Finite Group Databasebeta

← #34200370

Subgroups — #34200370

Order 20224.

476 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 67 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#23774175C211yesmaximal, characteristic, complemented
2#23765329C212yesmaximal, complemented
3#23761411C212yesmaximal, complemented
4#23761324C211yesmaximal, characteristic, complemented
5C79×#1999C211yesmaximal, characteristic
6C2×#14579140—21no—
7C2×#14579105—22no—
8#19098899—22no—
9#19098804—21no—
10#19098804C2212yes—
11#19098721—22no—
12#19098364—22no—
13C22⋊(C79⋊C16)C2212yes—
14C22×D8×C79C2211yescharacteristic
15C2×#121×C79—21no—
16C2×#102×C79—21no—
17C2×C22⋊C8×C79C2211yescharacteristic
18#340×C79C2211yescharacteristic
19C22×Dic158—21no—
20C2×Dic316—21no—
21C2×Dic316D412yescomplemented
22#14579140—22no—
23#14579140—22no—
24#14579140—41no—
25#14579140—41no—
26C2×C4⋊Dic79—21no—
27C2×Dic79⋊C4—41no—
28C2×C22⋊Dic79—41no—
29#14579105—44no—
30C2×C79⋊C16—22no—
31#14579095—21no—
32#14579095D412yescomplemented
33#14579086—22no—
34#14579086—22no—
35C22×D4×C79—21no—
36C22×D4×C79—41no—
37C23×C4×C79D411yescharacteristic
38C2×D8×C79—22no—
39C2×D8×C79—42no—
40C22×C8×C79D411yescharacteristic
41C22×C8×C79C2311yescharacteristic
42#121×C79—42no—
43#121×C79—42no—
44C2×C4⋊C4×C79—21no—
45C2×C22⋊C4×C79—41no—
46#102×C79—42no—
47C22⋊C8×C79—22no—
48C2×Dic158—22no—
49C2×Dic158—41no—
50C2×Dic158—81no—
51C2×Dic158—81no—
52C22×Dic79—41no—
53C22×Dic79—41no—
54Dic316—42no—
55Dic316—81no—
56C4⋊Dic79—22no—
57C4⋊Dic79—41no—
58Dic79⋊C4—42no—
59Dic79⋊C4—42no—
60C22⋊Dic79—42no—
61C22⋊Dic79—42no—
62C79⋊C16—42no—
63C24×C79—41no—
64C24×C79D811yescharacteristic
65C2×D4×C79—22no—
66C2×D4×C79—41no—
67C2×D4×C79—41no—
68C2×D4×C79—41no—
69C2×D4×C79—44no—
70C2×D4×C79—41no—
71C2×D4×C79—41no—
72C2×D4×C79—82no—
73C2×D4×C79—82no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—41no—
77C22×C4×C79—41no—
78C22×C4×C79—41no—
79C22×C4×C79D811yescharacteristic
80C22×C4×C79C2×D411yescharacteristic
81D8×C79—24no—
82D8×C79—42no—
83C2×C8×C79—21no—
84C2×C8×C79—41no—
85C2×C8×C79—41no—
86C2×C8×C79C2×D412yes—
87C4⋊C4×C79—22no—
88C4⋊C4×C79—41no—
89C22⋊C4×C79—44no—
90C23×C79—21no—
91C23×C79—21no—
92C23×C79—41no—
93C23×C79—41no—
94C23×C79—42no—
95C23×C79—81no—
96C23×C79—81no—
97C23×C79—81no—
98C23×C79—81no—
99C23×C79—81no—
100C23×C79—81no—
101C23×C79SD3212yescomplemented
102C23×C79C2×D811yescharacteristic
103D4×C79—42no—
104D4×C79—42no—
105D4×C79—81no—
106D4×C79—88no—
107D4×C79—81no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—42no—
115C2×C4×C79—81no—
116C2×C4×C79—81no—
117C2×C4×C79—81no—
118C2×C4×C79#12011yescharacteristic
119C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
120C2×Dic79—42no—
121C2×Dic79—42no—
122C2×Dic79—82no—
123C2×Dic79—82no—
124Dic158—42no—
125Dic158—42no—
126Dic158—82no—
127C8×C79—22no—
128C8×C79—41no—
129C22×C79—21no—
130C22×C79—21no—
131C22×C79—22no—
132C22×C79—42no—
133C22×C79—41no—
134C22×C79—41no—
135C22×C79—81no—
136C22×C79—82no—
137C22×C79—82no—
138C22×C79—82no—
139C22×C79—161no—
140C22×C79—161no—
141C22×C79—161no—
142C22×C79—161no—
143C22×C79#44711yescharacteristic
144C22×C79C2×SD3211yescharacteristic
145C22×C79#51011yescharacteristic
146C4×C79—21no—
147C4×C79—41no—
148C4×C79—82no—
149C4×C79#45311yescharacteristic
150C4×C79B(4,𝔽2)11yescharacteristic
151Dic79—82no—
152Dic79—82no—
153#12103—791nomaximal, Sylow
154C2×C79—21no—
155C2×C79—21no—
156C2×C79—42no—
157C2×C79—162no—
158C2×C79#159511yescharacteristic
159C2×C79#218511yescharacteristic
160C2×C79#219011yescharacteristic
161#3189—791no—
162#2187—792no—
163#1999D7911yescharacteristic
164#1348—792no—
165#1314—791no—
166C79#1210311yescharacteristic, complemented, Sylow
167C22×D8C2×D7911yescharacteristic
168C2×C22⋊Q8—1581no—
169C2×#121—21no—
170C2×SD32—1582no—
171#484—1582no—
172#467—792no—
173#467—1581no—
174#451—1582no—
175C2×#102—21no—
176C2×C22⋊C8C2×D7911yescharacteristic
177#357—1582no—
178C22⋊C16—792no—
179#340C2×D7911yescharacteristic
180C22×Q8—1581no—
181C22×D4—21no—
182C22×D4—41no—
183C23×C4C22⋊D7911yescharacteristic
184C2×Q16—792no—
185C2×Q16—1581no—
186C2×D8—22no—
187C2×D8—42no—
188C22×C8C22⋊D7911yescharacteristic
189C22×C8C22×D7911yescharacteristic
190C22⋊Q8—3161no—
191C22⋊Q8—3161no—
192C22⋊Q8—1582no—
193C22⋊Q8—1582no—
194#121—42no—
195#121—42no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—3161no—
198C2×C4⋊C4—1581no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—3161no—
201SD32—3164no—
202C2×C16—1582no—
203#107—792no—
204#107—1581no—
205#103—1582no—
206#103—1582no—
207#102—42no—
208C22⋊C8—22no—
209C24—41no—
210C24#1192960911yescharacteristic
211C2×Q8—3161no—
212C2×Q8—6321no—
213C2×Q8—6321no—
214C2×Q8—1582no—
215C2×D4—22no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—44no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—82no—
223C2×D4—82no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—3161no—
230C22×C4—3161no—
231C22×C4#1192960911yescharacteristic
232C22×C4C2×C22⋊D7911yescharacteristic
233Q16—3162no—
234Q16—6321no—
235D8—24no—
236D8—42no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8D4×D7912yes—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—3162no—
244C4⋊C4—3161no—
245C4⋊C4—3162no—
246C4⋊C4—1582no—
247C22⋊C4—44no—
248C22⋊C4—3162no—
249C22⋊C4—3162no—
250C16—3162no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23#1457910512yescomplemented
263C23C2×#1192960911yescharacteristic
264Q8—3162no—
265Q8—3162no—
266Q8—6322no—
267D4—42no—
268D4—42no—
269D4—81no—
270D4—88no—
271D4—81no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—3162no—
283C2×C4—3162no—
284C2×C4—6322no—
285C2×C4—6322no—
286C2×C4#1457912911yescharacteristic
287C2×C4#1457918611yescharacteristic
288C8—22no—
289C8—41no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—161no—
301C22—161no—
302C22—161no—
303C22—161no—
304C22#1909869611yescharacteristic
305C22C2×#1457910511yescharacteristic
306C22#1909900311yescharacteristic
307C4—21no—
308C4—41no—
309C4—82no—
310C4—6322no—
311C4—6322no—
312C4#1909873011yescharacteristic
313C4#1909874911yescharacteristic
314C2—21no—
315C2—21no—
316C2—42no—
317C2—162no—
318C2#2376231711yescharacteristic
319C2#2376531611yescharacteristic
320C2#2376534511yescharacteristic
subgroupG/H
centerC22#19098696
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
Fitting, prime radicalC79×#1999C2
socleC22×C79#447
hypercenter#1999D79
hypocenterC79#12103
LCSG ▷ C22×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. #23774175C2semidirect
2. #23765329C2semidirect
3. #23761411C2semidirect
4. #23761324C2semidirect
21. C2×Dic316D4semidirect
32. #14579095D4semidirect
101. C23×C79SD32semidirect
166. C79#12103semidirect
262. C23#14579105semidirect

Maximal subgroups

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

subgroupindex
#237741752
#237653292
#237614112
#237613242
C79×#19992
#1210379