Finite Group Databasebeta

← #34176773

Subgroups — #34176773

Order 20224.

444 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 107 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#23761224C212yesmaximal, complemented
4#23761220C212yesmaximal
5C79×#1744C211yesmaximal, 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—24no—
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—42no—
72(C2×D79)⋊C4—42no—
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22×C411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80C2×C8×C79—24no—
81C2×C8×C79—42no—
82C2×C8×C79—42no—
83C42×C79—22no—
84C42×C79—22no—
85C42×C79—22no—
86C42×C79—22no—
87C42×C79—22no—
88C42×C79—41no—
89C42×C79—41no—
90C23×C79C2×C22⋊C411yescharacteristic
91C22×D79—41no—
92C22×D79—41no—
93C22×D79—41no—
94C22×D79—41no—
95C22×D79—41no—
96C22×D79—41no—
97C22×D79—41no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
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#10412yes—
115C2×C4×C79#10412yes—
116C2×C4×C79#13112yes—
117C2×C4×C79#13512yes—
118C2×C4×C79AGL(1,ℤ/8)12yes—
119C2×Dic79—41no—
120C2×Dic79—41no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128D316—82no—
129C8×C79—44no—
130C79⋊C8—44no—
131C22×C79#40811yescharacteristic
132C22×C79#41011yescharacteristic
133C22×C79#41311yescharacteristic
134C22×C79#41611yescharacteristic
135C22×C79#41811yescharacteristic
136C22×C79C2×#10411yescharacteristic
137C22×C79C2×#10411yescharacteristic
138C2×D79—81no—
139C2×D79—81no—
140C2×D79—81no—
141C2×D79—81no—
142C2×D79—81no—
143C2×D79—81no—
144C2×D79—81no—
145C4×C79—22no—
146C4×C79—22no—
147C4×C79—22no—
148C4×C79—22no—
149C4×C79—22no—
150C4×C79—41no—
151C4×C79—41no—
152Dic79—81no—
153Dic79—81no—
154Dic79—81no—
155#7692—791nomaximal, Sylow
156C2×C79#147311yescharacteristic
157C2×C79#148411yescharacteristic
158C2×C79#149411yescharacteristic
159C2×C79#150011yescharacteristic
160C2×C79#150511yescharacteristic
161C2×C79#152411yescharacteristic
162C2×C79#155311yescharacteristic
163D79—161no—
164#2860—791no—
165#1745—791no—
166#1744D7911yescharacteristic, complemented
167#1271—792no—
168#1270—792no—
169C79#769211yescharacteristic, complemented, Sylow
170C2×#126—1581no—
171C2×#97—1581no—
172C2×C4×C8—21no—
173#390—791no—
174#390—791no—
175#390—791no—
176#389—791no—
177#389—791no—
178#389—791no—
179#388—1581no—
180#388—1581no—
181#388—1581no—
182C43C2×D7911yescharacteristic
183#336—21no—
184#336—792no—
185#336—1581no—
186#336C2×D7912yes—
187C22×D4—791no—
188C22×C8—22no—
189C22×C8—1582no—
190#126—3164no—
191C2×C4⋊C4—791no—
192C2×C4⋊C4—791no—
193C2×C4⋊C4—791no—
194C2×C4⋊C4—1581no—
195C2×C4⋊C4—1581no—
196C2×C4⋊C4—1581no—
197C2×C22⋊C4—1581no—
198C2×C22⋊C4—1581no—
199C2×C22⋊C4—1581no—
200C2×C22⋊C4—1581no—
201C2×C22⋊C4—1581no—
202C2×C22⋊C4—1581no—
203C2×C42C4×D7911yescharacteristic
204C2×C42C4×D7911yescharacteristic
205C2×C42D31611yescharacteristic
206C2×C42D31611yescharacteristic
207C2×C42C22⋊D7911yescharacteristic
208C2×C42C22⋊D7911yescharacteristic
209C2×C42C22×D7911yescharacteristic
210#97—1584no—
211C4×C8—24no—
212#95—1581no—
213#95—1581no—
214#95—1581no—
215C24—1581no—
216C2×D4—3161no—
217C2×D4—3161no—
218C2×D4—1582no—
219C22×C4—1581no—
220C22×C4—1581no—
221C22×C4—1581no—
222C22×C4(C2×D79)⋊C411yescharacteristic
223C22×C4(C2×D79)⋊C411yescharacteristic
224C22×C4(C2×D79)⋊C411yescharacteristic
225C22×C4(C2×D79)⋊C411yescharacteristic
226C22×C4C2×C4×D7911yescharacteristic
227C22×C4C2×C22⋊D7911yescharacteristic
228C22×C4C2×D31611yescharacteristic
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—3162no—
233C2×C8—3162no—
234C2×C8—1584no—
235C4⋊C4—3161no—
236C4⋊C4—3161no—
237C4⋊C4—3162no—
238C4⋊C4—3162no—
239C4⋊C4—3162no—
240C4⋊C4—1582no—
241C22⋊C4—3162no—
242C22⋊C4—3162no—
243C22⋊C4—3162no—
244C22⋊C4—3162no—
245C22⋊C4—3162no—
246C22⋊C4—3162no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—41no—
253C42—41no—
254C23—3161no—
255C23—3161no—
256C23—3161no—
257C23—3161no—
258C23—3161no—
259C23—3161no—
260C23—3161no—
261C23C2×(C2×D79)⋊C411yescharacteristic
262D4—6322no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
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—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—3161no—
280C2×C4—3161no—
281C2×C4—3161no—
282C2×C4—3161no—
283C2×C4—3161no—
284C2×C4—3161no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4#1457908712yes—
289C2×C4#1457908912yes—
290C2×C4#1457916712yes—
291C2×C4#1457918312yes—
292C2×C4#1457918612yes—
293C8—44no—
294C8—3164no—
295C22—6321no—
296C22—6321no—
297C22—6321no—
298C22—6321no—
299C22—6321no—
300C22—6321no—
301C22—6321no—
302C22#1909852911yescharacteristic
303C22#1909853611yescharacteristic
304C22#1909854511yescharacteristic
305C22#1909856011yescharacteristic
306C22#1909856711yescharacteristic
307C22C2×#1457908711yescharacteristic
308C22C2×#1457908911yescharacteristic
309C4—22no—
310C4—22no—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—6321no—
317C4—6321no—
318C4—6321no—
319C2—12641no—
320C2#2376176711yescharacteristic
321C2#2376181611yescharacteristic
322C2#2376186111yescharacteristic
323C2#2376188711yescharacteristic
324C2#2376190711yescharacteristic
325C2#2376198611yescharacteristic
326C2#2376210311yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1744C2
socleC23×C79C2×C22⋊C4
prime radicalC22×D316C2×C4
cosocleC2×C42×C79C23
hypercenter#1744D79
hypocenterC79#7692
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C43 ◁ #1744

Complements

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

NH ≅ G/Ntype
2. #23763149C2semidirect
3. #23761224C2semidirect
5. C79×#1744C2semidirect
166. #1744D79semidirect
169. C79#7692semidirect

Maximal subgroups

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

subgroupindex
#237717672
#237631492
#237612242
#237612202
C79×#17442
#769279