Finite Group Databasebeta

← #34184961

Subgroups — #34184961

Order 20224.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774338C211yesmaximal, characteristic
2#23764224C211yesmaximal, characteristic
3#23762069C211yesmaximal, characteristic
4#23761964C211yesmaximal, characteristic, complemented
5#23761260C211yesmaximal, characteristic, complemented
6#23761231C211yesmaximal, characteristic, complemented
7C79×#2961C211yesmaximal, characteristic, complemented
8C2×#14579158C2211yescharacteristic
9#19098937C412yescomplemented
10#19098911C412yescomplemented
11#19098763—22no—
12#19098713—22no—
13#19098616C2211yescharacteristic
14#19098616C2211yescharacteristic
15C2×#14579083—21no—
16C2×#14579083—21no—
17#19098429C2211yescharacteristic
18#19098306—22no—
19#19098300—22no—
20#533×C79C2211yescharacteristic
21#446×C79—22no—
22#446×C79—22no—
23#424×C79—21no—
24#423×C79C2211yescharacteristic
25#405×C79—21no—
26C2×#97×C79C2211yescharacteristic
27C22×D316—21no—
28C22×D316—21no—
29C22×D316—21no—
30C2×#11929611—24no—
31#14579158—21no—
32#14579158—21no—
33#14579158—21no—
34#14579158C2×C412yes—
35C2×C4⋊Dic79—21no—
36C4⋊(C79⋊C8)—21no—
37C4⋊(C79⋊C8)—21no—
38#14579083—42no—
39#14579083—42no—
40#14579081—24no—
41C22⋊(C79⋊C8)—21no—
42C22⋊(C79⋊C8)—21no—
43C4×C79⋊C8—21no—
44C4×C79⋊C8C2×C412yes—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C2×M16×C79—21no—
48C22×C8×C79D411yescharacteristic
49C22×C8×C79D411yescharacteristic
50C4⋊Q8×C79C2×C411yescharacteristic
51C4⋊Q8×C79C2×C411yescharacteristic
52#125×C79—21no—
53C22⋊Q8×C79—21no—
54C22⋊Q8×C79D412yes—
55C4×Q8×C79—21no—
56C4×Q8×C79D412yes—
57#117×C79—21no—
58C2×C42×C79C2311yescharacteristic
59C4⋊C8×C79—24no—
60C4⋊C8×C79—21no—
61C4⋊C8×C79D412yes—
62#97×C79—22no—
63#97×C79—21no—
64#97×C79—21no—
65C4×C8×C79—22no—
66C23×D79—41no—
67C2×D316—22no—
68C2×D316—22no—
69C2×D316—22no—
70C2×D316—41no—
71C2×D316—41no—
72C2×D316—41no—
73C2×D316—41no—
74C2×D316—41no—
75C2×D316—41no—
76C2×D316—41no—
77C2×D316—82no—
78C2×D316—82no—
79C22×Dic79—41no—
80#11929611—84no—
81C2×C79⋊C8—22no—
82C2×C79⋊C8—41no—
83C4⋊Dic79—41no—
84C4⋊Dic79—41no—
85C2×Q8×C79—21no—
86C2×Q8×C79C22⋊C412yes—
87C22×C4×C79C2×D411yescharacteristic
88C22×C4×C79C2×D411yescharacteristic
89C22×C4×C79#4111yescharacteristic
90M16×C79—41no—
91M16×C79—41no—
92C2×C8×C79—21no—
93C2×C8×C79—22no—
94C2×C8×C79—21no—
95C2×C8×C79—21no—
96C2×C8×C79—21no—
97C2×C8×C79—21no—
98C2×C8×C79SD1612yescomplemented
99C4⋊C4×C79—21no—
100C4⋊C4×C79—21no—
101C4⋊C4×C79—21no—
102C4⋊C4×C79—22no—
103C4⋊C4×C79—22no—
104C4⋊C4×C79C22⋊C412yes—
105C22⋊C4×C79—22no—
106C42×C79—22no—
107C42×C79C22×C411yescharacteristic
108C42×C79C2×D411yescharacteristic
109C42×C79C2×D411yescharacteristic
110C42×C79#4111yescharacteristic
111C23×C79#12111yescharacteristic
112C22×D79—42no—
113C22×D79—82no—
114C22×D79—82no—
115C22×D79—81no—
116C22×D79—81no—
117Q8×C79—42no—
118Q8×C79—81no—
119C2×C4×C79—22no—
120C2×C4×C79—22no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79C2×C22⋊C411yescharacteristic
124C2×C4×C79C4×D411yescharacteristic
125C2×C4×C79C4×D411yescharacteristic
126C2×C4×C79#12011yescharacteristic
127C2×C4×C79#12111yescharacteristic
128C2×C4×C79#12311yescharacteristic
129C2×C4×C79C2×SD1611yescharacteristic
130C2×C4×C79#13511yescharacteristic
131C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C79#13711yescharacteristic
133C2×Dic79—81no—
134C2×Dic79—81no—
135C2×Dic79—81no—
136D316—44no—
137D316—42no—
138D316—42no—
139D316—82no—
140D316—82no—
141D316—82no—
142D316—81no—
143D316—81no—
144C8×C79—22no—
145C8×C79—41no—
146C8×C79—42no—
147C79⋊C8—42no—
148C79⋊C8—81no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79#38611yescharacteristic
152C22×C79#46011yescharacteristic
153C22×C79#46411yescharacteristic
154C2×D79—82no—
155C2×D79—82no—
156C2×D79—82no—
157C2×D79—162no—
158C2×D79—162no—
159C4×C79—21no—
160C4×C79—21no—
161C4×C79—21no—
162C4×C79—21no—
163C4×C79—42no—
164C4×C79—42no—
165C4×C79C4×SD1611yescharacteristic
166C4×C79#44011yescharacteristic
167C4×C79#44811yescharacteristic
168C4×C79#44911yescharacteristic
169Dic79—161no—
170#9294—791nomaximal, Sylow
171C2×C79—21no—
172C2×C79—21no—
173C2×C79#188011yescharacteristic
174C2×C79#190311yescharacteristic
175C2×C79#191011yescharacteristic
176D79—162no—
177#3208—791no—
178#2961D7911yescharacteristic, complemented
179#1961—791no—
180#1545—791no—
181#1518—791no—
182#1286—791no—
183#1275—791no—
184C79#929411yescharacteristic, complemented, Sylow
185#533C2×D7911yescharacteristic
186C2×#127—791no—
187#492—792no—
188#486—792no—
189#460—1582no—
190#450—1582no—
191#446—22no—
192#446—22no—
193#431—791no—
194#431—791no—
195#424—21no—
196#423C2×D7911yescharacteristic
197C2×#102—1581no—
198C2×#102—1581no—
199#405—21no—
200C2×#97C2×D7911yescharacteristic
201#382—791no—
202#331—1582no—
203#328—1582no—
204C22×D4—1581no—
205C22×D4—1581no—
206C22×D4—1581no—
207C2×SD16—1584no—
208C2×M16—21no—
209C22×C8C22⋊D7911yescharacteristic
210C22×C8C22⋊D7911yescharacteristic
211C4⋊Q8C4×D7911yescharacteristic
212C4⋊Q8C4×D7911yescharacteristic
213#127—792no—
214#127—1581no—
215#127—1581no—
216#127—1581no—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8D31612yes—
220C4×Q8—21no—
221C4×Q8C22⋊D7912yes—
222#117—21no—
223C2×C4⋊C4—1581no—
224C2×C42C22×D7911yescharacteristic
225C4⋊C8—24no—
226C4⋊C8—21no—
227C4⋊C8—1581no—
228C4⋊C8—1581no—
229C4⋊C8C22⋊D7912yes—
230#102—3162no—
231#102—3162no—
232#102—1584no—
233C22⋊C8—1581no—
234C22⋊C8—1581no—
235#97—22no—
236#97—21no—
237#97—21no—
238C4×C8—22no—
239C4×C8—792no—
240C4×C8—1581no—
241#95—1581no—
242#95—1581no—
243C24—3161no—
244C2×Q8—21no—
245C2×Q8(C2×D79)⋊C412yes—
246C2×D4—3161no—
247C2×D4—3161no—
248C2×D4—3161no—
249C2×D4—3161no—
250C2×D4—3161no—
251C2×D4—3161no—
252C2×D4—3161no—
253C2×D4—6322no—
254C2×D4—6322no—
255C2×D4—1582no—
256C2×D4—1582no—
257C2×D4—1582no—
258C22×C4—3161no—
259C22×C4C2×C22⋊D7911yescharacteristic
260C22×C4C2×D31611yescharacteristic
261C22×C4#1192962411yescharacteristic
262SD16—6324no—
263M16—41no—
264M16—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—3161no—
272C2×C8—1582no—
273C2×C8#1192961112yescomplemented
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—3161no—
280C4⋊C4—3161no—
281C4⋊C4(C2×D79)⋊C412yes—
282C22⋊C4—22no—
283C42—22no—
284C42C2×C4×D7911yescharacteristic
285C42C2×C22⋊D7911yescharacteristic
286C42C2×C22⋊D7911yescharacteristic
287C42#1192962411yescharacteristic
288C23—3162no—
289C23—6322no—
290C23—6322no—
291C23—6321no—
292C23—6321no—
293C23#1457913511yescharacteristic
294Q8—42no—
295Q8—81no—
296D4—3164no—
297D4—3162no—
298D4—3162no—
299D4—6322no—
300D4—6322no—
301D4—6322no—
302D4—6321no—
303D4—6321no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—6321no—
309C2×C4—6321no—
310C2×C4—6321no—
311C2×C4C2×(C2×D79)⋊C411yescharacteristic
312C2×C4C4×C22⋊D7911yescharacteristic
313C2×C4C4×C22⋊D7911yescharacteristic
314C2×C4#1457913011yescharacteristic
315C2×C4#1457913511yescharacteristic
316C2×C4#1457914411yescharacteristic
317C2×C4C2×#1192961111yescharacteristic
318C2×C4#1457918111yescharacteristic
319C2×C4#1457918611yescharacteristic
320C2×C4#1457919211yescharacteristic
321C8—22no—
322C8—41no—
323C8—42no—
324C8—3162no—
325C8—6321no—
326C22—21no—
327C22—21no—
328C22—6322no—
329C22—6322no—
330C22—6322no—
331C22—12642no—
332C22—12642no—
333C22#1909844211yescharacteristic
334C22#1909876311yescharacteristic
335C22#1909878511yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—12641no—
343C4C4×#1192961111yescharacteristic
344C4#1909866111yescharacteristic
345C4#1909870211yescharacteristic
346C4#1909870811yescharacteristic
347C2—21no—
348C2—21no—
349C2—12642no—
350C2#2376375511yescharacteristic
351C2#2376388011yescharacteristic
352C2#2376392711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2961C2
socleC22×C79#386
prime radicalC2×#14579158C22
cosocleC2×C42×C79C23
hypercenter#2961D79
hypocenterC79#9294
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2961

Complements

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

NH ≅ G/Ntype
4. #23761964C2semidirect
5. #23761260C2semidirect
6. #23761231C2semidirect
7. C79×#2961C2semidirect
9. #19098937C4semidirect
10. #19098911C4semidirect
98. C2×C8×C79SD16semidirect
178. #2961D79semidirect
184. C79#9294semidirect
273. C2×C8#11929611semidirect

Maximal subgroups

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

subgroupindex
#237743382
#237642242
#237620692
#237619642
#237612602
#237612312
C79×#29612
#929479