Finite Group Databasebeta

← #34191261

Subgroups — #34191261

Order 20224.

640 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 109 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#23763334C212yesmaximal
2#23761670C211yesmaximal, characteristic
3#23761278C212yesmaximal
4#23761243C211yesmaximal, characteristic
5C79×#2646C211yesmaximal, characteristic
6C23×Dic158C2211yescharacteristic
7C2×#14579084—22no—
8C2×#14579084—22no—
9C2×C22⋊(C79⋊C8)C2212yes—
10C2×C79⋊#95—22no—
11C2×C79⋊#95C2211yescharacteristic
12#400×C79—22no—
13#398×C79—22no—
14#385×C79—22no—
15#380×C79C2212yes—
16C2×#95×C79C2211yescharacteristic
17C22×Dic158—21no—
18C22×Dic158—22no—
19C22×Dic158—22no—
20C22×Dic158—44no—
21C22×Dic158Q812yes—
22C23×Dic79—21no—
23C23×Dic79—21no—
24C22×C79⋊C8—22no—
25#14579084—44no—
26#14579084—44no—
27C22⋊(C79⋊C8)—22no—
28C22⋊(C79⋊C8)—22no—
29C79⋊#95—22no—
30C79⋊#95—44no—
31C23×C4×C79D412yes—
32C23×C4×C79C2311yescharacteristic
33C2×C4⋊C4×C79—22no—
34C2×C4⋊C4×C79—22no—
35C2×C4⋊C4×C79D412yes—
36C2×C4⋊C4×C79D412yes—
37C2×C22⋊C4×C79—22no—
38#95×C79—22no—
39#95×C79—22no—
40#95×C79—22no—
41#95×C79—22no—
42C2×Dic158—24no—
43C2×Dic158—24no—
44C2×Dic158—24no—
45C2×Dic158—42no—
46C2×Dic158—42no—
47C2×Dic158—44no—
48C2×Dic158—48no—
49C2×Dic158—44no—
50C2×Dic158—42no—
51C22×Dic79—24no—
52C22×Dic79—22no—
53C22×Dic79—22no—
54C22×Dic79—42no—
55C22×Dic79—42no—
56C22×Dic79—41no—
57C22×Dic79—41no—
58C22×Dic79—41no—
59C22×Dic79—41no—
60C22×Dic79—42no—
61C2×C79⋊C8—42no—
62C2×C79⋊C8—42no—
63C2×C79⋊C8—42no—
64C24×C79C2×D411yescharacteristic
65C22×C4×C79—22no—
66C22×C4×C79—22no—
67C22×C4×C79—22no—
68C22×C4×C79—22no—
69C22×C4×C79—22no—
70C22×C4×C79—42no—
71C22×C4×C79—42no—
72C22×C4×C79—42no—
73C22×C4×C79C2×D412yes—
74C22×C4×C79C2×Q811yescharacteristic
75C22×C4×C79#4112yes—
76C22×C4×C79#4111yescharacteristic
77C4⋊C4×C79—42no—
78C4⋊C4×C79—42no—
79C4⋊C4×C79—82no—
80C4⋊C4×C79—82no—
81C22⋊C4×C79—42no—
82C22⋊C4×C79—42no—
83C23×C79—21no—
84C23×C79—21no—
85C23×C79—22no—
86C23×C79#12011yescharacteristic
87C23×C79C22⋊Q811yescharacteristic
88C23×C79#12411yescharacteristic
89C23×C79#13512yes—
90C23×C79#13712yes—
91C2×C4×C79—22no—
92C2×C4×C79—22no—
93C2×C4×C79—41no—
94C2×C4×C79—42no—
95C2×C4×C79—41no—
96C2×C4×C79—44no—
97C2×C4×C79—42no—
98C2×C4×C79—42no—
99C2×C4×C79—42no—
100C2×C4×C79—42no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—82no—
107C2×C4×C79C22⋊Q812yes—
108C2×C4×C79#12412yes—
109C2×Dic79—24no—
110C2×Dic79—42no—
111C2×Dic79—44no—
112C2×Dic79—42no—
113C2×Dic79—44no—
114C2×Dic79—44no—
115C2×Dic79—42no—
116C2×Dic79—82no—
117C2×Dic79—81no—
118C2×Dic79—81no—
119Dic158—28no—
120Dic158—44no—
121Dic158—44no—
122Dic158—44no—
123C79⋊C8—82no—
124C22×C79—22no—
125C22×C79—22no—
126C22×C79—22no—
127C22×C79—22no—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79#39311yescharacteristic
133C22×C79#45212yes—
134C22×C79#45511yescharacteristic
135C22×C79#45611yescharacteristic
136C22×C79B(4,𝔽2)12yes—
137C22×C79#47912yes—
138C22×C79#48811yescharacteristic
139C22×C79#49111yescharacteristic
140C4×C79—22no—
141C4×C79—22no—
142C4×C79—82no—
143C4×C79—82no—
144C4×C79—82no—
145Dic79—44no—
146Dic79—82no—
147#10345—791nomaximal, Sylow
148C2×C79—21no—
149C2×C79—22no—
150C2×C79—21no—
151C2×C79#160312yes—
152C2×C79#160612yes—
153C2×C79#202611yescharacteristic
154C2×C79#202811yescharacteristic
155C2×C79#202911yescharacteristic
156#2646D7911yescharacteristic
157#1787—792no—
158#1446—791no—
159#1294—792no—
160#1280—791no—
161C79#1034511yescharacteristic, complemented, Sylow
162C23×Q8—791no—
163C2×#103—1582no—
164C2×#103—1582no—
165C2×C22⋊C8—792no—
166#400—22no—
167#398—22no—
168#385—22no—
169#380C2×D7912yes—
170C2×#95—791no—
171C2×#95—1582no—
172C2×#95C2×D7911yescharacteristic
173C22×Q8—3164no—
174C22×Q8—792no—
175C22×Q8—1582no—
176C22×Q8—1582no—
177C22×Q8—1581no—
178C23×C4—1581no—
179C23×C4—1581no—
180C23×C4C22⋊D7912yes—
181C23×C4C22×D7911yescharacteristic
182C22×C8—1582no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C22⋊D7912yes—
186C2×C4⋊C4C22⋊D7912yes—
187C2×C22⋊C4—22no—
188#103—3164no—
189#103—3164no—
190C22⋊C8—1582no—
191C22⋊C8—1582no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—3164no—
197#95—1582no—
198C24C2×C22⋊D7911yescharacteristic
199C2×Q8—3162no—
200C2×Q8—3162no—
201C2×Q8—3168no—
202C2×Q8—3164no—
203C2×Q8—3164no—
204C2×Q8—3162no—
205C2×Q8—1584no—
206C2×Q8—1584no—
207C2×Q8—1584no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—3162no—
217C22×C4—3161no—
218C22×C4—3161no—
219C22×C4—3162no—
220C22×C4—3161no—
221C22×C4—3161no—
222C22×C4—3162no—
223C22×C4—1582no—
224C22×C4—1584no—
225C22×C4—1582no—
226C22×C4C2×C22⋊D7912yes—
227C22×C4Q8×D7911yescharacteristic
228C22×C4#1192962512yes—
229C22×C4#1192962511yescharacteristic
230C2×C8—3162no—
231C2×C8—3162no—
232C2×C8—3162no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23#1457913011yescharacteristic
243C23#1457913711yescharacteristic
244C23#1457915011yescharacteristic
245C23#1457918112yes—
246C23#1457919112yes—
247Q8—3164no—
248Q8—3164no—
249Q8—3164no—
250Q8—1588no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—3164no—
268C2×C4—3162no—
269C2×C4—3162no—
270C2×C4—3164no—
271C2×C4—3162no—
272C2×C4—3164no—
273C2×C4—6321no—
274C2×C4—6322no—
275C2×C4—6321no—
276C2×C4—1584no—
277C2×C4#1457913712yes—
278C2×C4#1457915012yes—
279C8—6322no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22#1909847411yescharacteristic
289C22#1909872512yes—
290C22#1909874111yescharacteristic
291C22#1909874611yescharacteristic
292C22#1909875012yes—
293C22#1909886712yes—
294C22#1909892011yescharacteristic
295C22#1909893311yescharacteristic
296C4—22no—
297C4—22no—
298C4—82no—
299C4—82no—
300C4—82no—
301C4—3164no—
302C4—6322no—
303C2—21no—
304C2—21no—
305C2—22no—
306C2#2376236312yes—
307C2#2376238012yes—
308C2#2376454811yescharacteristic
309C2#2376455811yescharacteristic
310C2#2376456311yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC23×C4×C79C23
FrattiniC23×C4C22×D79
FittingC79×#2646C2
socleC23×C79#120
prime radicalC24×C79C2×D4
hypercenter#2646D79
hypocenterC79#10345
LCSG ▷ C23×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2646

Complements

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

NH ≅ G/Ntype
161. C79#10345semidirect

Maximal subgroups

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

subgroupindex
#237633342
#237616702
#237612782
#237612432
C79×#26462
#1034579