Finite Group Databasebeta

← #34198954

Subgroups — #34198954

Order 20224.

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

normal subgroups only

This group has 478 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
1C2×#19098493C211yesmaximal, characteristic
2#23761730C211yesmaximal, characteristic
3#23761405C214yesmaximal, complemented
4C79×#1821C211yesmaximal, characteristic
5C22×C4⋊Dic79C411yescharacteristic
6C22×Dic79⋊C4—21no—
7#19098527—21no—
8#19098493—21no—
9#19098493—21no—
10#19098493C412yes—
11#19098493C2212yes—
12C2×C79⋊#95—21no—
13C2×C79⋊#95C411yescharacteristic
14#19098289—21no—
15#19098289C2212yes—
16C22×C4⋊C4×C79C2211yescharacteristic
17#407×C79—21no—
18#343×C79—21no—
19#343×C79C2212yes—
20C23×Dic79—21no—
21C23×Dic79—21no—
22C2×#11929608—41no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—41no—
25C2×C4⋊Dic79—41no—
26C2×C4⋊Dic79C2×C412yes—
27C2×Dic79⋊C4—22no—
28C2×Dic79⋊C4—22no—
29C2×Dic79⋊C4—44no—
30C22⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—41no—
32C79⋊#95—22no—
33C79⋊#95—21no—
34C79⋊#95—22no—
35C79⋊#95C2×C412yes—
36C23×C4×C79C2×C411yescharacteristic
37C23×C4×C79D411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C2×M16×C79—22no—
40C2×C4⋊C4×C79—41no—
41C2×C4⋊C4×C79—41no—
42C2×C4⋊C4×C79C2×C411yescharacteristic
43C2×C4⋊C4×C79D411yescharacteristic
44C2×C4⋊C4×C79D411yescharacteristic
45C2×C4⋊C4×C79C2311yescharacteristic
46C22⋊C8×C79—41no—
47C22⋊C8×C79—41no—
48C22×Dic79—24no—
49C22×Dic79—22no—
50C22×Dic79—22no—
51C22×Dic79—42no—
52C22×Dic79—42no—
53C22×Dic79—42no—
54C22×Dic79—42no—
55C22×Dic79—41no—
56C22×Dic79—41no—
57#11929608—82no—
58C2×C79⋊C8—42no—
59C4⋊Dic79—42no—
60C4⋊Dic79—42no—
61Dic79⋊C4—44no—
62Dic79⋊C4—44no—
63C24×C79C22⋊C411yescharacteristic
64C22×C4×C79—21no—
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—22no—
71C22×C4×C79—41no—
72C22×C4×C79—42no—
73C22×C4×C79—42no—
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22×C411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80M16×C79—42no—
81M16×C79—42no—
82C2×C8×C79—42no—
83C4⋊C4×C79—41no—
84C4⋊C4×C79—41no—
85C4⋊C4×C79—41no—
86C4⋊C4×C79—41no—
87C23×C79—22no—
88C23×C79—22no—
89C23×C79—21no—
90C23×C79—21no—
91C23×C79#9911yescharacteristic
92C23×C79#9911yescharacteristic
93C23×C79C2×C22⋊C411yescharacteristic
94C2×C4×C79—21no—
95C2×C4×C79—21no—
96C2×C4×C79—21no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—42no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—81no—
113C2×C4×C79AGL(1,ℤ/8)12yes—
114C2×Dic79—24no—
115C2×Dic79—48no—
116C2×Dic79—42no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119C2×Dic79—42no—
120C2×Dic79—42no—
121C2×Dic79—82no—
122C2×Dic79—82no—
123C8×C79—82no—
124C79⋊C8—82no—
125C22×C79—24no—
126C22×C79—24no—
127C22×C79—21no—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79#35212yes—
134C22×C79#35612yes—
135C22×C79C2×#9911yescharacteristic
136C22×C79#41811yescharacteristic
137C22×C79#42111yescharacteristic
138C4×C79—22no—
139C4×C79—41no—
140C4×C79—41no—
141C4×C79—41no—
142C4×C79—41no—
143C4×C79—41no—
144Dic79—44no—
145Dic79—82no—
146#11803—791nomaximal, Sylow
147C2×C79—21no—
148C2×C79—22no—
149C2×C79—22no—
150C2×C79—21no—
151C2×C79#151011yescharacteristic
152C2×C79C2×#35211yescharacteristic
153C2×C79C2×#35611yescharacteristic
154C2×#398—791no—
155#1821D7911yescharacteristic
156#1463—791no—
157#1346—794no—
158C79#1180311yescharacteristic, complemented, Sylow
159C22×C4⋊C4—791no—
160C22×C4⋊C4—1581no—
161C22×C4⋊C4C2×D7911yescharacteristic
162#407—21no—
163#407—1581no—
164#398—792no—
165#398—792no—
166#398—1581no—
167#398—1581no—
168C2×#95—791no—
169C2×#95—1581no—
170#343—21no—
171#343C2×D7912yes—
172#324—792no—
173#324—1581no—
174C23×C4—1581no—
175C23×C4—1581no—
176C23×C4C4×D7911yescharacteristic
177C23×C4D31611yescharacteristic
178C23×C4C22⋊D7911yescharacteristic
179C2×M16—22no—
180C2×M16—3161no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—3161no—
184C2×C4⋊C4—3161no—
185C2×C4⋊C4—3164no—
186C2×C4⋊C4—792no—
187C2×C4⋊C4—1581no—
188C2×C4⋊C4—1582no—
189C2×C4⋊C4—1582no—
190C2×C4⋊C4C4×D7911yescharacteristic
191C2×C4⋊C4D31611yescharacteristic
192C2×C4⋊C4C22⋊D7911yescharacteristic
193C2×C4⋊C4C22×D7911yescharacteristic
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—3161no—
197C22⋊C8—1582no—
198#95—792no—
199#95—1581no—
200#95—1582no—
201#95—1582no—
202C24(C2×D79)⋊C411yescharacteristic
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—3162no—
214C22×C4—3162no—
215C22×C4—3162no—
216C22×C4—3162no—
217C22×C4—3161no—
218C22×C4—3161no—
219C22×C4—1582no—
220C22×C4—1584no—
221C22×C4—1582no—
222C22×C4(C2×D79)⋊C411yescharacteristic
223C22×C4(C2×D79)⋊C411yescharacteristic
224C22×C4(C2×D79)⋊C411yescharacteristic
225C22×C4C2×C4×D7911yescharacteristic
226C22×C4C2×C22⋊D7911yescharacteristic
227C22×C4C2×D31611yescharacteristic
228M16—42no—
229M16—42no—
230M16—6322no—
231C2×C8—42no—
232C2×C8—3162no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—3162no—
238C4⋊C4—3164no—
239C4⋊C4—3162no—
240C4⋊C4—3164no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23#1457907511yescharacteristic
246C23#1457907511yescharacteristic
247C23C2×(C2×D79)⋊C411yescharacteristic
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—3168no—
268C2×C4—3162no—
269C2×C4—3162no—
270C2×C4—3162no—
271C2×C4—3162no—
272C2×C4—3162no—
273C2×C4—6322no—
274C2×C4—6322no—
275C2×C4—1584no—
276C2×C4#1457918612yes—
277C8—82no—
278C8—6322no—
279C22—24no—
280C22—24no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22#1909835412yes—
288C22#1909836112yes—
289C22C2×#1457907511yescharacteristic
290C22#1909856711yescharacteristic
291C22#1909858111yescharacteristic
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—3164no—
299C4—6322no—
300C2—21no—
301C2—21no—
302C2—22no—
303C2—22no—
304C2#2376192811yescharacteristic
305C2C2×#1909835411yescharacteristic
306C2C2×#1909836111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1821C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1821D79
hypocenterC79#11803
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #23761405C2semidirect
158. C79#11803semidirect

Maximal subgroups

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

subgroupindex
C2×#190984932
#237617302
#237614052
C79×#18212
#1180379