Finite Group Databasebeta

← #34200327

Subgroups — #34200327

Order 20224.

458 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764531C211yesmaximal, characteristic
2#23762385C212yesmaximal
3#23761400C212yesmaximal, complemented
4#23761330C211yesmaximal, characteristic, complemented
5C79×#2817C211yesmaximal, characteristic, complemented
6#19099125C2211yescharacteristic
7C2×#14579144—21no—
8#19098876—22no—
9C2×#14579075—21no—
10#19098330C2211yescharacteristic
11#19098296—22no—
12#19098287C2212yes—
13C2×C22⋊Q8×C79—21no—
14#400×C79—22no—
15#397×C79—21no—
16#397×C79C2212yes—
17#395×C79—22no—
18#381×C79C2211yescharacteristic
19C22×C22⋊D79—21no—
20#14579145—41no—
21#14579144—42no—
22#14579144—42no—
23#14579135—22no—
24#14579131—41no—
25C2×Dic79⋊C4—41no—
26C2×C22⋊Dic79—21no—
27C2×(C2×D79)⋊C4—41no—
28#14579094—42no—
29#14579081—42no—
30#14579075—42no—
31C22⋊(C79⋊C8)—22no—
32C22×Q8×C79—21no—
33C23×C4×C79D411yescharacteristic
34C22⋊Q8×C79—42no—
35C22⋊Q8×C79—42no—
36C2×C4⋊C4×C79—22no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79—21no—
39C2×C4⋊C4×C79D412yes—
40C2×C22⋊C4×C79—22no—
41C2×C22⋊C4×C79D411yescharacteristic
42C2×C22⋊C4×C79C2311yescharacteristic
43#95×C79—22no—
44#95×C79—21no—
45#95×C79—22no—
46#95×C79—22no—
47#95×C79D412yes—
48C23×D79—21no—
49C2×D316—22no—
50C2×C22⋊D79—41no—
51C2×C22⋊D79—41no—
52C2×C22⋊D79—41no—
53C2×C22⋊D79—41no—
54C2×C22⋊D79—41no—
55C2×C22⋊D79—41no—
56C22×Dic79—21no—
57C22×Dic79—41no—
58C2×C79⋊C8—42no—
59C4⋊Dic79—22no—
60Dic79⋊C4—44no—
61C22⋊Dic79—22no—
62C22⋊Dic79—41no—
63(C2×D79)⋊C4—42no—
64(C2×D79)⋊C4—44no—
65C24×C79C2×D411yescharacteristic
66C2×Q8×C79—42no—
67C2×Q8×C79—81no—
68C2×Q8×C79—81no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—22no—
73C22×C4×C79—22no—
74C22×C4×C79—41no—
75C22×C4×C79—41no—
76C22×C4×C79D812yes—
77C22×C4×C79C2×D412yes—
78C4⋊C4×C79—42no—
79C4⋊C4×C79—44no—
80C4⋊C4×C79—42no—
81C4⋊C4×C79—81no—
82C22⋊C4×C79—41no—
83C22⋊C4×C79—42no—
84C22⋊C4×C79—42no—
85C22⋊C4×C79—41no—
86C23×C79—21no—
87C23×C79—21no—
88C23×C79—42no—
89C23×C79#12011yescharacteristic
90C23×C79C2×D811yescharacteristic
91C23×C79AGL(1,ℤ/8)11yescharacteristic
92C22×D79—41no—
93C22×D79—41no—
94C22×D79—41no—
95C22×D79—82no—
96Q8×C79—84no—
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—41no—
100C2×C4×C79—41no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79—42no—
104C2×C4×C79—44no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—41no—
108C2×C4×C79—42no—
109C2×C4×C79—42no—
110C2×C4×C79—81no—
111C22⋊D79—81no—
112C22⋊D79—81no—
113C22⋊D79—81no—
114C22⋊D79—81no—
115C2×Dic79—42no—
116C2×Dic79—41no—
117C2×Dic79—41no—
118C2×Dic79—41no—
119C2×Dic79—82no—
120D316—82no—
121C79⋊C8—82no—
122C22×C79—22no—
123C22×C79—21no—
124C22×C79—21no—
125C22×C79—41no—
126C22×C79—41no—
127C22×C79—42no—
128C22×C79—81no—
129C22×C79#44711yescharacteristic
130C22×C79#45311yescharacteristic
131C22×C79B(4,𝔽2)11yescharacteristic
132C2×D79—82no—
133C2×D79—81no—
134C2×D79—82no—
135C2×D79—161no—
136C4×C79—42no—
137C4×C79—42no—
138C4×C79—82no—
139C4×C79—81no—
140C4×C79—82no—
141Dic79—82no—
142Dic79—81no—
143#12095—791nomaximal, Sylow
144C2×C79—21no—
145C2×C79—21no—
146C2×C79—41no—
147C2×C79—41no—
148C2×C79#159511yescharacteristic
149C2×C79#220112yes—
150D79—161no—
151#2817D7911yescharacteristic, complemented
152#2022—791no—
153#1607—792no—
154#1344—792no—
155#1317—791no—
156C79#1209511yescharacteristic, complemented, Sylow
157#534—791no—
158C2×#123—1581no—
159C2×C22⋊Q8—21no—
160#480—1582no—
161C2×#99—1581no—
162#400—22no—
163#397—21no—
164#397C2×D7912yes—
165#395—22no—
166#381C2×D7911yescharacteristic
167#342—791no—
168#327—1582no—
169#323—792no—
170C22×Q8—21no—
171C22×D4—1581no—
172C23×C4C22⋊D7911yescharacteristic
173#123—3161no—
174#123—3162no—
175#123—3162no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178#121—1582no—
179#120—3161no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—3161no—
184C2×C4⋊C4C22⋊D7912yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—3161no—
187C2×C22⋊C4—1581no—
188C2×C22⋊C4C22⋊D7911yescharacteristic
189C2×C22⋊C4C22×D7911yescharacteristic
190#107—3162no—
191#102—3162no—
192#99—3162no—
193C22⋊C8—1582no—
194#95—22no—
195#95—21no—
196#95—22no—
197#95—22no—
198#95C22⋊D7912yes—
199C24—1581no—
200C24C2×C22⋊D7911yescharacteristic
201C2×Q8—42no—
202C2×Q8—81no—
203C2×Q8—81no—
204C2×D4—3161no—
205C2×D4—3161no—
206C2×D4—3161no—
207C2×D4—3161no—
208C2×D4—3161no—
209C2×D4—3161no—
210C2×D4—1582no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—3161no—
219C22×C4—1581no—
220C22×C4#1192960912yes—
221C22×C4C2×C22⋊D7912yes—
222C2×C8—3162no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—81no—
227C4⋊C4—3164no—
228C4⋊C4—1582no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—41no—
233C22⋊C4—3162no—
234C22⋊C4—3164no—
235C22⋊C4—3161no—
236C22⋊C4—1582no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—3161no—
241C23—3161no—
242C23—3161no—
243C23—6322no—
244C23#1457913011yescharacteristic
245C23C2×#1192960911yescharacteristic
246C23#1457918711yescharacteristic
247Q8—84no—
248D4—6322no—
249D4—6321no—
250D4—6321no—
251D4—6321no—
252D4—6321no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—81no—
267C2×C4—3162no—
268C2×C4—3161no—
269C2×C4—3161no—
270C2×C4—3161no—
271C2×C4—6322no—
272C8—6322no—
273C22—22no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—81no—
280C22—6322no—
281C22—6321no—
282C22—6322no—
283C22—12641no—
284C22#1909869711yescharacteristic
285C22#1909873111yescharacteristic
286C22#1909875011yescharacteristic
287C4—42no—
288C4—42no—
289C4—82no—
290C4—81no—
291C4—82no—
292C4—6322no—
293C4—6321no—
294C2—21no—
295C2—21no—
296C2—41no—
297C2—41no—
298C2—12641no—
299C2#2376231811yescharacteristic
300C2#2376541312yes—
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2817C2
socleC22×C79B(4,𝔽2)
prime radical#19099125C22
hypercenter#2817D79
hypocenterC79#12095
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
3. #23761400C2semidirect
4. #23761330C2semidirect
5. C79×#2817C2semidirect
151. #2817D79semidirect
156. C79#12095semidirect

Maximal subgroups

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

subgroupindex
#237645312
#237623852
#237614002
#237613302
C79×#28172
#1209579