Finite Group Databasebeta

← #34177424

Subgroups — #34177424

Order 20224.

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

normal subgroups only

This group has 350 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#23771355C211yesmaximal, characteristic
2#23764957C211yesmaximal, characteristic
3#23764872C211yesmaximal, characteristic
4#23761226C211yesmaximal, characteristic
5#23761224C211yesmaximal, characteristic
6#23761222C211yesmaximal, characteristic
7C79×#2072C211yesmaximal, characteristic
8C2×#14579094—21no—
9#19098505—21no—
10#19098503—21no—
11#19098482—21no—
12#19098461C2211yescharacteristic
13#19098461C2211yescharacteristic
14#19098460—21no—
15#19098446—21no—
16#19098444—21no—
17#19098325—21no—
18#19098325—21no—
19#19098325—21no—
20#19098318C2211yescharacteristic
21#19098318C2211yescharacteristic
22C2×#103×C79—21no—
23#391×C79C2211yescharacteristic
24#384×C79C2211yescharacteristic
25#340×C79—21no—
26#336×C79C2211yescharacteristic
27C22×C79⋊C8—21no—
28C22×C79⋊C8—21no—
29C2×C4⋊Dic79D411yescharacteristic
30C2×C4⋊Dic79D411yescharacteristic
31C2×C4⋊Dic79D411yescharacteristic
32C2×C4⋊Dic79D411yescharacteristic
33C2×Dic79⋊C4—21no—
34C2×Dic79⋊C4—21no—
35C2×Dic79⋊C4—21no—
36C2×Dic79⋊C4—21no—
37C2×C4×Dic79—21no—
38#14579094—42no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C22×Q8×C79D411yescharacteristic
47C22×C8×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79—21no—
50C2×C4⋊C4×C79—21no—
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C42×C79C2311yescharacteristic
53#103×C79—42no—
54#95×C79—21no—
55#95×C79—21no—
56C22×Dic79—21no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C2×C79⋊C8—22no—
61C2×C79⋊C8—41no—
62C2×C79⋊C8—41no—
63C2×C79⋊C8—41no—
64C2×C79⋊C8—41no—
65C2×C79⋊C8—41no—
66C4⋊Dic79—21no—
67C4⋊Dic79—21no—
68C4⋊Dic79—21no—
69C4⋊Dic79—41no—
70C4⋊Dic79—41no—
71C4⋊Dic79Q1612yes—
72Dic79⋊C4—41no—
73Dic79⋊C4—41no—
74Dic79⋊C4—41no—
75Dic79⋊C4—41no—
76Dic79⋊C4—81no—
77Dic79⋊C4—81no—
78C4×Dic79—81no—
79C2×Q8×C79—21no—
80C2×Q8×C79—21no—
81C2×Q8×C79—41no—
82C2×Q8×C79—41no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—41no—
86C22×C4×C79C2×D411yescharacteristic
87C22×C4×C79C2×D411yescharacteristic
88C22×C4×C79C2×D411yescharacteristic
89C2×C8×C79—41no—
90C2×C8×C79—41no—
91C2×C8×C79—41no—
92C4⋊C4×C79—41no—
93C4⋊C4×C79—42no—
94C4⋊C4×C79—41no—
95C4⋊C4×C79—41no—
96C4⋊C4×C79—41no—
97C4⋊C4×C79—41no—
98C42×C79—41no—
99C23×C79#12011yescharacteristic
100Q8×C79—42no—
101Q8×C79—81no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—81no—
115C2×C4×C79—81no—
116C2×C4×C79—81no—
117C2×C4×C79C2×Q1611yescharacteristic
118C2×C4×C79#13511yescharacteristic
119C2×C4×C79#13511yescharacteristic
120C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C79#13711yescharacteristic
123C2×Dic79—22no—
124C2×Dic79—41no—
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—
135C8×C79—81no—
136C79⋊C8—42no—
137C79⋊C8—81no—
138C22×C79#44911yescharacteristic
139C22×C79#44911yescharacteristic
140C22×C79#45111yescharacteristic
141C22×C79#45411yescharacteristic
142C22×C79#45511yescharacteristic
143C22×C79#45611yescharacteristic
144C22×C79#45811yescharacteristic
145C4×C79—21no—
146C4×C79—21no—
147C4×C79—21no—
148C4×C79—21no—
149C4×C79—21no—
150C4×C79—21no—
151C4×C79—81no—
152C4×C79—81no—
153C4×C79—161no—
154Dic79—42no—
155Dic79—81no—
156Dic79—81no—
157Dic79—81no—
158#7806—791nomaximal, Sylow
159C2×C79#160911yescharacteristic
160C2×C79#161211yescharacteristic
161C2×C79#161711yescharacteristic
162C2×C79#162811yescharacteristic
163C2×C79#164411yescharacteristic
164C2×C79#164811yescharacteristic
165C2×C79#166411yescharacteristic
166#2805—791no—
167#2099—791no—
168#2083—791no—
169#2072D7911yescharacteristic
170#1272—791no—
171#1271—791no—
172#1271—791no—
173C79#780611yescharacteristic, complemented, Sylow
174C2×#107—1581no—
175C2×#103—21no—
176#400—1581no—
177#400—1581no—
178#395—1581no—
179#391C2×D7911yescharacteristic
180#389—791no—
181#389—791no—
182#389—1581no—
183#387—1581no—
184#387—1581no—
185#384C2×D7911yescharacteristic
186#340—21no—
187#340—1581no—
188#340—1581no—
189#340—1581no—
190#336—791no—
191#336—791no—
192#336C2×D7911yescharacteristic
193C22×Q8C22⋊D7911yescharacteristic
194C22×C8—21no—
195C22×C8—1581no—
196C22×C8—1581no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—791no—
201C2×C4⋊C4—791no—
202C2×C4⋊C4—791no—
203C2×C4⋊C4—791no—
204C2×C4⋊C4—1581no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4—1581no—
207C2×C4⋊C4—1581no—
208C2×C4⋊C4C22⋊D7911yescharacteristic
209C2×C42—1581no—
210C2×C42C22×D7911yescharacteristic
211#107—3162no—
212#103—42no—
213#95—21no—
214#95—21no—
215#95—1581no—
216#95—1581no—
217#95—1581no—
218#95—1581no—
219#95—1581no—
220#95—1581no—
221#95—1581no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—41no—
225C2×Q8—41no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—1581no—
230C22×C4—1581no—
231C22×C4—1581no—
232C22×C4—1581no—
233C22×C4D4×D7911yescharacteristic
234C22×C4D4×D7911yescharacteristic
235C22×C4C2×C22⋊D7911yescharacteristic
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—3161no—
240C2×C8—3161no—
241C2×C8—3161no—
242C2×C8—3161no—
243C2×C8—3161no—
244C2×C8—1582no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—3161no—
252C4⋊C4—3161no—
253C4⋊C4—3161no—
254C4⋊C4—3161no—
255C4⋊C4—3161no—
256C4⋊C4—3161no—
257C4⋊C4—792no—
258C4⋊C4—6321no—
259C4⋊C4—6321no—
260C4⋊C4—1581no—
261C4⋊C4—1581no—
262C4⋊C4—1581no—
263C42—41no—
264C42—6321no—
265C23#1457912911yescharacteristic
266Q8—42no—
267Q8—81no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—3161no—
284C2×C4—3161no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4—3161no—
289C2×C4—3161no—
290C2×C4—3161no—
291C2×C4—3161no—
292C2×C4—3161no—
293C2×C4—3161no—
294C2×C4—1582no—
295C2×C4Q16×D7911yescharacteristic
296C2×C4#1457918111yescharacteristic
297C2×C4#1457918211yescharacteristic
298C2×C4#1457918711yescharacteristic
299C2×C4#1457918811yescharacteristic
300C2×C4#1457919311yescharacteristic
301C8—81no—
302C8—3162no—
303C8—6321no—
304C22#1909870611yescharacteristic
305C22#1909870711yescharacteristic
306C22#1909871711yescharacteristic
307C22#1909873411yescharacteristic
308C22#1909874011yescharacteristic
309C22#1909874411yescharacteristic
310C22#1909875211yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—81no—
318C4—81no—
319C4—161no—
320C4—3162no—
321C4—6321no—
322C4—6321no—
323C4—6321no—
324C2#2376239611yescharacteristic
325C2#2376241211yescharacteristic
326C2#2376244011yescharacteristic
327C2#2376250911yescharacteristic
328C2#2376261111yescharacteristic
329C2#2376264011yescharacteristic
330C2#2376273311yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2072C2
socle, prime radicalC23×C79#120
hypercenter#2072D79
hypocenterC79#7806
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2072

Complements

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

NH ≅ G/Ntype
173. C79#7806semidirect

Maximal subgroups

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

subgroupindex
#237713552
#237649572
#237648722
#237612262
#237612242
#237612222
C79×#20722
#780679