Finite Group Databasebeta

← #34186406

Subgroups — #34186406

Order 20224.

372 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764116C211yesmaximal, characteristic
2C4×#14579095C211yesmaximal, characteristic
3#23763265C211yesmaximal, characteristic
4#23761239C211yesmaximal, characteristic
5#23761239C211yesmaximal, characteristic
6C79⋊#1273C211yesmaximal, characteristic
7C79×#1277C211yesmaximal, characteristic
8C2×#14579095C411yescharacteristic
9C2×#14579095C411yescharacteristic
10C2×Dic79⋊C8—21no—
11C2×C4×C79⋊C8C2211yescharacteristic
12C2×C4×C79⋊C8C2211yescharacteristic
13C2×C8×Dic79—21no—
14C4×C4⋊Dic79C2211yescharacteristic
15C4×C4⋊Dic79C2211yescharacteristic
16#19098326C411yescharacteristic
17#19098326C411yescharacteristic
18#19098317—21no—
19#19098317—21no—
20C2×C4⋊C8×C79C2211yescharacteristic
21C2×C4⋊C8×C79C2211yescharacteristic
22C2×C4×C8×C79C2211yescharacteristic
23C22×C79⋊C8—21no—
24C22×C79⋊C8D411yescharacteristic
25C22×C79⋊C8D411yescharacteristic
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×C4×Dic79—21no—
31C2×C4×Dic79—21no—
32#14579095—22no—
33#14579095C814yescomplemented
34Dic79⋊C8—42no—
35C4×C79⋊C8—21no—
36C4×C79⋊C8—21no—
37C4×C79⋊C8—21no—
38C4×C79⋊C8—21no—
39C8×Dic79—42no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C22×C8×C79—21no—
43C22×C8×C79—21no—
44C22×C8×C79C2×C411yescharacteristic
45C22×C8×C79C2×C411yescharacteristic
46C2×C42×C79C2311yescharacteristic
47C4⋊C8×C79—21no—
48C4⋊C8×C79—21no—
49C4⋊C8×C79—21no—
50C4⋊C8×C79Q812yes—
51C4×C8×C79—21no—
52C4×C8×C79—21no—
53C22×Dic79—21no—
54C22×Dic79—21no—
55C22×Dic79—21no—
56C22×Dic79—21no—
57C2×C79⋊C8—22no—
58C2×C79⋊C8—21no—
59C2×C79⋊C8—21no—
60C2×C79⋊C8—21no—
61C2×C79⋊C8—21no—
62C2×C79⋊C8—21no—
63C2×C79⋊C8—41no—
64C2×C79⋊C8—41no—
65C2×C79⋊C8Q1612yes—
66C4⋊Dic79—21no—
67C4⋊Dic79—21no—
68C4⋊Dic79—21no—
69C4⋊Dic79—21no—
70C4⋊Dic79—21no—
71C4⋊Dic79C2×C812yes—
72C4⋊Dic79C2×C812yes—
73C4⋊Dic79M1612yes—
74C4×Dic79—41no—
75C4×Dic79—41no—
76C4×Dic79—41no—
77C4×Dic79—41no—
78C22×C4×C79C22×C411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80C22×C4×C79#4111yescharacteristic
81C2×C8×C79—21no—
82C2×C8×C79—21no—
83C2×C8×C79—22no—
84C2×C8×C79—21no—
85C2×C8×C79—21no—
86C2×C8×C79—21no—
87C2×C8×C79—41no—
88C2×C8×C79—41no—
89C2×C8×C79—41no—
90C2×C8×C79—41no—
91C2×C8×C79—41no—
92C2×C8×C79C2×C812yes—
93C42×C79C2×Q811yescharacteristic
94C42×C79#4111yescharacteristic
95C42×C79#4111yescharacteristic
96C42×C79#4111yescharacteristic
97C23×C79C4×D411yescharacteristic
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79#11711yescharacteristic
101C2×C4×C79C4×Q811yescharacteristic
102C2×C4×C79C22⋊Q811yescharacteristic
103C2×C4×C79#12311yescharacteristic
104C2×C4×C79#12511yescharacteristic
105C2×C4×C79#12611yescharacteristic
106C2×C4×C79C22×C811yescharacteristic
107C2×C4×C79C2×M1611yescharacteristic
108C2×C4×C79#13111yescharacteristic
109C2×C4×C79#13111yescharacteristic
110C2×C4×C79C2×Q1611yescharacteristic
111C2×C4×C79#13511yescharacteristic
112C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
113C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
114C2×Dic79—22no—
115C2×Dic79—22no—
116C2×Dic79—22no—
117C2×Dic79—41no—
118C2×Dic79—41no—
119C2×Dic79—41no—
120C2×Dic79—41no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C8×C79—22no—
127C8×C79—41no—
128C8×C79—42no—
129C8×C79—81no—
130C79⋊C8—22no—
131C79⋊C8—42no—
132C79⋊C8—41no—
133C22×C79#38711yescharacteristic
134C22×C79C8×D411yescharacteristic
135C22×C79#43511yescharacteristic
136C22×C79C4×Q1611yescharacteristic
137C22×C79#44211yescharacteristic
138C22×C79#44311yescharacteristic
139C22×C79#44411yescharacteristic
140C4×C79—21no—
141C4×C79—21no—
142C4×C79#43111yescharacteristic
143C4×C79#43311yescharacteristic
144C4×C79C8×Q811yescharacteristic
145C4×C79#44611yescharacteristic
146C4×C79#47711yescharacteristic
147C4×C79#47811yescharacteristic
148C4×C79#48211yescharacteristic
149C4×C79#48411yescharacteristic
150Dic79—42no—
151Dic79—42no—
152Dic79—42no—
153Dic79—81no—
154#9536—791nomaximal, Sylow
155C2×C79C8×Q1611yescharacteristic
156C2×C79#158011yescharacteristic
157C2×C79#158411yescharacteristic
158C2×C79#158611yescharacteristic
159C2×C79#191611yescharacteristic
160C2×C79#192111yescharacteristic
161C2×C79#192211yescharacteristic
162#1939—791no—
163C4×#107—791no—
164C8×C4⋊C4—791no—
165#1279—791no—
166#1279—791no—
167#1277D7911yescharacteristic
168#1273—791no—
169C79#953611yescharacteristic, complemented, Sylow
170C2×#107—791no—
171C2×#107—791no—
172C2×C4⋊C8—1581no—
173C2×C4⋊C8C2×D7911yescharacteristic
174C2×C4⋊C8C2×D7911yescharacteristic
175C2×C4×C8—791no—
176C2×C4×C8—791no—
177C2×C4×C8—1581no—
178C2×C4×C8C2×D7911yescharacteristic
179C4×C4⋊C4—791no—
180C4×C4⋊C4—791no—
181#340—791no—
182#340—791no—
183#336—1581no—
184#336—1581no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—791no—
188C22×C8—791no—
189C22×C8—1581no—
190C22×C8C4×D7911yescharacteristic
191C22×C8C4×D7911yescharacteristic
192C2×C4⋊C4—791no—
193C2×C4⋊C4—791no—
194C2×C4⋊C4—791no—
195C2×C4⋊C4—791no—
196C2×C42—1581no—
197C2×C42—1581no—
198C2×C42C22×D7911yescharacteristic
199#107—794no—
200#107—1582no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—3162no—
205C4⋊C8Dic15812yes—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—3162no—
209C4×C8—1581no—
210C4×C8—1581no—
211C4×C8—1581no—
212C4×C8—1581no—
213#95—1581no—
214#95—1581no—
215C22×C4—1581no—
216C22×C4—1581no—
217C22×C4—1581no—
218C22×C4—1581no—
219C22×C4C2×C4×D7911yescharacteristic
220C22×C4D4×D7911yescharacteristic
221C22×C4#1192962511yescharacteristic
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—22no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—3161no—
234C2×C8—3161no—
235C2×C8—792no—
236C2×C8—1582no—
237C2×C8—1581no—
238C2×C8—1581no—
239C2×C8—1581no—
240C2×C8—1581no—
241C2×C8—1581no—
242C2×C8C8×D7912yes—
243C4⋊C4—792no—
244C4⋊C4—792no—
245C4⋊C4—792no—
246C4⋊C4—1581no—
247C4⋊C4—1581no—
248C4⋊C4—1581no—
249C4⋊C4—1581no—
250C4⋊C4—1581no—
251C42—3161no—
252C42—3161no—
253C42—3161no—
254C42—3161no—
255C42C2×Dic15811yescharacteristic
256C42#1192962311yescharacteristic
257C42#1192962311yescharacteristic
258C42#1192962411yescharacteristic
259C23#1457912311yescharacteristic
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—3161no—
263C2×C4—3161no—
264C2×C4—3161no—
265C2×C4—3161no—
266C2×C4—3161no—
267C2×C4—3161no—
268C2×C4—3161no—
269C2×C4—3161no—
270C2×C4—3161no—
271C2×C4—1582no—
272C2×C4—1582no—
273C2×C4—1582no—
274C2×C4#1457911711yescharacteristic
275C2×C4C4×Dic15811yescharacteristic
276C2×C4#1457913811yescharacteristic
277C2×C4#1457914311yescharacteristic
278C2×C4#1457915211yescharacteristic
279C2×C4#1457915611yescharacteristic
280C2×C4C2×C8×D7911yescharacteristic
281C2×C4M16×D7911yescharacteristic
282C2×C4#1457916711yescharacteristic
283C2×C4#1457916911yescharacteristic
284C2×C4Q16×D7911yescharacteristic
285C2×C4#1457918011yescharacteristic
286C2×C4#1457918811yescharacteristic
287C2×C4#1457918811yescharacteristic
288C8—22no—
289C8—42no—
290C8—41no—
291C8—81no—
292C8—3162no—
293C8—3161no—
294C8—1582no—
295C22#1909844611yescharacteristic
296C22#1909862811yescharacteristic
297C22#1909863311yescharacteristic
298C22#1909865711yescharacteristic
299C22#1909867411yescharacteristic
300C22#1909867911yescharacteristic
301C22#1909868411yescharacteristic
302C4—21no—
303C4—21no—
304C4—3162no—
305C4—3162no—
306C4—3162no—
307C4—6321no—
308C4#1909861311yescharacteristic
309C4#1909862111yescharacteristic
310C4C8×Dic15811yescharacteristic
311C4#1909869111yescharacteristic
312C4#1909885611yescharacteristic
313C4#1909886111yescharacteristic
314C4#1909888511yescharacteristic
315C4#1909889711yescharacteristic
316C2#2376220611yescharacteristic
317C2#2376222311yescharacteristic
318C2#2376224911yescharacteristic
319C2#2376226311yescharacteristic
320C2#2376395611yescharacteristic
321C2#2376399111yescharacteristic
322C2#2376399811yescharacteristic
subgroupG/H
centerC22×C4D4×D79
commutatorC2×C4×C79C22×C8
FrattiniC2×C42C22×D79
FittingC79×#1277C2
socle, prime radicalC23×C79C4×D4
cosocleC2×C42×C79C23
hypercenter#1277D79
hypocenterC79#9536
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
33. #14579095C8semidirect
169. C79#9536semidirect

Maximal subgroups

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

subgroupindex
#237641162
C4×#145790952
#237632652
#237612392
C79⋊#12732
C79×#12772
#953679