Finite Group Databasebeta

← #34197552

Subgroups — #34197552

Order 20224.

524 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 87 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#23766124C211yesmaximal, characteristic
2#23761717C212yesmaximal, complemented
3#23761343C212yesmaximal, complemented
4#23761332C211yesmaximal, characteristic, complemented
5C79×#1318C211yesmaximal, characteristic, complemented
6C2×#14579157C411yescharacteristic
7C2×D4×Dic79C411yescharacteristic
8C2×#14579123C2211yescharacteristic
9C22⋊C8×D79—22no—
10#19098465C411yescharacteristic
11#19098440—22no—
12#19098439—22no—
13#19098428C2211yescharacteristic
14#19098409C411yescharacteristic
15#19098289C2212yes—
16#19098286—22no—
17#390×C79C2211yescharacteristic
18#324×C79C2212yes—
19C22×D316C2×C411yescharacteristic
20C22×C22⋊D79—22no—
21C23×Dic79—21no—
22C22×C4×D79D412yes—
23C2×C8×D79—42no—
24#14579157—22no—
25#14579157C814yescomplemented
26D4×Dic79—42no—
27#14579123—42no—
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×C22⋊Dic79—21no—
30C2×(C2×D79)⋊C4—22no—
31C2×(C2×D79)⋊C4D412yes—
32C2×C4×Dic79C2×C411yescharacteristic
33C2×C4×Dic79C2×C411yescharacteristic
34C22⋊(C79⋊C8)—22no—
35(C2×D79)⋊C8—42no—
36C79⋊#95—22no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C22×D4×C79C2×C411yescharacteristic
40C2×C4⋊C4×C79C2311yescharacteristic
41C2×C22⋊C4×C79—22no—
42C2×C42×C79C2×C411yescharacteristic
43C22⋊C8×C79—22no—
44C23×D79C22⋊C412yes—
45C2×D316—41no—
46C2×D316—41no—
47C2×D316C2×C812yes—
48C2×D316M1612yes—
49C2×C22⋊D79—28no—
50C2×C22⋊D79—82no—
51C2×C22⋊D79—82no—
52C22×Dic79—21no—
53C22×Dic79—21no—
54C22×Dic79—42no—
55C22×Dic79—82no—
56C22×Dic79C22⋊C411yescharacteristic
57C22×Dic79C22⋊C411yescharacteristic
58C2×C4×D79—22no—
59C2×C4×D79—22no—
60C2×C4×D79—82no—
61C2×C79⋊C8—42no—
62C8×D79—84no—
63C4⋊Dic79—41no—
64C22⋊Dic79—42no—
65(C2×D79)⋊C4—42no—
66(C2×D79)⋊C4—82no—
67C4×Dic79—22no—
68C4×Dic79—21no—
69C4×Dic79C2×C812yescomplemented
70C24×C79—21no—
71C2×D4×C79—21no—
72C2×D4×C79—41no—
73C2×D4×C79—41no—
74C2×D4×C79C2×C812yescomplemented
75C22×C4×C79—21no—
76C22×C4×C79C22×C411yescharacteristic
77C22×C4×C79C2×D412yes—
78C2×C8×C79—42no—
79C4⋊C4×C79—41no—
80C22⋊C4×C79—82no—
81C42×C79—41no—
82C23×C79—22no—
83C23×C79—42no—
84C23×C79—42no—
85C23×C79—41no—
86C23×C79—41no—
87C23×C79C2×C22⋊C411yescharacteristic
88C22×D79—22no—
89C22×D79—22no—
90C22×D79—42no—
91C22×D79—42no—
92C22×D79C22⋊C814yes—
93D4×C79—42no—
94D4×C79—42no—
95C2×C4×C79—21no—
96C2×C4×C79—21no—
97C2×C4×C79—22no—
98C2×C4×C79—42no—
99C2×C4×C79—41no—
100C2×C4×C79—41no—
101C2×C4×C79—41no—
102C2×C4×C79C22×C811yescharacteristic
103C2×C4×C79C2×M1611yescharacteristic
104C22⋊D79—88no—
105C2×Dic79—22no—
106C2×Dic79—21no—
107C2×Dic79—41no—
108C2×Dic79—41no—
109C2×Dic79—41no—
110C2×Dic79—41no—
111C2×Dic79—42no—
112C2×Dic79—41no—
113C2×Dic79—84no—
114C2×Dic79#9912yes—
115D316—42no—
116D316—42no—
117C4×D79—44no—
118C4×D79—82no—
119C8×C79—82no—
120C79⋊C8—82no—
121C22×C79—21no—
122C22×C79—21no—
123C22×C79—44no—
124C22×C79—42no—
125C22×C79—82no—
126C22×C79—81no—
127C22×C79—81no—
128C22×C79C2×C22⋊C811yescharacteristic
129C22×C79C2×#9911yescharacteristic
130C22×C79#41311yescharacteristic
131C2×D79—24no—
132C2×D79—42no—
133C2×D79—42no—
134C2×D79—44no—
135C2×D79—82no—
136C4×C79—21no—
137C4×C79—21no—
138C4×C79—42no—
139C4×C79—81no—
140Dic79—22no—
141Dic79—42no—
142Dic79—41no—
143Dic79—81no—
144#11479—791nomaximal, Sylow
145C2×C79—21no—
146C2×C79—21no—
147C2×C79—82no—
148C2×C79#145911yescharacteristic
149C2×C79#212711yescharacteristic
150C2×C79#213411yescharacteristic
151D79—44no—
152#2328—791no—
153#1459—792no—
154#1324—792no—
155#1318—791no—
156#1318D7911yescharacteristic, complemented
157C79#1147911yescharacteristic, complemented, Sylow
158C2×#127—791no—
159C2×C4×D4—791no—
160C2×C4×D4—791no—
161C2×C22⋊C8—1582no—
162#390—791no—
163#390C2×D7911yescharacteristic
164#386—1582no—
165#386—1582no—
166#382—791no—
167#376—791no—
168#324—792no—
169#324C2×D7912yes—
170#323—1582no—
171C22×D4—791no—
172C22×D4—1582no—
173C22×D4C4×D7911yescharacteristic
174C23×C4—792no—
175C23×C4—1581no—
176C22×C8—3162no—
177#127—794no—
178#127—1582no—
179C4×D4—3162no—
180C4×D4—3162no—
181C2×C4⋊C4—791no—
182C2×C4⋊C4C22×D7911yescharacteristic
183C2×C22⋊C4—22no—
184C2×C22⋊C4—792no—
185C2×C22⋊C4—1581no—
186C2×C22⋊C4—1582no—
187C2×C42—791no—
188C2×C42—791no—
189C2×C42C4×D7911yescharacteristic
190C22⋊C8—22no—
191C22⋊C8—3162no—
192C22⋊C8—1582no—
193#95—1581no—
194#95—1581no—
195#95—1582no—
196C24—21no—
197C24—792no—
198C2×D4—21no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—3161no—
202C2×D4—3161no—
203C2×D4—792no—
204C2×D4—792no—
205C2×D4—6322no—
206C2×D4—6322no—
207C2×D4—1588no—
208C2×D4C8×D7912yescomplemented
209C22×C4—21no—
210C22×C4—3162no—
211C22×C4—791no—
212C22×C4—791no—
213C22×C4—6322no—
214C22×C4—6322no—
215C22×C4—1582no—
216C22×C4—1582no—
217C22×C4—1581no—
218C22×C4—1581no—
219C22×C4C2×C4×D7911yescharacteristic
220C22×C4D4×D7912yes—
221C2×C8—42no—
222C2×C8—3162no—
223C2×C8—6324no—
224C4⋊C4—41no—
225C4⋊C4—3161no—
226C22⋊C4—82no—
227C22⋊C4—3162no—
228C22⋊C4—3162no—
229C22⋊C4—6322no—
230C42—41no—
231C42—792no—
232C42—1582no—
233C42—1581no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23—41no—
238C23—41no—
239C23—3162no—
240C23—3162no—
241C23—794no—
242C23—1582no—
243C23—1582no—
244C23C22⋊C4×D7911yescharacteristic
245D4—42no—
246D4—42no—
247D4—3162no—
248D4—3162no—
249D4—6328no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—3164no—
258C2×C4—3161no—
259C2×C4—3161no—
260C2×C4—3161no—
261C2×C4—3162no—
262C2×C4—3161no—
263C2×C4—3161no—
264C2×C4—792no—
265C2×C4—6322no—
266C2×C4—6324no—
267C2×C4—1582no—
268C2×C4—1581no—
269C2×C4C2×C8×D7911yescharacteristic
270C2×C4M16×D7911yescharacteristic
271C8—82no—
272C8—6322no—
273C22—21no—
274C22—21no—
275C22—44no—
276C22—42no—
277C22—82no—
278C22—81no—
279C22—81no—
280C22—3164no—
281C22—3162no—
282C22—3162no—
283C22—6322no—
284C22—1584no—
285C22C22⋊C8×D7911yescharacteristic
286C22#99×D7911yescharacteristic
287C22#1909854411yescharacteristic
288C4—21no—
289C4—21no—
290C4—42no—
291C4—81no—
292C4—3162no—
293C4—3161no—
294C4—6321no—
295C4—1582no—
296C2—21no—
297C2—21no—
298C2—82no—
299C2—3164no—
300C2#2376171511yescharacteristic
301C2#2376506611yescharacteristic
302C2#2376509011yescharacteristic
subgroupG/H
centerC22#99×D79
commutatorC2×C4×C79C22×C8
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1318C2
socleC22×C79C2×#99
prime radicalC2×#14579157C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1318D79
hypocenterC79#11479
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1318

Complements

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

NH ≅ G/Ntype
2. #23761717C2semidirect
3. #23761343C2semidirect
4. #23761332C2semidirect
5. C79×#1318C2semidirect
25. #14579157C8semidirect
69. C4×Dic79C2×C8semidirect
74. C2×D4×C79C2×C8semidirect
156. #1318D79semidirect
157. C79#11479semidirect
208. C2×D4C8×D79semidirect

Maximal subgroups

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

subgroupindex
#237661242
#237617172
#237613432
#237613322
C79×#13182
#1147979