Finite Group Databasebeta

← #34201335

Subgroups — #34201335

Order 20224.

352 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764750C211yesmaximal, characteristic
2#23762441C211yesmaximal, characteristic
3#23762373C211yesmaximal, characteristic
4#23761430C211yesmaximal, characteristic
5#23761427C211yesmaximal, characteristic
6#23761344C211yesmaximal, characteristic
7C79×#2612C211yesmaximal, characteristic
8#19099194C2211yescharacteristic
9#19098898—21no—
10#19098719—21no—
11#19098603—21no—
12#19098332C2211yescharacteristic
13#19098323—21no—
14#19098299—21no—
15#19098299—21no—
16#19098289C2211yescharacteristic
17#19098289C2211yescharacteristic
18#400×C79C2211yescharacteristic
19#398×C79—21no—
20#395×C79C2211yescharacteristic
21#391×C79—21no—
22#389×C79—21no—
23#387×C79—21no—
24#387×C79—21no—
25#382×C79C2211yescharacteristic
26#376×C79—21no—
27C2×#11929614—41no—
28C2×#11929608—21no—
29#14579160—41no—
30#14579152—41no—
31#14579140—21no—
32#14579140—21no—
33#14579120—21no—
34#14579094—41no—
35#14579094—41no—
36#14579092—41no—
37#14579085—41no—
38#14579084—41no—
39C22⋊(C79⋊C8)—21no—
40C22⋊(C79⋊C8)—21no—
41C22×Q8×C79D411yescharacteristic
42C2×C4⋊C4×C79—21no—
43C2×C4⋊C4×C79—21no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79—21no—
46C2×C4⋊C4×C79D411yescharacteristic
47C2×C4⋊C4×C79D411yescharacteristic
48C2×C4⋊C4×C79C2311yescharacteristic
49C2×C42×C79—21no—
50C2×C42×C79—21no—
51C2×C42×C79D411yescharacteristic
52#95×C79—21no—
53#95×C79—21no—
54#95×C79—21no—
55#95×C79—21no—
56#95×C79—21no—
57#95×C79—21no—
58#95×C79—21no—
59#95×C79—21no—
60#95×C79D411yescharacteristic
61#95×C79D411yescharacteristic
62C2×Dic158—21no—
63C2×Dic158—21no—
64#11929614—82no—
65#11929608—22no—
66#11929608—41no—
67C2×C79⋊C8—41no—
68C2×C79⋊C8—41no—
69C2×C79⋊C8—41no—
70C4⋊Dic79—21no—
71C4⋊Dic79—21no—
72C4⋊Dic79—21no—
73C4⋊Dic79—21no—
74Dic79⋊C4—41no—
75Dic79⋊C4—41no—
76C22⋊Dic79—41no—
77C4×Dic79—41no—
78C2×Q8×C79—42no—
79C2×Q8×C79—41no—
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79C2×D411yescharacteristic
87C22×C4×C79C2×D411yescharacteristic
88C22×C4×C79C2×D411yescharacteristic
89C4⋊C4×C79—41no—
90C4⋊C4×C79—42no—
91C4⋊C4×C79—42no—
92C4⋊C4×C79—42no—
93C4⋊C4×C79—41no—
94C4⋊C4×C79—41no—
95C4⋊C4×C79—81no—
96C42×C79—21no—
97C42×C79—42no—
98C42×C79—42no—
99C42×C79Q1612yescomplemented
100C23×C79#12011yescharacteristic
101Q8×C79—42no—
102Q8×C79—81no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—22no—
106C2×C4×C79—22no—
107C2×C4×C79—22no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79C2×Q1611yescharacteristic
128C2×C4×C79#13711yescharacteristic
129C2×Dic79—41no—
130C2×Dic79—41no—
131Dic158—81no—
132Dic158—81no—
133C79⋊C8—42no—
134C79⋊C8—81no—
135C79⋊C8—81no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79#45111yescharacteristic
139C22×C79#45611yescharacteristic
140C22×C79B(4,𝔽2)11yescharacteristic
141C4×C79—21no—
142C4×C79—21no—
143C4×C79—42no—
144C4×C79—42no—
145C4×C79—42no—
146C4×C79—41no—
147C4×C79—41no—
148C4×C79—81no—
149C4×C79—81no—
150C4×C79—81no—
151Dic79—81no—
152Dic79—81no—
153#12258—791nomaximal, Sylow
154C2×C79—21no—
155C2×C79—21no—
156C2×C79#160511yescharacteristic
157C2×C79#219711yescharacteristic
158C2×C79#220411yescharacteristic
159#2612D7911yescharacteristic
160#2061—791no—
161#1617—791no—
162#1605—791no—
163#1354—791no—
164#1353—791no—
165#1324—791no—
166C79#1225811yescharacteristic, complemented, Sylow
167#544—791no—
168#484—1581no—
169#451—1581no—
170#428—1581no—
171#400C2×D7911yescharacteristic
172#398—21no—
173#395C2×D7911yescharacteristic
174#391—21no—
175#389—21no—
176#387—21no—
177#387—21no—
178#382C2×D7911yescharacteristic
179#376—21no—
180#343—791no—
181#339—1581no—
182#328—1581no—
183#328—1581no—
184#324—791no—
185#324—791no—
186C22×Q8C22⋊D7911yescharacteristic
187C2×Q16—3161no—
188C2×M16—1581no—
189C4⋊Q8—3161no—
190#125—3161no—
191C22⋊Q8—1581no—
192C22⋊Q8—1581no—
193#117—1581no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22⋊D7911yescharacteristic
199C2×C4⋊C4C22⋊D7911yescharacteristic
200C2×C4⋊C4C22×D7911yescharacteristic
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22⋊D7911yescharacteristic
204#107—3161no—
205#107—3161no—
206#106—3161no—
207#103—3161no—
208#103—3161no—
209C22⋊C8—1581no—
210C22⋊C8—1581no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95C22⋊D7911yescharacteristic
220#95C22⋊D7911yescharacteristic
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—1581no—
224C2×Q8—1581no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4C2×C22⋊D7911yescharacteristic
232C22×C4C2×C22⋊D7911yescharacteristic
233C22×C4C2×C22⋊D7911yescharacteristic
234Q16—6322no—
235M16—3161no—
236M16—1582no—
237C2×C8—3161no—
238C2×C8—3161no—
239C2×C8—3161no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C4⋊C4—3161no—
248C4⋊C4—3161no—
249C4⋊C4—1581no—
250C4⋊C4—1581no—
251C4⋊C4—1581no—
252C4⋊C4—1581no—
253C22⋊C4—3161no—
254C42—21no—
255C42—42no—
256C42—42no—
257C42—3161no—
258C42#1192961412yescomplemented
259C23#1457913011yescharacteristic
260Q8—42no—
261Q8—81no—
262Q8—6321no—
263Q8—6321no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—3161no—
289C2×C4—3161no—
290C2×C4C2×#1192961411yescharacteristic
291C2×C4#1457919211yescharacteristic
292C8—3162no—
293C8—6321no—
294C8—6321no—
295C22—21no—
296C22—21no—
297C22#1909871911yescharacteristic
298C22#1909874611yescharacteristic
299C22#1909875011yescharacteristic
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—42no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—6321no—
311C4—6321no—
312C2—21no—
313C2—21no—
314C2#2376237411yescharacteristic
315C2#2376539011yescharacteristic
316C2#2376542611yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2612C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2612D79
hypocenterC79#12258
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2612

Complements

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

NH ≅ G/Ntype
99. C42×C79Q16semidirect
166. C79#12258semidirect
258. C42#11929614semidirect

Maximal subgroups

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

subgroupindex
#237647502
#237624412
#237623732
#237614302
#237614272
#237613442
C79×#26122
#1225879