Finite Group Databasebeta

← #34198855

Subgroups — #34198855

Order 20224.

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

normal subgroups only

This group has 464 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#23767708C211yesmaximal, characteristic
2#23761738C211yesmaximal, characteristic
3#23761405C214yesmaximal, complemented
4C4×#100×C79C211yesmaximal, characteristic
5C2×#14579142—21no—
6C22×C4⋊Dic79C411yescharacteristic
7#19098527—21no—
8#19098493C2212yes—
9#19098488C412yes—
10#19098461—21no—
11#19098424—21no—
12#19098423—21no—
13#19098423C411yescharacteristic
14#19098289C2212yes—
15#19098287—21no—
16C2×C4×D4×C79C2211yescharacteristic
17C2×#100×C79—21no—
18C4×M16×C79—21no—
19#343×C79C2212yes—
20C23×Dic79—21no—
21C2×#11929608—41no—
22#14579142—44no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—41no—
26C2×C4⋊Dic79—41no—
27C2×C4⋊Dic79C2×C412yes—
28C2×Dic79⋊C4—22no—
29C2×C22⋊Dic79—22no—
30C2×C22⋊Dic79—21no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—41no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C79⋊#95C2×C412yes—
36C22×D4×C79D411yescharacteristic
37C23×C4×C79C2×C411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C2×M16×C79—22no—
40C4×D4×C79—41no—
41C4×D4×C79—41no—
42C2×C4⋊C4×C79C2311yescharacteristic
43C2×C22⋊C4×C79C2×C411yescharacteristic
44C2×C22⋊C4×C79D411yescharacteristic
45C2×C42×C79D411yescharacteristic
46#100×C79—24no—
47#97×C79—41no—
48C4×C8×C79—41no—
49C22×Dic79—22no—
50C22×Dic79—22no—
51C22×Dic79—42no—
52C22×Dic79—42no—
53C22×Dic79—42no—
54#11929608—82no—
55C2×C79⋊C8—42no—
56C4⋊Dic79—42no—
57C4⋊Dic79—42no—
58C4⋊Dic79—41no—
59C4⋊Dic79—41no—
60Dic79⋊C4—44no—
61C22⋊Dic79—42no—
62C22⋊Dic79—44no—
63C24×C79C22⋊C411yescharacteristic
64C24×C79C22⋊C411yescharacteristic
65C2×D4×C79—22no—
66C2×D4×C79—22no—
67C2×D4×C79—41no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—42no—
73C22×C4×C79—42no—
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22×C411yescharacteristic
77C22×C4×C79C2×D411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79M16×C79—24no—
80M16×C79—42no—
81C2×C8×C79—42no—
82C4⋊C4×C79—41no—
83C22⋊C4×C79—41no—
84C22⋊C4×C79—41no—
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—81no—
112C2×C4×C79#9911yescharacteristic
113C2×C4×C79#9911yescharacteristic
114C2×C4×C79AGL(1,ℤ/8)12yes—
115C2×Dic79—42no—
116C2×Dic79—42no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119C2×Dic79—42no—
120C2×Dic79—42no—
121C2×Dic79—84no—
122C8×C79—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#35212yes—
145Dic79—82no—
146Dic79—82no—
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#2486—791no—
156C4×#100D7911yescharacteristic
157#1465—791no—
158#1346—794no—
159C79#1178011yescharacteristic, complemented, Sylow
160C2×#123—1581no—
161C2×C4×D4C2×D7911yescharacteristic
162C22×C4⋊C4—791no—
163C2×#100—21no—
164#407—1581no—
165C4×M16—21no—
166#398—792no—
167#397—792no—
168#389—1581no—
169#381—1581no—
170#380—791no—
171#380—1581no—
172#343C2×D7912yes—
173#324—792no—
174#323—1581no—
175C22×D4C22⋊D7911yescharacteristic
176C23×C4—1581no—
177C23×C4C4×D7911yescharacteristic
178C23×C4D31611yescharacteristic
179C2×M16—22no—
180C2×M16—3161no—
181#123—3164no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—3161no—
185C2×C4⋊C4—3161no—
186C2×C4⋊C4—792no—
187C2×C4⋊C4—1581no—
188C2×C4⋊C4—1581no—
189C2×C4⋊C4—1582no—
190C2×C4⋊C4C22×D7911yescharacteristic
191C2×C22⋊C4—1581no—
192C2×C22⋊C4—1582no—
193C2×C22⋊C4C4×D7911yescharacteristic
194C2×C22⋊C4D31611yescharacteristic
195C2×C42C22⋊D7911yescharacteristic
196#100—24no—
197C22⋊C8—3161no—
198C22⋊C8—1582no—
199#97—41no—
200C4×C8—41no—
201#95—792no—
202#95—1582no—
203#95—1582no—
204C24(C2×D79)⋊C411yescharacteristic
205C24(C2×D79)⋊C411yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—3162no—
216C22×C4—3162no—
217C22×C4—3162no—
218C22×C4—1582no—
219C22×C4—1582no—
220C22×C4(C2×D79)⋊C411yescharacteristic
221C22×C4(C2×D79)⋊C411yescharacteristic
222C22×C4C2×C4×D7911yescharacteristic
223C22×C4C2×C22⋊D7911yescharacteristic
224C22×C4C2×D31611yescharacteristic
225M16—24no—
226M16—42no—
227M16—6322no—
228C2×C8—42no—
229C2×C8—3162no—
230C4⋊C4—41no—
231C4⋊C4—3162no—
232C4⋊C4—3162no—
233C4⋊C4—3161no—
234C4⋊C4—3161no—
235C4⋊C4—3164no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—3162no—
239C22⋊C4—3164no—
240C42—21no—
241C42—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1457908912yes—
252C23C2×(C2×D79)⋊C411yescharacteristic
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—81no—
267C2×C4—3162no—
268C2×C4—3162no—
269C2×C4—3162no—
270C2×C4—3162no—
271C2×C4—3162no—
272C2×C4—3162no—
273C2×C4—6324no—
274C2×C4#1457907511yescharacteristic
275C2×C4#1457907511yescharacteristic
276C2×C4#1457918612yes—
277C8—44no—
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×#1457907511yescharacteristic
292C22#1909856711yescharacteristic
293C22C2×#1457908911yescharacteristic
294C4—21no—
295C4—22no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—6322no—
300C4—6322no—
301C4#1909835412yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2#2376191511yescharacteristic
307C2C2×#1909835411yescharacteristic
308C2#2376509411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC4×#100×C79C2
socleC22×C79C2×#99
prime radicalC22×D4×C79D4
cosocleC2×C4⋊C4×C79C23
hypercenterC4×#100D79
hypocenterC79#11780
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
3. #23761405C2semidirect
159. C79#11780semidirect

Maximal subgroups

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

subgroupindex
#237677082
#237617382
#237614052
C4×#100×C792
#1178079