Finite Group Databasebeta

← #34205454

Subgroups — #34205454

Order 20224.

476 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772284C211yesmaximal, characteristic, complemented
2#23765136C211yesmaximal, characteristic, complemented
3C2×#19098373C211yesmaximal, characteristic, complemented
4#23764299C211yesmaximal, characteristic, complemented
5#23761506C211yesmaximal, characteristic, complemented
6#23761487C211yesmaximal, characteristic, complemented
7C79×#3414C211yesmaximal, characteristic
8#19099046—21no—
9C2×#14579101C2211yescharacteristic
10#19098675—21no—
11#19098670—21no—
12SD16×Dic79—22no—
13#19098601C2211yescharacteristic
14C2×#14579093C2211yescharacteristic
15#19098580—21no—
16#19098564—21no—
17#19098537—21no—
18#19098518C2211yescharacteristic
19#19098373—22no—
20#19098373—22no—
21C2×#135×C79C2211yescharacteristic
22C2×#135×C79C2211yescharacteristic
23#510×C79—21no—
24#510×C79C412yescomplemented
25#509×C79—21no—
26#509×C79C412yescomplemented
27#508×C79—22no—
28C2×M32×C79C2211yescharacteristic
29C2×#11929608D411yescharacteristic
30Q8×Dic79—22no—
31D4×Dic79—22no—
32D4×Dic79—42no—
33#14579120—21no—
34#14579120—41no—
35#14579120D411yescharacteristic
36C2×C4⋊Dic79—21no—
37C2×C4⋊Dic79—41no—
38#14579101—21no—
39#14579101—21no—
40C2×C79⋊C16—21no—
41#14579095—21no—
42#14579093—21no—
43#14579093—21no—
44#14579093D412yes—
45#14579088—42no—
46#14579085—22no—
47#14579082—22no—
48#14579076—42no—
49#14579071—21no—
50C8×Dic79—21no—
51C2×#41×C79—21no—
52C2×#41×C79—21no—
53#135×C79—21no—
54#135×C79—21no—
55#135×C79—21no—
56#135×C79—21no—
57#135×C79—21no—
58#135×C79—21no—
59#135×C79C2×C412yes—
60#135×C79D412yes—
61C2×Q16×C79C2×C411yescharacteristic, complemented
62C2×Q16×C79D411yescharacteristic
63C2×SD16×C79—21no—
64C2×SD16×C79—21no—
65C2×D8×C79C2×C411yescharacteristic, complemented
66C2×D8×C79D411yescharacteristic
67C22×C8×C79C2311yescharacteristic
68Q32×C79—22no—
69SD32×C79—22no—
70SD32×C79—22no—
71D16×C79—22no—
72M32×C79—21no—
73M32×C79C2×C412yescomplemented
74C2×C16×C79—21no—
75C22×Dic79—42no—
76C22×Dic79—41no—
77#11929608—21no—
78#11929608—21no—
79C2×C79⋊C8—21no—
80C4⋊Dic79—22no—
81C4⋊Dic79—21no—
82C4⋊Dic79—42no—
83C4⋊Dic79—41no—
84C4⋊Dic79—41no—
85C22⋊Dic79—42no—
86C22⋊Dic79—41no—
87C22⋊Dic79D812yes—
88C4×Dic79—21no—
89C4×Dic79—42no—
90C4×Dic79—41no—
91C79⋊C16—41no—
92#41×C79—22no—
93#41×C79—22no—
94#41×C79—41no—
95#41×C79—41no—
96#41×C79—41no—
97#41×C79—41no—
98#41×C79—41no—
99#41×C79—41no—
100C2×Q8×C79—21no—
101C2×Q8×C79—21no—
102C2×D4×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79—41no—
105C2×D4×C79—41no—
106C22×C4×C79—41no—
107C22×C4×C79—41no—
108C22×C4×C79C2×D411yescharacteristic
109Q16×C79—21no—
110Q16×C79—21no—
111Q16×C79—21no—
112Q16×C79C22⋊C412yes—
113SD16×C79—22no—
114SD16×C79—22no—
115SD16×C79—41no—
116SD16×C79—41no—
117D8×C79—21no—
118D8×C79—21no—
119D8×C79—21no—
120D8×C79C22⋊C412yes—
121C2×C8×C79C22×C411yescharacteristic
122C2×C8×C79C2×D411yescharacteristic
123C2×C8×C79C2×D411yescharacteristic
124C2×C8×C79C2×D411yescharacteristic
125C2×C8×C79#4111yescharacteristic
126C2×C8×C79#4111yescharacteristic
127C16×C79—22no—
128C23×C79—41no—
129C23×C79—41no—
130C23×C79C2×D811yescharacteristic
131Q8×C79—22no—
132Q8×C79—41no—
133Q8×C79—41no—
134Q8×C79—41no—
135D4×C79—22no—
136D4×C79—42no—
137D4×C79—42no—
138D4×C79—41no—
139D4×C79—41no—
140D4×C79—41no—
141D4×C79—81no—
142D4×C79—81no—
143C2×C4×C79—41no—
144C2×C4×C79—42no—
145C2×C4×C79—42no—
146C2×C4×C79—41no—
147C2×C4×C79—81no—
148C2×C4×C79—81no—
149C2×C4×C79C4×D411yescharacteristic
150C2×C4×C79#12011yescharacteristic
151C2×C4×C79#12111yescharacteristic
152C2×C4×C79#13511yescharacteristic
153C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
155C2×Dic79—22no—
156C2×Dic79—42no—
157C2×Dic79—81no—
158C2×Dic79—162no—
159C2×Dic79—161no—
160C8×C79C2×C22⋊C411yescharacteristic
161C8×C79C4×D411yescharacteristic
162C8×C79#12111yescharacteristic
163C8×C79#12311yescharacteristic
164C79⋊C8—41no—
165C22×C79—41no—
166C22×C79—42no—
167C22×C79—81no—
168C22×C79—81no—
169C22×C79—81no—
170C22×C79—161no—
171C22×C79C4×D811yescharacteristic
172C22×C79#44711yescharacteristic
173C22×C79#45911yescharacteristic
174C4×C79—42no—
175C4×C79—81no—
176C4×C79#38611yescharacteristic
177C4×C79#44211yescharacteristic
178C4×C79#44911yescharacteristic
179C4×C79#46311yescharacteristic
180Dic79—42no—
181Dic79—82no—
182Dic79—161no—
183#13030—791nomaximal, Sylow
184C2×C79—21no—
185C2×C79—21no—
186C2×C79—21no—
187C2×C79—82no—
188C2×C79—161no—
189C2×C79#187911yescharacteristic
190#3414D7911yescharacteristic
191#2952—791no—
192#2145—791no—
193C2×#360—791no—
194#1976—791no—
195#1386—791no—
196#1378—791no—
197C79#1303011yescharacteristic, complemented, Sylow
198C2×#135C2×D7911yescharacteristic
199C2×#135C2×D7911yescharacteristic
200#520—1581no—
201#510—21no—
202#510Dic7912yescomplemented
203#509—21no—
204#509Dic7912yescomplemented
205#508—22no—
206C2×M32—791no—
207C2×M32C2×D7911yescharacteristic
208#442—1581no—
209#441—1581no—
210C4×SD16—1582no—
211#427—791no—
212C2×#106—791no—
213#421—1581no—
214#417—1581no—
215#410—1581no—
216#405—791no—
217#360—1582no—
218#360—1582no—
219C2×#41—21no—
220C2×#41—21no—
221#135—21no—
222#135—21no—
223#135—21no—
224#135—21no—
225#135—21no—
226#135—21no—
227#135C2×Dic7912yes—
228#135C22⋊D7912yes—
229C2×Q16C2×Dic7911yescharacteristic, complemented
230C2×Q16C22⋊D7911yescharacteristic
231C2×SD16—21no—
232C2×SD16—21no—
233C2×D8C2×Dic7911yescharacteristic, complemented
234C2×D8C22⋊D7911yescharacteristic
235C2×M16—791no—
236C22×C8C22×D7911yescharacteristic
237C4×Q8—1582no—
238C4×D4—3162no—
239C4×D4—1582no—
240#117—3161no—
241#117—791no—
242#117—1581no—
243C2×C4⋊C4—3161no—
244C2×C4⋊C4—1581no—
245Q32—22no—
246SD32—22no—
247SD32—22no—
248D16—22no—
249M32—21no—
250M32—1581no—
251M32—1581no—
252M32C2×Dic7912yescomplemented
253C2×C16—21no—
254C2×C16—1581no—
255#107—1581no—
256#106—792no—
257#106—1581no—
258#106—1581no—
259#104—3162no—
260#103—1582no—
261#102—1582no—
262#99—3162no—
263#97—1581no—
264C4×C8—1581no—
265#41—22no—
266#41—22no—
267#41—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—3161no—
282C22×C4—3162no—
283C22×C4D4×D7911yescharacteristic
284Q16—21no—
285Q16—21no—
286Q16—21no—
287Q16C22⋊Dic7912yes—
288SD16—22no—
289SD16—22no—
290SD16—41no—
291SD16—41no—
292D8—21no—
293D8—21no—
294D8—21no—
295D8C22⋊Dic7912yes—
296M16—1581no—
297M16—1581no—
298C2×C8—1581no—
299C2×C8C22×Dic7911yescharacteristic
300C2×C8D4×D7911yescharacteristic
301C2×C8C2×C22⋊D7911yescharacteristic
302C2×C8C2×C22⋊D7911yescharacteristic
303C2×C8#1192962311yescharacteristic
304C2×C8#1192962311yescharacteristic
305C4⋊C4—3162no—
306C4⋊C4—3161no—
307C4⋊C4—3161no—
308C4⋊C4—1582no—
309C4⋊C4—1581no—
310C22⋊C4—3162no—
311C22⋊C4—3161no—
312C22⋊C4—792no—
313C42—3162no—
314C42—3161no—
315C42—1581no—
316C16—22no—
317C16—3161no—
318C23—41no—
319C23—41no—
320C23D8×D7911yescharacteristic
321Q8—22no—
322Q8—41no—
323Q8—41no—
324Q8—41no—
325D4—22no—
326D4—42no—
327D4—42no—
328D4—41no—
329D4—41no—
330D4—41no—
331D4—81no—
332D4—81no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—3162no—
340C2×C4—6321no—
341C2×C4—1582no—
342C2×C4—12642no—
343C2×C4—12641no—
344C2×C4D4×Dic7911yescharacteristic
345C2×C4#1457912911yescharacteristic
346C2×C4#1457913311yescharacteristic
347C2×C4#1457918011yescharacteristic
348C2×C4#1457918511yescharacteristic
349C2×C4#1457918511yescharacteristic
350C8—3161no—
351C8C2×C22⋊Dic7911yescharacteristic
352C8D4×Dic7911yescharacteristic
353C8#1457913311yescharacteristic
354C8#1457914211yescharacteristic
355C22—41no—
356C22—42no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—161no—
361C22D8×Dic7911yescharacteristic
362C22#1909869511yescharacteristic
363C22#1909875611yescharacteristic
364C4—42no—
365C4—81no—
366C4—3162no—
367C4—6322no—
368C4—12641no—
369C4#1909844311yescharacteristic
370C4#1909867511yescharacteristic
371C4#1909870511yescharacteristic
372C4#1909877611yescharacteristic
373C2—21no—
374C2—21no—
375C2—21no—
376C2—82no—
377C2—161no—
378C2#2376375011yescharacteristic
subgroupG/H
centerC2#23763750
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
Fitting, prime radicalC79×#3414C2
socleC2×C79#1879
cosocleC22×C8×C79C23
hypercenter#3414D79
hypocenterC79#13030
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. #23772284C2semidirect
2. #23765136C2semidirect
3. C2×#19098373C2semidirect
4. #23764299C2semidirect
5. #23761506C2semidirect
6. #23761487C2semidirect
24. #510×C79C4semidirect
26. #509×C79C4semidirect
61. C2×Q16×C79C2×C4semidirect
65. C2×D8×C79C2×C4semidirect
73. M32×C79C2×C4semidirect
197. C79#13030semidirect
202. #510Dic79semidirect
204. #509Dic79semidirect
229. C2×Q16C2×Dic79semidirect
233. C2×D8C2×Dic79semidirect
252. M32C2×Dic79semidirect

Maximal subgroups

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

subgroupindex
#237722842
#237651362
C2×#190983732
#237642992
#237615062
#237614872
C79×#34142
#1303079