Finite Group Databasebeta

← #34182934

Subgroups — #34182934

Order 20224.

558 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 123 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×#14579148C211yesmaximal, characteristic
2#23761794C212yesmaximal, complemented
3#23761721C211yesmaximal, characteristic
4#23761232C212yesmaximal, complemented
5C79×#2975C211yesmaximal, characteristic
6C2×#14579148C411yescharacteristic
7C2×C4×Dic158C2211yescharacteristic
8#19098616C2212yes—
9#19098468C411yescharacteristic
10#19098466C411yescharacteristic
11#19098457—22no—
12#19098428C411yescharacteristic
13C4×C22⋊Dic79—22no—
14C42×Dic79C2211yescharacteristic
15#19098293—24no—
16#19098287—22no—
17C2×C4×D4×C79C2211yescharacteristic
18#436×C79—24no—
19#431×C79C2212yes—
20#407×C79—22no—
21C22×Dic158C2×C411yescharacteristic
22#14579148—22no—
23#14579148—22no—
24C4×Dic158—22no—
25C4×Dic158—22no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×C22⋊Dic79—22no—
29C2×C22⋊Dic79—22no—
30C2×C4×Dic79—21no—
31C2×C4×Dic79—21no—
32C2×C4×Dic79—21no—
33C2×C4×Dic79C2×C411yescharacteristic
34C2×C4×Dic79C2×C411yescharacteristic
35C4⋊(C79⋊C8)—22no—
36C22⋊(C79⋊C8)—22no—
37C4×C79⋊C8—22no—
38C79⋊#95—22no—
39C79⋊#95—22no—
40C22×D4×C79C2×C411yescharacteristic
41C23×C4×C79D412yes—
42C2×M16×C79—24no—
43C4×D4×C79—22no—
44C4×D4×C79—22no—
45C2×C4⋊C4×C79C2×C411yescharacteristic
46C2×C22⋊C4×C79D412yes—
47C2×C42×C79C2311yescharacteristic
48C4⋊C8×C79D414yes—
49C22⋊C8×C79—24no—
50C22⋊C8×C79—22no—
51C4×C8×C79—22no—
52C2×Dic158—22no—
53C2×Dic158—21no—
54C2×Dic158—22no—
55C2×Dic158M1612yescomplemented
56C22×Dic79—21no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C2×C79⋊C8—42no—
61C4⋊Dic79—22no—
62Dic79⋊C4—42no—
63C22⋊Dic79—28no—
64C22⋊Dic79—44no—
65C4×Dic79—21no—
66C4×Dic79—22no—
67C4×Dic79—42no—
68C4×Dic79—42no—
69C4×Dic79—41no—
70C4×Dic79—41no—
71C4×Dic79M1612yescomplemented
72C24×C79C22⋊C412yes—
73C2×D4×C79—22no—
74C2×D4×C79—21no—
75C2×D4×C79—21no—
76C2×D4×C79—22no—
77C22×C4×C79—22no—
78C22×C4×C79—22no—
79C22×C4×C79—44no—
80C22×C4×C79C22⋊C412yes—
81C22×C4×C79C22×C411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84M16×C79—84no—
85C2×C8×C79—24no—
86C4⋊C4×C79—22no—
87C22⋊C4×C79—24no—
88C42×C79C2×D412yes—
89C42×C79#4112yes—
90C23×C79—22no—
91C23×C79—22no—
92C23×C79—22no—
93C23×C79—42no—
94C23×C79—42no—
95C23×C79C2×C22⋊C411yescharacteristic
96D4×C79—42no—
97D4×C79—42no—
98C2×C4×C79—22no—
99C2×C4×C79—22no—
100C2×C4×C79—41no—
101C2×C4×C79—44no—
102C2×C4×C79—42no—
103C2×C4×C79—82no—
104C2×C4×C79C4×D412yes—
105C2×C4×C79#12011yescharacteristic
106C2×C4×C79#12112yes—
107C2×C4×C79#12311yescharacteristic
108C2×C4×C79C2×M1611yescharacteristic
109C2×C4×C79C2×M1611yescharacteristic
110C2×C4×C79#13112yes—
111C2×Dic79—22no—
112C2×Dic79—22no—
113C2×Dic79—22no—
114C2×Dic79—41no—
115C2×Dic79—41no—
116C2×Dic79—41no—
117C2×Dic79—41no—
118C2×Dic79—41no—
119C2×Dic79—41no—
120C2×Dic79—42no—
121C2×Dic79—41no—
122Dic158—24no—
123Dic158—42no—
124C8×C79—44no—
125C79⋊C8—82no—
126C22×C79—21no—
127C22×C79—21no—
128C22×C79—44no—
129C22×C79—42no—
130C22×C79—42no—
131C22×C79—42no—
132C22×C79—82no—
133C22×C79#38611yescharacteristic
134C22×C79#40711yescharacteristic
135C22×C79#40811yescharacteristic
136C4×C79—22no—
137C4×C79—21no—
138C4×C79—42no—
139C4×C79#43512yes—
140C4×C79#43612yes—
141C4×C79#45512yes—
142Dic79—42no—
143Dic79—42no—
144Dic79—42no—
145Dic79—81no—
146#8961—791nomaximal, Sylow
147C2×C79—21no—
148C2×C79—21no—
149C2×C79—82no—
150C2×C79#187011yescharacteristic
151C2×C79#188411yescharacteristic
152C2×C79#188511yescharacteristic
153#2975D7911yescharacteristic
154C4×#124—791no—
155#1479—792no—
156#1460—791no—
157#1276—792no—
158C79#896111yescharacteristic, complemented, Sylow
159C2×#124—791no—
160C2×C4×Q8—791no—
161C2×C4×D4C2×D7911yescharacteristic
162#436—24no—
163#431—792no—
164#431C2×D7912yes—
165#407—22no—
166#391—791no—
167#390—791no—
168#388—1582no—
169#382—791no—
170C4×C22⋊C4—1582no—
171C43—791no—
172Q8⋊C8—1584no—
173#323—1582no—
174C22×Q8—791no—
175C22×D4C4×D7911yescharacteristic
176C23×C4C22⋊D7912yes—
177C2×M16—24no—
178#124—1582no—
179#124—1582no—
180C4×Q8—1582no—
181C4×Q8—1582no—
182C4×D4—22no—
183C4×D4—22no—
184C2×C4⋊C4—791no—
185C2×C4⋊C4—1581no—
186C2×C4⋊C4C4×D7911yescharacteristic
187C2×C22⋊C4—1582no—
188C2×C22⋊C4—1582no—
189C2×C22⋊C4D31612yes—
190C2×C42—791no—
191C2×C42—791no—
192C2×C42—1581no—
193C2×C42—1581no—
194C2×C42—1581no—
195C2×C42C22×D7911yescharacteristic
196C4⋊C8—1582no—
197C4⋊C8C22⋊D7914yes—
198C22⋊C8—24no—
199C22⋊C8—22no—
200C22⋊C8—1582no—
201C4×C8—22no—
202C4×C8—1582no—
203#95—1582no—
204#95—1582no—
205C24(C2×D79)⋊C412yes—
206C2×Q8—792no—
207C2×Q8—1582no—
208C2×Q8—1582no—
209C2×Q8—1581no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—44no—
217C22×C4—1581no—
218C22×C4—1581no—
219C22×C4—1581no—
220C22×C4—1581no—
221C22×C4(C2×D79)⋊C412yes—
222C22×C4C2×C4×D7911yescharacteristic
223C22×C4C2×C22⋊D7911yescharacteristic
224C22×C4C2×D31611yescharacteristic
225M16—84no—
226C2×C8—24no—
227C2×C8—3162no—
228C4⋊C4—22no—
229C4⋊C4—3162no—
230C4⋊C4—1582no—
231C22⋊C4—24no—
232C22⋊C4—3164no—
233C22⋊C4—1588no—
234C42—3162no—
235C42—3162no—
236C42—3161no—
237C42—3161no—
238C42—792no—
239C42—1582no—
240C42—1581no—
241C42C2×C22⋊D7912yes—
242C42#1192962412yes—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23C2×(C2×D79)⋊C411yescharacteristic
249Q8—3162no—
250Q8—1584no—
251D4—42no—
252D4—42no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4—3161no—
260C2×C4—3161no—
261C2×C4—3161no—
262C2×C4—3161no—
263C2×C4—3161no—
264C2×C4—3161no—
265C2×C4—3162no—
266C2×C4—3161no—
267C2×C4—1582no—
268C2×C4—1582no—
269C2×C4—1582no—
270C2×C4C4×C22⋊D7912yes—
271C2×C4#1457913011yescharacteristic
272C2×C4#1457913512yes—
273C2×C4#1457914411yescharacteristic
274C2×C4M16×D7911yescharacteristic
275C2×C4M16×D7911yescharacteristic
276C2×C4#1457916912yes—
277C8—44no—
278C8—6322no—
279C22—21no—
280C22—21no—
281C22—44no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—82no—
286C22#1909844211yescharacteristic
287C22#1909852611yescharacteristic
288C22#1909853111yescharacteristic
289C4—22no—
290C4—21no—
291C4—42no—
292C4—3162no—
293C4—3162no—
294C4—3162no—
295C4—6321no—
296C4#1909863212yes—
297C4#1909863912yes—
298C4#1909874112yes—
299C2—21no—
300C2—21no—
301C2—82no—
302C2#2376369711yescharacteristic
303C2#2376378311yescharacteristic
304C2#2376378811yescharacteristic
subgroupG/H
centerC2×C4#14579130
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2975C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC2×C42×C79C23
hypercenter#2975D79
hypocenterC79#8961
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #23761794C2semidirect
4. #23761232C2semidirect
55. C2×Dic158M16semidirect
71. C4×Dic79M16semidirect
158. C79#8961semidirect

Maximal subgroups

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

subgroupindex
C4×#145791482
#237617942
#237617212
#237612322
C79×#29752
#896179