Finite Group Databasebeta

← #34177431

Subgroups — #34177431

Order 20224.

366 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771350C211yesmaximal, characteristic
2#23764944C211yesmaximal, characteristic
3#23764872C211yesmaximal, characteristic
4#23761225C211yesmaximal, characteristic
5#23761224C211yesmaximal, characteristic
6#23761223C211yesmaximal, characteristic
7C79×#2084C211yesmaximal, characteristic
8C2×#14579094—21no—
9C2×#14579092—21no—
10C2×#14579084—21no—
11#19098505—21no—
12#19098501—21no—
13#19098483—21no—
14#19098469C2211yescharacteristic
15#19098461C2211yescharacteristic
16#19098458—21no—
17#19098449—21no—
18#19098448—21no—
19#19098325—21no—
20#19098318C2211yescharacteristic
21#19098318C2211yescharacteristic
22C2×#106×C79—21no—
23#389×C79C2211yescharacteristic
24#384×C79C2211yescharacteristic
25#340×C79—21no—
26#336×C79C2211yescharacteristic
27C22×Dic158D411yescharacteristic
28C22×C79⋊C8—21no—
29C22×C79⋊C8—21no—
30C2×C4⋊Dic79D411yescharacteristic
31C2×C4⋊Dic79D411yescharacteristic
32C2×C4⋊Dic79D411yescharacteristic
33C2×Dic79⋊C4—21no—
34C2×Dic79⋊C4—21no—
35C2×Dic79⋊C4—21no—
36C2×C4×Dic79—21no—
37#14579094—42no—
38#14579092—42no—
39#14579084—42no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C22×C8×C79—21no—
49C2×C4⋊C4×C79—21no—
50C2×C4⋊C4×C79—21no—
51C2×C4⋊C4×C79—21no—
52C2×C4⋊C4×C79—21no—
53C2×C4⋊C4×C79D411yescharacteristic
54C2×C4⋊C4×C79D411yescharacteristic
55C2×C42×C79C2311yescharacteristic
56#106×C79—42no—
57#95×C79—21no—
58C2×Dic158—21no—
59C2×Dic158—21no—
60C2×Dic158—41no—
61C2×Dic158—41no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C2×C79⋊C8—22no—
67C2×C79⋊C8—22no—
68C2×C79⋊C8—41no—
69C2×C79⋊C8—41no—
70C2×C79⋊C8—41no—
71C2×C79⋊C8—41no—
72C4⋊Dic79—21no—
73C4⋊Dic79—41no—
74C4⋊Dic79—41no—
75C4⋊Dic79SD1612yes—
76Dic79⋊C4—41no—
77Dic79⋊C4—41no—
78Dic79⋊C4—81no—
79Dic79⋊C4—81no—
80C4×Dic79—81no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—41no—
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C2×C8×C79—22no—
88C2×C8×C79—41no—
89C2×C8×C79—41no—
90C4⋊C4×C79—21no—
91C4⋊C4×C79—21no—
92C4⋊C4×C79—41no—
93C4⋊C4×C79—41no—
94C4⋊C4×C79—41no—
95C4⋊C4×C79—41no—
96C4⋊C4×C79—41no—
97C4⋊C4×C79—41no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—41no—
100C4⋊C4×C79—41no—
101C42×C79—41no—
102C23×C79#12011yescharacteristic
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—41no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—81no—
116C2×C4×C79—81no—
117C2×C4×C79—81no—
118C2×C4×C79C2×SD1611yescharacteristic
119C2×C4×C79#13511yescharacteristic
120C2×C4×C79#13511yescharacteristic
121C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C79#13711yescharacteristic
123C2×C4×C79#13711yescharacteristic
124C2×Dic79—22no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136Dic158—42no—
137Dic158—81no—
138C8×C79—42no—
139C79⋊C8—42no—
140C79⋊C8—42no—
141C22×C79#44811yescharacteristic
142C22×C79#44911yescharacteristic
143C22×C79#45211yescharacteristic
144C22×C79#45311yescharacteristic
145C22×C79#45611yescharacteristic
146C22×C79#45611yescharacteristic
147C22×C79#45811yescharacteristic
148C4×C79—21no—
149C4×C79—21no—
150C4×C79—21no—
151C4×C79—21no—
152C4×C79—21no—
153C4×C79—21no—
154C4×C79—81no—
155C4×C79—81no—
156C4×C79—161no—
157Dic79—42no—
158Dic79—81no—
159Dic79—81no—
160Dic79—81no—
161#7807—791nomaximal, Sylow
162C2×C79#160911yescharacteristic
163C2×C79#161111yescharacteristic
164C2×C79#161811yescharacteristic
165C2×C79#162211yescharacteristic
166C2×C79#165011yescharacteristic
167C2×C79#165311yescharacteristic
168C2×C79#166011yescharacteristic
169#2805—791no—
170#2096—791no—
171#2084D7911yescharacteristic
172#2083—791no—
173#1272—791no—
174#1271—791no—
175#1271—791no—
176C79#780711yescharacteristic, complemented, Sylow
177C2×#107—1581no—
178C2×#106—21no—
179C2×#106—1581no—
180C2×#103—1581no—
181#400—1581no—
182#400—1581no—
183#395—1581no—
184#391—791no—
185#389—791no—
186#389—1581no—
187#389C2×D7911yescharacteristic
188#387—1581no—
189#387—1581no—
190#384C2×D7911yescharacteristic
191#340—21no—
192#340—1581no—
193#336—791no—
194#336—791no—
195#336C2×D7911yescharacteristic
196C22×Q8—791no—
197C22×C8—21no—
198C22×C8—1581no—
199C22×C8—1581no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—791no—
205C2×C4⋊C4—791no—
206C2×C4⋊C4—791no—
207C2×C4⋊C4—1581no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4C22⋊D7911yescharacteristic
211C2×C4⋊C4C22⋊D7911yescharacteristic
212C2×C42—1581no—
213C2×C42C22×D7911yescharacteristic
214#107—3162no—
215#106—42no—
216#106—3162no—
217#103—3162no—
218#95—21no—
219#95—1581no—
220#95—1581no—
221#95—1581no—
222#95—1581no—
223#95—1581no—
224#95—1581no—
225#95—1581no—
226#95—1581no—
227C2×Q8—3161no—
228C2×Q8—3161no—
229C2×Q8—1581no—
230C2×Q8—1581no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—1581no—
235C22×C4—1581no—
236C22×C4—1581no—
237C22×C4—1581no—
238C22×C4D4×D7911yescharacteristic
239C22×C4D4×D7911yescharacteristic
240C22×C4C2×C22⋊D7911yescharacteristic
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—3161no—
245C2×C8—3161no—
246C2×C8—3161no—
247C2×C8—3161no—
248C2×C8—1582no—
249C2×C8—1582no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—3161no—
262C4⋊C4—3161no—
263C4⋊C4—3161no—
264C4⋊C4—3161no—
265C4⋊C4—792no—
266C4⋊C4—6321no—
267C4⋊C4—6321no—
268C4⋊C4—1581no—
269C42—41no—
270C42—6321no—
271C23#1457912911yescharacteristic
272Q8—3162no—
273Q8—6321no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
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—3161no—
298C2×C4—3161no—
299C2×C4—3161no—
300C2×C4—1582no—
301C2×C4SD16×D7911yescharacteristic
302C2×C4#1457918111yescharacteristic
303C2×C4#1457918211yescharacteristic
304C2×C4#1457918711yescharacteristic
305C2×C4#1457919011yescharacteristic
306C2×C4#1457919011yescharacteristic
307C8—42no—
308C8—3162no—
309C8—3162no—
310C22#1909870011yescharacteristic
311C22#1909870611yescharacteristic
312C22#1909872211yescharacteristic
313C22#1909872911yescharacteristic
314C22#1909874411yescharacteristic
315C22#1909874511yescharacteristic
316C22#1909875211yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—81no—
324C4—81no—
325C4—161no—
326C4—3162no—
327C4—6321no—
328C4—6321no—
329C4—6321no—
330C2#2376239511yescharacteristic
331C2#2376240611yescharacteristic
332C2#2376244511yescharacteristic
333C2#2376247111yescharacteristic
334C2#2376265411yescharacteristic
335C2#2376267411yescharacteristic
336C2#2376271011yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2084C2
socle, prime radicalC23×C79#120
hypercenter#2084D79
hypocenterC79#7807
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
176. C79#7807semidirect

Maximal subgroups

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

subgroupindex
#237713502
#237649442
#237648722
#237612252
#237612242
#237612232
C79×#20842
#780779