Finite Group Databasebeta

← #34184963

Subgroups — #34184963

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#23772341C211yesmaximal, characteristic, complemented
2#23764225C211yesmaximal, characteristic
3#23762068C211yesmaximal, characteristic
4#23761964C211yesmaximal, characteristic, complemented
5#23761259C211yesmaximal, characteristic, complemented
6#23761230C211yesmaximal, characteristic, complemented
7C79×#3208C211yesmaximal, characteristic
8#19098694—22no—
9#19098694—22no—
10#19098616C2211yescharacteristic
11#19098590—21no—
12#19098587C2211yescharacteristic
13C2×#14579082—21no—
14C2×#14579082—21no—
15#19098519—21no—
16C2×#14579072C2211yescharacteristic
17#19098429C2211yescharacteristic
18#19098307—22no—
19#19098300—22no—
20#533×C79C2211yescharacteristic
21C2×#127×C79C2211yescharacteristic
22#492×C79C412yescomplemented
23#486×C79C412yescomplemented
24#460×C79—22no—
25#450×C79—22no—
26#431×C79C2211yescharacteristic
27C2×#11929608—21no—
28C22×C79⋊C8D411yescharacteristic
29C22×C79⋊C8D411yescharacteristic
30C2×C4⋊Dic79—21no—
31C4⋊(C79⋊C8)—24no—
32C4⋊(C79⋊C8)—21no—
33C4⋊(C79⋊C8)—21no—
34C4⋊(C79⋊C8)D412yes—
35#14579082—42no—
36#14579082—42no—
37C22⋊(C79⋊C8)—21no—
38#14579072—22no—
39#14579072—21no—
40#14579072—21no—
41C4×C79⋊C8—22no—
42C4×C79⋊C8—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C22×D4×C79—21no—
46C22×D4×C79—21no—
47C22×D4×C79—21no—
48C2×SD16×C79—24no—
49C4⋊Q8×C79C2×C411yescharacteristic
50C4⋊Q8×C79C2×C411yescharacteristic
51#127×C79—21no—
52#127×C79—21no—
53#127×C79—21no—
54#127×C79C2×C412yes—
55#125×C79—21no—
56C22⋊Q8×C79—21no—
57C22⋊Q8×C79D412yes—
58C4×Q8×C79—21no—
59C4×Q8×C79D412yes—
60#117×C79—21no—
61C2×C42×C79C2311yescharacteristic
62C4⋊C8×C79—21no—
63#102×C79—24no—
64C22⋊C8×C79—21no—
65C4×C8×C79C2×C412yes—
66C22×Dic79—41no—
67#11929608—41no—
68#11929608—41no—
69C2×C79⋊C8—21no—
70C2×C79⋊C8—22no—
71C2×C79⋊C8—21no—
72C2×C79⋊C8—21no—
73C2×C79⋊C8—21no—
74C2×C79⋊C8—21no—
75C2×C79⋊C8—41no—
76C2×C79⋊C8SD1612yescomplemented
77C4⋊Dic79—41no—
78C4⋊Dic79—41no—
79C24×C79—41no—
80C2×Q8×C79—21no—
81C2×Q8×C79C22⋊C412yes—
82C2×D4×C79—22no—
83C2×D4×C79—22no—
84C2×D4×C79—22no—
85C2×D4×C79—41no—
86C2×D4×C79—41no—
87C2×D4×C79—41no—
88C2×D4×C79—41no—
89C2×D4×C79—41no—
90C2×D4×C79—41no—
91C2×D4×C79—41no—
92C2×D4×C79—82no—
93C2×D4×C79—82no—
94C22×C4×C79C2×D411yescharacteristic
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79#4111yescharacteristic
97SD16×C79—84no—
98C2×C8×C79—22no—
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—42no—
112C23×C79—82no—
113C23×C79—82no—
114C23×C79—81no—
115C23×C79—81no—
116C23×C79#12111yescharacteristic
117Q8×C79—42no—
118Q8×C79—81no—
119D4×C79—44no—
120D4×C79—42no—
121D4×C79—42no—
122D4×C79—82no—
123D4×C79—82no—
124D4×C79—82no—
125D4×C79—81no—
126D4×C79—81no—
127C2×C4×C79—22no—
128C2×C4×C79—22no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79C2×C22⋊C411yescharacteristic
132C2×C4×C79C4×D411yescharacteristic
133C2×C4×C79C4×D411yescharacteristic
134C2×C4×C79#12011yescharacteristic
135C2×C4×C79#12111yescharacteristic
136C2×C4×C79#12311yescharacteristic
137C2×C4×C79C2×SD1611yescharacteristic
138C2×C4×C79#13511yescharacteristic
139C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C79#13711yescharacteristic
141C2×Dic79—81no—
142C2×Dic79—81no—
143C2×Dic79—81no—
144C8×C79—42no—
145C79⋊C8—22no—
146C79⋊C8—41no—
147C79⋊C8—42no—
148C79⋊C8—81no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—82no—
152C22×C79—82no—
153C22×C79—82no—
154C22×C79—162no—
155C22×C79—162no—
156C22×C79#38611yescharacteristic
157C22×C79#46011yescharacteristic
158C22×C79#46411yescharacteristic
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—162no—
174C2×C79#188011yescharacteristic
175C2×C79#190311yescharacteristic
176C2×C79#191011yescharacteristic
177#3208D7911yescharacteristic
178#2961—791no—
179#1961—791no—
180#1545—791no—
181#1518—791no—
182#1286—791no—
183#1275—791no—
184C79#929411yescharacteristic, complemented, Sylow
185#533C2×D7911yescharacteristic
186C2×#127C2×D7911yescharacteristic
187#492Dic7912yescomplemented
188#486Dic7912yescomplemented
189#460—22no—
190#450—22no—
191#446—1582no—
192#446—1582no—
193#431—791no—
194#431C2×D7911yescharacteristic
195#424—1581no—
196#423—791no—
197C2×#102—1581no—
198C2×#102—1581no—
199#405—1581no—
200C2×#97—791no—
201#382—791no—
202#331—1582no—
203#328—1582no—
204C22×D4—21no—
205C22×D4—21no—
206C22×D4—21no—
207C2×SD16—24no—
208C2×M16—1581no—
209C22×C8—791no—
210C22×C8—791no—
211C4⋊Q8C2×Dic7911yescharacteristic
212C4⋊Q8C2×Dic7911yescharacteristic
213#127—21no—
214#127—21no—
215#127—21no—
216#127C2×Dic7912yes—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8C22⋊D7912yes—
220C4×Q8—21no—
221C4×Q8C22⋊D7912yes—
222#117—21no—
223C2×C4⋊C4—1581no—
224C2×C42C22×D7911yescharacteristic
225C4⋊C8—21no—
226C4⋊C8—792no—
227C4⋊C8—1584no—
228C4⋊C8—1581no—
229C4⋊C8—1581no—
230#102—24no—
231#102—3162no—
232#102—3162no—
233C22⋊C8—21no—
234C22⋊C8—1581no—
235#97—1582no—
236#97—1581no—
237#97—1581no—
238C4×C8—1582no—
239C4×C8—1581no—
240C4×C8C2×Dic7912yes—
241#95—1581no—
242#95—1581no—
243C24—41no—
244C2×Q8—21no—
245C2×Q8C22⋊Dic7912yes—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—82no—
257C2×D4—82no—
258C22×C4—3161no—
259C22×C4D4×D7911yescharacteristic
260C22×C4C2×C22⋊D7911yescharacteristic
261C22×C4#1192962311yescharacteristic
262SD16—84no—
263M16—3161no—
264M16—3161no—
265C2×C8—22no—
266C2×C8—3161no—
267C2×C8—792no—
268C2×C8—1582no—
269C2×C8—1581no—
270C2×C8—1581no—
271C2×C8—1581no—
272C2×C8—1581no—
273C2×C8—1581no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—3161no—
280C4⋊C4—3161no—
281C4⋊C4C22⋊Dic7912yes—
282C22⋊C4—22no—
283C42—22no—
284C42C22×Dic7911yescharacteristic
285C42D4×D7911yescharacteristic
286C42C2×C22⋊D7911yescharacteristic
287C42#1192962311yescharacteristic
288C23—42no—
289C23—82no—
290C23—82no—
291C23—81no—
292C23—81no—
293C23#1457913311yescharacteristic
294Q8—42no—
295Q8—81no—
296D4—44no—
297D4—42no—
298D4—42no—
299D4—82no—
300D4—82no—
301D4—82no—
302D4—81no—
303D4—81no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—6321no—
309C2×C4—6321no—
310C2×C4—6321no—
311C2×C4C2×C22⋊Dic7911yescharacteristic
312C2×C4D4×Dic7911yescharacteristic
313C2×C4D4×Dic7911yescharacteristic
314C2×C4#1457912911yescharacteristic
315C2×C4#1457913311yescharacteristic
316C2×C4#1457914211yescharacteristic
317C2×C4SD16×D7911yescharacteristic
318C2×C4#1457918211yescharacteristic
319C2×C4#1457918811yescharacteristic
320C2×C4#1457919011yescharacteristic
321C8—42no—
322C8—3161no—
323C8—3162no—
324C8—6321no—
325C8—1582no—
326C22—21no—
327C22—21no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—162no—
332C22—162no—
333C22#1909844311yescharacteristic
334C22#1909876111yescharacteristic
335C22#1909878311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—12641no—
343C4SD16×Dic7911yescharacteristic
344C4#1909866411yescharacteristic
345C4#1909870011yescharacteristic
346C4#1909870711yescharacteristic
347C2—21no—
348C2—21no—
349C2—162no—
350C2#2376375711yescharacteristic
351C2#2376388211yescharacteristic
352C2#2376392811yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3208C2
socleC22×C79#386
prime radicalC2×#127×C79C22
cosocleC2×C42×C79C23
hypercenter#3208D79
hypocenterC79#9294
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #23772341C2semidirect
4. #23761964C2semidirect
5. #23761259C2semidirect
6. #23761230C2semidirect
22. #492×C79C4semidirect
23. #486×C79C4semidirect
76. C2×C79⋊C8SD16semidirect
184. C79#9294semidirect
187. #492Dic79semidirect
188. #486Dic79semidirect

Maximal subgroups

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

subgroupindex
#237723412
#237642252
#237620682
#237619642
#237612592
#237612302
C79×#32082
#929479