Finite Group Databasebeta

← #34292674

Subgroups — #34292674

Order 20224.

722 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14579181C211yesmaximal, characteristic
2#23772052C211yesmaximal, characteristic, complemented
3#23772051C211yesmaximal, characteristic, complemented
4#23764285C212yesmaximal
5#23764147C212yesmaximal, complemented
6#23764112C212yesmaximal, complemented
7#23764108C212yesmaximal, complemented
8C4×#14579096C211yesmaximal, characteristic, complemented
9#23763211C211yesmaximal, characteristic, complemented
10#23763087C211yesmaximal, characteristic, complemented
11C79×#1759C211yesmaximal, characteristic, complemented
12C2×#14579181C2211yescharacteristic
13C4×#11929624C2211yescharacteristic
14C4×#11929614C412yescomplemented
15C4×#11929611C412yescomplemented
16C4×#11929612C412yescomplemented
17C4×#11929609C412yescomplemented
18#19098624—22no—
19#19098622—22no—
20#19098620—22no—
21#19098618—22no—
22C2×#14579096C2212yes—
23#19098599C2211yescharacteristic
24#19098591C2212yes—
25#19098590C2212yes—
26C2×#14579089C2212yes—
27#19098559C2211yescharacteristic
28#19098557C2211yescharacteristic
29#19098531C2212yes—
30#19098530C2212yes—
31C4×#11929608C2211yescharacteristic
32C2×#14579072C2212yes—
33C2×#14579072C2211yescharacteristic
34C2×C4×C79⋊C8C2212yes—
35C2×C4×C79⋊C8C2211yescharacteristic
36#19098327C2212yes—
37#19098323C2212yes—
38C4×#41×C79C2211yescharacteristic
39C2×#104×C79C2212yes—
40C4×M16×C79C2211yescharacteristic
41#376×C79C2211yescharacteristic
42#339×C79C2212yes—
43C2×#11929624C2311yescharacteristic
44#14579181—24no—
45C2×#11929614C2×C411yescharacteristic
46C2×#11929612C2×C411yescharacteristic
47C2×#11929611C2×C411yescharacteristic
48C2×#11929609C2×C411yescharacteristic
49C2×#11929608C2312yes—
50C22×C79⋊C8C2311yescharacteristic
51C22×C79⋊C8C2312yes—
52C22×C79⋊C8C2311yescharacteristic
53C4×Dic158C2×C412yes—
54C4×D316C2×C412yes—
55C4×C22⋊D79—22no—
56#14579120C2311yescharacteristic
57#14579118—21no—
58C42×D79—21no—
59#14579096—24no—
60#14579094C2×C412yes—
61#14579092C2×C412yes—
62C4⋊(C79⋊C8)—24no—
63Dic79⋊C8—24no—
64#14579089—24no—
65#14579085C2×C412yes—
66#14579084C2×C412yes—
67#14579082C2×C412yes—
68#14579081C2×C412yes—
69C22⋊(C79⋊C8)—24no—
70(C2×D79)⋊C8—24no—
71#14579072—21no—
72#14579072—24no—
73#14579072D412yes—
74#14579072D412yes—
75C4×C79⋊C8—21no—
76C4×C79⋊C8—24no—
77C4×C79⋊C8C2×C412yes—
78C4×C79⋊C8C2×C412yes—
79C2×#41×C79C2311yescharacteristic
80C2×M16×C79C2312yes—
81C4×Q8×C79C2×C412yes—
82C4×D4×C79—22no—
83C4×D4×C79C2×C412yes—
84#117×C79—21no—
85#117×C79C2311yescharacteristic
86C2×C42×C79—21no—
87C2×C42×C79C2312yes—
88C2×C42×C79C2311yescharacteristic
89#104×C79—24no—
90#97×C79—21no—
91C4×C8×C79—21no—
92#95×C79—22no—
93#11929624—22no—
94#11929624—22no—
95C2×Dic158C22×C411yescharacteristic
96C2×D316C22×C411yescharacteristic
97C2×C22⋊D79—21no—
98C2×C4×D79—22no—
99C2×C4×D79—21no—
100#11929614—22no—
101#11929612—22no—
102#11929611—22no—
103#11929609—22no—
104#11929608—22no—
105#11929608—22no—
106C2×C79⋊C8—22no—
107C2×C79⋊C8—22no—
108C2×C79⋊C8—22no—
109C2×C79⋊C8—22no—
110C2×C79⋊C8C22×C412yes—
111C2×C79⋊C8C22×C411yescharacteristic
112C2×C79⋊C8C22×C411yescharacteristic
113C2×C79⋊C8C2×D414yes—
114C2×C79⋊C8#4114yes—
115C4⋊Dic79C22×C412yes—
116Dic79⋊C4—22no—
117C22⋊Dic79—21no—
118(C2×D79)⋊C4—22no—
119C4×Dic79—22no—
120C4×Dic79—21no—
121#41×C79—22no—
122#41×C79—22no—
123C2×Q8×C79C22×C411yescharacteristic
124C2×D4×C79—21no—
125C2×D4×C79C22×C411yescharacteristic
126C22×C4×C79—22no—
127C22×C4×C79—21no—
128C22×C4×C79—22no—
129C22×C4×C79#4112yes—
130C22×C4×C79C2411yescharacteristic
131M16×C79—22no—
132M16×C79—22no—
133C2×C8×C79—22no—
134C4⋊C4×C79—22no—
135C4⋊C4×C79C22×C412yes—
136C22⋊C4×C79—21no—
137C22⋊C4×C79—22no—
138C42×C79—22no—
139C42×C79—21no—
140C42×C79—22no—
141C42×C79C22×C412yes—
142C42×C79C2×D411yescharacteristic
143C42×C79C2×D411yescharacteristic
144C42×C79#4114yes—
145C23×C79—21no—
146C23×C79C2×#4111yescharacteristic
147C22×D79—21no—
148Q8×C79—21no—
149Q8×C79—21no—
150D4×C79—21no—
151D4×C79—21no—
152D4×C79—42no—
153C2×C4×C79—22no—
154C2×C4×C79—21no—
155C2×C4×C79—24no—
156C2×C4×C79—42no—
157C2×C4×C79—44no—
158C2×C4×C79—42no—
159C2×C4×C79—82no—
160C2×C4×C79C4×D412yes—
161C2×C4×C79C4×D412yes—
162C2×C4×C79C23×C411yescharacteristic
163C2×C4×C79C22×D411yescharacteristic
164C2×C4×C79C2×#4112yes—
165C2×C4×C79C2×#4112yes—
166C22⋊D79—42no—
167C2×Dic79—22no—
168C2×Dic79—21no—
169D316—21no—
170D316—21no—
171C4×D79—42no—
172C4×D79—82no—
173Dic158—21no—
174Dic158—21no—
175C8×C79—42no—
176C79⋊C8—22no—
177C79⋊C8—24no—
178C79⋊C8—42no—
179C79⋊C8#11712yes—
180C79⋊C8#11712yes—
181C22×C79—22no—
182C22×C79—42no—
183C22×C79—41no—
184C22×C79C2×C4×D411yescharacteristic
185C22×C79#52912yes—
186C2×D79—42no—
187C2×D79—41no—
188C4×C79—22no—
189C4×C79—22no—
190C4×C79—44no—
191C4×C79—42no—
192C4×C79—41no—
193C4×C79C2×#11711yescharacteristic
194C4×C79C4×#4111yescharacteristic
195C4×C79#53212yes—
196Dic79—42no—
197Dic79—41no—
198#21294—791nomaximal, Sylow
199C2×C79—22no—
200C2×C79—81no—
201C2×C79#229311yescharacteristic
202C2×C79#295712yes—
203D79—81no—
204C4×#135—791no—
205#2923—791no—
206#2923—791no—
207#1974—792no—
208#1945—792no—
209#1938—792no—
210#1938—792no—
211C4×#108—791no—
212#1759—791no—
213#1759D7911yescharacteristic, complemented
214#1726—791no—
215C79#2129411yescharacteristic, complemented, Sylow
216C2×#135—791no—
217C4×#41—791no—
218C4×#41C2×D7911yescharacteristic
219C4×Q16—792no—
220C4×SD16—792no—
221C4×SD16—792no—
222C4×D8—792no—
223#433—1582no—
224#433—1582no—
225#432—1582no—
226#432—1582no—
227C2×#108—792no—
228#427—791no—
229#424—792no—
230#424—792no—
231C2×#104—792no—
232C2×#104C2×D7912yes—
233#416—791no—
234#416—791no—
235#408—792no—
236#408—792no—
237C4×M16—791no—
238C4×M16C2×D7911yescharacteristic
239C2×#97—791no—
240C2×#97—792no—
241C2×C4×C8—791no—
242C2×C4×C8—792no—
243#376C2×D7911yescharacteristic
244#341—792no—
245#339—792no—
246#339C2×D7912yes—
247C2×#41—791no—
248C2×#41C22×D7911yescharacteristic
249#135—1584no—
250C2×Q16—791no—
251C2×SD16—791no—
252C2×SD16—791no—
253C2×D8—791no—
254C2×M16—792no—
255C2×M16C22×D7912yes—
256C22×C8—791no—
257C22×C8—791no—
258C22×C8—792no—
259C4×Q8—792no—
260C4×Q8C4×D7912yes—
261C4×D4—22no—
262C4×D4—792no—
263C4×D4—1582no—
264C4×D4C4×D7912yes—
265#117—21no—
266#117—791no—
267#117—1581no—
268#117C22×D7911yescharacteristic
269C2×C42—21no—
270C2×C42—1581no—
271C2×C42C22×D7912yes—
272C2×C42C22×D7911yescharacteristic
273#108—1584no—
274#107—792no—
275#106—792no—
276C4⋊C8—1584no—
277C4⋊C8—1584no—
278#104—24no—
279#104—1584no—
280#103—792no—
281#103—792no—
282#102—792no—
283#102—792no—
284C22⋊C8—1584no—
285C22⋊C8—1584no—
286#97—21no—
287#97—792no—
288#97—792no—
289#97—1581no—
290#97—1584no—
291C4×C8—21no—
292C4×C8—792no—
293C4×C8—792no—
294C4×C8—1584no—
295C4×C8—1581no—
296#95—22no—
297#41—22no—
298#41—22no—
299#41—1582no—
300#41—1582no—
301C2×Q8—791no—
302C2×Q8C2×C4×D7911yescharacteristic
303C2×D4—21no—
304C2×D4—791no—
305C2×D4—1581no—
306C2×D4C2×C4×D7911yescharacteristic
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—1581no—
311C22×C4—1582no—
312C22×C4#1192962312yes—
313C22×C4C23×D7911yescharacteristic
314Q16—1582no—
315SD16—1582no—
316SD16—1582no—
317D8—1582no—
318M16—22no—
319M16—22no—
320M16—1582no—
321M16—1582no—
322C2×C8—22no—
323C2×C8—794no—
324C2×C8—791no—
325C2×C8—791no—
326C2×C8—792no—
327C2×C8—794no—
328C2×C8—1582no—
329C2×C8—1582no—
330C2×C8—1582no—
331C2×C8—1582no—
332C4⋊C4—22no—
333C4⋊C4—792no—
334C4⋊C4—1582no—
335C4⋊C4C2×C4×D7912yes—
336C22⋊C4—21no—
337C22⋊C4—22no—
338C22⋊C4—1581no—
339C22⋊C4—1582no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—1582no—
344C42—1581no—
345C42C2×C4×D7912yes—
346C42D4×D7911yescharacteristic
347C42D4×D7911yescharacteristic
348C42#1192962414yes—
349C23—21no—
350C23—1581no—
351C23C2×#1192962311yescharacteristic
352Q8—21no—
353Q8—21no—
354Q8—1581no—
355Q8—1581no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—3162no—
360D4—1581no—
361D4—1581no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—24no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—3162no—
370C2×C4—6322no—
371C2×C4—1582no—
372C2×C4—1581no—
373C2×C4(C22⋊D79)⋊C412yes—
374C2×C4(C22⋊D79)⋊C412yes—
375C2×C4C22×C4×D7911yescharacteristic
376C2×C4C2×D4×D7911yescharacteristic
377C2×C4#41×D7912yes—
378C2×C4C2×#1192962412yes—
379C8—42no—
380C8—3162no—
381C8—792no—
382C8—792no—
383C8—1582no—
384C8—1584no—
385C22—22no—
386C22—41no—
387C22—42no—
388C22—3161no—
389C22—3162no—
390C22C2×(C22⋊D79)⋊C411yescharacteristic
391C22#1909909312yes—
392C4—22no—
393C4—22no—
394C4—44no—
395C4—41no—
396C4—42no—
397C4—3161no—
398C4—3162no—
399C4#117×D7911yescharacteristic
400C4C4×#1192962411yescharacteristic
401C4#1909911012yes—
402C2—22no—
403C2—81no—
404C2—6321no—
405C2#2376586311yescharacteristic
406C2#2377231312yes—
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1759C2
socleC22×C79C2×C4×D4
prime radicalC2×#14579181C22
cosocleC22×C4×C79C24
hypercenter#1759D79
hypocenterC79#21294
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1759

Complements

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

NH ≅ G/Ntype
2. #23772052C2semidirect
3. #23772051C2semidirect
5. #23764147C2semidirect
6. #23764112C2semidirect
7. #23764108C2semidirect
8. C4×#14579096C2semidirect
9. #23763211C2semidirect
10. #23763087C2semidirect
11. C79×#1759C2semidirect
14. C4×#11929614C4semidirect
15. C4×#11929611C4semidirect
16. C4×#11929612C4semidirect
17. C4×#11929609C4semidirect
213. #1759D79semidirect
215. C79#21294semidirect

Maximal subgroups

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

subgroupindex
C4×#145791812
#237720522
#237720512
#237642852
#237641472
#237641122
#237641082
C4×#145790962
#237632112
#237630872
C79×#17592
#2129479