Finite Group Databasebeta

← #34198856

Subgroups — #34198856

Order 20224.

464 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#14579078C211yesmaximal, characteristic
2#23761738C211yesmaximal, characteristic
3#23761406C214yesmaximal, complemented
4C79×#2486C211yesmaximal, characteristic
5C2×#14579078—21no—
6#19098527—21no—
7C4×#11929608—21no—
8#19098332C2212yes—
9#19098289C2212yes—
10#19098287—21no—
11C2×#123×C79—21no—
12C2×C4×D4×C79C2211yescharacteristic
13C22×C4⋊C4×C79C411yescharacteristic
14#398×C79C2212yes—
15#397×C79C412yes—
16#389×C79—21no—
17#381×C79—21no—
18#380×C79—21no—
19#380×C79C411yescharacteristic
20C2×#11929608—22no—
21C2×#11929608—41no—
22#14579078—24no—
23C22⋊(C79⋊C8)—22no—
24C22⋊(C79⋊C8)—41no—
25#14579072—41no—
26C4×C79⋊C8—41no—
27C22×D4×C79D411yescharacteristic
28C23×C4×C79—21no—
29C23×C4×C79C2×C411yescharacteristic
30C23×C4×C79D411yescharacteristic
31#123×C79—44no—
32C4×D4×C79—41no—
33C4×D4×C79—41no—
34C2×C4⋊C4×C79—21no—
35C2×C4⋊C4×C79—21no—
36C2×C4⋊C4×C79—22no—
37C2×C4⋊C4×C79—41no—
38C2×C4⋊C4×C79—41no—
39C2×C4⋊C4×C79C2×C412yes—
40C2×C4⋊C4×C79C2311yescharacteristic
41C2×C22⋊C4×C79—22no—
42C2×C22⋊C4×C79—21no—
43C2×C22⋊C4×C79C2×C411yescharacteristic
44C2×C22⋊C4×C79D411yescharacteristic
45C2×C42×C79D411yescharacteristic
46#95×C79—22no—
47#95×C79—22no—
48#95×C79C2×C412yes—
49#11929608—24no—
50#11929608—42no—
51#11929608—82no—
52C2×C79⋊C8—42no—
53C2×C79⋊C8—42no—
54C24×C79C22⋊C411yescharacteristic
55C24×C79C22⋊C411yescharacteristic
56C2×D4×C79—22no—
57C2×D4×C79—22no—
58C2×D4×C79—41no—
59C22×C4×C79—21no—
60C22×C4×C79—21no—
61C22×C4×C79—21no—
62C22×C4×C79—21no—
63C22×C4×C79—22no—
64C22×C4×C79—22no—
65C22×C4×C79—42no—
66C22×C4×C79—42no—
67C22×C4×C79—42no—
68C22×C4×C79—42no—
69C22×C4×C79—42no—
70C22×C4×C79C22⋊C411yescharacteristic
71C22×C4×C79C22⋊C411yescharacteristic
72C22×C4×C79C22×C411yescharacteristic
73C22×C4×C79C2×D411yescharacteristic
74C22×C4×C79C2×D411yescharacteristic
75C4⋊C4×C79—41no—
76C4⋊C4×C79—42no—
77C4⋊C4×C79—42no—
78C4⋊C4×C79—41no—
79C4⋊C4×C79—41no—
80C4⋊C4×C79—44no—
81C22⋊C4×C79—41no—
82C22⋊C4×C79—41no—
83C22⋊C4×C79—42no—
84C22⋊C4×C79—44no—
85C42×C79—21no—
86C42×C79—21no—
87C23×C79—21no—
88C23×C79—21no—
89C23×C79—21no—
90C23×C79—21no—
91C23×C79—21no—
92C23×C79—41no—
93C23×C79—41no—
94C23×C79—41no—
95C23×C79—41no—
96C23×C79#10412yes—
97C23×C79C2×C22⋊C411yescharacteristic
98D4×C79—44no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
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—42no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—81no—
118C2×C4×C79—84no—
119C2×C4×C79#9911yescharacteristic
120C2×C4×C79#9911yescharacteristic
121C2×C4×C79AGL(1,ℤ/8)12yes—
122C79⋊C8—44no—
123C79⋊C8—82no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79—22no—
127C22×C79—42no—
128C22×C79—42no—
129C22×C79—42no—
130C22×C79—41no—
131C22×C79—41no—
132C22×C79—41no—
133C22×C79—41no—
134C22×C79—41no—
135C22×C79—81no—
136C22×C79C2×#9911yescharacteristic
137C22×C79#41811yescharacteristic
138C22×C79C2×#10411yescharacteristic
139C4×C79—21no—
140C4×C79—22no—
141C4×C79—41no—
142C4×C79—41no—
143C4×C79—41no—
144C4×C79—82no—
145C4×C79—82no—
146C4×C79#35212yes—
147#11780—791nomaximal, Sylow
148C2×C79—21no—
149C2×C79—21no—
150C2×C79—42no—
151C2×C79—81no—
152C2×C79#150711yescharacteristic
153C2×C79C2×#35211yescharacteristic
154C2×C79#213511yescharacteristic
155#2486D7911yescharacteristic
156C4×#100—791no—
157#1465—791no—
158#1346—794no—
159C79#1178011yescharacteristic, complemented, Sylow
160C2×#123—21no—
161C2×C4×D4C2×D7911yescharacteristic
162C22×C4⋊C4Dic7911yescharacteristic
163C2×#100—1581no—
164#407—1581no—
165C4×M16—1581no—
166#398C2×D7912yes—
167#397Dic7912yes—
168#389—21no—
169#381—21no—
170#380—21no—
171#380Dic7911yescharacteristic
172#343—792no—
173#324—792no—
174#323—1581no—
175C22×D4C22⋊D7911yescharacteristic
176C23×C4—21no—
177C23×C4C2×Dic7911yescharacteristic
178C23×C4C22⋊D7911yescharacteristic
179C2×M16—3161no—
180C2×M16—1582no—
181#123—44no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C2×Dic7912yes—
190C2×C4⋊C4C22×D7911yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4C2×Dic7911yescharacteristic
194C2×C22⋊C4C22⋊D7911yescharacteristic
195C2×C42C22⋊D7911yescharacteristic
196#100—1584no—
197C22⋊C8—3161no—
198C22⋊C8—1582no—
199#97—3161no—
200C4×C8—3161no—
201#95—22no—
202#95—22no—
203#95C2×Dic7912yes—
204C24C22⋊Dic7911yescharacteristic
205C24C22⋊Dic7911yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4C22⋊Dic7911yescharacteristic
221C22×C4C22⋊Dic7911yescharacteristic
222C22×C4C22×Dic7911yescharacteristic
223C22×C4C2×C22⋊D7911yescharacteristic
224C22×C4C2×C22⋊D7911yescharacteristic
225M16—3162no—
226M16—6322no—
227M16—1584no—
228C2×C8—3162no—
229C2×C8—3162no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—44no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C42—21no—
241C42—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1457908812yes—
252C23C2×C22⋊Dic7911yescharacteristic
253D4—44no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—84no—
274C2×C4#1457907611yescharacteristic
275C2×C4#1457907611yescharacteristic
276C2×C4#1457918712yes—
277C8—3164no—
278C8—6322no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22C2×#1457907611yescharacteristic
292C22#1909856811yescharacteristic
293C22C2×#1457908811yescharacteristic
294C4—21no—
295C4—22no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—82no—
300C4—82no—
301C4#1909835312yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2#2376191611yescharacteristic
307C2C2×#1909835311yescharacteristic
308C2#2376509511yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2486C2
socleC22×C79C2×#99
prime radicalC22×D4×C79D4
cosocleC2×C4⋊C4×C79C23
hypercenter#2486D79
hypocenterC79#11780
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #23761406C2semidirect
159. C79#11780semidirect

Maximal subgroups

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

subgroupindex
C4×#145790782
#237617382
#237614062
C79×#24862
#1178079