Finite Group Databasebeta

← #34182947

Subgroups — #34182947

Order 20224.

542 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 119 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#23772485C211yesmaximal, characteristic, complemented
2C4×#14579150C211yesmaximal, characteristic
3#23761792C212yesmaximal
4#23761232C212yesmaximal, complemented
5C79×#1461C211yesmaximal, characteristic, complemented
6C2×#14579150C411yescharacteristic
7C2×C4×D316C2211yescharacteristic
8#19098694—24no—
9#19098620—22no—
10#19098616C2212yes—
11#19098470C411yescharacteristic
12#19098464C411yescharacteristic
13#19098453—22no—
14#19098428C411yescharacteristic
15C4×(C2×D79)⋊C4—22no—
16C42×Dic79C2211yescharacteristic
17#19098292—24no—
18C2×C4×Q8×C79C2211yescharacteristic
19#431×C79C2212yes—
20#324×C79—22no—
21C22×D316C2×C411yescharacteristic
22C22×C4×D79—21no—
23#14579150—22no—
24#14579150—22no—
25C4×D316—22no—
26C4×D316—22no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×(C2×D79)⋊C4—22no—
29C2×(C2×D79)⋊C4—22no—
30C2×(C2×D79)⋊C4—21no—
31C2×C4×Dic79—21no—
32C2×C4×Dic79—21no—
33C2×C4×Dic79C2×C411yescharacteristic
34C2×C4×Dic79C2×C411yescharacteristic
35C4⋊(C79⋊C8)—24no—
36C4⋊(C79⋊C8)D414yes—
37C22⋊(C79⋊C8)—22no—
38#14579072—24no—
39C4×C79⋊C8—22no—
40C79⋊#95—22no—
41C79⋊#95—22no—
42C22×Q8×C79C2×C411yescharacteristic
43C4×Q8×C79—22no—
44C4×Q8×C79—22no—
45C2×C4⋊C4×C79C2×C411yescharacteristic
46C2×C4⋊C4×C79D412yes—
47C2×C42×C79D412yes—
48C2×C42×C79C2311yescharacteristic
49C4⋊C8×C79—22no—
50C22⋊C8×C79—22no—
51C4×C8×C79—22no—
52C23×D79—21no—
53C2×D316—22no—
54C2×D316—21no—
55C2×D316—22no—
56C2×D316M1612yes—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C2×C4×D79—44no—
61C2×C4×D79—41no—
62C2×C4×D79—41no—
63C2×C79⋊C8—24no—
64C4⋊Dic79—22no—
65(C2×D79)⋊C4—28no—
66(C2×D79)⋊C4—44no—
67(C2×D79)⋊C4—42no—
68C4×Dic79—21no—
69C4×Dic79—22no—
70C4×Dic79—42no—
71C4×Dic79—42no—
72C4×Dic79M1612yes—
73C2×Q8×C79—22no—
74C2×Q8×C79—21no—
75C2×Q8×C79—21no—
76C2×Q8×C79—22no—
77C22×C4×C79C22⋊C412yes—
78C22×C4×C79C22⋊C412yes—
79C22×C4×C79C22×C411yescharacteristic
80C22×C4×C79C2×D411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C2×C8×C79—42no—
83C4⋊C4×C79—22no—
84C4⋊C4×C79—24no—
85C42×C79—22no—
86C42×C79—22no—
87C42×C79C2×D412yes—
88C42×C79#4112yes—
89C23×C79C2×C22⋊C411yescharacteristic
90C22×D79—22no—
91C22×D79—42no—
92C22×D79—42no—
93C22×D79—41no—
94C22×D79—41no—
95Q8×C79—42no—
96Q8×C79—42no—
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—22no—
100C2×C4×C79—22no—
101C2×C4×C79—22no—
102C2×C4×C79—41no—
103C2×C4×C79—42no—
104C2×C4×C79C4×D412yes—
105C2×C4×C79#12011yescharacteristic
106C2×C4×C79#12112yes—
107C2×C4×C79#12311yescharacteristic
108C2×C4×C79C2×M1611yescharacteristic
109C2×C4×C79C2×M1611yescharacteristic
110C2×C4×C79#13112yes—
111C2×Dic79—22no—
112C2×Dic79—22no—
113C2×Dic79—41no—
114C2×Dic79—41no—
115C2×Dic79—41no—
116C2×Dic79—41no—
117C2×Dic79—42no—
118C2×Dic79—41no—
119D316—24no—
120D316—42no—
121C4×D79—82no—
122C4×D79—82no—
123C8×C79—82no—
124C79⋊C8—44no—
125C22×C79—21no—
126C22×C79—21no—
127C22×C79#38611yescharacteristic
128C22×C79#40711yescharacteristic
129C22×C79#40811yescharacteristic
130C2×D79—44no—
131C2×D79—42no—
132C2×D79—82no—
133C2×D79—81no—
134C2×D79—81no—
135C4×C79—22no—
136C4×C79—21no—
137C4×C79—42no—
138C4×C79—42no—
139C4×C79#43512yes—
140C4×C79#43612yes—
141C4×C79#45412yes—
142Dic79—42no—
143Dic79—42no—
144Dic79—81no—
145#8964—791nomaximal, Sylow
146C2×C79—21no—
147C2×C79—21no—
148C2×C79#187011yescharacteristic
149C2×C79#188411yescharacteristic
150C2×C79#188811yescharacteristic
151D79—82no—
152#2983—791no—
153C4×#124—791no—
154#1478—792no—
155#1461D7911yescharacteristic, complemented
156#1276—792no—
157C79#896411yescharacteristic, complemented, Sylow
158C2×#124—791no—
159C2×C4×Q8C2×D7911yescharacteristic
160C2×C4×D4—791no—
161#446—1584no—
162#432—1582no—
163#431—792no—
164#431C2×D7912yes—
165#391—791no—
166#390—791no—
167#388—1582no—
168#382—791no—
169C4×C22⋊C4—1582no—
170C43—791no—
171D4⋊C8—1584no—
172#324—22no—
173C22×Q8C4×D7911yescharacteristic
174C22×D4—791no—
175C23×C4—1581no—
176#124—1582no—
177#124—1582no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×D4—1582no—
181C4×D4—1582no—
182C2×C4⋊C4—791no—
183C2×C4⋊C4C4×D7911yescharacteristic
184C2×C4⋊C4C22⋊D7912yes—
185C2×C22⋊C4—1582no—
186C2×C22⋊C4—1581no—
187C2×C22⋊C4—1582no—
188C2×C42—791no—
189C2×C42—791no—
190C2×C42—1581no—
191C2×C42—1581no—
192C2×C42D31612yes—
193C2×C42C22×D7911yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—794no—
196C4⋊C8—1584no—
197C22⋊C8—22no—
198C22⋊C8—1582no—
199#97—1584no—
200C4×C8—22no—
201C4×C8—1582no—
202#95—1582no—
203#95—1582no—
204C24—1581no—
205C2×Q8—22no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×D4—792no—
210C2×D4—1582no—
211C2×D4—1581no—
212C2×D4—1582no—
213C22×C4—3164no—
214C22×C4—3161no—
215C22×C4—3161no—
216C22×C4—1581no—
217C22×C4—1581no—
218C22×C4—1581no—
219C22×C4(C2×D79)⋊C412yes—
220C22×C4(C2×D79)⋊C412yes—
221C22×C4C2×C4×D7911yescharacteristic
222C22×C4C2×C22⋊D7911yescharacteristic
223C22×C4C2×D31611yescharacteristic
224C2×C8—42no—
225C2×C8—1584no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—1582no—
229C22⋊C4—3164no—
230C22⋊C4—3162no—
231C22⋊C4—1588no—
232C42—22no—
233C42—22no—
234C42—3162no—
235C42—3162no—
236C42—792no—
237C42—1582no—
238C42—1581no—
239C42D4×D7912yes—
240C42#1192962312yes—
241C23—3162no—
242C23—3162no—
243C23—3161no—
244C23—3161no—
245C23—1582no—
246C23C2×(C2×D79)⋊C411yescharacteristic
247Q8—42no—
248Q8—42no—
249D4—3162no—
250D4—1584no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—3161no—
259C2×C4—3161no—
260C2×C4—3161no—
261C2×C4—3161no—
262C2×C4—3162no—
263C2×C4—3161no—
264C2×C4—6322no—
265C2×C4—6322no—
266C2×C4—1582no—
267C2×C4—1582no—
268C2×C4(C22⋊D79)⋊C412yes—
269C2×C4#1457913111yescharacteristic
270C2×C4#1457913312yes—
271C2×C4#1457914511yescharacteristic
272C2×C4M16×D7911yescharacteristic
273C2×C4M16×D7911yescharacteristic
274C2×C4#1457916712yes—
275C8—82no—
276C8—3164no—
277C22—21no—
278C22—21no—
279C22—3164no—
280C22—3162no—
281C22—6322no—
282C22—6321no—
283C22—6321no—
284C22#1909844011yescharacteristic
285C22#1909852611yescharacteristic
286C22#1909852911yescharacteristic
287C4—22no—
288C4—21no—
289C4—42no—
290C4—42no—
291C4—3162no—
292C4—3162no—
293C4—6321no—
294C4#1909863712yes—
295C4#1909863812yes—
296C4#1909873812yes—
297C2—21no—
298C2—21no—
299C2—6322no—
300C2#2376369511yescharacteristic
301C2#2376378111yescharacteristic
302C2#2376380111yescharacteristic
subgroupG/H
centerC2×C4#14579131
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1461C2
socleC22×C79#386
prime radicalC22×D316C2×C4
cosocleC2×C42×C79C23
hypercenter#1461D79
hypocenterC79#8964
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. #23772485C2semidirect
4. #23761232C2semidirect
5. C79×#1461C2semidirect
155. #1461D79semidirect
157. C79#8964semidirect

Maximal subgroups

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

subgroupindex
#237724852
C4×#145791502
#237617922
#237612322
C79×#14612
#896479