Finite Group Databasebeta

← #34177525

Subgroups — #34177525

Order 20224.

422 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 83 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#23771346C211yesmaximal, characteristic
2#23764928C211yesmaximal, characteristic
3#23764810C212yesmaximal, complemented
4#23761225C212yesmaximal, complemented
5C79×#1270C211yesmaximal, characteristic
6C2×#14579094—21no—
7C2×#14579086—22no—
8C2×#14579082—21no—
9#19098502—21no—
10#19098475—22no—
11#19098469C2212yes—
12#19098443—21no—
13#19098438—22no—
14#19098433C2211yescharacteristic
15#19098326—22no—
16#19098318C2211yescharacteristic
17#390×C79C2211yescharacteristic
18#336×C79C2212yes—
19C22×Dic158D412yes—
20C23×Dic79—21no—
21C22×C79⋊C8—21no—
22C2×C4⋊Dic79—21no—
23C2×C4⋊Dic79—22no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79D412yes—
26C2×C22⋊Dic79—22no—
27C2×C22⋊Dic79—21no—
28#14579094—42no—
29#14579086—44no—
30#14579082—42no—
31C79⋊#95—22no—
32C79⋊#95—21no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C22×D4×C79D411yescharacteristic
36C22×C8×C79—22no—
37C2×C4⋊C4×C79D411yescharacteristic
38C2×C22⋊C4×C79—22no—
39C2×C42×C79C2311yescharacteristic
40C2×Dic158—22no—
41C2×Dic158—22no—
42C2×Dic158—42no—
43C2×Dic158—42no—
44C22×Dic79—22no—
45C22×Dic79—22no—
46C22×Dic79—41no—
47C22×Dic79—81no—
48C22×Dic79—82no—
49C2×C79⋊C8—22no—
50C2×C79⋊C8—41no—
51C2×C79⋊C8—41no—
52C4⋊Dic79—42no—
53C4⋊Dic79—41no—
54C4⋊Dic79—41no—
55C4⋊Dic79—42no—
56C4⋊Dic79—42no—
57C4⋊Dic79—81no—
58C22⋊Dic79—82no—
59C22⋊Dic79—81no—
60C24×C79—21no—
61C2×D4×C79—21no—
62C2×D4×C79—41no—
63C2×D4×C79—41no—
64C2×D4×C79Q1612yes—
65C22×C4×C79—21no—
66C22×C4×C79C2×D411yescharacteristic
67C22×C4×C79C2×D412yes—
68C2×C8×C79—42no—
69C2×C8×C79—42no—
70C2×C8×C79—42no—
71C4⋊C4×C79—41no—
72C22⋊C4×C79—42no—
73C22⋊C4×C79—42no—
74C42×C79—41no—
75C23×C79—22no—
76C23×C79—42no—
77C23×C79—41no—
78C23×C79—42no—
79C23×C79—41no—
80C23×C79#12011yescharacteristic
81D4×C79—42no—
82D4×C79—81no—
83C2×C4×C79—21no—
84C2×C4×C79—22no—
85C2×C4×C79—21no—
86C2×C4×C79—22no—
87C2×C4×C79—42no—
88C2×C4×C79—41no—
89C2×C4×C79C2×Q1611yescharacteristic
90C2×C4×C79#13512yes—
91C2×C4×C79AGL(1,ℤ/8)12yes—
92C2×C4×C79#13711yescharacteristic
93C2×Dic79—42no—
94C2×Dic79—42no—
95C2×Dic79—42no—
96C2×Dic79—42no—
97C2×Dic79—42no—
98C2×Dic79—42no—
99C2×Dic79—84no—
100C2×Dic79—82no—
101C2×Dic79—81no—
102Dic158—44no—
103Dic158—82no—
104C8×C79—82no—
105C79⋊C8—42no—
106C22×C79—42no—
107C22×C79—44no—
108C22×C79—82no—
109C22×C79—81no—
110C22×C79—81no—
111C22×C79#44912yes—
112C22×C79#45111yescharacteristic
113C22×C79#45311yescharacteristic
114C22×C79#45512yes—
115C22×C79#45811yescharacteristic
116C4×C79—22no—
117C4×C79—22no—
118C4×C79—21no—
119C4×C79—21no—
120C4×C79—81no—
121Dic79—82no—
122Dic79—82no—
123Dic79—161no—
124#7821—791nomaximal, Sylow
125C2×C79—82no—
126C2×C79#161212yes—
127C2×C79#161411yescharacteristic
128C2×C79#162011yescharacteristic
129C2×C79#164412yes—
130C2×C79#165911yescharacteristic
131#2804—791no—
132#2093—791no—
133#2072—792no—
134#1272—792no—
135#1270D7911yescharacteristic
136C79#782111yescharacteristic, complemented, Sylow
137C2×#107—1581no—
138C2×#103—1582no—
139C2×#102—1581no—
140#400—1581no—
141#393—1582no—
142#391—792no—
143#390C2×D7911yescharacteristic
144#386—1581no—
145#385—1582no—
146#384—791no—
147#340—1582no—
148#336—791no—
149#336C2×D7912yes—
150C22×Q8—792no—
151C22×D4D31611yescharacteristic
152C23×C4—1581no—
153C22×C8—22no—
154C22×C8—1581no—
155C2×C4⋊C4—792no—
156C2×C4⋊C4—1582no—
157C2×C4⋊C4—1581no—
158C2×C4⋊C4—1581no—
159C2×C4⋊C4D31611yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—1582no—
162C2×C22⋊C4—1581no—
163C2×C42C22×D7911yescharacteristic
164#107—3162no—
165#103—3164no—
166#102—3162no—
167#95—1582no—
168#95—1581no—
169#95—1582no—
170#95—1582no—
171C24—21no—
172C2×Q8—3162no—
173C2×Q8—3162no—
174C2×Q8—1582no—
175C2×Q8—1582no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4Dic31612yes—
180C22×C4—21no—
181C22×C4—3161no—
182C22×C4—6321no—
183C22×C4—6322no—
184C22×C4—1582no—
185C22×C4—1582no—
186C22×C4D4×D7912yes—
187C22×C4C2×D31611yescharacteristic
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—3161no—
192C2×C8—3161no—
193C2×C8—1582no—
194C4⋊C4—41no—
195C4⋊C4—3162no—
196C4⋊C4—3161no—
197C4⋊C4—3161no—
198C4⋊C4—3162no—
199C4⋊C4—3162no—
200C4⋊C4—6321no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—6322no—
204C22⋊C4—6321no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#1457913111yescharacteristic
212Q8—3164no—
213Q8—6322no—
214D4—42no—
215D4—81no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—3162no—
223C2×C4—3162no—
224C2×C4—3162no—
225C2×C4—3162no—
226C2×C4—3162no—
227C2×C4—3162no—
228C2×C4—6324no—
229C2×C4—6322no—
230C2×C4—6321no—
231C2×C4C2×Dic31611yescharacteristic
232C2×C4#1457918212yes—
233C2×C4#1457918512yes—
234C2×C4#1457919411yescharacteristic
235C8—82no—
236C8—3162no—
237C22—44no—
238C22—42no—
239C22—81no—
240C22—81no—
241C22—82no—
242C22#1909870912yes—
243C22#1909872111yescharacteristic
244C22#1909873311yescharacteristic
245C22#1909874212yes—
246C22#1909875411yescharacteristic
247C4—22no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—81no—
252C4—6322no—
253C4—6322no—
254C4—12641no—
255C2—82no—
256C2#2376241512yes—
257C2#2376242711yescharacteristic
258C2#2376246011yescharacteristic
259C2#2376261712yes—
260C2#2376270711yescharacteristic
subgroupG/H
centerC23#14579131
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1270C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#1270D79
hypocenterC79#7821
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
3. #23764810C2semidirect
4. #23761225C2semidirect
136. C79#7821semidirect

Maximal subgroups

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

subgroupindex
#237713462
#237649282
#237648102
#237612252
C79×#12702
#782179