Finite Group Databasebeta

← #34177292

Subgroups — #34177292

Order 20224.

682 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764928C211yesmaximal, characteristic, complemented
2#23764803C212yesmaximal, complemented
3#23761221C211yesmaximal, characteristic, complemented
4#23761221C212yesmaximal, complemented
5C79×#2679C211yesmaximal, characteristic
6C2×#14579094—21no—
7C2×#14579082—22no—
8C2×#14579082—21no—
9#19098433C2211yescharacteristic, complemented
10#19098325—22no—
11#19098318C2212yescomplemented
12#19098318C2211yescharacteristic, complemented
13C2×#120×C79—22no—
14C2×#120×C79—21no—
15#394×C79—22no—
16#394×C79—21no—
17#390×C79C2212yes—
18#390×C79C2211yescharacteristic
19C22×C79⋊C8—21no—
20C22×C79⋊C8—22no—
21C2×C4⋊Dic79—22no—
22C2×C4⋊Dic79—21no—
23#14579094—42no—
24#14579082—42no—
25#14579082—44no—
26C25×C79—21no—
27C22×D4×C79—21no—
28C22×D4×C79—22no—
29C22×D4×C79D412yes—
30C22×D4×C79D411yescharacteristic
31#120×C79—44no—
32#120×C79—42no—
33#120×C79—44no—
34#120×C79—42no—
35C2×C4⋊C4×C79D412yes—
36C2×C4⋊C4×C79D411yescharacteristic
37C2×C22⋊C4×C79—22no—
38C2×C22⋊C4×C79—21no—
39C2×C22⋊C4×C79—22no—
40C2×C22⋊C4×C79—22no—
41C2×C22⋊C4×C79—22no—
42C2×C42×C79C2311yescharacteristic
43#95×C79—21no—
44C22×Dic79—41no—
45C2×C79⋊C8—22no—
46C2×C79⋊C8—41no—
47C2×C79⋊C8—42no—
48C2×C79⋊C8—42no—
49C2×C79⋊C8—42no—
50C2×C79⋊C8—41no—
51C4⋊Dic79—41no—
52C4⋊Dic79—41no—
53C4⋊Dic79—42no—
54C4⋊Dic79—42no—
55C24×C79—22no—
56C24×C79—21no—
57C24×C79—44no—
58C24×C79—42no—
59C24×C79—81no—
60C24×C79—82no—
61C24×C79—81no—
62C2×D4×C79—22no—
63C2×D4×C79—21no—
64C2×D4×C79—42no—
65C2×D4×C79—42no—
66C2×D4×C79—41no—
67C2×D4×C79—41no—
68C2×D4×C79—44no—
69C2×D4×C79—44no—
70C2×D4×C79—44no—
71C2×D4×C79—81no—
72C2×D4×C79—82no—
73C2×D4×C79D814yes—
74C2×D4×C79SD1612yes—
75C22×C4×C79—21no—
76C22×C4×C79—22no—
77C22×C4×C79C2×D412yes—
78C22×C4×C79C2×D411yescharacteristic
79C4⋊C4×C79—42no—
80C4⋊C4×C79—41no—
81C22⋊C4×C79—44no—
82C22⋊C4×C79—42no—
83C22⋊C4×C79—42no—
84C22⋊C4×C79—42no—
85C22⋊C4×C79—42no—
86C22⋊C4×C79—42no—
87C22⋊C4×C79—42no—
88C22⋊C4×C79—42no—
89C42×C79—41no—
90C23×C79—24no—
91C23×C79—22no—
92C23×C79—42no—
93C23×C79—42no—
94C23×C79—42no—
95C23×C79—41no—
96C23×C79—42no—
97C23×C79—42no—
98C23×C79—42no—
99C23×C79—42no—
100C23×C79—42no—
101C23×C79—41no—
102C23×C79—84no—
103C23×C79—88no—
104C23×C79—88no—
105C23×C79—84no—
106C23×C79—84no—
107C23×C79#12011yescharacteristic
108D4×C79—44no—
109D4×C79—42no—
110D4×C79—82no—
111D4×C79—81no—
112D4×C79—84no—
113D4×C79—88no—
114C2×C4×C79—21no—
115C2×C4×C79—22no—
116C2×C4×C79—22no—
117C2×C4×C79—21no—
118C2×C4×C79—42no—
119C2×C4×C79—42no—
120C2×C4×C79—42no—
121C2×C4×C79—41no—
122C2×C4×C79—42no—
123C2×C4×C79C2×D812yes—
124C2×C4×C79C2×SD1611yescharacteristic
125C2×C4×C79AGL(1,ℤ/8)12yes—
126C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
127C2×Dic79—82no—
128C2×Dic79—81no—
129C79⋊C8—42no—
130C79⋊C8—82no—
131C22×C79—44no—
132C22×C79—42no—
133C22×C79—44no—
134C22×C79—44no—
135C22×C79—44no—
136C22×C79—816no—
137C22×C79—82no—
138C22×C79—82no—
139C22×C79—82no—
140C22×C79—81no—
141C22×C79—82no—
142C22×C79—81no—
143C22×C79—82no—
144C22×C79#44712yes—
145C22×C79#45011yescharacteristic
146C22×C79#45311yescharacteristic
147C22×C79#45512yes—
148C22×C79#45811yescharacteristic
149C4×C79—22no—
150C4×C79—22no—
151C4×C79—21no—
152C4×C79—21no—
153C4×C79—81no—
154C4×C79—82no—
155Dic79—161no—
156#7786—791nomaximal, Sylow
157C2×C79—84no—
158C2×C79—82no—
159C2×C79#160712yes—
160C2×C79#161311yescharacteristic
161C2×C79#161911yescharacteristic
162C2×C79#163912yes—
163C2×C79#165911yescharacteristic
164#2679D7911yescharacteristic
165#2093—791no—
166#2071—792no—
167#1270—791no—
168#1270—792no—
169C79#778611yescharacteristic, complemented, Sylow
170C2×#120—22no—
171C2×#120—21no—
172C2×#107—1581no—
173C2×#102—1582no—
174C2×#102—1581no—
175#394—22no—
176#394—21no—
177#390C2×D7912yes—
178#390C2×D7911yescharacteristic
179#384—791no—
180#340—1582no—
181#336—792no—
182#336—791no—
183C25—21no—
184C22×D4—21no—
185C22×D4—22no—
186C22×D4C22⋊D7912yes—
187C22×D4C22⋊D7911yescharacteristic
188C22×C8—1582no—
189C22×C8—1581no—
190#120—44no—
191#120—42no—
192#120—44no—
193#120—42no—
194C2×C4⋊C4—1582no—
195C2×C4⋊C4—1581no—
196C2×C4⋊C4C22⋊D7912yes—
197C2×C4⋊C4C22⋊D7911yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C42C22×D7911yescharacteristic
204#107—3162no—
205#102—3164no—
206#102—3162no—
207#95—21no—
208C24—22no—
209C24—21no—
210C24—44no—
211C24—42no—
212C24—81no—
213C24—82no—
214C24—81no—
215C2×D4—22no—
216C2×D4—21no—
217C2×D4—41no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—81no—
225C2×D4—82no—
226C2×D4#1192960914yes—
227C2×D4#1192961212yes—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—3161no—
231C22×C4C2×C22⋊D7912yes—
232C22×C4C2×C22⋊D7911yescharacteristic
233C2×C8—3161no—
234C2×C8—3161no—
235C2×C8—3162no—
236C2×C8—3162no—
237C2×C8—3162no—
238C2×C8—1582no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—3161no—
242C4⋊C4—3161no—
243C4⋊C4—3162no—
244C4⋊C4—3162no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C42—41no—
254C23—24no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23—84no—
267C23—88no—
268C23—88no—
269C23—84no—
270C23—84no—
271C23#1457913011yescharacteristic
272D4—44no—
273D4—42no—
274D4—82no—
275D4—81no—
276D4—84no—
277D4—88no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—6322no—
288C2×C4—6321no—
289C2×C4C2×#1192960912yes—
290C2×C4C2×#1192961211yescharacteristic
291C2×C4#1457918712yes—
292C2×C4#1457918711yescharacteristic
293C8—3162no—
294C8—6322no—
295C22—44no—
296C22—44no—
297C22—44no—
298C22—44no—
299C22—42no—
300C22—816no—
301C22—81no—
302C22—82no—
303C22—81no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22#1909869712yes—
309C22#1909871411yescharacteristic
310C22#1909873111yescharacteristic
311C22#1909874112yes—
312C22#1909875311yescharacteristic
313C4—22no—
314C4—22no—
315C4—21no—
316C4—21no—
317C4—81no—
318C4—82no—
319C4—12641no—
320C2—84no—
321C2—82no—
322C2#2376238612yes—
323C2#2376242011yescharacteristic
324C2#2376245411yescharacteristic
325C2#2376257912yes—
326C2#2376270611yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#2679C2
socleC23×C79#120
hypercenter#2679D79
hypocenterC79#7786
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2679

Complements

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

NH ≅ G/Ntype
1. #23764928C2semidirect
2. #23764803C2semidirect
3. #23761221C2semidirect
4. #23761221C2semidirect
9. #19098433C22semidirect
11. #19098318C22semidirect
12. #19098318C22semidirect
169. C79#7786semidirect

Maximal subgroups

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

subgroupindex
#237649282
#237648032
#237612212
C79×#26792
#778679