Finite Group Databasebeta

← #34183264

Subgroups — #34183264

Order 20224.

756 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 135 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#23766003C211yesmaximal, characteristic
2#23763335C212yesmaximal
3C2×#19098289C211yesmaximal, characteristic
4#23761279C212yesmaximal
5C2×#395×C79C211yesmaximal, characteristic
6C2×#14579085—24no—
7C2×C22⋊(C79⋊C8)C2212yes—
8#19098449—24no—
9#19098426—21no—
10#19098423C2212yes—
11C4×C22⋊Dic79—21no—
12#19098289—22no—
13C23×Q8×C79C2211yescharacteristic
14C22×C4⋊C4×C79—21no—
15C22×C4⋊C4×C79C2211yescharacteristic
16#395×C79—22no—
17#395×C79C414yescomplemented
18C2×#95×C79—21no—
19C2×#95×C79C2211yescharacteristic
20C22×C79⋊C8—22no—
21C2×C4⋊Dic79D414yes—
22C2×C22⋊Dic79—22no—
23C2×C4×Dic79—22no—
24#14579085—48no—
25C22⋊(C79⋊C8)—22no—
26C22⋊(C79⋊C8)—22no—
27C79⋊#95—22no—
28C79⋊#95—24no—
29C79⋊#95—22no—
30C22×Q8×C79—21no—
31C22×Q8×C79—22no—
32C22×Q8×C79—22no—
33C22×Q8×C79—44no—
34C22×Q8×C79C2×C412yes—
35C23×C4×C79—21no—
36C23×C4×C79—21no—
37C23×C4×C79D411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C23×C4×C79C2311yescharacteristic
40C2×C4⋊C4×C79—22no—
41C2×C4⋊C4×C79—22no—
42C2×C4⋊C4×C79—24no—
43C2×C4⋊C4×C79—44no—
44C2×C4⋊C4×C79C2×C412yes—
45C2×C4⋊C4×C79D412yes—
46#95×C79—24no—
47#95×C79—21no—
48#95×C79C2×C412yes—
49C22×Dic79—24no—
50C2×C79⋊C8—44no—
51C2×C79⋊C8—42no—
52C4⋊Dic79—44no—
53C22⋊Dic79—42no—
54C22⋊Dic79—42no—
55C4×Dic79—82no—
56C24×C79C2×D411yescharacteristic
57C2×Q8×C79—28no—
58C2×Q8×C79—24no—
59C2×Q8×C79—44no—
60C2×Q8×C79—44no—
61C2×Q8×C79—44no—
62C2×Q8×C79—48no—
63C2×Q8×C79—42no—
64C22×C4×C79—24no—
65C22×C4×C79—24no—
66C22×C4×C79—22no—
67C22×C4×C79—22no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—22no—
72C22×C4×C79—22no—
73C22×C4×C79—21no—
74C22×C4×C79—42no—
75C22×C4×C79—42no—
76C22×C4×C79—42no—
77C22×C4×C79—41no—
78C22×C4×C79—41no—
79C22×C4×C79—42no—
80C22×C4×C79—42no—
81C22×C4×C79C22⋊C412yes—
82C22×C4×C79C22⋊C412yes—
83C22×C4×C79C22×C411yescharacteristic
84C22×C4×C79C2×D412yes—
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79#4112yes—
87C4⋊C4×C79—24no—
88C4⋊C4×C79—24no—
89C4⋊C4×C79—48no—
90C23×C79—22no—
91C23×C79—22no—
92C23×C79C2×C22⋊C411yescharacteristic
93C23×C79#12011yescharacteristic
94C23×C79#12311yescharacteristic
95C23×C79#13512yes—
96C23×C79#13712yes—
97Q8×C79—28no—
98Q8×C79—44no—
99Q8×C79—48no—
100C2×C4×C79—24no—
101C2×C4×C79—24no—
102C2×C4×C79—24no—
103C2×C4×C79—22no—
104C2×C4×C79—22no—
105C2×C4×C79—22no—
106C2×C4×C79—22no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—42no—
110C2×C4×C79—42no—
111C2×C4×C79—48no—
112C2×C4×C79—42no—
113C2×C4×C79—44no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—42no—
118C2×C4×C79—44no—
119C2×C4×C79—42no—
120C2×C4×C79—42no—
121C2×C4×C79—81no—
122C2×C4×C79C4×D412yes—
123C2×C4×C79#12112yes—
124C2×Dic79—44no—
125C2×Dic79—44no—
126C2×Dic79—44no—
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—44no—
144C4×C79—42no—
145C4×C79—44no—
146C4×C79—42no—
147Dic79—84no—
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×#395D7911yescharacteristic
157#2313—791no—
158#1787—792no—
159C2×#324—791no—
160#1294—792no—
161C79#901411yescharacteristic, complemented, Sylow
162C23×Q8C2×D7911yescharacteristic
163C22×C4⋊C4—21no—
164C22×C4⋊C4C2×D7911yescharacteristic
165C2×#103—1584no—
166C2×C22⋊C8—792no—
167#395—22no—
168#395Dic7914yescomplemented
169#387—1584no—
170#381—1581no—
171#380—792no—
172C4×C22⋊C4—1581no—
173C2×#95—21no—
174C2×#95C2×D7911yescharacteristic
175#324—1582no—
176C22×Q8—21no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—44no—
180C22×Q8C2×Dic7912yes—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C22⋊D7911yescharacteristic
184C23×C4C22⋊D7911yescharacteristic
185C23×C4C22×D7911yescharacteristic
186C22×C8—1582no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—794no—
192C2×C4⋊C4C2×Dic7912yes—
193C2×C4⋊C4C22⋊D7912yes—
194C2×C22⋊C4—1582no—
195C2×C42—1582no—
196#103—3168no—
197C22⋊C8—1582no—
198C22⋊C8—1582no—
199#95—24no—
200#95—21no—
201#95—1582no—
202#95—1584no—
203#95—1582no—
204#95C2×Dic7912yes—
205C24C2×C22⋊D7911yescharacteristic
206C2×Q8—28no—
207C2×Q8—24no—
208C2×Q8—44no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—48no—
212C2×Q8—42no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—1584no—
231C22×C4C22⋊Dic7912yes—
232C22×C4C22⋊Dic7912yes—
233C22×C4C22×Dic7911yescharacteristic
234C22×C4D4×D7912yes—
235C22×C4C2×C22⋊D7911yescharacteristic
236C22×C4#1192962312yes—
237C2×C8—3164no—
238C2×C8—3162no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—48no—
242C4⋊C4—3164no—
243C22⋊C4—3162no—
244C22⋊C4—3162no—
245C42—6322no—
246C23—22no—
247C23—22no—
248C23C2×C22⋊Dic7911yescharacteristic
249C23#1457912911yescharacteristic
250C23#1457914211yescharacteristic
251C23#1457918112yes—
252C23#1457919212yes—
253Q8—28no—
254Q8—48no—
255Q8—44no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—48no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—3164no—
279C2×C4—3164no—
280C2×C4—3164no—
281C2×C4D4×Dic7912yes—
282C2×C4#1457913312yes—
283C8—6322no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—24no—
289C22—24no—
290C22#1909844311yescharacteristic
291C22#1909855911yescharacteristic
292C22#1909857211yescharacteristic
293C22#1909872312yes—
294C22#1909874512yes—
295C22#1909874912yes—
296C22#1909890112yes—
297C4—22no—
298C4—22no—
299C4—44no—
300C4—42no—
301C4—44no—
302C4—42no—
303C4—6324no—
304C2—21no—
305C2—22no—
306C2—21no—
307C2#2376236114yes—
308C2#2376371311yescharacteristic
309C2#2376379411yescharacteristic
310C2#2376380911yescharacteristic
subgroupG/H
centerC23#14579129
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#395×C79C2
socleC23×C79#120
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenterC2×#395D79
hypocenterC79#9014
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
17. #395×C79C4semidirect
161. C79#9014semidirect
168. #395Dic79semidirect

Maximal subgroups

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

subgroupindex
#237660032
#237633352
C2×#190982892
#237612792
C2×#395×C792
#901479