Finite Group Databasebeta

← #34176653

Subgroups — #34176653

Order 20224.

438 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771767C211yesmaximal, characteristic
2#23763517C211yesmaximal, characteristic, complemented
3#23761222C212yesmaximal, complemented
4#23761220C212yesmaximal
5C79×#1744C211yesmaximal, characteristic, complemented
6C2×#14579155—21no—
7C2×C4⋊(C79⋊C8)—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—42no—
26#14579155—42no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊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—
38C2×(C2×D79)⋊C4—21no—
39C4⋊(C79⋊C8)—42no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C22×C8×C79—22no—
44C2×C42×C79C2×C411yescharacteristic
45C2×C42×C79C2×C411yescharacteristic
46C2×C42×C79D411yescharacteristic
47C2×C42×C79D411yescharacteristic
48C2×C42×C79D411yescharacteristic
49C2×C42×C79D411yescharacteristic
50C2×C42×C79C2311yescharacteristic
51C4×C8×C79—24no—
52C23×D79—21no—
53C2×D316—22no—
54C2×D316—41no—
55C2×D316—41no—
56C22×Dic79—21no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C2×C79⋊C8—42no—
60C2×C79⋊C8—42no—
61C2×C79⋊C8—42no—
62C4⋊Dic79—22no—
63C4⋊Dic79—41no—
64C4⋊Dic79—41no—
65Dic79⋊C4—42no—
66Dic79⋊C4—42no—
67Dic79⋊C4—41no—
68Dic79⋊C4—41no—
69(C2×D79)⋊C4—42no—
70(C2×D79)⋊C4—42no—
71(C2×D79)⋊C4—42no—
72(C2×D79)⋊C4—42no—
73(C2×D79)⋊C4—41no—
74(C2×D79)⋊C4—41no—
75(C2×D79)⋊C4—42no—
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22×C411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83C2×C8×C79—24no—
84C2×C8×C79—42no—
85C2×C8×C79—42no—
86C42×C79—22no—
87C42×C79—22no—
88C42×C79—22no—
89C42×C79—41no—
90C42×C79—41no—
91C42×C79—41no—
92C42×C79D812yes—
93C42×C79SD1612yes—
94C23×C79C2×C22⋊C411yescharacteristic
95C22×D79—41no—
96C22×D79—41no—
97C22×D79—41no—
98C22×D79—41no—
99C22×D79—41no—
100C22×D79—41no—
101C22×D79—41no—
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#10212yes—
118C2×C4×C79#10212yes—
119C2×C4×C79#10412yes—
120C2×C4×C79#13112yes—
121C2×C4×C79C2×D811yescharacteristic
122C2×C4×C79C2×SD1611yescharacteristic
123C2×C4×C79#13512yes—
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—
133D316—82no—
134C8×C79—44no—
135C79⋊C8—82no—
136C22×C79#40811yescharacteristic
137C22×C79#41011yescharacteristic
138C22×C79#41311yescharacteristic
139C22×C79C2×#10211yescharacteristic
140C22×C79#41611yescharacteristic
141C22×C79C2×#10411yescharacteristic
142C22×C79#42111yescharacteristic
143C2×D79—81no—
144C2×D79—81no—
145C2×D79—81no—
146C2×D79—81no—
147C2×D79—81no—
148C2×D79—81no—
149C2×D79—81no—
150C4×C79—22no—
151C4×C79—22no—
152C4×C79—22no—
153C4×C79—22no—
154C4×C79—21no—
155C4×C79—21no—
156C4×C79—22no—
157C4×C79—41no—
158Dic79—81no—
159Dic79—81no—
160Dic79—81no—
161#7666—791nomaximal, Sylow
162C2×C79#147311yescharacteristic
163C2×C79#148211yescharacteristic
164C2×C79#149211yescharacteristic
165C2×C79#150011yescharacteristic
166C2×C79#151711yescharacteristic
167C2×C79#153211yescharacteristic
168C2×C79#154011yescharacteristic
169D79—161no—
170#2860—791no—
171#1831—791no—
172#1744D7911yescharacteristic, complemented
173#1271—792no—
174#1270—792no—
175C79#766611yescharacteristic, complemented, Sylow
176C2×#126—1581no—
177C2×C4⋊C8—1581no—
178C2×C4×C8—21no—
179#390—791no—
180#390—791no—
181#390—791no—
182#389—791no—
183#389—791no—
184#389—791no—
185#388—1581no—
186#388—1581no—
187#388—1581no—
188C43C2×D7911yescharacteristic
189#336—21no—
190#336—792no—
191#336—1581no—
192#336C2×D7912yes—
193C22×D4—791no—
194C22×C8—22no—
195C22×C8—1582no—
196#126—3162no—
197#126—3162no—
198C2×C4⋊C4—791no—
199C2×C4⋊C4—791no—
200C2×C4⋊C4—791no—
201C2×C4⋊C4—1581no—
202C2×C4⋊C4—1581no—
203C2×C4⋊C4—1581no—
204C2×C22⋊C4—1581no—
205C2×C22⋊C4—1581no—
206C2×C22⋊C4—1581no—
207C2×C22⋊C4—1581no—
208C2×C22⋊C4—1581no—
209C2×C22⋊C4—1581no—
210C2×C42C4×D7911yescharacteristic
211C2×C42C4×D7911yescharacteristic
212C2×C42D31611yescharacteristic
213C2×C42D31611yescharacteristic
214C2×C42C22⋊D7911yescharacteristic
215C2×C42C22⋊D7911yescharacteristic
216C2×C42C22×D7911yescharacteristic
217C4⋊C8—3162no—
218C4×C8—24no—
219#95—1581no—
220#95—1581no—
221#95—1581no—
222C24—1581no—
223C2×D4—3161no—
224C2×D4—3161no—
225C2×D4—1582no—
226C22×C4—1581no—
227C22×C4—1581no—
228C22×C4—1581no—
229C22×C4(C2×D79)⋊C411yescharacteristic
230C22×C4(C2×D79)⋊C411yescharacteristic
231C22×C4(C2×D79)⋊C411yescharacteristic
232C22×C4(C2×D79)⋊C411yescharacteristic
233C22×C4C2×C4×D7911yescharacteristic
234C22×C4C2×C22⋊D7911yescharacteristic
235C22×C4C2×D31611yescharacteristic
236C2×C8—24no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—3162no—
240C2×C8—3162no—
241C2×C8—3162no—
242C4⋊C4—3161no—
243C4⋊C4—3161no—
244C4⋊C4—3162no—
245C4⋊C4—3162no—
246C4⋊C4—3161no—
247C4⋊C4—3161no—
248C4⋊C4—1582no—
249C22⋊C4—3162no—
250C22⋊C4—3162no—
251C22⋊C4—3162no—
252C22⋊C4—3162no—
253C22⋊C4—3161no—
254C22⋊C4—3161no—
255C22⋊C4—3162no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42D63212yes—
263C42#1192961312yes—
264C23—3161no—
265C23—3161no—
266C23—3161no—
267C23—3161no—
268C23—3161no—
269C23—3161no—
270C23—3161no—
271C23C2×(C2×D79)⋊C411yescharacteristic
272D4—6322no—
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—21no—
287C2×C4—21no—
288C2×C4—3161no—
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#1457908312yes—
298C2×C4#1457908312yes—
299C2×C4#1457908912yes—
300C2×C4#1457916712yes—
301C2×C4C2×D63211yescharacteristic
302C2×C4C2×#1192961311yescharacteristic
303C2×C4#1457918112yes—
304C8—44no—
305C8—6322no—
306C22—6321no—
307C22—6321no—
308C22—6321no—
309C22—6321no—
310C22—6321no—
311C22—6321no—
312C22—6321no—
313C22#1909852911yescharacteristic
314C22#1909853611yescharacteristic
315C22#1909854511yescharacteristic
316C22C2×#1457908311yescharacteristic
317C22#1909855711yescharacteristic
318C22C2×#1457908911yescharacteristic
319C22#1909857911yescharacteristic
320C4—21no—
321C4—21no—
322C4—22no—
323C4—22no—
324C4—22no—
325C4—22no—
326C4—22no—
327C4—41no—
328C4—6321no—
329C4—6321no—
330C4—6321no—
331C2—12641no—
332C2#2376176711yescharacteristic
333C2#2376180611yescharacteristic
334C2#2376185211yescharacteristic
335C2#2376188411yescharacteristic
336C2#2376196111yescharacteristic
337C2#2376201811yescharacteristic
338C2#2376204711yescharacteristic
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#7666
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. #23763517C2semidirect
3. #23761222C2semidirect
5. C79×#1744C2semidirect
172. #1744D79semidirect
175. C79#7666semidirect

Maximal subgroups

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

subgroupindex
#237717672
#237635172
#237612222
#237612202
C79×#17442
#766679