Finite Group Databasebeta

← #34198970

Subgroups — #34198970

Order 20224.

488 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 79 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#23769150C211yesmaximal, characteristic
2#23763483C211yesmaximal, characteristic, complemented
3#23761404C214yesmaximal, complemented
4C79×#1464C211yesmaximal, characteristic, complemented
5C2×#14579158C411yescharacteristic
6C2×#14579135—21no—
7#19098527—21no—
8#19098480—21no—
9#19098480C2212yes—
10#19098472—21no—
11#19098472C412yes—
12#19098465—21no—
13#19098429C411yescharacteristic
14#19098332—21no—
15#19098332C2212yes—
16C2×#117×C79C2211yescharacteristic
17#432×C79—21no—
18#324×C79C2212yes—
19#323×C79—21no—
20C22×D316—21no—
21C22×D316—21no—
22C22×D316C2×C412yes—
23C22×C22⋊D79—22no—
24C2×#11929608—22no—
25#14579158—41no—
26#14579158—41no—
27#14579135—44no—
28C2×C4⋊Dic79—21no—
29C2×(C2×D79)⋊C4—22no—
30C2×(C2×D79)⋊C4—22no—
31C2×(C2×D79)⋊C4—22no—
32C22⋊(C79⋊C8)—41no—
33C22⋊(C79⋊C8)—41no—
34C79⋊#95—21no—
35C79⋊#95C2×C412yes—
36C23×C4×C79D411yescharacteristic
37#117×C79—41no—
38#117×C79—41no—
39C2×C4⋊C4×C79D411yescharacteristic
40C2×C4⋊C4×C79C2311yescharacteristic
41C2×C22⋊C4×C79C2×C411yescharacteristic
42C2×C22⋊C4×C79D411yescharacteristic
43C2×C42×C79C2×C411yescharacteristic
44C2×C42×C79D411yescharacteristic
45C4⋊C8×C79—41no—
46C22⋊C8×C79—22no—
47#97×C79—41no—
48C23×D79—22no—
49C2×D316—42no—
50C2×D316—42no—
51C2×D316—42no—
52C2×D316—42no—
53C2×D316—41no—
54C2×D316—41no—
55C2×D316—82no—
56C2×D316—82no—
57C2×C22⋊D79—44no—
58C2×C22⋊D79—82no—
59C2×C22⋊D79—82no—
60C22×Dic79—22no—
61#11929608—42no—
62#11929608—42no—
63C2×C79⋊C8—42no—
64C4⋊Dic79—42no—
65(C2×D79)⋊C4—44no—
66(C2×D79)⋊C4—82no—
67(C2×D79)⋊C4—82no—
68C24×C79C22⋊C411yescharacteristic
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—22no—
72C22×C4×C79—41no—
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22×C411yescharacteristic
77C22×C4×C79C2×D411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79C2×C8×C79—42no—
80C4⋊C4×C79—41no—
81C4⋊C4×C79—41no—
82C22⋊C4×C79—41no—
83C22⋊C4×C79—41no—
84C42×C79—21no—
85C42×C79—21no—
86C42×C79—21no—
87C42×C79—21no—
88C23×C79—21no—
89C23×C79—21no—
90C23×C79—21no—
91C23×C79—41no—
92C23×C79—41no—
93C23×C79C2×C22⋊C411yescharacteristic
94C22×D79—42no—
95C22×D79—42no—
96C22×D79—42no—
97C22×D79—42no—
98C22×D79—42no—
99C22×D79—42no—
100C22×D79—42no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—42no—
110C2×C4×C79—42no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79#9911yescharacteristic
116C2×C4×C79#9911yescharacteristic
117C2×C4×C79AGL(1,ℤ/8)12yes—
118C22⋊D79—88no—
119C2×Dic79—42no—
120C2×Dic79—42no—
121C2×Dic79—42no—
122D316—82no—
123D316—82no—
124D316—84no—
125D316—84no—
126C8×C79—82no—
127C79⋊C8—82no—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79—42no—
131C22×C79—41no—
132C22×C79—41no—
133C22×C79—41no—
134C22×C79—81no—
135C22×C79C2×#9911yescharacteristic
136C22×C79#41811yescharacteristic
137C22×C79#42111yescharacteristic
138C2×D79—82no—
139C2×D79—82no—
140C2×D79—82no—
141C2×D79—82no—
142C2×D79—82no—
143C2×D79—82no—
144C2×D79—82no—
145C4×C79—21no—
146C4×C79—21no—
147C4×C79—22no—
148C4×C79—41no—
149C4×C79—41no—
150C4×C79—41no—
151C4×C79—41no—
152Dic79—82no—
153#11806—791nomaximal, Sylow
154C2×C79—21no—
155C2×C79—21no—
156C2×C79—81no—
157C2×C79#151011yescharacteristic
158C2×C79#212211yescharacteristic
159C2×C79#213411yescharacteristic
160D79—162no—
161#2619—791no—
162#1822—791no—
163#1464D7911yescharacteristic, complemented
164#1345—794no—
165C79#1180611yescharacteristic, complemented, Sylow
166C2×#127—791no—
167C2×#121—1581no—
168C2×#117C2×D7911yescharacteristic
169#432—21no—
170#407—1581no—
171#394—792no—
172#394—1581no—
173#392—792no—
174#392—1581no—
175#390—1581no—
176#382—791no—
177#343—792no—
178#343—1581no—
179#324C2×D7912yes—
180#323—21no—
181C22×D4—792no—
182C22×D4—1582no—
183C22×D4—1581no—
184C22×D4—1581no—
185C23×C4D31611yescharacteristic
186C2×M16—1582no—
187#127—3161no—
188#127—3161no—
189#121—3164no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—1581no—
193C2×C4⋊C4D31611yescharacteristic
194C2×C4⋊C4C22×D7911yescharacteristic
195C2×C22⋊C4—1582no—
196C2×C22⋊C4—1582no—
197C2×C22⋊C4—1582no—
198C2×C22⋊C4C4×D7911yescharacteristic
199C2×C22⋊C4C22⋊D7911yescharacteristic
200C2×C42C4×D7911yescharacteristic
201C2×C42C22⋊D7911yescharacteristic
202C4⋊C8—41no—
203C22⋊C8—22no—
204C22⋊C8—3161no—
205C22⋊C8—3161no—
206#97—41no—
207#95—792no—
208#95—1581no—
209C24—1582no—
210C24(C2×D79)⋊C411yescharacteristic
211C2×D4—3162no—
212C2×D4—3162no—
213C2×D4—3162no—
214C2×D4—3162no—
215C2×D4—3164no—
216C2×D4—3161no—
217C2×D4—3161no—
218C2×D4—6322no—
219C2×D4—6322no—
220C2×D4—6322no—
221C2×D4—6322no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4—1582no—
227C22×C4(C2×D79)⋊C411yescharacteristic
228C22×C4(C2×D79)⋊C411yescharacteristic
229C22×C4(C2×D79)⋊C411yescharacteristic
230C22×C4C2×C4×D7911yescharacteristic
231C22×C4C2×C22⋊D7911yescharacteristic
232C22×C4C2×D31611yescharacteristic
233M16—3162no—
234M16—3162no—
235C2×C8—42no—
236C2×C8—3162no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—3162no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—3164no—
243C22⋊C4—6322no—
244C22⋊C4—6322no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—3162no—
255C23—3162no—
256C23—3162no—
257C23—3162no—
258C23—3162no—
259C23—3162no—
260C23—3162no—
261C23C2×(C2×D79)⋊C411yescharacteristic
262D4—6322no—
263D4—6322no—
264D4—6324no—
265D4—6324no—
266D4—6328no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—3162no—
282C2×C4—3162no—
283C2×C4—3162no—
284C2×C4#1457907511yescharacteristic
285C2×C4#1457907511yescharacteristic
286C2×C4#1457918912yes—
287C8—82no—
288C8—6322no—
289C22—21no—
290C22—21no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—6322no—
297C22—6322no—
298C22—6322no—
299C22—6322no—
300C22—6322no—
301C22—6322no—
302C22—6322no—
303C22C2×#1457907511yescharacteristic
304C22#1909856911yescharacteristic
305C22#1909857911yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—6322no—
314C2—21no—
315C2—21no—
316C2—81no—
317C2—12642no—
318C2#2376193011yescharacteristic
319C2#2376504911yescharacteristic
320C2#2376509111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1464C2
socleC22×C79C2×#99
prime radical#23769150C2
cosocleC2×C4⋊C4×C79C23
hypercenter#1464D79
hypocenterC79#11806
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #23763483C2semidirect
3. #23761404C2semidirect
4. C79×#1464C2semidirect
163. #1464D79semidirect
165. C79#11806semidirect

Maximal subgroups

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

subgroupindex
#237691502
#237634832
#237614042
C79×#14642
#1180679