Finite Group Databasebeta

← #34177690

Subgroups — #34177690

Order 20224.

626 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772033C211yesmaximal, characteristic
2C22×#14579072C211yesmaximal, characteristic
3#23761230C214yesmaximal, complemented
4C79×#2920C211yesmaximal, characteristic
5C23×C79⋊C8C412yes—
6#19098620—21no—
7#19098620—21no—
8#19098616C412yes—
9#19098616C2212yes—
10#19098587—21no—
11#19098527—21no—
12C4×#11929608C412yes—
13C2×#14579072—21no—
14C2×#14579072—21no—
15C2×#14579072—21no—
16C2×#14579072—21no—
17C2×#14579072C412yes—
18C2×#14579072C2212yes—
19C22×C42×C79C2211yescharacteristic
20#432×C79—21no—
21#432×C79—21no—
22#431×C79C412yes—
23#431×C79C2212yes—
24#423×C79—21no—
25#407×C79—21no—
26C4×M16×C79C412yes—
27C2×#11929608—22no—
28C22×C79⋊C8—22no—
29C22×C79⋊C8—22no—
30C22×C79⋊C8—28no—
31C22×C79⋊C8C2×C414yes—
32C4⋊(C79⋊C8)—22no—
33C4⋊(C79⋊C8)—22no—
34C22⋊(C79⋊C8)—22no—
35C22⋊(C79⋊C8)—22no—
36#14579072—22no—
37#14579072—22no—
38#14579072—22no—
39#14579072—22no—
40#14579072—22no—
41C4×C79⋊C8C2×C414yes—
42C23×C4×C79C2×C411yescharacteristic
43C23×C4×C79D411yescharacteristic
44C23×C4×C79Q811yescharacteristic
45C2×M16×C79—22no—
46C2×C42×C79C2×C411yescharacteristic
47C2×C42×C79C2×C411yescharacteristic
48C2×C42×C79C2×C411yescharacteristic
49C2×C42×C79C2×C411yescharacteristic
50C2×C42×C79C2×C411yescharacteristic
51C2×C42×C79D411yescharacteristic
52C2×C42×C79D411yescharacteristic
53C2×C42×C79D411yescharacteristic
54C2×C42×C79D411yescharacteristic
55C2×C42×C79D411yescharacteristic
56C2×C42×C79Q811yescharacteristic
57C2×C42×C79C2311yescharacteristic
58C4⋊C8×C79—22no—
59C4⋊C8×C79—22no—
60C22⋊C8×C79—22no—
61C22⋊C8×C79—22no—
62#97×C79—22no—
63C4×C8×C79C2×C414yes—
64#11929608—44no—
65C2×C79⋊C8—28no—
66C2×C79⋊C8—28no—
67C2×C79⋊C8—24no—
68C2×C79⋊C8—24no—
69C2×C79⋊C8—24no—
70C2×C79⋊C8M1618yescomplemented
71C24×C79C4⋊C411yescharacteristic
72C22×C4×C79—24no—
73C22×C4×C79—42no—
74C22×C4×C79—42no—
75C22×C4×C79C4211yescharacteristic
76C22×C4×C79C4211yescharacteristic
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C4⋊C411yescharacteristic
84C22×C4×C79C4⋊C411yescharacteristic
85C22×C4×C79C4⋊C411yescharacteristic
86C22×C4×C79C4⋊C411yescharacteristic
87C22×C4×C79C4⋊C411yescharacteristic
88C22×C4×C79C4⋊C411yescharacteristic
89C22×C4×C79C4⋊C411yescharacteristic
90C22×C4×C79C22×C411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×Q811yescharacteristic
93M16×C79—44no—
94C2×C8×C79—24no—
95C42×C79C4211yescharacteristic
96C42×C79C4211yescharacteristic
97C42×C79C22⋊C411yescharacteristic
98C42×C79C22⋊C411yescharacteristic
99C42×C79C22⋊C411yescharacteristic
100C42×C79C22⋊C411yescharacteristic
101C42×C79C22⋊C411yescharacteristic
102C42×C79C22⋊C411yescharacteristic
103C42×C79C4⋊C411yescharacteristic
104C42×C79C4⋊C411yescharacteristic
105C42×C79C4⋊C411yescharacteristic
106C42×C79C4⋊C411yescharacteristic
107C42×C79C22×C411yescharacteristic
108C42×C79C22×C411yescharacteristic
109C42×C79C2×D411yescharacteristic
110C42×C79C2×D411yescharacteristic
111C23×C79—22no—
112C23×C79—22no—
113C23×C79—22no—
114C23×C79#9511yescharacteristic
115C23×C79#9511yescharacteristic
116C23×C79C2×C4⋊C411yescharacteristic
117C2×C4×C79—22no—
118C2×C4×C79—22no—
119C2×C4×C79—22no—
120C2×C4×C79—22no—
121C2×C4×C79—22no—
122C2×C4×C79—22no—
123C2×C4×C79—22no—
124C2×C4×C79—22no—
125C2×C4×C79—22no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—41no—
133C2×C4×C79—41no—
134C2×C4×C79#9511yescharacteristic
135C2×C4×C79#9511yescharacteristic
136C2×C4×C79#9511yescharacteristic
137C2×C4×C79#9511yescharacteristic
138C2×C4×C79#9511yescharacteristic
139C2×C4×C79#9511yescharacteristic
140C2×C4×C79C2×C4211yescharacteristic
141C2×C4×C79C2×C22⋊C411yescharacteristic
142C2×C4×C79C2×C22⋊C411yescharacteristic
143C2×C4×C79C2×C22⋊C411yescharacteristic
144C2×C4×C79C2×C4⋊C411yescharacteristic
145C2×C4×C79C2×C4⋊C411yescharacteristic
146C2×C4×C79C2×M1614yes—
147C8×C79—44no—
148C79⋊C8—28no—
149C79⋊C8—44no—
150C22×C79—28no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—21no—
154C22×C79—21no—
155C22×C79—21no—
156C22×C79—21no—
157C22×C79H(3,ℤ/4)12yes—
158C22×C79#33812yes—
159C22×C79C2×#9511yescharacteristic
160C22×C79#42312yes—
161C4×C79—21no—
162C4×C79—21no—
163C4×C79—21no—
164C4×C79—21no—
165C4×C79—21no—
166C4×C79—21no—
167C4×C79—21no—
168C4×C79—21no—
169C4×C79—21no—
170C4×C79—21no—
171C4×C79C4×M1612yes—
172C4×C79#40712yes—
173#7861—791nomaximal, Sylow
174C2×C79—22no—
175C2×C79—22no—
176C2×C79—22no—
177C2×C79#172811yescharacteristic
178C2×C79C2×H(3,ℤ/4)11yescharacteristic
179C2×C79C2×#33811yescharacteristic
180#2920—791no—
181#2920D7911yescharacteristic
182C22×#97—791no—
183#1275—794no—
184C79#786111yescharacteristic, complemented, Sylow
185C23×C8—792no—
186C22×C42C2×D7911yescharacteristic
187#432—21no—
188#432—21no—
189#432—1581no—
190#432—1581no—
191#431—792no—
192#431—792no—
193#431Dic7912yes—
194#431C2×D7912yes—
195#423—21no—
196#423—1581no—
197#407—21no—
198#407—1581no—
199C4×M16—792no—
200C4×M16Dic7912yes—
201C2×#97—792no—
202C2×#97—792no—
203C2×#97—1581no—
204C2×#97—1581no—
205C2×#97—1581no—
206C2×#97—1581no—
207C23×C4Dic15811yescharacteristic
208C23×C4C4×D7911yescharacteristic
209C23×C4C22⋊D7911yescharacteristic
210C2×M16—22no—
211C2×M16—1582no—
212C22×C8—794no—
213C22×C8—1588no—
214C22×C8—1582no—
215C22×C8—1582no—
216C2×C42Dic15811yescharacteristic
217C2×C42C4×D7911yescharacteristic
218C2×C42C4×D7911yescharacteristic
219C2×C42C4×D7911yescharacteristic
220C2×C42D31611yescharacteristic
221C2×C42D31611yescharacteristic
222C2×C42C2×Dic7911yescharacteristic
223C2×C42C2×Dic7911yescharacteristic
224C2×C42C22⋊D7911yescharacteristic
225C2×C42C22⋊D7911yescharacteristic
226C2×C42C22⋊D7911yescharacteristic
227C2×C42C22×D7911yescharacteristic
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—1582no—
231C4⋊C8—1582no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—1582no—
235C22⋊C8—1582no—
236#97—22no—
237#97—1582no—
238#97—1582no—
239#97—1582no—
240#97—1582no—
241#97—1582no—
242C4×C8—794no—
243C4×C8C2×Dic7914yes—
244C24Dic79⋊C411yescharacteristic
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4C4×Dic7911yescharacteristic
249C22×C4C4×Dic7911yescharacteristic
250C22×C4(C2×D79)⋊C411yescharacteristic
251C22×C4(C2×D79)⋊C411yescharacteristic
252C22×C4(C2×D79)⋊C411yescharacteristic
253C22×C4(C2×D79)⋊C411yescharacteristic
254C22×C4C22⋊Dic7911yescharacteristic
255C22×C4C22⋊Dic7911yescharacteristic
256C22×C4Dic79⋊C411yescharacteristic
257C22×C4Dic79⋊C411yescharacteristic
258C22×C4Dic79⋊C411yescharacteristic
259C22×C4Dic79⋊C411yescharacteristic
260C22×C4Dic79⋊C411yescharacteristic
261C22×C4C4⋊Dic7911yescharacteristic
262C22×C4C4⋊Dic7911yescharacteristic
263C22×C4C2×C4×D7911yescharacteristic
264C22×C4C2×C22⋊D7911yescharacteristic
265C22×C4C2×Dic15811yescharacteristic
266M16—44no—
267M16—3164no—
268C2×C8—24no—
269C2×C8—798no—
270C2×C8—1584no—
271C2×C8—1588no—
272C2×C8—1588no—
273C2×C8—1584no—
274C2×C8—1584no—
275C42C4×Dic7911yescharacteristic
276C42C4×Dic7911yescharacteristic
277C42(C2×D79)⋊C411yescharacteristic
278C42(C2×D79)⋊C411yescharacteristic
279C42(C2×D79)⋊C411yescharacteristic
280C42(C2×D79)⋊C411yescharacteristic
281C42C22⋊Dic7911yescharacteristic
282C42C22⋊Dic7911yescharacteristic
283C42Dic79⋊C411yescharacteristic
284C42Dic79⋊C411yescharacteristic
285C42C4⋊Dic7911yescharacteristic
286C42C4⋊Dic7911yescharacteristic
287C42C2×C4×D7911yescharacteristic
288C42C22×Dic7911yescharacteristic
289C42C2×C22⋊D7911yescharacteristic
290C42C2×D31611yescharacteristic
291C23—22no—
292C23—22no—
293C23—22no—
294C23C79⋊#9511yescharacteristic
295C23C79⋊#9511yescharacteristic
296C23C2×Dic79⋊C411yescharacteristic
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4C79⋊#9511yescharacteristic
315C2×C4C79⋊#9511yescharacteristic
316C2×C4C79⋊#9511yescharacteristic
317C2×C4C79⋊#9511yescharacteristic
318C2×C4C79⋊#9511yescharacteristic
319C2×C4C79⋊#9511yescharacteristic
320C2×C4C2×C4×Dic7911yescharacteristic
321C2×C4C2×(C2×D79)⋊C411yescharacteristic
322C2×C4C2×(C2×D79)⋊C411yescharacteristic
323C2×C4C2×C22⋊Dic7911yescharacteristic
324C2×C4C2×Dic79⋊C411yescharacteristic
325C2×C4C2×C4⋊Dic7911yescharacteristic
326C2×C4M16×D7914yes—
327C8—44no—
328C8—3164no—
329C8—1588no—
330C22—28no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22#1909831912yes—
338C22C79⋊#33812yes—
339C22C2×C79⋊#9511yescharacteristic
340C22#1909858612yes—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4M16×Dic7912yes—
352C4#1909852612yes—
353C2—22no—
354C2—22no—
355C2—22no—
356C2#2376309311yescharacteristic
357C2C2×#1909831911yescharacteristic
358C2C2×C79⋊#33811yescharacteristic
subgroupG/H
centerC2×C4C2×C4×Dic79
commutatorC2×C4×C79C2×C42
FrattiniC2×C42C22×D79
FittingC79×#2920C2
socleC22×C79C2×#95
prime radicalC24×C79C4⋊C4
cosocleC2×C42×C79C23
hypercenter#2920D79
hypocenterC79#7861
LCSG ▷ C2×C4×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #2920

Complements

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

NH ≅ G/Ntype
3. #23761230C2semidirect
70. C2×C79⋊C8M16semidirect
184. C79#7861semidirect

Maximal subgroups

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

subgroupindex
#237720332
C22×#145790722
#237612302
C79×#29202
#786179