Finite Group Databasebeta

← #34441613

Subgroups — #34441613

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×#19099001C211yesmaximal, characteristic
2#23772277C211yesmaximal, characteristic, complemented
3#23772253C211yesmaximal, characteristic, complemented
4#23765277C212yesmaximal, complemented
5#23765266C212yesmaximal, complemented
6C4×#14579106C212yesmaximal, complemented
7C4×D1264C212yesmaximal, complemented
8#23765188C211yesmaximal, characteristic, complemented
9#23765108C211yesmaximal, characteristic, complemented
10C2×#19098365C211yesmaximal, characteristic, complemented
11C79×#2108C211yesmaximal, characteristic, complemented
12C2×#14579183C2211yescharacteristic
13C22×D632C2211yescharacteristic
14#19099043—21no—
15#19099037—21no—
16#19099001C418yescomplemented
17C2×#14579106C2212yes—
18C2×D1264C2212yes—
19#19098676—22no—
20#19098665—22no—
21C4×Dic316C2212yescomplemented
22C4×D632C2212yes—
23C4×D632C2212yes—
24C4×D632C2212yescomplemented
25#19098601C2211yescharacteristic
26C2×#14579095C2211yescharacteristic
27#19098569—21no—
28#19098565—21no—
29#19098391C2212yescomplemented
30#19098389C2212yescomplemented
31#19098368C2211yescharacteristic, complemented
32#19098368C2211yescharacteristic, complemented
33#19098365C2212yes—
34#19098365C2212yes—
35#19098365C2211yescharacteristic, complemented
36#19098365C2211yescharacteristic, complemented
37C2×M32×C79C2211yescharacteristic
38C22×C16×C79C2211yescharacteristic
39#405×C79C2211yescharacteristic
40#346×C79C2212yescomplemented
41C4×C16×C79C2212yescomplemented
42C2×#11929624—21no—
43C22×D316—21no—
44#14579183—22no—
45#14579183C2×C414yescomplemented
46C2×Dic316C2311yescharacteristic
47C2×#11929613—21no—
48C2×D632—21no—
49C2×D632—22no—
50C2×D632C2×C414yescomplemented
51C2×D632C2312yes—
52C2×D632C2311yescharacteristic
53C4×Dic158—22no—
54C4×D316—22no—
55C4×D316—22no—
56C4×D316—22no—
57#14579123—41no—
58#14579123—41no—
59(C22⋊D79)⋊C4—42no—
60#14579120—21no—
61#14579119—41no—
62C2×C4⋊Dic79—21no—
63C4⋊C4×D79—41no—
64C22⋊C4×D79—42no—
65#14579106C2×C418yescomplemented
66D1264C2×C418yescomplemented
67#14579095—21no—
68#14579095C2311yescharacteristic
69#14579095C2311yescharacteristic
70#14579095C2312yes—
71#14579093—21no—
72#14579086—22no—
73#14579083—22no—
74#14579083—22no—
75#14579083—22no—
76C2×M16×C79D411yescharacteristic
77C22×C8×C79C2311yescharacteristic
78#117×C79D411yescharacteristic
79M32×C79C2×C414yescomplemented
80C2×C16×C79—22no—
81C2×C16×C79C2311yescharacteristic
82C2×C16×C79C2311yescharacteristic
83C2×C16×C79C2312yes—
84#97×C79D412yes—
85C4×C8×C79C2312yes—
86C23×D79—41no—
87#11929624—24no—
88#11929624—42no—
89C2×Dic158—21no—
90C2×D316—24no—
91C2×D316—21no—
92C2×D316—22no—
93C2×D316—41no—
94C2×D316—42no—
95C2×C22⋊D79—41no—
96C22×Dic79—41no—
97C2×C4×D79—41no—
98C2×C4×D79—42no—
99C2×C4×D79—42no—
100C2×C4×D79—42no—
101Dic316—21no—
102Dic316C22×C412yes—
103#11929613—24no—
104D632—24no—
105D632—22no—
106D632—21no—
107D632C22×C412yes—
108D632C22×C414yes—
109C4⋊Dic79—21no—
110C4⋊Dic79—21no—
111C4⋊Dic79—22no—
112C4⋊Dic79—41no—
113Dic79⋊C4—42no—
114C22⋊Dic79—41no—
115(C2×D79)⋊C4—42no—
116(C2×D79)⋊C4—42no—
117(C2×D79)⋊C4—42no—
118C4×Dic79—42no—
119C4×Dic79—41no—
120C22×C4×C79C2×D411yescharacteristic
121M16×C79—22no—
122C2×C8×C79C22×C412yes—
123C2×C8×C79C2×D412yes—
124C2×C8×C79C2×D411yescharacteristic
125C2×C8×C79#4112yes—
126C2×C8×C79C2411yescharacteristic
127C4⋊C4×C79D811yescharacteristic
128C4⋊C4×C79D811yescharacteristic
129C22⋊C4×C79D812yes—
130C42×C79C2×D412yes—
131C16×C79—21no—
132C16×C79—21no—
133C16×C79C22×C414yes—
134C23×C79C2×D811yescharacteristic
135C22×D79—41no—
136C22×D79—42no—
137C22×D79—44no—
138C22×D79—161no—
139C22×D79—161no—
140C2×C4×C79C4×D412yes—
141C2×C4×C79C2×D812yes—
142C2×C4×C79C2×D812yes—
143C2×C4×C79C2×D811yescharacteristic
144C2×C4×C79#13512yes—
145C2×C4×C79C22×D411yescharacteristic
146C22⋊D79—44no—
147C2×Dic79—41no—
148C2×Dic79—41no—
149C2×Dic79—41no—
150C2×Dic79—42no—
151C2×Dic79—161no—
152D316—24no—
153D316—22no—
154D316—42no—
155D316—41no—
156D316—44no—
157C4×D79—44no—
158C4×D79—162no—
159C4×D79—162no—
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×C79C4×D812yes—
170C22×C79C22×D811yescharacteristic
171C2×D79—42no—
172C2×D79—44no—
173C2×D79—82no—
174C2×D79—81no—
175C2×D79—81no—
176C2×D79—82no—
177C2×D79—164no—
178C4×C79—22no—
179C4×C79—22no—
180C4×C79C4×D812yes—
181C4×C79C2×C4×D411yescharacteristic
182C4×C79C2×#13511yescharacteristic
183Dic79—42no—
184Dic79—81no—
185Dic79—82no—
186Dic79—81no—
187#33454—791nomaximal, Sylow
188C2×C79—22no—
189C2×C79C2×C4×D811yescharacteristic
190C2×C79#341511yescharacteristic
191C2×C79#341611yescharacteristic
192D79—82no—
193D79—84no—
194C2×#509—791no—
195#2951—791no—
196#2948—791no—
197#2177—792no—
198#2175—792no—
199C4×SD32—792no—
200C4×D16—792no—
201#2159—791no—
202#2138—791no—
203C2×#357—791no—
204#2108D7911yescharacteristic, complemented
205C79#3345411yescharacteristic, complemented, Sylow
206C2×#135—791no—
207C22×D8—791no—
208#520—1581no—
209#518—1581no—
210#509—798no—
211C2×SD32—792no—
212C2×D16—792no—
213C2×M32C2×D7911yescharacteristic
214C22×C16C2×D7911yescharacteristic
215#442—1582no—
216#440—1582no—
217C4×Q16—792no—
218C4×D8—792no—
219C4×D8—792no—
220C4×D8—792no—
221#427—791no—
222C2×#107—791no—
223#418—1581no—
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#346C2×D7912yescomplemented
235C4×C16C2×D7912yescomplemented
236C2×#41—1581no—
237C22×D4—1581no—
238#135—794no—
239#135—1582no—
240C2×Q16—791no—
241C2×SD16—1581no—
242C2×D8—794no—
243C2×D8—792no—
244C2×D8—791no—
245C2×D8—1581no—
246C2×D8—1582no—
247C2×M16D31611yescharacteristic
248C22×C8C22×D7911yescharacteristic
249C4×Q8—1582no—
250C4×D4—3161no—
251C4×D4—3161no—
252C4×D4—3162no—
253C4×D4—1582no—
254C4×D4—1582no—
255C4×D4—1582no—
256#117—3161no—
257#117—1581no—
258#117D31611yescharacteristic
259C2×C4⋊C4—3161no—
260C2×C4⋊C4—1581no—
261C2×C22⋊C4—3162no—
262SD32—798no—
263D16—798no—
264M32C4×D7914yescomplemented
265C2×C16—22no—
266C2×C16C22×D7911yescharacteristic
267C2×C16C22×D7911yescharacteristic
268C2×C16C22×D7912yes—
269#107—792no—
270#107—791no—
271#107—791no—
272#107—1581no—
273#106—1581no—
274#103—1582no—
275#102—1582no—
276#102—1582no—
277#102—1582no—
278#97D31612yes—
279C4×C8C22×D7912yes—
280C24—3161no—
281#41—3162no—
282#41—1584no—
283C2×Q8—1581no—
284C2×D4—3161no—
285C2×D4—3161no—
286C2×D4—3162no—
287C2×D4—1584no—
288C2×D4—1582no—
289C2×D4—1581no—
290C22×C4—3161no—
291C22×C4—3161no—
292C22×C4—3162no—
293C22×C4—3162no—
294C22×C4—3162no—
295C22×C4C2×D31611yescharacteristic
296Q16—792no—
297Q16—1581no—
298SD16—1584no—
299D8—792no—
300D8—794no—
301D8—1582no—
302D8—1581no—
303D8—1584no—
304M16—22no—
305C2×C8C2×C4×D7912yes—
306C2×C8C2×D31612yes—
307C2×C8C2×D31611yescharacteristic
308C2×C8#1192962412yes—
309C2×C8C23×D7911yescharacteristic
310C4⋊C4—3162no—
311C4⋊C4—3161no—
312C4⋊C4—1581no—
313C4⋊C4—1581no—
314C4⋊C4—1582no—
315C4⋊C4D63211yescharacteristic
316C4⋊C4D63211yescharacteristic
317C22⋊C4—3161no—
318C22⋊C4—3162no—
319C22⋊C4—3162no—
320C22⋊C4—3162no—
321C22⋊C4D63212yes—
322C42—3162no—
323C42—3161no—
324C42C2×D31612yes—
325C16—21no—
326C16—21no—
327C16C2×C4×D7914yes—
328C23—3161no—
329C23—3162no—
330C23—3164no—
331C23—12641no—
332C23—12641no—
333C23C2×D63211yescharacteristic
334Q8—3161no—
335Q8—1582no—
336D4—3164no—
337D4—3164no—
338D4—3162no—
339D4—3161no—
340D4—1584no—
341D4—1582no—
342C2×C4—3161no—
343C2×C4—3164no—
344C2×C4—3161no—
345C2×C4—3161no—
346C2×C4—3162no—
347C2×C4—12642no—
348C2×C4—12642no—
349C2×C4—12642no—
350C2×C4—12641no—
351C2×C4C4×D31612yes—
352C2×C4C2×D63212yes—
353C2×C4C2×D63212yes—
354C2×C4C2×D63211yescharacteristic
355C2×C4#1457918312yes—
356C2×C4C22×D31611yescharacteristic
357C8—22no—
358C8C4×D31612yes—
359C8C22×C4×D7911yescharacteristic
360C8C2×#1192962411yescharacteristic
361C22—21no—
362C22—21no—
363C22—3162no—
364C22—3164no—
365C22—6321no—
366C22—6322no—
367C22—6322no—
368C22—6321no—
369C22—12644no—
370C22C4×D63212yes—
371C22C22×D63211yescharacteristic
372C4—22no—
373C4—22no—
374C4—3162no—
375C4—6321no—
376C4—6322no—
377C4—6321no—
378C4C4×D63212yes—
379C4C2×C4×D31611yescharacteristic
380C4C2×#1457918311yescharacteristic
381C2—22no—
382C2—6324no—
383C2—6322no—
384C2C2×C4×D63211yescharacteristic
385C2#2377654111yescharacteristic
386C2#2377654811yescharacteristic
subgroupG/H
centerC22C22×D632
commutatorC8×C79C23×C4
FrattiniC2×C8C23×D79
FittingC79×#2108C2
socleC22×C79C22×D8
prime radicalC2×#19099001C2
cosocleC2×C8×C79C24
hypercenter#2108D79
hypocenterC79#33454
LCSG ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108

Complements

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

NH ≅ G/Ntype
2. #23772277C2semidirect
3. #23772253C2semidirect
4. #23765277C2semidirect
5. #23765266C2semidirect
6. C4×#14579106C2semidirect
7. C4×D1264C2semidirect
8. #23765188C2semidirect
9. #23765108C2semidirect
10. C2×#19098365C2semidirect
11. C79×#2108C2semidirect
16. #19099001C4semidirect
21. C4×Dic316C22semidirect
24. C4×D632C22semidirect
29. #19098391C22semidirect
30. #19098389C22semidirect
31. #19098368C22semidirect
32. #19098368C22semidirect
35. #19098365C22semidirect
36. #19098365C22semidirect
40. #346×C79C22semidirect
41. C4×C16×C79C22semidirect
45. #14579183C2×C4semidirect
50. C2×D632C2×C4semidirect
65. #14579106C2×C4semidirect
66. D1264C2×C4semidirect
79. M32×C79C2×C4semidirect
204. #2108D79semidirect
205. C79#33454semidirect
234. #346C2×D79semidirect
235. C4×C16C2×D79semidirect
264. M32C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190990012
#237722772
#237722532
#237652772
#237652662
C4×#145791062
C4×D12642
#237651882
#237651082
C2×#190983652
C79×#21082
#3345479