Finite Group Databasebeta

← #34199185

Subgroups — #34199185

Order 20224.

418 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 79 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#23769087C211yesmaximal, characteristic
2#23761714C211yesmaximal, characteristic, complemented
3#23761429C212yesmaximal, complemented
4#23761402C212yesmaximal, complemented
5C79×#1822C211yesmaximal, characteristic, complemented
6C2×#14579149C411yescharacteristic
7#19098616—21no—
8#19098483C2211yescharacteristic
9#19098480C2211yescharacteristic
10#19098475C411yescharacteristic
11#19098472C411yescharacteristic
12#19098463—21no—
13#19098454—21no—
14#19098453—21no—
15#19098442—21no—
16#19098409C411yescharacteristic
17#19098289C2212yes—
18#19098287—21no—
19C2×#117×C79C2211yescharacteristic
20#407×C79—21no—
21#343×C79—21no—
22#343×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C22×D316C2×C411yescharacteristic
25C22×C22⋊D79—21no—
26#14579149—41no—
27#14579149—41no—
28C2×Dic79⋊C4—21no—
29C2×C22⋊Dic79—21no—
30C2×(C2×D79)⋊C4—21no—
31C2×(C2×D79)⋊C4—21no—
32C2×(C2×D79)⋊C4—21no—
33C2×(C2×D79)⋊C4—21no—
34C2×(C2×D79)⋊C4—21no—
35C2×C4×Dic79—21no—
36C4⋊(C79⋊C8)—41no—
37C22⋊(C79⋊C8)—22no—
38C4×C79⋊C8—41no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C23×C4×C79D411yescharacteristic
45C2×M16×C79—22no—
46#117×C79—41no—
47#117×C79—41no—
48C2×C4⋊C4×C79D411yescharacteristic
49C2×C4⋊C4×C79C2311yescharacteristic
50C2×C22⋊C4×C79C2×C411yescharacteristic
51C2×C22⋊C4×C79D411yescharacteristic
52C2×C42×C79C2×C411yescharacteristic
53C2×C42×C79D411yescharacteristic
54C22⋊C8×C79—41no—
55C22⋊C8×C79—41no—
56C23×D79—21no—
57C2×Dic158—41no—
58C2×Dic158—81no—
59C2×D316—41no—
60C2×D316—81no—
61C2×C22⋊D79—42no—
62C2×C22⋊D79—42no—
63C2×C22⋊D79—81no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C22×Dic79—21no—
67C2×C79⋊C8—42no—
68Dic79⋊C4—81no—
69C22⋊Dic79—81no—
70(C2×D79)⋊C4—42no—
71(C2×D79)⋊C4—42no—
72(C2×D79)⋊C4—41no—
73(C2×D79)⋊C4—41no—
74(C2×D79)⋊C4—42no—
75(C2×D79)⋊C4—81no—
76C4×Dic79—42no—
77C24×C79C22⋊C411yescharacteristic
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—22no—
81C22×C4×C79—41no—
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22×C411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C22×C4×C79C2×D411yescharacteristic
88M16×C79—42no—
89M16×C79—42no—
90C2×C8×C79—42no—
91C4⋊C4×C79—41no—
92C4⋊C4×C79—41no—
93C22⋊C4×C79—41no—
94C22⋊C4×C79—41no—
95C42×C79—21no—
96C42×C79—21no—
97C42×C79—21no—
98C42×C79—21no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—21no—
102C23×C79—41no—
103C23×C79—41no—
104C23×C79C2×C22⋊C411yescharacteristic
105C22×D79—41no—
106C22×D79—41no—
107C22×D79—41no—
108C22×D79—41no—
109C22×D79—41no—
110C22×D79—41no—
111C22×D79—41no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—42no—
121C2×C4×C79—42no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79#9911yescharacteristic
127C2×C4×C79#9911yescharacteristic
128C2×C4×C79#13512yes—
129C22⋊D79—84no—
130C2×Dic79—22no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139D316—82no—
140Dic158—82no—
141C8×C79—82no—
142C79⋊C8—82no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—42no—
146C22×C79—41no—
147C22×C79—41no—
148C22×C79—41no—
149C22×C79—81no—
150C22×C79C2×#9911yescharacteristic
151C22×C79#41611yescharacteristic
152C22×C79#42111yescharacteristic
153C2×D79—81no—
154C2×D79—81no—
155C2×D79—81no—
156C2×D79—81no—
157C2×D79—81no—
158C2×D79—81no—
159C2×D79—81no—
160C4×C79—21no—
161C4×C79—21no—
162C4×C79—22no—
163C4×C79—41no—
164C4×C79—41no—
165C4×C79—41no—
166C4×C79—41no—
167Dic79—42no—
168Dic79—81no—
169Dic79—81no—
170#11851—791nomaximal, Sylow
171C2×C79—21no—
172C2×C79—21no—
173C2×C79—81no—
174C2×C79#151611yescharacteristic
175C2×C79#212211yescharacteristic
176C2×C79#213211yescharacteristic
177D79—161no—
178#2610—791no—
179#1822D7911yescharacteristic, complemented
180#1458—791no—
181#1354—792no—
182#1345—792no—
183C79#1185111yescharacteristic, complemented, Sylow
184C2×#124—791no—
185C2×#117C2×D7911yescharacteristic
186#431—1581no—
187#407—21no—
188#395—791no—
189#394—791no—
190#393—791no—
191#392—791no—
192#390—1581no—
193#388—1581no—
194#388—1581no—
195#386—1581no—
196#376—791no—
197#343—21no—
198#343C2×D7912yes—
199#324—792no—
200#323—1581no—
201C22×Q8—791no—
202C22×D4—791no—
203C22×D4—1581no—
204C23×C4C22⋊D7911yescharacteristic
205C2×M16—22no—
206#124—3161no—
207#124—3161no—
208#117—41no—
209#117—41no—
210C2×C4⋊C4—1581no—
211C2×C4⋊C4C22⋊D7911yescharacteristic
212C2×C4⋊C4C22×D7911yescharacteristic
213C2×C22⋊C4—1581no—
214C2×C22⋊C4—1581no—
215C2×C22⋊C4—1581no—
216C2×C22⋊C4—1581no—
217C2×C22⋊C4—1581no—
218C2×C22⋊C4—1581no—
219C2×C22⋊C4C4×D7911yescharacteristic
220C2×C22⋊C4D31611yescharacteristic
221C2×C42—1581no—
222C2×C42C4×D7911yescharacteristic
223C2×C42D31611yescharacteristic
224C4⋊C8—3161no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8—1582no—
228C4×C8—3161no—
229#95—791no—
230#95—791no—
231#95—1581no—
232#95—1581no—
233#95—1581no—
234C24—1581no—
235C24(C2×D79)⋊C411yescharacteristic
236C2×Q8—3161no—
237C2×Q8—6321no—
238C2×D4—3161no—
239C2×D4—3162no—
240C2×D4—3162no—
241C2×D4—6321no—
242C2×D4—6321no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—1581no—
248C22×C4—1581no—
249C22×C4—1581no—
250C22×C4(C2×D79)⋊C411yescharacteristic
251C22×C4(C2×D79)⋊C411yescharacteristic
252C22×C4(C2×D79)⋊C411yescharacteristic
253C22×C4C2×C4×D7911yescharacteristic
254C22×C4C2×C22⋊D7911yescharacteristic
255C22×C4C2×D31611yescharacteristic
256M16—42no—
257M16—42no—
258C2×C8—42no—
259C2×C8—3162no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—6321no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—3162no—
266C22⋊C4—3162no—
267C22⋊C4—3161no—
268C22⋊C4—3161no—
269C22⋊C4—3162no—
270C22⋊C4—6321no—
271C22⋊C4—6321no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—3162no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—3161no—
283C23—3161no—
284C23—3161no—
285C23—3161no—
286C23—3161no—
287C23—3161no—
288C23—3161no—
289C23C2×(C2×D79)⋊C411yescharacteristic
290Q8—6322no—
291D4—6322no—
292D4—6324no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—3161no—
308C2×C4—3161no—
309C2×C4—3161no—
310C2×C4—3161no—
311C2×C4—3161no—
312C2×C4—3161no—
313C2×C4—3161no—
314C2×C4—3161no—
315C2×C4—1582no—
316C2×C4#1457907511yescharacteristic
317C2×C4#1457907511yescharacteristic
318C2×C4#1457918112yes—
319C8—82no—
320C8—6322no—
321C22—21no—
322C22—21no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—6321no—
329C22—6321no—
330C22—6321no—
331C22—6321no—
332C22—6321no—
333C22—6321no—
334C22—6321no—
335C22C2×#1457907511yescharacteristic
336C22#1909855711yescharacteristic
337C22#1909858111yescharacteristic
338C4—22no—
339C4—21no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—3162no—
346C4—6321no—
347C4—6321no—
348C2—21no—
349C2—21no—
350C2—81no—
351C2—12641no—
352C2#2376195411yescharacteristic
353C2#2376505111yescharacteristic
354C2#2376508311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1822C2
socleC22×C79C2×#99
prime radical#19098472C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1822D79
hypocenterC79#11851
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
2. #23761714C2semidirect
3. #23761429C2semidirect
4. #23761402C2semidirect
5. C79×#1822C2semidirect
179. #1822D79semidirect
183. C79#11851semidirect

Maximal subgroups

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

subgroupindex
#237690872
#237617142
#237614292
#237614022
C79×#18222
#1185179