Finite Group Databasebeta

← #34198944

Subgroups — #34198944

Order 20224.

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

normal subgroups only

This group has 502 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#23769150C211yesmaximal, characteristic
2#23763477C211yesmaximal, characteristic, complemented
3#23761398C214yesmaximal, complemented
4C79×#1464C211yesmaximal, characteristic, complemented
5C2×#14579158C411yescharacteristic, complemented
6C2×#14579135—21no—
7#19098480—21no—
8#19098480C412yescomplemented
9#19098472—21no—
10#19098472C2212yes—
11#19098465—21no—
12#19098429C411yescharacteristic, complemented
13#19098426—21no—
14#19098330—21no—
15#19098330C2212yes—
16C2×#117×C79C2211yescharacteristic
17#432×C79—21no—
18#324×C79—21no—
19#323×C79C2212yes—
20C22×D316—21no—
21C22×D316—21no—
22C22×D316C2×C412yes—
23C22×C22⋊D79—22no—
24#14579158—41no—
25#14579158—41no—
26#14579135—44no—
27C2×C4⋊Dic79—21no—
28C2×C22⋊Dic79—22no—
29C2×(C2×D79)⋊C4—22no—
30C2×(C2×D79)⋊C4—22no—
31C2×(C2×D79)⋊C4—22no—
32C79⋊#95—21no—
33C79⋊#95—41no—
34C79⋊#95—41no—
35C79⋊#95C2×C412yes—
36C23×C4×C79D411yescharacteristic
37#117×C79—41no—
38#117×C79—41no—
39C2×C4⋊C4×C79C2×C411yescharacteristic
40C2×C4⋊C4×C79D411yescharacteristic
41C2×C22⋊C4×C79D411yescharacteristic
42C2×C22⋊C4×C79C2311yescharacteristic
43C2×C42×C79C2×C411yescharacteristic
44C2×C42×C79D411yescharacteristic
45C4⋊C8×C79—41no—
46C22⋊C8×C79—22no—
47#97×C79—41no—
48C23×D79—22no—
49C2×D316—42no—
50C2×D316—42no—
51C2×D316—42no—
52C2×D316—42no—
53C2×D316—41no—
54C2×D316—41no—
55C2×D316—82no—
56C2×D316—82no—
57C2×C22⋊D79—44no—
58C2×C22⋊D79—82no—
59C2×C22⋊D79—82no—
60C22×Dic79—22no—
61C22×Dic79—42no—
62C4⋊Dic79—42no—
63C22⋊Dic79—42no—
64C22⋊Dic79—42no—
65(C2×D79)⋊C4—44no—
66(C2×D79)⋊C4—82no—
67(C2×D79)⋊C4—82no—
68C24×C79C2×D411yescharacteristic
69C22×C4×C79—22no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—41no—
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22×C411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79C2×C8×C79—42no—
80C4⋊C4×C79—41no—
81C4⋊C4×C79—41no—
82C22⋊C4×C79—41no—
83C22⋊C4×C79—41no—
84C42×C79—21no—
85C42×C79—21no—
86C42×C79—21no—
87C42×C79—21no—
88C23×C79—21no—
89C23×C79—21no—
90C23×C79—41no—
91C23×C79—41no—
92C23×C79C2×C22⋊C411yescharacteristic
93C23×C79AGL(1,ℤ/8)12yes—
94C22×D79—42no—
95C22×D79—42no—
96C22×D79—42no—
97C22×D79—42no—
98C22×D79—42no—
99C22×D79—42no—
100C22×D79—42no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79#9911yescharacteristic
117C2×C4×C79#9911yescharacteristic
118C22⋊D79—88no—
119C2×Dic79—42no—
120C2×Dic79—42no—
121C2×Dic79—42no—
122C2×Dic79—82no—
123C2×Dic79—82no—
124C2×Dic79—82no—
125D316—82no—
126D316—82no—
127D316—84no—
128D316—84no—
129C8×C79—82no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79—22no—
133C22×C79—42no—
134C22×C79—41no—
135C22×C79—41no—
136C22×C79—81no—
137C22×C79C2×#9911yescharacteristic
138C22×C79#41811yescharacteristic
139C22×C79#42111yescharacteristic
140C2×D79—82no—
141C2×D79—82no—
142C2×D79—82no—
143C2×D79—82no—
144C2×D79—82no—
145C2×D79—82no—
146C2×D79—82no—
147C4×C79—21no—
148C4×C79—21no—
149C4×C79—41no—
150C4×C79—41no—
151C4×C79—41no—
152C4×C79—41no—
153C4×C79—41no—
154Dic79—82no—
155Dic79—162no—
156#11800—791nomaximal, Sylow
157C2×C79—21no—
158C2×C79—21no—
159C2×C79—42no—
160C2×C79#151011yescharacteristic
161C2×C79#212211yescharacteristic
162C2×C79#212811yescharacteristic
163D79—162no—
164#2619—791no—
165#1820—791no—
166#1464D7911yescharacteristic, complemented
167#1343—794no—
168C79#1180011yescharacteristic, complemented, Sylow
169C2×#127—791no—
170C2×#121—1581no—
171C2×#117C2×D7911yescharacteristic
172#432—21no—
173#394—792no—
174#394—1581no—
175#392—792no—
176#392—1581no—
177#390—1581no—
178#382—791no—
179#381—1581no—
180#342—792no—
181#342—1581no—
182#324—21no—
183#323C2×D7912yes—
184C22×D4—792no—
185C22×D4—1582no—
186C22×D4—1581no—
187C22×D4—1581no—
188C23×C4D31611yescharacteristic
189#127—3161no—
190#127—3161no—
191#121—3164no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—1581no—
195C2×C4⋊C4C4×D7911yescharacteristic
196C2×C4⋊C4C22⋊D7911yescharacteristic
197C2×C22⋊C4—1582no—
198C2×C22⋊C4—1582no—
199C2×C22⋊C4—1582no—
200C2×C22⋊C4—1582no—
201C2×C22⋊C4D31611yescharacteristic
202C2×C22⋊C4C22×D7911yescharacteristic
203C2×C42C4×D7911yescharacteristic
204C2×C42C22⋊D7911yescharacteristic
205C4⋊C8—41no—
206C22⋊C8—22no—
207#97—41no—
208#95—3161no—
209#95—3161no—
210#95—792no—
211#95—1581no—
212C24—1582no—
213C24C2×D31611yescharacteristic
214C2×D4—3162no—
215C2×D4—3162no—
216C2×D4—3162no—
217C2×D4—3162no—
218C2×D4—3164no—
219C2×D4—3161no—
220C2×D4—3161no—
221C2×D4—6322no—
222C2×D4—6322no—
223C2×D4—6322no—
224C2×D4—6322no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—3162no—
230C22×C4—1582no—
231C22×C4(C2×D79)⋊C411yescharacteristic
232C22×C4(C2×D79)⋊C411yescharacteristic
233C22×C4(C2×D79)⋊C411yescharacteristic
234C22×C4(C2×D79)⋊C411yescharacteristic
235C22×C4C2×C4×D7911yescharacteristic
236C22×C4C2×C22⋊D7911yescharacteristic
237C2×C8—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—3162no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—3164no—
244C22⋊C4—3162no—
245C22⋊C4—3162no—
246C22⋊C4—6322no—
247C22⋊C4—6322no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—3162no—
257C23—3162no—
258C23—3162no—
259C23—3162no—
260C23—3162no—
261C23—3162no—
262C23—3162no—
263C23C2×(C2×D79)⋊C411yescharacteristic
264C23#1457918912yes—
265D4—6322no—
266D4—6322no—
267D4—6324no—
268D4—6324no—
269D4—6328no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—3162no—
286C2×C4—3162no—
287C2×C4—3162no—
288C2×C4—6322no—
289C2×C4—6322no—
290C2×C4—6322no—
291C2×C4#1457907511yescharacteristic
292C2×C4#1457907511yescharacteristic
293C8—82no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—6322no—
302C22—6322no—
303C22—6322no—
304C22—6322no—
305C22—6322no—
306C22—6322no—
307C22—6322no—
308C22C2×#1457907511yescharacteristic
309C22#1909856911yescharacteristic
310C22#1909857911yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—6322no—
319C4—12642no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—12642no—
324C2#2376193011yescharacteristic
325C2#2376505111yescharacteristic
326C2#2376507111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1464C2
socleC22×C79C2×#99
prime radical#23769150C2
cosocleC2×C22⋊C4×C79C23
hypercenter#1464D79
hypocenterC79#11800
LCSG ▷ C22×C4×C79 ▷ C23×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. #23763477C2semidirect
3. #23761398C2semidirect
4. C79×#1464C2semidirect
5. C2×#14579158C4semidirect
8. #19098480C4semidirect
12. #19098429C4semidirect
166. #1464D79semidirect
168. C79#11800semidirect

Maximal subgroups

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

subgroupindex
#237691502
#237634772
#237613982
C79×#14642
#1180079