Finite Group Databasebeta

← #34197851

Subgroups — #34197851

Order 20224.

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

normal subgroups only

This group has 408 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#23767858C211yesmaximal, characteristic
2#23761714C211yesmaximal, characteristic, complemented
3#23761345C212yesmaximal, complemented
4#23761331C212yesmaximal, complemented
5C79×#1464C211yesmaximal, characteristic, complemented
6C2×#14579135C411yescharacteristic
7#19098616—21no—
8#19098478—21no—
9#19098478—21no—
10#19098464C411yescharacteristic
11#19098464C2211yescharacteristic
12#19098454—21no—
13#19098454—21no—
14#19098432C2211yescharacteristic
15#19098428C411yescharacteristic
16C4×C22⋊Dic79C411yescharacteristic
17#19098289C2212yes—
18#19098287—21no—
19C2×#117×C79C2211yescharacteristic
20#432×C79—21no—
21#324×C79C2212yes—
22#323×C79—21no—
23C22×D316C2×C411yescharacteristic
24C22×C22⋊D79—21no—
25#14579135—41no—
26#14579135—41no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×C22⋊Dic79—21no—
31C2×(C2×D79)⋊C4—21no—
32C2×(C2×D79)⋊C4—21no—
33C2×(C2×D79)⋊C4—21no—
34C2×(C2×D79)⋊C4—21no—
35C2×(C2×D79)⋊C4—21no—
36C2×C4×Dic79C2×C411yescharacteristic
37C2×C4×Dic79C2×C411yescharacteristic
38C4⋊(C79⋊C8)—41no—
39C22⋊(C79⋊C8)—22no—
40C4×C79⋊C8—41no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C23×C4×C79C2×C411yescharacteristic
45#117×C79—41no—
46#117×C79—41no—
47C2×C4⋊C4×C79C2×C411yescharacteristic
48C2×C4⋊C4×C79C2311yescharacteristic
49C2×C22⋊C4×C79D411yescharacteristic
50C2×C22⋊C4×C79D411yescharacteristic
51C2×C42×C79D411yescharacteristic
52C2×C42×C79D411yescharacteristic
53C4⋊C8×C79—41no—
54C22⋊C8×C79—22no—
55#97×C79—41no—
56C23×D79—21no—
57C2×D316—22no—
58C2×D316—41no—
59C2×D316—41no—
60C2×C22⋊D79—42no—
61C2×C22⋊D79—81no—
62C2×C22⋊D79—81no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C2×C79⋊C8—42no—
67C4⋊Dic79—41no—
68Dic79⋊C4—81no—
69Dic79⋊C4—81no—
70C22⋊Dic79—42no—
71(C2×D79)⋊C4—42no—
72(C2×D79)⋊C4—42no—
73(C2×D79)⋊C4—41no—
74(C2×D79)⋊C4—41no—
75(C2×D79)⋊C4—81no—
76(C2×D79)⋊C4—81no—
77C4×Dic79—22no—
78C4×Dic79—41no—
79C24×C79C22⋊C411yescharacteristic
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—22no—
83C22×C4×C79—41no—
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22⋊C411yescharacteristic
86C22×C4×C79C22⋊C411yescharacteristic
87C22×C4×C79C22×C411yescharacteristic
88C22×C4×C79C2×D411yescharacteristic
89C22×C4×C79C2×D411yescharacteristic
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#13112yes—
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—
140C8×C79—82no—
141C79⋊C8—82no—
142C22×C79—21no—
143C22×C79—21no—
144C22×C79—41no—
145C22×C79—41no—
146C22×C79—41no—
147C22×C79—81no—
148C22×C79—81no—
149C22×C79#40811yescharacteristic
150C22×C79C2×#9911yescharacteristic
151C22×C79#41311yescharacteristic
152C2×D79—81no—
153C2×D79—81no—
154C2×D79—81no—
155C2×D79—81no—
156C2×D79—81no—
157C2×D79—81no—
158C2×D79—81no—
159C4×C79—21no—
160C4×C79—21no—
161C4×C79—22no—
162C4×C79—41no—
163C4×C79—41no—
164C4×C79—41no—
165C4×C79—41no—
166Dic79—42no—
167Dic79—81no—
168Dic79—81no—
169#11554—791nomaximal, Sylow
170C2×C79—21no—
171C2×C79—21no—
172C2×C79—81no—
173C2×C79#146811yescharacteristic
174C2×C79#212811yescharacteristic
175C2×C79#213211yescharacteristic
176D79—161no—
177#2500—791no—
178#1464D7911yescharacteristic, complemented
179#1458—791no—
180#1325—792no—
181#1318—792no—
182C79#1155411yescharacteristic, complemented, Sylow
183C2×#121—791no—
184C2×#117C2×D7911yescharacteristic
185#432—21no—
186#431—1581no—
187#394—1581no—
188#394—1581no—
189#390—791no—
190#390—791no—
191#388—1581no—
192#388—1581no—
193#384—791no—
194#382—791no—
195C4×C22⋊C4—791no—
196#324—792no—
197#324C2×D7912yes—
198#323—21no—
199#323—1581no—
200C22×D4—791no—
201C22×D4—1581no—
202C23×C4C4×D7911yescharacteristic
203#121—3161no—
204#121—3161no—
205#117—41no—
206#117—41no—
207C2×C4⋊C4—791no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4C4×D7911yescharacteristic
211C2×C4⋊C4C22×D7911yescharacteristic
212C2×C22⋊C4—1581no—
213C2×C22⋊C4—1581no—
214C2×C22⋊C4—1581no—
215C2×C22⋊C4—1581no—
216C2×C22⋊C4—1581no—
217C2×C22⋊C4—1581no—
218C2×C22⋊C4D31611yescharacteristic
219C2×C22⋊C4C22⋊D7911yescharacteristic
220C2×C42—791no—
221C2×C42—791no—
222C2×C42D31611yescharacteristic
223C2×C42C22⋊D7911yescharacteristic
224C4⋊C8—41no—
225C4⋊C8—3161no—
226C22⋊C8—22no—
227C22⋊C8—1582no—
228#97—41no—
229C4×C8—3161no—
230#95—1581no—
231#95—1581no—
232#95—1581no—
233C24—1581no—
234C24(C2×D79)⋊C411yescharacteristic
235C2×D4—3161no—
236C2×D4—3161no—
237C2×D4—3162no—
238C2×D4—6321no—
239C2×D4—6321no—
240C2×D4—1582no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—1581no—
246C22×C4—1581no—
247C22×C4—1581no—
248C22×C4(C2×D79)⋊C411yescharacteristic
249C22×C4(C2×D79)⋊C411yescharacteristic
250C22×C4(C2×D79)⋊C411yescharacteristic
251C22×C4C2×C4×D7911yescharacteristic
252C22×C4C2×C22⋊D7911yescharacteristic
253C22×C4C2×D31611yescharacteristic
254C2×C8—42no—
255C2×C8—3162no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—3161no—
259C4⋊C4—6321no—
260C4⋊C4—6321no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—3162no—
264C22⋊C4—3162no—
265C22⋊C4—3162no—
266C22⋊C4—3161no—
267C22⋊C4—3161no—
268C22⋊C4—6321no—
269C22⋊C4—6321no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—3161no—
275C42—1582no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—3161no—
282C23—3161no—
283C23—3161no—
284C23—3161no—
285C23—3161no—
286C23—3161no—
287C23—3161no—
288C23C2×(C2×D79)⋊C411yescharacteristic
289D4—6322no—
290D4—6324no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—3161no—
306C2×C4—3161no—
307C2×C4—3161no—
308C2×C4—3161no—
309C2×C4—3161no—
310C2×C4—3161no—
311C2×C4—3161no—
312C2×C4—3161no—
313C2×C4—1582no—
314C2×C4#1457907511yescharacteristic
315C2×C4#1457907511yescharacteristic
316C2×C4#1457916912yes—
317C8—82no—
318C8—6322no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—6321no—
327C22—6321no—
328C22—6321no—
329C22—6321no—
330C22—6321no—
331C22—6321no—
332C22—6321no—
333C22#1909853111yescharacteristic
334C22C2×#1457907511yescharacteristic
335C22#1909854511yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3162no—
344C4—6321no—
345C4—6321no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—12641no—
350C2#2376174911yescharacteristic
351C2#2376507111yescharacteristic
352C2#2376508311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1464C2
socleC22×C79C2×#99
prime radicalC2×#14579135C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1464D79
hypocenterC79#11554
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #23761714C2semidirect
3. #23761345C2semidirect
4. #23761331C2semidirect
5. C79×#1464C2semidirect
178. #1464D79semidirect
182. C79#11554semidirect

Maximal subgroups

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

subgroupindex
#237678582
#237617142
#237613452
#237613312
C79×#14642
#1155479