Finite Group Databasebeta

← #34197758

Subgroups — #34197758

Order 20224.

440 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 75 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#23766114C211yesmaximal, characteristic
2#23761737C212yesmaximal
3#23761345C212yesmaximal, complemented
4#23761334C211yesmaximal, characteristic, complemented
5C79×#1319C211yesmaximal, characteristic, complemented
6C2×#14579150C411yescharacteristic
7C2×Q8×Dic79C411yescharacteristic
8C2×#14579123C2211yescharacteristic
9#19098525—22no—
10#19098465C411yescharacteristic
11#19098452—22no—
12#19098437—22no—
13#19098432C2211yescharacteristic
14#19098409C411yescharacteristic
15#19098289C2212yes—
16#19098286—22no—
17#391×C79C2211yescharacteristic
18#324×C79C2212yes—
19C22×D316C2×C411yescharacteristic
20C22×C4×D79D412yes—
21C2×#11929607—42no—
22#14579150—22no—
23#14579150—22no—
24Q8×Dic79—42no—
25#14579123—42no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—22no—
29C2×(C2×D79)⋊C4—22no—
30C2×(C2×D79)⋊C4—22no—
31C2×(C2×D79)⋊C4D412yes—
32C2×C4×Dic79—21no—
33C2×C4×Dic79C2×C411yescharacteristic
34C2×C4×Dic79C2×C411yescharacteristic
35C22⋊(C79⋊C8)—22no—
36(C2×D79)⋊C8—42no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C22×Q8×C79C2×C411yescharacteristic
40C2×C4⋊C4×C79C2311yescharacteristic
41C2×C42×C79C2×C411yescharacteristic
42C22⋊C8×C79—22no—
43#95×C79—22no—
44C23×D79C22⋊C412yes—
45C2×D316—21no—
46C2×D316—21no—
47C2×D316—41no—
48C2×D316—41no—
49C22×Dic79—21no—
50C22×Dic79—21no—
51C22×Dic79C22⋊C411yescharacteristic
52C22×Dic79C22⋊C411yescharacteristic
53C2×C4×D79—22no—
54C2×C4×D79—22no—
55C2×C4×D79—82no—
56#11929607—84no—
57C2×C79⋊C8—42no—
58C4⋊Dic79—41no—
59C4⋊Dic79—42no—
60Dic79⋊C4—82no—
61(C2×D79)⋊C4—28no—
62(C2×D79)⋊C4—42no—
63(C2×D79)⋊C4—82no—
64C4×Dic79—22no—
65C4×Dic79—21no—
66C4×Dic79—42no—
67C4×Dic79M1612yescomplemented
68C2×Q8×C79—21no—
69C2×Q8×C79—41no—
70C2×Q8×C79—41no—
71C2×Q8×C79M1612yescomplemented
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79C22×C411yescharacteristic
75C22×C4×C79C2×D412yes—
76C2×C8×C79—42no—
77C4⋊C4×C79—41no—
78C42×C79—41no—
79C23×C79C2×C22⋊C411yescharacteristic
80C22×D79—22no—
81C22×D79—22no—
82C22×D79—22no—
83C22×D79—42no—
84C22×D79—42no—
85Q8×C79—42no—
86Q8×C79—42no—
87C2×C4×C79—21no—
88C2×C4×C79—21no—
89C2×C4×C79—22no—
90C2×C4×C79—22no—
91C2×C4×C79—41no—
92C2×C4×C79—41no—
93C2×C4×C79—42no—
94C2×C4×C79—41no—
95C2×C4×C79—41no—
96C2×C4×C79—41no—
97C2×C4×C79C2×M1611yescharacteristic
98C2×C4×C79C2×M1611yescharacteristic
99C2×Dic79—22no—
100C2×Dic79—21no—
101C2×Dic79—41no—
102C2×Dic79—41no—
103C2×Dic79—41no—
104C2×Dic79—41no—
105C2×Dic79—42no—
106C2×Dic79—41no—
107C2×Dic79#9912yes—
108D316—42no—
109D316—42no—
110C4×D79—44no—
111C4×D79—82no—
112C8×C79—82no—
113C79⋊C8—82no—
114C22×C79—21no—
115C22×C79—21no—
116C22×C79#40711yescharacteristic
117C22×C79C2×#9911yescharacteristic
118C22×C79#41311yescharacteristic
119C2×D79—44no—
120C2×D79—42no—
121C2×D79—42no—
122C2×D79—42no—
123C2×D79—82no—
124C4×C79—21no—
125C4×C79—21no—
126C4×C79—42no—
127C4×C79—42no—
128C4×C79—81no—
129Dic79—22no—
130Dic79—42no—
131Dic79—41no—
132Dic79—81no—
133#11535—791nomaximal, Sylow
134C2×C79—21no—
135C2×C79—21no—
136C2×C79#146511yescharacteristic
137C2×C79#212711yescharacteristic
138C2×C79#213511yescharacteristic
139D79—82no—
140#2326—791no—
141#1465—792no—
142#1325—792no—
143#1319—791no—
144#1319D7911yescharacteristic, complemented
145C79#1153511yescharacteristic, complemented, Sylow
146C2×#124—791no—
147C2×C4×Q8—791no—
148C2×C4×D4—791no—
149#407—1582no—
150#391C2×D7911yescharacteristic
151#390—791no—
152#388—1582no—
153#385—1582no—
154#384—791no—
155#376—791no—
156#324—792no—
157#324C2×D7912yes—
158#323—1582no—
159C22×Q8C4×D7911yescharacteristic
160C22×D4—791no—
161C23×C4—792no—
162C2×M16—3162no—
163#124—1582no—
164#124—1582no—
165C4×Q8—3162no—
166C4×D4—3162no—
167C2×C4⋊C4—791no—
168C2×C4⋊C4—1581no—
169C2×C4⋊C4—1582no—
170C2×C4⋊C4C22×D7911yescharacteristic
171C2×C22⋊C4—792no—
172C2×C22⋊C4—1582no—
173C2×C22⋊C4—1582no—
174C2×C42—791no—
175C2×C42—791no—
176C2×C42—1581no—
177C2×C42C4×D7911yescharacteristic
178C22⋊C8—22no—
179C22⋊C8—3162no—
180C22⋊C8—1582no—
181#95—22no—
182#95—1581no—
183#95—1581no—
184C24—792no—
185C2×Q8—21no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8#1192960712yescomplemented
189C2×D4—3161no—
190C2×D4—3161no—
191C2×D4—1581no—
192C2×D4—1581no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—791no—
196C22×C4—791no—
197C22×C4—6322no—
198C22×C4—1582no—
199C22×C4—1582no—
200C22×C4—1581no—
201C22×C4—1581no—
202C22×C4C2×C4×D7911yescharacteristic
203C22×C4D4×D7912yes—
204M16—6324no—
205C2×C8—42no—
206C2×C8—3162no—
207C4⋊C4—41no—
208C4⋊C4—3161no—
209C4⋊C4—3162no—
210C4⋊C4—6322no—
211C22⋊C4—3162no—
212C22⋊C4—6322no—
213C22⋊C4—1588no—
214C42—41no—
215C42—3162no—
216C42—792no—
217C42—1582no—
218C42—1581no—
219C23—3162no—
220C23—3162no—
221C23—1582no—
222C23—1582no—
223C23—1582no—
224C23C22⋊C4×D7911yescharacteristic
225Q8—42no—
226Q8—42no—
227D4—3162no—
228D4—3162no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—3164no—
240C2×C4—3161no—
241C2×C4—3161no—
242C2×C4—3161no—
243C2×C4—3162no—
244C2×C4—3161no—
245C2×C4—3161no—
246C2×C4—792no—
247C2×C4—6322no—
248C2×C4—1582no—
249C2×C4—1581no—
250C2×C4M16×D7911yescharacteristic
251C2×C4C2×#1192960711yescharacteristic
252C8—82no—
253C8—6322no—
254C22—21no—
255C22—21no—
256C22—3164no—
257C22—3162no—
258C22—3162no—
259C22—3162no—
260C22—6322no—
261C22#1909852511yescharacteristic
262C22#99×D7911yescharacteristic
263C22#1909854411yescharacteristic
264C4—21no—
265C4—21no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4—3162no—
270C4—3161no—
271C4—6321no—
272C4—1582no—
273C2—21no—
274C2—21no—
275C2—6322no—
276C2#2376173511yescharacteristic
277C2#2376506811yescharacteristic
278C2#2376509311yescharacteristic
subgroupG/H
centerC22#99×D79
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1319C2
socleC22×C79C2×#99
prime radicalC22×D316C2×C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1319D79
hypocenterC79#11535
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
3. #23761345C2semidirect
4. #23761334C2semidirect
5. C79×#1319C2semidirect
67. C4×Dic79M16semidirect
71. C2×Q8×C79M16semidirect
144. #1319D79semidirect
145. C79#11535semidirect
188. C2×Q8#11929607semidirect

Maximal subgroups

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

subgroupindex
#237661142
#237617372
#237613452
#237613342
C79×#13192
#1153579