C2×#19098364C2semidirect
10. #23764997C2semidirect
11. C79×#2948C2semidirect
14. #19098999C4semidirect
20. C4×Dic316C22semidirect
21. C4×D632C22semidirect
24. #19098390C22semidirect
25. #19098388C22semidirect
26. #19098366C22semidirect
27. #19098366C22semidirect
30. #19098363C22semidirect
31. #19098363C22semidirect
32. #19098339C22semidirect
33. C4×C79⋊C16C22semidirect
44. #14579183C2×C4semidirect
54. #14579105C2×C4semidirect
55. #14579102C2×C4semidirect
56. #14579101C2×C4semidirect
68. C2×D8×C79C2×C4semidirect
196. #2948D79semidirect
205. C79#33454semidirect
245. C2×D8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190989992
#237722772
#237652742
#237652642
C4×#145791052
C4×#145791022
#237651892
#237651052
C2×#190983642
#237649972
C79×#29482
#3345479