Finite Group Databasebeta

← #34201207

Subgroups — #34201207

Order 20224.

368 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 59 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, complemented
2#23762441C211yesmaximal, characteristic, complemented
3#23762339C211yesmaximal, characteristic
4#23761427C211yesmaximal, characteristic, complemented
5#23761403C211yesmaximal, characteristic
6#23761346C211yesmaximal, characteristic, complemented
7C79×#2795C211yesmaximal, characteristic
8#19099194C2211yescharacteristic
9#19098898—21no—
10#19098714—21no—
11#19098603—21no—
12#19098332C2211yescharacteristic
13#19098323—21no—
14#19098299—21no—
15#19098299—21no—
16#19098289C2211yescharacteristic
17#19098289C2211yescharacteristic
18#400×C79C2211yescharacteristic
19#396×C79—21no—
20#394×C79—21no—
21#394×C79C2211yescharacteristic
22#388×C79—21no—
23#388×C79—21no—
24#388×C79—21no—
25#387×C79—21no—
26#384×C79C2211yescharacteristic
27C2×#11929612—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#14579084—41no—
38#14579082—41no—
39C22⋊(C79⋊C8)—21no—
40C22⋊(C79⋊C8)—21no—
41C22×D4×C79D411yescharacteristic
42C2×C4⋊C4×C79—21no—
43C2×C4⋊C4×C79—21no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C4⋊C4×C79D411yescharacteristic
47C2×C4⋊C4×C79C2311yescharacteristic
48C2×C22⋊C4×C79—21no—
49C2×C22⋊C4×C79—21no—
50C2×C22⋊C4×C79—21no—
51C2×C22⋊C4×C79—21no—
52C2×C22⋊C4×C79—21no—
53C2×C22⋊C4×C79—21no—
54C2×C42×C79—21no—
55C2×C42×C79D411yescharacteristic
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#11929612—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—
78C24×C79—21no—
79C2×D4×C79—41no—
80C2×D4×C79—81no—
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—41no—
92C4⋊C4×C79—41no—
93C4⋊C4×C79—81no—
94C4⋊C4×C79—81no—
95C22⋊C4×C79—42no—
96C22⋊C4×C79—42no—
97C22⋊C4×C79—41no—
98C22⋊C4×C79—41no—
99C22⋊C4×C79—81no—
100C22⋊C4×C79—81no—
101C22⋊C4×C79—81no—
102C42×C79—21no—
103C42×C79—21no—
104C42×C79—81no—
105C23×C79—41no—
106C23×C79—41no—
107C23×C79—41no—
108C23×C79—41no—
109C23×C79—41no—
110C23×C79—41no—
111C23×C79—41no—
112C23×C79#12011yescharacteristic
113D4×C79—42no—
114D4×C79—81no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—22no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
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×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—41no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79#13511yescharacteristic
137C2×C4×C79#13711yescharacteristic
138C2×Dic79—41no—
139C2×Dic79—41no—
140Dic158—81no—
141Dic158—81no—
142C79⋊C8—42no—
143C79⋊C8—81no—
144C79⋊C8—81no—
145C22×C79—21no—
146C22×C79—21no—
147C22×C79—81no—
148C22×C79—81no—
149C22×C79—81no—
150C22×C79—81no—
151C22×C79—81no—
152C22×C79—81no—
153C22×C79—81no—
154C22×C79#45211yescharacteristic
155C22×C79#45611yescharacteristic
156C22×C79B(4,𝔽2)11yescharacteristic
157C4×C79—21no—
158C4×C79—21no—
159C4×C79—42no—
160C4×C79—41no—
161C4×C79—41no—
162C4×C79—81no—
163C4×C79—81no—
164C4×C79—81no—
165C4×C79—81no—
166Dic79—81no—
167Dic79—81no—
168#12238—791nomaximal, Sylow
169C2×C79—21no—
170C2×C79—21no—
171C2×C79—161no—
172C2×C79#160311yescharacteristic
173C2×C79#220011yescharacteristic
174C2×C79#220411yescharacteristic
175#2795D7911yescharacteristic
176#2060—791no—
177#1617—791no—
178#1599—791no—
179#1353—791no—
180#1345—791no—
181#1325—791no—
182C79#1223811yescharacteristic, complemented, Sylow
183#544—791no—
184#484—1581no—
185#450—1581no—
186#428—1581no—
187#400C2×D7911yescharacteristic
188#396—21no—
189#394—21no—
190#394C2×D7911yescharacteristic
191#388—21no—
192#388—21no—
193#388—21no—
194#387—21no—
195#384C2×D7911yescharacteristic
196#343—791no—
197#339—1581no—
198#328—1581no—
199#328—1581no—
200#324—791no—
201#324—791no—
202C22×D4C22⋊D7911yescharacteristic
203C2×SD16—3161no—
204C2×M16—1581no—
205C4⋊Q8—3161no—
206#125—3161no—
207C22⋊Q8—1581no—
208C22⋊Q8—1581no—
209#117—1581no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C22⋊D7911yescharacteristic
214C2×C4⋊C4C22⋊D7911yescharacteristic
215C2×C4⋊C4C22×D7911yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42—21no—
223C2×C42C22⋊D7911yescharacteristic
224#107—3161no—
225#107—3161no—
226#106—3161no—
227#103—3161no—
228#102—3161no—
229C22⋊C8—1581no—
230C22⋊C8—1581no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95C22⋊D7911yescharacteristic
236#95C22⋊D7911yescharacteristic
237C24—21no—
238C2×Q8—1581no—
239C2×Q8—1581no—
240C2×D4—41no—
241C2×D4—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4C2×C22⋊D7911yescharacteristic
248C22×C4C2×C22⋊D7911yescharacteristic
249C22×C4C2×C22⋊D7911yescharacteristic
250SD16—6322no—
251M16—3161no—
252M16—1582no—
253C2×C8—3161no—
254C2×C8—3161no—
255C2×C8—3161no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C4⋊C4—3161no—
263C4⋊C4—3161no—
264C4⋊C4—1581no—
265C4⋊C4—1581no—
266C4⋊C4—1581no—
267C4⋊C4—1581no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—81no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C22⋊C4—3161no—
276C42—21no—
277C42—21no—
278C42—81no—
279C42—3161no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23#1457913011yescharacteristic
288Q8—6321no—
289Q8—6321no—
290D4—42no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—3161no—
314C2×C4—3161no—
315C2×C4#1457918111yescharacteristic
316C2×C4#1457919111yescharacteristic
317C8—3162no—
318C8—6321no—
319C8—6321no—
320C22—21no—
321C22—21no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22#1909872511yescharacteristic
330C22#1909874611yescharacteristic
331C22#1909875011yescharacteristic
332C4—21no—
333C4—21no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—6321no—
342C4—6321no—
343C2—21no—
344C2—21no—
345C2—161no—
346C2#2376236311yescharacteristic
347C2#2376540711yescharacteristic
348C2#2376542611yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2795C2
socleC22×C79B(4,𝔽2)
prime radicalC22×D4×C79D4
hypercenter#2795D79
hypocenterC79#12238
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2795

Complements

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

NH ≅ G/Ntype
1. #23764745C2semidirect
2. #23762441C2semidirect
4. #23761427C2semidirect
6. #23761346C2semidirect
182. C79#12238semidirect

Maximal subgroups

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

subgroupindex
#237647452
#237624412
#237623392
#237614272
#237614032
#237613462
C79×#27952
#1223879