Finite Group Databasebeta

← #34183263

Subgroups — #34183263

Order 20224.

756 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#19098483C211yesmaximal, characteristic
2#23763334C212yesmaximal
3C2×#19098289C211yesmaximal, characteristic
4#23761278C212yesmaximal
5C79×#2313C211yesmaximal, characteristic
6C23×Dic158C2211yescharacteristic
7C22×Dic79⋊C4—21no—
8C2×#14579084—24no—
9C2×C22⋊(C79⋊C8)C2212yes—
10#19098483—22no—
11#19098483C414yescomplemented
12C2×C79⋊#95—21no—
13C2×C79⋊#95C2211yescharacteristic
14#19098289—22no—
15C22×C4⋊C4×C79C2211yescharacteristic
16#387×C79—24no—
17#381×C79—21no—
18#380×C79C2212yes—
19C4×C22⋊C4×C79—21no—
20C22×Dic158—21no—
21C22×Dic158—22no—
22C22×Dic158—22no—
23C22×Dic158—44no—
24C22×Dic158C2×C412yes—
25C23×Dic79—21no—
26C23×Dic79—21no—
27C22×C79⋊C8—22no—
28C2×Dic79⋊C4—24no—
29C2×Dic79⋊C4—44no—
30#14579084—48no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—22no—
33C79⋊#95—24no—
34C79⋊#95—21no—
35C79⋊#95C2×C412yes—
36C23×C4×C79D411yescharacteristic
37C23×C4×C79D411yescharacteristic
38C23×C4×C79C2311yescharacteristic
39C2×C4⋊C4×C79—22no—
40C2×C4⋊C4×C79—22no—
41C2×C4⋊C4×C79C2×C412yes—
42C2×C4⋊C4×C79D414yes—
43C2×C4⋊C4×C79D412yes—
44C2×C22⋊C4×C79—22no—
45C2×C42×C79—22no—
46#95×C79—22no—
47#95×C79—24no—
48#95×C79—22no—
49C2×Dic158—28no—
50C2×Dic158—24no—
51C2×Dic158—44no—
52C2×Dic158—44no—
53C2×Dic158—48no—
54C2×Dic158—44no—
55C2×Dic158—42no—
56C22×Dic79—24no—
57C22×Dic79—24no—
58C22×Dic79—22no—
59C22×Dic79—22no—
60C22×Dic79—42no—
61C22×Dic79—42no—
62C22×Dic79—41no—
63C22×Dic79—41no—
64C22×Dic79—42no—
65C2×C79⋊C8—44no—
66C2×C79⋊C8—42no—
67Dic79⋊C4—48no—
68C24×C79C2×D411yescharacteristic
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—22no—
73C22×C4×C79—22no—
74C22×C4×C79—21no—
75C22×C4×C79—24no—
76C22×C4×C79—42no—
77C22×C4×C79—42no—
78C22×C4×C79C22⋊C412yes—
79C22×C4×C79C22⋊C412yes—
80C22×C4×C79C22×C411yescharacteristic
81C22×C4×C79C2×D412yes—
82C22×C4×C79C2×D411yescharacteristic
83C22×C4×C79#4112yes—
84C4⋊C4×C79—24no—
85C4⋊C4×C79—24no—
86C4⋊C4×C79—44no—
87C22⋊C4×C79—42no—
88C22⋊C4×C79—42no—
89C42×C79—82no—
90C23×C79—22no—
91C23×C79—22no—
92C23×C79C2×C22⋊C411yescharacteristic
93C23×C79#12011yescharacteristic
94C23×C79#12311yescharacteristic
95C23×C79#13512yes—
96C23×C79#13712yes—
97C2×C4×C79—24no—
98C2×C4×C79—22no—
99C2×C4×C79—22no—
100C2×C4×C79—22no—
101C2×C4×C79—22no—
102C2×C4×C79—41no—
103C2×C4×C79—41no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—42no—
108C2×C4×C79—42no—
109C2×C4×C79—44no—
110C2×C4×C79—44no—
111C2×C4×C79—44no—
112C2×C4×C79—81no—
113C2×C4×C79C4×D412yes—
114C2×C4×C79#12112yes—
115C2×Dic79—24no—
116C2×Dic79—24no—
117C2×Dic79—42no—
118C2×Dic79—48no—
119C2×Dic79—42no—
120C2×Dic79—44no—
121C2×Dic79—42no—
122C2×Dic79—42no—
123C2×Dic79—44no—
124Dic158—28no—
125Dic158—44no—
126Dic158—48no—
127C79⋊C8—82no—
128C22×C79—24no—
129C22×C79—24no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79#38611yescharacteristic
135C22×C79#41611yescharacteristic
136C22×C79#41911yescharacteristic
137C22×C79#45212yes—
138C22×C79#45612yes—
139C22×C79B(4,𝔽2)12yes—
140C22×C79#48512yes—
141C4×C79—22no—
142C4×C79—22no—
143C4×C79—42no—
144C4×C79—42no—
145C4×C79—84no—
146Dic79—44no—
147Dic79—44no—
148#9014—791nomaximal, Sylow
149C2×C79—21no—
150C2×C79—21no—
151C2×C79—22no—
152C2×C79#160314yes—
153C2×C79#187311yescharacteristic
154C2×C79#188611yescharacteristic
155C2×C79#188911yescharacteristic
156C2×#395—791no—
157#2313D7911yescharacteristic
158#1787—792no—
159C2×#324—791no—
160#1294—792no—
161C79#901411yescharacteristic, complemented, Sylow
162C23×Q8—791no—
163C22×C4⋊C4—1581no—
164C22×C4⋊C4C2×D7911yescharacteristic
165C2×#103—1584no—
166C2×C22⋊C8—792no—
167#395—794no—
168#395—1582no—
169#387—24no—
170#381—21no—
171#380C2×D7912yes—
172C4×C22⋊C4—21no—
173C2×#95—791no—
174C2×#95—1581no—
175#324—1582no—
176C22×Q8—3164no—
177C22×Q8—792no—
178C22×Q8—1582no—
179C22×Q8—1582no—
180C22×Q8—1581no—
181C23×C4—1581no—
182C23×C4—1581no—
183C23×C4C22⋊D7911yescharacteristic
184C23×C4C22⋊D7911yescharacteristic
185C23×C4C22×D7911yescharacteristic
186C22×C8—1582no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—3164no—
190C2×C4⋊C4—1584no—
191C2×C4⋊C4C4×D7912yes—
192C2×C4⋊C4D31612yes—
193C2×C4⋊C4C22⋊D7914yes—
194C2×C22⋊C4—22no—
195C2×C42—22no—
196#103—3168no—
197C22⋊C8—1582no—
198C22⋊C8—1582no—
199#95—22no—
200#95—24no—
201#95—22no—
202#95—792no—
203#95—1584no—
204#95—1581no—
205C24C2×C22⋊D7911yescharacteristic
206C2×Q8—3164no—
207C2×Q8—3168no—
208C2×Q8—3164no—
209C2×Q8—3164no—
210C2×Q8—3162no—
211C2×Q8—1588no—
212C2×Q8—1584no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—3162no—
223C22×C4—3162no—
224C22×C4—3161no—
225C22×C4—3161no—
226C22×C4—3162no—
227C22×C4—1584no—
228C22×C4—1584no—
229C22×C4—1582no—
230C22×C4—1582no—
231C22×C4(C2×D79)⋊C412yes—
232C22×C4(C2×D79)⋊C412yes—
233C22×C4C2×C4×D7911yescharacteristic
234C22×C4C2×C22⋊D7912yes—
235C22×C4C2×D31611yescharacteristic
236C22×C4#1192962412yes—
237C2×C8—3164no—
238C2×C8—3162no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—3168no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C42—82no—
246C23—22no—
247C23—22no—
248C23C2×(C2×D79)⋊C411yescharacteristic
249C23#1457913011yescharacteristic
250C23#1457914411yescharacteristic
251C23#1457918112yes—
252C23#1457919112yes—
253Q8—3168no—
254Q8—3164no—
255Q8—1588no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—81no—
272C2×C4—3162no—
273C2×C4—3162no—
274C2×C4—3164no—
275C2×C4—3168no—
276C2×C4—3162no—
277C2×C4—3164no—
278C2×C4—3162no—
279C2×C4—1584no—
280C2×C4—1584no—
281C2×C4C4×C22⋊D7912yes—
282C2×C4#1457913512yes—
283C8—6322no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—24no—
289C22—24no—
290C22#1909844211yescharacteristic
291C22#1909855711yescharacteristic
292C22#1909857111yescharacteristic
293C22#1909872512yes—
294C22#1909874612yes—
295C22#1909875012yes—
296C22#1909890412yes—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—42no—
301C4—84no—
302C4—3164no—
303C4—3164no—
304C2—21no—
305C2—22no—
306C2—21no—
307C2#2376236314yes—
308C2#2376371211yescharacteristic
309C2#2376379311yescharacteristic
310C2#2376380811yescharacteristic
subgroupG/H
centerC23#14579130
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2313C2
socleC23×C79#120
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#2313D79
hypocenterC79#9014
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
11. #19098483C4semidirect
161. C79#9014semidirect

Maximal subgroups

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

subgroupindex
C2×#190984832
#237633342
C2×#190982892
#237612782
C79×#23132
#901479