Finite Group Databasebeta

← #34176837

Subgroups — #34176837

Order 20224.

426 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 103 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#23771767C211yesmaximal, characteristic
2#23763149C211yesmaximal, characteristic, complemented
3#23761222C212yesmaximal, complemented
4#23761220C212yesmaximal
5C79×#1831C211yesmaximal, characteristic, complemented
6C2×#14579155—21no—
7C2×#14579072—21no—
8#19098465C411yescharacteristic
9#19098465C411yescharacteristic
10#19098465C2211yescharacteristic
11#19098461C411yescharacteristic
12#19098461C411yescharacteristic
13#19098461C2211yescharacteristic
14#19098454—21no—
15#19098454—21no—
16#19098454—21no—
17#19098318—21no—
18#19098318C2212yes—
19C2×C4⋊C8×C79—21no—
20C43×C79C2211yescharacteristic
21#336×C79—21no—
22#336×C79C2212yes—
23C22×D316C2×C411yescharacteristic
24C22×C79⋊C8—22no—
25#14579155—44no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×(C2×D79)⋊C4—21no—
33C2×(C2×D79)⋊C4—21no—
34C2×(C2×D79)⋊C4—21no—
35C2×(C2×D79)⋊C4—21no—
36C2×(C2×D79)⋊C4—21no—
37C2×(C2×D79)⋊C4—21no—
38#14579072—24no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C22×C8×C79—22no—
43C2×C42×C79C2×C411yescharacteristic
44C2×C42×C79C2×C411yescharacteristic
45C2×C42×C79D411yescharacteristic
46C2×C42×C79D411yescharacteristic
47C2×C42×C79D411yescharacteristic
48C2×C42×C79D411yescharacteristic
49C2×C42×C79C2311yescharacteristic
50C4⋊C8×C79—42no—
51C23×D79—21no—
52C2×D316—22no—
53C2×D316—41no—
54C2×D316—41no—
55C22×Dic79—21no—
56C22×Dic79—21no—
57C22×Dic79—21no—
58C2×C79⋊C8—24no—
59C2×C79⋊C8—42no—
60C2×C79⋊C8—42no—
61C4⋊Dic79—22no—
62C4⋊Dic79—41no—
63C4⋊Dic79—41no—
64Dic79⋊C4—42no—
65Dic79⋊C4—42no—
66Dic79⋊C4—42no—
67(C2×D79)⋊C4—42no—
68(C2×D79)⋊C4—42no—
69(C2×D79)⋊C4—42no—
70(C2×D79)⋊C4—42no—
71(C2×D79)⋊C4—41no—
72(C2×D79)⋊C4—42no—
73(C2×D79)⋊C4—41no—
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22×C411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80C22×C4×C79C2×D411yescharacteristic
81C2×C8×C79—42no—
82C2×C8×C79—42no—
83C2×C8×C79—42no—
84C42×C79—22no—
85C42×C79—22no—
86C42×C79—21no—
87C42×C79—22no—
88C42×C79—21no—
89C42×C79—41no—
90C42×C79—41no—
91C42×C79—41no—
92C23×C79C2×C22⋊C411yescharacteristic
93C22×D79—41no—
94C22×D79—41no—
95C22×D79—41no—
96C22×D79—41no—
97C22×D79—41no—
98C22×D79—41no—
99C22×D79—41no—
100C2×C4×C79—21no—
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—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79#10412yes—
118C2×C4×C79#13112yes—
119C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C79AGL(1,ℤ/8)12yes—
121C2×C4×C79#13711yescharacteristic
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131D316—82no—
132C8×C79—82no—
133C79⋊C8—44no—
134C22×C79#40811yescharacteristic
135C22×C79#41011yescharacteristic
136C22×C79#41311yescharacteristic
137C22×C79#41711yescharacteristic
138C22×C79#41811yescharacteristic
139C22×C79C2×#10411yescharacteristic
140C22×C79#42111yescharacteristic
141C2×D79—81no—
142C2×D79—81no—
143C2×D79—81no—
144C2×D79—81no—
145C2×D79—81no—
146C2×D79—81no—
147C2×D79—81no—
148C4×C79—22no—
149C4×C79—22no—
150C4×C79—22no—
151C4×C79—21no—
152C4×C79—21no—
153C4×C79—41no—
154C4×C79—41no—
155C4×C79—41no—
156Dic79—81no—
157Dic79—81no—
158Dic79—81no—
159#7706—791nomaximal, Sylow
160C2×C79#147311yescharacteristic
161C2×C79#148311yescharacteristic
162C2×C79#149611yescharacteristic
163C2×C79#150511yescharacteristic
164C2×C79#151211yescharacteristic
165C2×C79#152911yescharacteristic
166C2×C79#155711yescharacteristic
167D79—161no—
168#2860—791no—
169#1831D7911yescharacteristic, complemented
170#1745—791no—
171#1271—792no—
172#1270—792no—
173C79#770611yescharacteristic, complemented, Sylow
174C2×#126—1581no—
175C2×C4⋊C8—21no—
176C2×#97—1581no—
177#390—791no—
178#390—791no—
179#390—791no—
180#389—791no—
181#389—791no—
182#389—791no—
183#388—1581no—
184#388—1581no—
185#388—1581no—
186C43C2×D7911yescharacteristic
187#336—21no—
188#336—792no—
189#336—1581no—
190#336C2×D7912yes—
191C22×D4—791no—
192C22×C8—22no—
193C22×C8—1582no—
194#126—3164no—
195C2×C4⋊C4—791no—
196C2×C4⋊C4—791no—
197C2×C4⋊C4—791no—
198C2×C4⋊C4—1581no—
199C2×C4⋊C4—1581no—
200C2×C4⋊C4—1581no—
201C2×C22⋊C4—1581no—
202C2×C22⋊C4—1581no—
203C2×C22⋊C4—1581no—
204C2×C22⋊C4—1581no—
205C2×C22⋊C4—1581no—
206C2×C22⋊C4—1581no—
207C2×C42C4×D7911yescharacteristic
208C2×C42C4×D7911yescharacteristic
209C2×C42D31611yescharacteristic
210C2×C42D31611yescharacteristic
211C2×C42C22⋊D7911yescharacteristic
212C2×C42C22⋊D7911yescharacteristic
213C2×C42C22×D7911yescharacteristic
214C4⋊C8—42no—
215#97—1584no—
216#95—1581no—
217#95—1581no—
218#95—1581no—
219C24—1581no—
220C2×D4—3161no—
221C2×D4—3161no—
222C2×D4—1582no—
223C22×C4—1581no—
224C22×C4—1581no—
225C22×C4—1581no—
226C22×C4(C2×D79)⋊C411yescharacteristic
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
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—3162no—
237C2×C8—3162no—
238C2×C8—1584no—
239C4⋊C4—3161no—
240C4⋊C4—3161no—
241C4⋊C4—3162no—
242C4⋊C4—3162no—
243C4⋊C4—3162no—
244C4⋊C4—1582no—
245C22⋊C4—3162no—
246C22⋊C4—3162no—
247C22⋊C4—3162no—
248C22⋊C4—3162no—
249C22⋊C4—3161no—
250C22⋊C4—3162no—
251C22⋊C4—3161no—
252C42—22no—
253C42—22no—
254C42—21no—
255C42—22no—
256C42—21no—
257C42—41no—
258C42—41no—
259C42—41no—
260C23—3161no—
261C23—3161no—
262C23—3161no—
263C23—3161no—
264C23—3161no—
265C23—3161no—
266C23—3161no—
267C23C2×(C2×D79)⋊C411yescharacteristic
268D4—6322no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4—3161no—
289C2×C4—3161no—
290C2×C4—3161no—
291C2×C4—3161no—
292C2×C4—3161no—
293C2×C4—3161no—
294C2×C4—3161no—
295C2×C4#1457908712yes—
296C2×C4#1457916712yes—
297C2×C4#1457918711yescharacteristic
298C2×C4#1457918912yes—
299C2×C4#1457919211yescharacteristic
300C8—82no—
301C8—3164no—
302C22—6321no—
303C22—6321no—
304C22—6321no—
305C22—6321no—
306C22—6321no—
307C22—6321no—
308C22—6321no—
309C22#1909852911yescharacteristic
310C22#1909853611yescharacteristic
311C22#1909854511yescharacteristic
312C22#1909856211yescharacteristic
313C22#1909856911yescharacteristic
314C22C2×#1457908711yescharacteristic
315C22#1909858111yescharacteristic
316C4—21no—
317C4—21no—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—6321no—
325C4—6321no—
326C4—6321no—
327C2—12641no—
328C2#2376176711yescharacteristic
329C2#2376181111yescharacteristic
330C2#2376187011yescharacteristic
331C2#2376190911yescharacteristic
332C2#2376193611yescharacteristic
333C2#2376200811yescharacteristic
334C2#2376212011yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1831C2
socleC23×C79C2×C22⋊C4
prime radicalC22×D316C2×C4
cosocleC2×C42×C79C23
hypercenter#1831D79
hypocenterC79#7706
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C43 ◁ #1831

Complements

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

NH ≅ G/Ntype
2. #23763149C2semidirect
3. #23761222C2semidirect
5. C79×#1831C2semidirect
169. #1831D79semidirect
173. C79#7706semidirect

Maximal subgroups

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

subgroupindex
#237717672
#237631492
#237612222
#237612202
C79×#18312
#770679