Finite Group Databasebeta

← #34197363

Subgroups — #34197363

Order 20224.

526 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 93 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#23765985C211yesmaximal, characteristic
2#23761711C212yesmaximal
3#23761330C211yesmaximal, characteristic
4#23761329C212yesmaximal
5C79×#1317C211yesmaximal, characteristic
6C23×Dic158C411yescharacteristic
7C2×#14579117C411yescharacteristic
8C2×#14579117C2211yescharacteristic
9#19098613—22no—
10#19098467—22no—
11#19098467—22no—
12#19098426C411yescharacteristic
13#19098424C411yescharacteristic
14#19098424C2211yescharacteristic
15#19098288—22no—
16#19098287C2212yes—
17#381×C79C2211yescharacteristic
18#323×C79C2212yes—
19C22×Dic158—21no—
20C22×Dic158—21no—
21C22×Dic158—22no—
22C22×Dic158—42no—
23C22×Dic158—42no—
24C22×Dic158C814yescomplemented
25C23×Dic79C2×C411yescharacteristic
26C23×Dic79C2×C411yescharacteristic
27#14579117—42no—
28#14579117—42no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4D412yes—
31C2×C22⋊Dic79C2×C411yescharacteristic
32C2×C22⋊Dic79C2×C411yescharacteristic
33C2×C4×Dic79—21no—
34C2×C4×Dic79D412yes—
35Dic79⋊C8—42no—
36C22⋊(C79⋊C8)—22no—
37C8×Dic79—42no—
38C79⋊#95—22no—
39C79⋊#95—22no—
40C79⋊#95—22no—
41C23×C4×C79C2×C411yescharacteristic
42C2×C22⋊C4×C79C2×C411yescharacteristic
43C2×C22⋊C4×C79C2311yescharacteristic
44C22⋊C8×C79—22no—
45#95×C79—22no—
46C2×Dic158—24no—
47C2×Dic158—24no—
48C2×Dic158—48no—
49C2×Dic158—42no—
50C2×Dic158—42no—
51C2×Dic158—42no—
52C2×Dic158—42no—
53C2×Dic158—48no—
54C22×Dic79—22no—
55C22×Dic79—22no—
56C22×Dic79—21no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—41no—
60C22×Dic79—41no—
61C22×Dic79C22⋊C411yescharacteristic
62C22×Dic79C22⋊C412yes—
63C22×Dic79C22⋊C411yescharacteristic
64C22×Dic79C2×C812yes—
65C22×Dic79C2×C812yes—
66C22×Dic79M1612yes—
67C2×C79⋊C8—42no—
68Dic79⋊C4—42no—
69Dic79⋊C4—42no—
70C22⋊Dic79—41no—
71C22⋊Dic79—41no—
72C4×Dic79—22no—
73C4×Dic79—22no—
74C4×Dic79—42no—
75C24×C79C22×C411yescharacteristic
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—41no—
80C22×C4×C79—41no—
81C22×C4×C79C2×C812yes—
82C22×C4×C79C2×D412yes—
83C2×C8×C79—42no—
84C22⋊C4×C79—41no—
85C22⋊C4×C79—41no—
86C23×C79—21no—
87C23×C79—21no—
88C23×C79—42no—
89C23×C79C2×C22⋊C411yescharacteristic
90C23×C79C22×C811yescharacteristic
91C23×C79C2×M1611yescharacteristic
92C2×C4×C79—22no—
93C2×C4×C79—22no—
94C2×C4×C79—42no—
95C2×C4×C79—41no—
96C2×C4×C79—41no—
97C2×C4×C79—42no—
98C2×C4×C79—42no—
99C2×C4×C79—42no—
100C2×C4×C79—41no—
101C2×Dic79—21no—
102C2×Dic79—21no—
103C2×Dic79—24no—
104C2×Dic79—24no—
105C2×Dic79—22no—
106C2×Dic79—22no—
107C2×Dic79—42no—
108C2×Dic79—41no—
109C2×Dic79—41no—
110C2×Dic79—42no—
111C2×Dic79—42no—
112C2×Dic79—42no—
113C2×Dic79—41no—
114C2×Dic79C22⋊C812yes—
115C2×Dic79C22⋊C812yes—
116C2×Dic79#9912yes—
117C2×Dic79#10012yes—
118Dic158—48no—
119Dic158—48no—
120C8×C79—82no—
121C79⋊C8—82no—
122C22×C79—22no—
123C22×C79—21no—
124C22×C79—21no—
125C22×C79—41no—
126C22×C79—41no—
127C22×C79—42no—
128C22×C79—42no—
129C22×C79C2×C22⋊C811yescharacteristic
130C22×C79C2×#9911yescharacteristic
131C22×C79C2×#10011yescharacteristic
132C4×C79—42no—
133C4×C79—42no—
134C4×C79—81no—
135Dic79—22no—
136Dic79—42no—
137Dic79—42no—
138Dic79—81no—
139Dic79#32314yes—
140#11428—791nomaximal, Sylow
141C2×C79—21no—
142C2×C79—21no—
143C2×C79—41no—
144C2×C79—41no—
145C2×C79C2×#32311yescharacteristic
146C2×C79#212312yes—
147#2310—791no—
148#1458—792no—
149#1317—791no—
150#1317—792no—
151#1317D7911yescharacteristic
152C79#1142811yescharacteristic, complemented, Sylow
153C23×Q8—791no—
154C2×#117—791no—
155C2×#117—791no—
156#431—1582no—
157#391—1582no—
158#391—1582no—
159#381—791no—
160#381—791no—
161#381—791no—
162#381C2×D7911yescharacteristic
163#324—1582no—
164#323—792no—
165#323C2×D7912yes—
166C22×Q8—3162no—
167C22×Q8—3162no—
168C22×Q8—794no—
169C22×Q8—1582no—
170C22×Q8—1581no—
171C22×Q8—1581no—
172C23×C4—791no—
173C23×C4—791no—
174C23×C4C4×D7911yescharacteristic
175#117—3162no—
176#117—3162no—
177C2×C4⋊C4—792no—
178C2×C4⋊C4—1581no—
179C2×C22⋊C4—791no—
180C2×C22⋊C4—791no—
181C2×C22⋊C4C4×D7911yescharacteristic
182C2×C22⋊C4C22×D7911yescharacteristic
183C2×C42—792no—
184C2×C42—1581no—
185C4⋊C8—3162no—
186C22⋊C8—22no—
187C22⋊C8—1582no—
188C4×C8—3162no—
189#95—22no—
190#95—1582no—
191#95—1582no—
192#95—1582no—
193C24C2×C4×D7911yescharacteristic
194C2×Q8—3162no—
195C2×Q8—3162no—
196C2×Q8—3168no—
197C2×Q8—3168no—
198C2×Q8—3162no—
199C2×Q8—3162no—
200C2×Q8—1584no—
201C2×Q8—1584no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—3161no—
208C22×C4—3161no—
209C22×C4—792no—
210C22×C4—792no—
211C22×C4—791no—
212C22×C4—791no—
213C22×C4—792no—
214C22×C4—792no—
215C22×C4—1581no—
216C22×C4—1581no—
217C22×C4—1581no—
218C22×C4—1582no—
219C22×C4—1582no—
220C22×C4C8×D7912yes—
221C22×C4D4×D7912yes—
222C2×C8—42no—
223C2×C8—3162no—
224C4⋊C4—3162no—
225C4⋊C4—3162no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—3161no—
229C22⋊C4—3161no—
230C42—3162no—
231C42—1582no—
232C42—1582no—
233C23—21no—
234C23—21no—
235C23—42no—
236C23C22⋊C4×D7911yescharacteristic
237C23C2×C8×D7911yescharacteristic
238C23M16×D7911yescharacteristic
239Q8—3168no—
240Q8—3168no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—3162no—
251C2×C4—3161no—
252C2×C4—3161no—
253C2×C4—3162no—
254C2×C4—3162no—
255C2×C4—3162no—
256C2×C4—3161no—
257C2×C4—792no—
258C2×C4—792no—
259C2×C4—792no—
260C2×C4—792no—
261C2×C4—1582no—
262C2×C4—1582no—
263C2×C4—1584no—
264C2×C4—1584no—
265C2×C4—1581no—
266C2×C4—1581no—
267C8—82no—
268C8—6322no—
269C22—22no—
270C22—21no—
271C22—21no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22C22⋊C8×D7911yescharacteristic
277C22#99×D7911yescharacteristic
278C22#100×D7911yescharacteristic
279C4—42no—
280C4—42no—
281C4—81no—
282C4—3162no—
283C4—3162no—
284C4—794no—
285C4—6321no—
286C4—1582no—
287C2—21no—
288C2—21no—
289C2—41no—
290C2—41no—
291C2#323×D7911yescharacteristic
292C2#2376505212yes—
subgroupG/H
centerC22#99×D79
commutatorC23×C79C22×C8
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1317C2
socleC22×C79C2×#99
prime radicalC24×C79C22×C4
cosocleC2×C22⋊C4×C79C23
hypercenter#1317D79
hypocenterC79#11428
LCSG ▷ C23×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1317

Complements

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

NH ≅ G/Ntype
24. C22×Dic158C8semidirect
152. C79#11428semidirect

Maximal subgroups

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

subgroupindex
#237659852
#237617112
#237613302
#237613292
C79×#13172
#1142879