Finite Group Databasebeta

← #34197544

Subgroups — #34197544

Order 20224.

480 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 89 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#23765943C211yesmaximal, characteristic
2#23761717C212yesmaximal
3#23761343C212yesmaximal, complemented
4#23761334C211yesmaximal, characteristic, complemented
5C79×#1319C211yesmaximal, characteristic, complemented
6C2×#14579150C411yescharacteristic
7C2×Q8×Dic79C411yescharacteristic
8C2×#14579123C2211yescharacteristic
9C22⋊C8×D79—22no—
10#19098465C411yescharacteristic
11#19098455—22no—
12#19098428C2211yescharacteristic
13#19098410—22no—
14C42×Dic79C411yescharacteristic
15#19098289C2212yes—
16#19098286—22no—
17#391×C79C2211yescharacteristic
18#324×C79C2212yes—
19C22×D316C2×C411yescharacteristic
20C22×C4×D79D412yes—
21C2×C8×D79—42no—
22#14579150—22no—
23#14579150C814yescomplemented
24Q8×Dic79—41no—
25Q8×Dic79—41no—
26#14579123—41no—
27#14579123—41no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×(C2×D79)⋊C4—22no—
31C2×(C2×D79)⋊C4—22no—
32C2×(C2×D79)⋊C4D412yes—
33C2×C4×Dic79—21no—
34C2×C4×Dic79—21no—
35C2×C4×Dic79—21no—
36C2×C4×Dic79C2×C411yescharacteristic
37C2×C4×Dic79C2×C411yescharacteristic
38C22⋊(C79⋊C8)—22no—
39(C2×D79)⋊C8—42no—
40C79⋊#95—22no—
41C22×Q8×C79C2×C411yescharacteristic
42C2×C4⋊C4×C79C2311yescharacteristic
43C2×C42×C79C2×C411yescharacteristic
44C22⋊C8×C79—22no—
45#95×C79—22no—
46C23×D79C22⋊C412yes—
47C2×D316—41no—
48C2×D316—41no—
49C2×D316C2×C812yes—
50C2×D316M1612yes—
51C22×Dic79—21no—
52C22×Dic79—21no—
53C22×Dic79C22⋊C411yescharacteristic
54C22×Dic79C22⋊C411yescharacteristic
55C2×C4×D79—22no—
56C2×C4×D79—22no—
57C2×C4×D79—44no—
58C2×C79⋊C8—42no—
59C8×D79—84no—
60C4⋊Dic79—41no—
61C4⋊Dic79—42no—
62(C2×D79)⋊C4—28no—
63(C2×D79)⋊C4—42no—
64(C2×D79)⋊C4—44no—
65C4×Dic79—21no—
66C4×Dic79—21no—
67C4×Dic79—21no—
68C4×Dic79—42no—
69C4×Dic79—42no—
70C4×Dic79—41no—
71C4×Dic79—41no—
72C4×Dic79C2×C812yescomplemented
73C2×Q8×C79—21no—
74C2×Q8×C79—41no—
75C2×Q8×C79—41no—
76C2×Q8×C79C2×C812yescomplemented
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79C22×C411yescharacteristic
80C22×C4×C79C2×D412yes—
81C2×C8×C79—42no—
82C4⋊C4×C79—41no—
83C42×C79—41no—
84C23×C79C2×C22⋊C411yescharacteristic
85C22×D79—22no—
86C22×D79—22no—
87C22×D79—42no—
88C22×D79—42no—
89C22×D79C22⋊C814yes—
90Q8×C79—42no—
91Q8×C79—42no—
92C2×C4×C79—21no—
93C2×C4×C79—21no—
94C2×C4×C79—22no—
95C2×C4×C79—22no—
96C2×C4×C79—22no—
97C2×C4×C79—41no—
98C2×C4×C79—41no—
99C2×C4×C79—41no—
100C2×C4×C79—41no—
101C2×C4×C79—42no—
102C2×C4×C79C22×C811yescharacteristic
103C2×C4×C79C2×M1611yescharacteristic
104C2×Dic79—22no—
105C2×Dic79—22no—
106C2×Dic79—21no—
107C2×Dic79—41no—
108C2×Dic79—41no—
109C2×Dic79—41no—
110C2×Dic79—42no—
111C2×Dic79—41no—
112C2×Dic79#9911yescharacteristic
113C2×Dic79#9911yescharacteristic
114D316—42no—
115D316—42no—
116C4×D79—44no—
117C4×D79—82no—
118C8×C79—82no—
119C79⋊C8—82no—
120C22×C79—21no—
121C22×C79—21no—
122C22×C79C2×C22⋊C811yescharacteristic
123C22×C79C2×#9911yescharacteristic
124C22×C79#41311yescharacteristic
125C2×D79—24no—
126C2×D79—42no—
127C2×D79—42no—
128C2×D79—44no—
129C2×D79—82no—
130C4×C79—21no—
131C4×C79—21no—
132C4×C79—42no—
133C4×C79—42no—
134C4×C79—42no—
135Dic79—21no—
136Dic79—42no—
137Dic79—41no—
138Dic79—81no—
139Dic79#35412yes—
140#11477—791nomaximal, Sylow
141C2×C79—21no—
142C2×C79—21no—
143C2×C79#145911yescharacteristic
144C2×C79C2×#35411yescharacteristic
145C2×C79#213211yescharacteristic
146D79—44no—
147C4×#124—791no—
148#1459—792no—
149#1324—792no—
150#1319—791no—
151#1319D7911yescharacteristic, complemented
152C79#1147711yescharacteristic, complemented, Sylow
153C2×#124—791no—
154C2×C4×Q8—791no—
155C2×C4×D4—791no—
156C2×C22⋊C8—1582no—
157#391C2×D7911yescharacteristic
158#390—791no—
159#388—1582no—
160#382—791no—
161C4×C22⋊C4—1582no—
162C43—791no—
163#324—792no—
164#324C2×D7912yes—
165#323—1582no—
166C22×Q8C4×D7911yescharacteristic
167C22×D4—791no—
168C23×C4—792no—
169C22×C8—3162no—
170#124—794no—
171#124—1582no—
172C4×Q8—3161no—
173C4×Q8—3161no—
174C4×D4—3161no—
175C4×D4—3161no—
176C2×C4⋊C4—791no—
177C2×C4⋊C4—1581no—
178C2×C4⋊C4C22×D7911yescharacteristic
179C2×C22⋊C4—792no—
180C2×C22⋊C4—1582no—
181C2×C22⋊C4—1582no—
182C2×C42—791no—
183C2×C42—791no—
184C2×C42—1581no—
185C2×C42—1581no—
186C2×C42—1581no—
187C2×C42C4×D7911yescharacteristic
188C22⋊C8—22no—
189C22⋊C8—3162no—
190C22⋊C8—1582no—
191#95—22no—
192#95—1582no—
193C24—792no—
194C2×Q8—21no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8C8×D7912yescomplemented
198C2×D4—3161no—
199C2×D4—3161no—
200C2×D4—792no—
201C2×D4—792no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—3164no—
205C22×C4—791no—
206C22×C4—791no—
207C22×C4—1582no—
208C22×C4—1582no—
209C22×C4—1581no—
210C22×C4—1581no—
211C22×C4C2×C4×D7911yescharacteristic
212C22×C4D4×D7912yes—
213C2×C8—42no—
214C2×C8—3162no—
215C2×C8—6324no—
216C4⋊C4—41no—
217C4⋊C4—3161no—
218C4⋊C4—3162no—
219C22⋊C4—3164no—
220C22⋊C4—3162no—
221C22⋊C4—1588no—
222C42—41no—
223C42—3162no—
224C42—3161no—
225C42—3161no—
226C42—3162no—
227C42—792no—
228C42—1581no—
229C42—1581no—
230C42—1581no—
231C23—3162no—
232C23—3162no—
233C23—794no—
234C23—1582no—
235C23—1582no—
236C23C22⋊C4×D7911yescharacteristic
237Q8—42no—
238Q8—42no—
239D4—3162no—
240D4—3162no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—3164no—
252C2×C4—3161no—
253C2×C4—3161no—
254C2×C4—3161no—
255C2×C4—3162no—
256C2×C4—3161no—
257C2×C4—791no—
258C2×C4—791no—
259C2×C4—6322no—
260C2×C4—1582no—
261C2×C4—1581no—
262C2×C4—1582no—
263C2×C4C2×C8×D7911yescharacteristic
264C2×C4M16×D7911yescharacteristic
265C8—82no—
266C8—6322no—
267C22—21no—
268C22—21no—
269C22—3164no—
270C22—3162no—
271C22—3162no—
272C22—6322no—
273C22—1584no—
274C22C22⋊C8×D7911yescharacteristic
275C22#99×D7911yescharacteristic
276C22#1909854411yescharacteristic
277C4—21no—
278C4—21no—
279C4—42no—
280C4—42no—
281C4—42no—
282C4—3162no—
283C4—3161no—
284C4—792no—
285C4—6321no—
286C4—1581no—
287C2—21no—
288C2—21no—
289C2—3164no—
290C2#2376171511yescharacteristic
291C2#354×D7911yescharacteristic
292C2#2376508411yescharacteristic
subgroupG/H
centerC22#99×D79
commutatorC2×C4×C79C22×C8
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1319C2
socleC22×C79C2×#99
prime radicalC22×D316C2×C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1319D79
hypocenterC79#11477
LCSG ▷ C2×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. #23761343C2semidirect
4. #23761334C2semidirect
5. C79×#1319C2semidirect
23. #14579150C8semidirect
72. C4×Dic79C2×C8semidirect
76. C2×Q8×C79C2×C8semidirect
151. #1319D79semidirect
152. C79#11477semidirect
197. C2×Q8C8×D79semidirect

Maximal subgroups

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

subgroupindex
#237659432
#237617172
#237613432
#237613342
C79×#13192
#1147779