Finite Group Databasebeta

← #34193032

Subgroups — #34193032

Order 20224.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764336C211yesmaximal, characteristic
2#23763528C211yesmaximal, characteristic
3#23763406C211yesmaximal, characteristic
4#23763406C211yesmaximal, characteristic
5#23761284C211yesmaximal, characteristic
6#23761284C211yesmaximal, characteristic
7C79×#1277C211yesmaximal, characteristic
8C2×C4×Dic158C2211yescharacteristic
9C2×C4⋊(C79⋊C8)C2211yescharacteristic
10C2×#14579086—21no—
11C2×#14579086—21no—
12#19098433C2211yescharacteristic
13#19098433C2211yescharacteristic
14#19098326—21no—
15#19098326—21no—
16#19098326—21no—
17#19098326—21no—
18#19098317—21no—
19#19098317—21no—
20C2×C4⋊C8×C79C2211yescharacteristic
21C2×C4⋊C8×C79C2211yescharacteristic
22C2×C4×C8×C79C2211yescharacteristic
23C22×Dic158D411yescharacteristic
24C22×C79⋊C8—21no—
25C4×Dic158—21no—
26C4×Dic158—21no—
27C4×Dic158—21no—
28C4×Dic158—21no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79—21no—
31C2×C4⋊Dic79—21no—
32C2×C4⋊Dic79—21no—
33C2×C4⋊Dic79—21no—
34C2×C4⋊Dic79—21no—
35C2×C4⋊Dic79D411yescharacteristic
36C2×Dic79⋊C4—21no—
37C2×C4×Dic79—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40#14579086—42no—
41#14579086—42no—
42C22×C8×C79—21no—
43C22×C8×C79—21no—
44C22×C8×C79D411yescharacteristic
45C22×C8×C79D411yescharacteristic
46C2×C42×C79C2311yescharacteristic
47C4⋊C8×C79—21no—
48C4⋊C8×C79—21no—
49C4⋊C8×C79—21no—
50C4⋊C8×C79—21no—
51C4×C8×C79—21no—
52C4×C8×C79—21no—
53C2×Dic158—21no—
54C2×Dic158—21no—
55C2×Dic158—21no—
56C2×Dic158—21no—
57C2×Dic158—21no—
58C2×Dic158SD1612yes—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—41no—
62C22×Dic79—41no—
63C2×C79⋊C8—41no—
64C2×C79⋊C8—41no—
65C2×C79⋊C8—41no—
66C4⋊Dic79—21no—
67C4⋊Dic79—21no—
68C4⋊Dic79—41no—
69C4⋊Dic79—41no—
70C4⋊Dic79—41no—
71C4⋊Dic79—41no—
72C4⋊Dic79—41no—
73C4⋊Dic79—41no—
74C4⋊Dic79—41no—
75C4⋊Dic79—41no—
76C4⋊Dic79—81no—
77C4⋊Dic79—81no—
78Dic79⋊C4—41no—
79Dic79⋊C4—41no—
80C4×Dic79—41no—
81C4×Dic79—41no—
82C22×C4×C79C2×D411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C22×C4×C79#4111yescharacteristic
85C2×C8×C79—21no—
86C2×C8×C79—21no—
87C2×C8×C79—21no—
88C2×C8×C79—21no—
89C2×C8×C79—21no—
90C2×C8×C79—21no—
91C2×C8×C79—41no—
92C2×C8×C79—41no—
93C2×C8×C79—41no—
94C2×C8×C79—41no—
95C2×C8×C79—41no—
96C2×C8×C79—41no—
97C42×C79#4111yescharacteristic
98C42×C79#4111yescharacteristic
99C42×C79#4111yescharacteristic
100C42×C79#4111yescharacteristic
101C23×C79#12111yescharacteristic
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79#12311yescharacteristic
105C2×C4×C79#12311yescharacteristic
106C2×C4×C79#12311yescharacteristic
107C2×C4×C79#12411yescharacteristic
108C2×C4×C79#12611yescharacteristic
109C2×C4×C79#12611yescharacteristic
110C2×C4×C79C2×SD1611yescharacteristic
111C2×C4×C79#13511yescharacteristic
112C2×C4×C79#13511yescharacteristic
113C2×C4×C79#13511yescharacteristic
114C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C79#13711yescharacteristic
116C2×C4×C79#13711yescharacteristic
117C2×C4×C79#13711yescharacteristic
118C2×Dic79—22no—
119C2×Dic79—41no—
120C2×Dic79—41no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—81no—
125C2×Dic79—81no—
126C2×Dic79—81no—
127C2×Dic79—81no—
128C2×Dic79—81no—
129C2×Dic79—81no—
130Dic158—22no—
131Dic158—22no—
132Dic158—41no—
133Dic158—41no—
134C8×C79—41no—
135C8×C79—41no—
136C8×C79—81no—
137C8×C79—81no—
138C79⋊C8—81no—
139C22×C79#39711yescharacteristic
140C22×C79#46411yescharacteristic
141C22×C79#46411yescharacteristic
142C22×C79#46511yescharacteristic
143C22×C79#46811yescharacteristic
144C22×C79#47211yescharacteristic
145C22×C79#47311yescharacteristic
146C4×C79—21no—
147C4×C79—21no—
148C4×C79#48111yescharacteristic
149C4×C79#48211yescharacteristic
150C4×C79#48311yescharacteristic
151C4×C79#48511yescharacteristic
152C4×C79#48511yescharacteristic
153C4×C79#48511yescharacteristic
154C4×C79#48811yescharacteristic
155C4×C79#49111yescharacteristic
156Dic79—42no—
157Dic79—81no—
158Dic79—161no—
159Dic79—161no—
160#10626—791nomaximal, Sylow
161C2×C79#167811yescharacteristic
162C2×C79#168011yescharacteristic
163C2×C79#169811yescharacteristic
164C2×C79#170011yescharacteristic
165C2×C79#207411yescharacteristic
166C2×C79#207711yescharacteristic
167C2×C79#207811yescharacteristic
168#1984—791no—
169#1834—791no—
170#1803—791no—
171#1803—791no—
172#1296—791no—
173#1296—791no—
174#1277D7911yescharacteristic
175C79#1062611yescharacteristic, complemented, Sylow
176C2×C4×Q8—791no—
177C2×C4⋊C8—791no—
178C2×C4⋊C8C2×D7911yescharacteristic
179C2×C4⋊C8C2×D7911yescharacteristic
180C2×#103—1581no—
181C2×#103—1581no—
182C2×C4×C8C2×D7911yescharacteristic
183#384—791no—
184#384—791no—
185#340—1581no—
186#340—1581no—
187#340—1581no—
188#340—1581no—
189#336—1581no—
190#336—1581no—
191C22×Q8—791no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—1581no—
195C22×C8D31611yescharacteristic
196C22×C8D31611yescharacteristic
197C4×Q8—1581no—
198C4×Q8—1581no—
199C4×Q8—1581no—
200C4×Q8—1581no—
201C2×C4⋊C4—791no—
202C2×C4⋊C4—1581no—
203C2×C4⋊C4—1581no—
204C2×C4⋊C4—1581no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4—1581no—
207C2×C4⋊C4—1581no—
208C2×C4⋊C4—1581no—
209C2×C42—1581no—
210C2×C42C22×D7911yescharacteristic
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—1581no—
216C4⋊C8—1581no—
217#103—3162no—
218#103—3162no—
219C4×C8—21no—
220C4×C8—21no—
221C2×Q8—792no—
222C2×Q8—1581no—
223C2×Q8—1581no—
224C2×Q8—1581no—
225C2×Q8—1581no—
226C2×Q8—1581no—
227C22×C4—3161no—
228C22×C4—3161no—
229C22×C4—1581no—
230C22×C4—1581no—
231C22×C4D4×D7911yescharacteristic
232C22×C4C2×D31611yescharacteristic
233C22×C4#1192962511yescharacteristic
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—3161no—
247C2×C8—3161no—
248C2×C8—3161no—
249C4⋊C4—3161no—
250C4⋊C4—3161no—
251C4⋊C4—3161no—
252C4⋊C4—3161no—
253C4⋊C4—3161no—
254C4⋊C4—3161no—
255C4⋊C4—3161no—
256C4⋊C4—3161no—
257C4⋊C4—3161no—
258C4⋊C4—3161no—
259C4⋊C4—6321no—
260C4⋊C4—6321no—
261C4⋊C4—1581no—
262C4⋊C4—1581no—
263C42—3161no—
264C42—3161no—
265C42#1192962311yescharacteristic
266C42#1192962311yescharacteristic
267C42#1192962411yescharacteristic
268C42#1192962411yescharacteristic
269C23#1457913611yescharacteristic
270Q8—3161no—
271Q8—3161no—
272Q8—1582no—
273Q8—1582no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—3161no—
277C2×C4—3161no—
278C2×C4—3161no—
279C2×C4—3161no—
280C2×C4—3161no—
281C2×C4—6321no—
282C2×C4—6321no—
283C2×C4—6321no—
284C2×C4—6321no—
285C2×C4—6321no—
286C2×C4—6321no—
287C2×C4—1582no—
288C2×C4#1457914311yescharacteristic
289C2×C4#1457914311yescharacteristic
290C2×C4#1457914511yescharacteristic
291C2×C4#1457914911yescharacteristic
292C2×C4#1457915611yescharacteristic
293C2×C4#1457915611yescharacteristic
294C2×C4SD16×D7911yescharacteristic
295C2×C4#1457918211yescharacteristic
296C2×C4#1457918311yescharacteristic
297C2×C4#1457918311yescharacteristic
298C2×C4#1457918511yescharacteristic
299C2×C4#1457919311yescharacteristic
300C2×C4#1457919411yescharacteristic
301C2×C4#1457919411yescharacteristic
302C8—41no—
303C8—41no—
304C8—81no—
305C8—81no—
306C8—6321no—
307C22#1909849011yescharacteristic
308C22#1909878611yescharacteristic
309C22#1909878611yescharacteristic
310C22#1909879311yescharacteristic
311C22#1909881011yescharacteristic
312C22#1909883411yescharacteristic
313C22#1909883911yescharacteristic
314C4—21no—
315C4—21no—
316C4—3162no—
317C4—6321no—
318C4—12641no—
319C4—12641no—
320C4#1909888011yescharacteristic
321C4#1909888611yescharacteristic
322C4#1909889211yescharacteristic
323C4#1909890211yescharacteristic
324C4#1909890511yescharacteristic
325C4#1909890511yescharacteristic
326C4#1909891811yescharacteristic
327C4#1909893211yescharacteristic
328C2#2376281711yescharacteristic
329C2#2376283111yescharacteristic
330C2#2376294111yescharacteristic
331C2#2376295511yescharacteristic
332C2#2376482011yescharacteristic
333C2#2376483511yescharacteristic
334C2#2376484211yescharacteristic
subgroupG/H
centerC23#14579136
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1277C2
socle, prime radicalC23×C79#121
hypercenter#1277D79
hypocenterC79#10626
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
175. C79#10626semidirect

Maximal subgroups

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

subgroupindex
#237643362
#237635282
#237634062
#237612842
C79×#12772
#1062679