Finite Group Databasebeta

← #34201342

Subgroups — #34201342

Order 20224.

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

normal subgroups only

This group has 332 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#23764745C211yesmaximal, characteristic
2#23762441C211yesmaximal, characteristic
3#23762373C211yesmaximal, characteristic
4#23761430C211yesmaximal, characteristic
5#23761427C211yesmaximal, characteristic
6#23761346C211yesmaximal, characteristic
7C79×#2843C211yesmaximal, 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×C79—21no—
19#400×C79C2211yescharacteristic
20#398×C79—21no—
21#398×C79—21no—
22#398×C79—21no—
23#395×C79—21no—
24#395×C79C2211yescharacteristic
25#384×C79C2211yescharacteristic
26#383×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×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79D411yescharacteristic
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C4⋊C4×C79C2311yescharacteristic
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—41no—
79C2×Q8×C79—81no—
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—41no—
91C4⋊C4×C79—41no—
92C4⋊C4×C79—81no—
93C4⋊C4×C79—81no—
94C4⋊C4×C79—81no—
95C4⋊C4×C79—81no—
96C4⋊C4×C79—81no—
97C4⋊C4×C79—81no—
98C42×C79—21no—
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—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—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—42no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79C2×Q1611yescharacteristic
127C2×C4×C79#13711yescharacteristic
128C2×Dic79—41no—
129C2×Dic79—41no—
130Dic158—81no—
131Dic158—81no—
132C79⋊C8—42no—
133C79⋊C8—81no—
134C79⋊C8—81no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79#45111yescharacteristic
138C22×C79#45611yescharacteristic
139C22×C79B(4,𝔽2)11yescharacteristic
140C4×C79—21no—
141C4×C79—21no—
142C4×C79—42no—
143C4×C79—41no—
144C4×C79—41no—
145C4×C79—81no—
146C4×C79—81no—
147C4×C79—81no—
148C4×C79—81no—
149C4×C79—81no—
150Dic79—81no—
151Dic79—81no—
152#12259—791nomaximal, Sylow
153C2×C79—21no—
154C2×C79—21no—
155C2×C79#160511yescharacteristic
156C2×C79#220011yescharacteristic
157C2×C79#220311yescharacteristic
158#2843D7911yescharacteristic
159#2060—791no—
160#1617—791no—
161#1605—791no—
162#1354—791no—
163#1353—791no—
164#1325—791no—
165C79#1225911yescharacteristic, complemented, Sylow
166#544—791no—
167#484—1581no—
168#451—1581no—
169#428—1581no—
170#400—21no—
171#400C2×D7911yescharacteristic
172#398—21no—
173#398—21no—
174#398—21no—
175#395—21no—
176#395C2×D7911yescharacteristic
177#384C2×D7911yescharacteristic
178#383—21no—
179#343—791no—
180#339—1581no—
181#328—1581no—
182#328—1581no—
183#324—791no—
184#324—791no—
185C22×Q8C22⋊D7911yescharacteristic
186C2×Q16—3161no—
187C2×M16—1581no—
188C4⋊Q8—3161no—
189#125—3161no—
190C22⋊Q8—1581no—
191C22⋊Q8—1581no—
192#117—1581no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C22⋊D7911yescharacteristic
200C2×C4⋊C4C22⋊D7911yescharacteristic
201C2×C4⋊C4C22×D7911yescharacteristic
202C2×C42C22⋊D7911yescharacteristic
203#107—3161no—
204#107—3161no—
205#106—3161no—
206#103—3161no—
207#103—3161no—
208C22⋊C8—1581no—
209C22⋊C8—1581no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95C22⋊D7911yescharacteristic
219#95C22⋊D7911yescharacteristic
220C2×Q8—41no—
221C2×Q8—81no—
222C2×Q8—1581no—
223C2×Q8—1581no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4C2×C22⋊D7911yescharacteristic
231C22×C4C2×C22⋊D7911yescharacteristic
232C22×C4C2×C22⋊D7911yescharacteristic
233Q16—6322no—
234M16—3161no—
235M16—1582no—
236C2×C8—3161no—
237C2×C8—3161no—
238C2×C8—3161no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C4⋊C4—3161no—
249C4⋊C4—3161no—
250C4⋊C4—1581no—
251C4⋊C4—1581no—
252C4⋊C4—1581no—
253C4⋊C4—1581no—
254C22⋊C4—3161no—
255C42—21no—
256C42—3161no—
257C42#1192961412yescomplemented
258C23#1457913011yescharacteristic
259Q8—42no—
260Q8—81no—
261Q8—6321no—
262Q8—6321no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
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—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4C2×#1192961411yescharacteristic
289C2×C4#1457919211yescharacteristic
290C8—3162no—
291C8—6321no—
292C8—6321no—
293C22—21no—
294C22—21no—
295C22#1909871911yescharacteristic
296C22#1909874611yescharacteristic
297C22#1909875011yescharacteristic
298C4—21no—
299C4—21no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—6321no—
309C4—6321no—
310C2—21no—
311C2—21no—
312C2#2376237411yescharacteristic
313C2#2376540711yescharacteristic
314C2#2376542111yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2843C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2843D79
hypocenterC79#12259
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2843

Complements

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

NH ≅ G/Ntype
99. C42×C79Q16semidirect
165. C79#12259semidirect
257. C42#11929614semidirect

Maximal subgroups

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

subgroupindex
#237647452
#237624412
#237623732
#237614302
#237614272
#237613462
C79×#28432
#1225979