Finite Group Databasebeta

← #34181803

Subgroups — #34181803

Order 20224.

438 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 99 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#23766018C211yesmaximal, characteristic
2#23761730C211yesmaximal, characteristic
3#23761307C212yesmaximal
4#23761305C212yesmaximal
5C79×#1463C211yesmaximal, characteristic
6C2×#14579120C411yescharacteristic
7C2×#14579120C411yescharacteristic
8#19098527C2212yes—
9#19098462—21no—
10#19098462—21no—
11#19098462—21no—
12#19098462—21no—
13#19098423C411yescharacteristic
14#19098423C411yescharacteristic
15#19098423C2211yescharacteristic
16#19098423C2211yescharacteristic
17#19098289—21no—
18#19098289—21no—
19C22×C4⋊C4×C79C2211yescharacteristic
20#407×C79C2212yes—
21#324×C79—21no—
22#324×C79—21no—
23C2×#11929608—22no—
24#14579120—42no—
25#14579120—42no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79—21no—
31C2×C4⋊Dic79—21no—
32C2×C22⋊Dic79C2×C411yescharacteristic
33C2×C22⋊Dic79C2×C411yescharacteristic
34C2×C22⋊Dic79C2×C411yescharacteristic
35C2×C22⋊Dic79C2×C411yescharacteristic
36C2×C4×Dic79—21no—
37C2×C4×Dic79—21no—
38C22⋊(C79⋊C8)—22no—
39C22⋊(C79⋊C8)—22no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C23×C4×C79C2×C411yescharacteristic
45C23×C4×C79C2×C411yescharacteristic
46C23×C4×C79C2311yescharacteristic
47C2×M16×C79—22no—
48C2×C4⋊C4×C79—22no—
49C2×C4⋊C4×C79—22no—
50C2×C4⋊C4×C79D411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C4⋊C4×C79D411yescharacteristic
54C22⋊C8×C79—22no—
55C22⋊C8×C79—22no—
56C22×Dic79—21no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60#11929608—82no—
61C2×C79⋊C8—42no—
62C4⋊Dic79—42no—
63C4⋊Dic79—42no—
64C4⋊Dic79—81no—
65C4⋊Dic79—81no—
66C4⋊Dic79—81no—
67C4⋊Dic79—81no—
68C22⋊Dic79—41no—
69C22⋊Dic79—41no—
70C22⋊Dic79—41no—
71C22⋊Dic79—41no—
72C4×Dic79—42no—
73C4×Dic79—42no—
74C24×C79C22×C411yescharacteristic
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—22no—
78C22×C4×C79—22no—
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79—81no—
82C22×C4×C79—81no—
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22⋊C411yescharacteristic
86C22×C4×C79C22⋊C411yescharacteristic
87C22×C4×C79C2×D411yescharacteristic
88C22×C4×C79C2×D411yescharacteristic
89C22×C4×C79#4112yes—
90C22×C4×C79#4112yes—
91M16×C79—82no—
92C2×C8×C79—42no—
93C4⋊C4×C79—22no—
94C4⋊C4×C79—22no—
95C4⋊C4×C79—22no—
96C4⋊C4×C79—22no—
97C23×C79—21no—
98C23×C79—21no—
99C23×C79—41no—
100C23×C79—41no—
101C23×C79C2×C22⋊C411yescharacteristic
102C23×C79#11711yescharacteristic
103C23×C79#11711yescharacteristic
104C2×C4×C79—24no—
105C2×C4×C79—24no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—81no—
119C2×C4×C79—81no—
120C2×C4×C79#10012yes—
121C2×C4×C79#10112yes—
122C2×C4×C79#12312yes—
123C2×C4×C79#12312yes—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C8×C79—82no—
137C79⋊C8—82no—
138C22×C79—21no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—21no—
146C22×C79—42no—
147C22×C79—42no—
148C22×C79#38111yescharacteristic
149C22×C79C2×#10011yescharacteristic
150C22×C79C2×#10111yescharacteristic
151C4×C79—22no—
152C4×C79—22no—
153C4×C79—22no—
154C4×C79—22no—
155C4×C79—41no—
156C4×C79—41no—
157Dic79—81no—
158Dic79—81no—
159Dic79—81no—
160Dic79—81no—
161#8757—791nomaximal, Sylow
162C2×C79—21no—
163C2×C79—21no—
164C2×C79—41no—
165C2×C79—41no—
166C2×C79#182111yescharacteristic
167C2×C79#182811yescharacteristic
168C2×C79#183011yescharacteristic
169#2316—791no—
170#1463—791no—
171#1463D7911yescharacteristic
172#1305—792no—
173#1305—792no—
174C79#875711yescharacteristic, complemented, Sylow
175C2×#117—791no—
176C2×#117—791no—
177C22×C4⋊C4C2×D7911yescharacteristic
178#407—792no—
179#407C2×D7912yes—
180#389—1581no—
181#389—1581no—
182#389—1581no—
183#389—1581no—
184#380—791no—
185#380—791no—
186#380—791no—
187#380—791no—
188#324—21no—
189#324—21no—
190#324—1581no—
191#324—1581no—
192C23×C4C4×D7911yescharacteristic
193C23×C4C4×D7911yescharacteristic
194C23×C4C22×D7911yescharacteristic
195C2×M16—22no—
196C2×M16—1582no—
197#117—3162no—
198#117—3162no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—1581no—
202C2×C4⋊C4—1581no—
203C2×C4⋊C4—1581no—
204C2×C4⋊C4—1581no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4—1581no—
207C2×C4⋊C4D31611yescharacteristic
208C2×C4⋊C4D31611yescharacteristic
209C2×C4⋊C4C22⋊D7911yescharacteristic
210C2×C4⋊C4C22⋊D7911yescharacteristic
211C2×C22⋊C4—791no—
212C2×C22⋊C4—791no—
213C2×C22⋊C4—791no—
214C2×C22⋊C4—791no—
215C2×C42—1581no—
216C2×C42—1581no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—1582no—
220C22⋊C8—1582no—
221#95—1581no—
222#95—1581no—
223#95—1581no—
224#95—1581no—
225C24C2×C4×D7911yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—1581no—
235C22×C4—1581no—
236C22×C4—1581no—
237C22×C4—1581no—
238C22×C4(C2×D79)⋊C411yescharacteristic
239C22×C4(C2×D79)⋊C411yescharacteristic
240C22×C4(C2×D79)⋊C411yescharacteristic
241C22×C4(C2×D79)⋊C411yescharacteristic
242C22×C4C2×C22⋊D7911yescharacteristic
243C22×C4C2×D31611yescharacteristic
244C22×C4#1192962312yes—
245C22×C4#1192962312yes—
246M16—82no—
247M16—6322no—
248C2×C8—42no—
249C2×C8—3162no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—3162no—
255C4⋊C4—3162no—
256C4⋊C4—6321no—
257C4⋊C4—6321no—
258C4⋊C4—6321no—
259C4⋊C4—6321no—
260C22⋊C4—3161no—
261C22⋊C4—3161no—
262C22⋊C4—3161no—
263C22⋊C4—3161no—
264C42—3162no—
265C42—3162no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23C2×(C2×D79)⋊C411yescharacteristic
271C23#1457911711yescharacteristic
272C23#1457911711yescharacteristic
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—3161no—
290C2×C4—3161no—
291C2×C4—3161no—
292C2×C4—3161no—
293C2×C4—3161no—
294C2×C4—3161no—
295C2×C4—3161no—
296C2×C4—3161no—
297C2×C4—3161no—
298C2×C4—3161no—
299C2×C4—3161no—
300C2×C4—3161no—
301C2×C4#1457907712yes—
302C2×C4#1457907912yes—
303C2×C4#1457914212yes—
304C2×C4#1457914512yes—
305C8—82no—
306C8—6322no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22#1909842411yescharacteristic
318C22C2×#1457907711yescharacteristic
319C22C2×#1457907911yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—41no—
326C4—6321no—
327C4—6321no—
328C4—6321no—
329C4—6321no—
330C2—21no—
331C2—21no—
332C2—41no—
333C2—41no—
334C2#2376347811yescharacteristic
335C2#2376350611yescharacteristic
336C2#2376351211yescharacteristic
subgroupG/H
centerC22#19098424
commutator, prime radicalC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1463C2
socleC22×C79#381
cosocleC23×C4×C79C23
hypercenter#1463D79
hypocenterC79#8757
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
174. C79#8757semidirect

Maximal subgroups

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

subgroupindex
#237660182
#237617302
#237613072
#237613052
C79×#14632
#875779