Finite Group Databasebeta

← #34177402

Subgroups — #34177402

Order 20224.

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

normal subgroups only

This group has 366 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#23771388C211yesmaximal, characteristic
2#23764943C211yesmaximal, characteristic
3#23764873C212yesmaximal
4#23761224C212yesmaximal
5C79×#1272C211yesmaximal, characteristic
6C2×#14579095—22no—
7C2×#14579092—21no—
8C2×#14579085—21no—
9#19098502—21no—
10#19098495—22no—
11#19098470—21no—
12#19098461C2212yes—
13#19098450—22no—
14#19098433C2211yescharacteristic
15#19098326—22no—
16#19098318C2211yescharacteristic
17#391×C79C2211yescharacteristic
18#336×C79C2212yes—
19C22×C79⋊C8—21no—
20C2×C4⋊Dic79—21no—
21C2×C4⋊Dic79—22no—
22C2×C4⋊Dic79—21no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79D412yes—
25C2×C4⋊Dic79D412yes—
26C2×Dic79⋊C4—22no—
27C2×C4×Dic79—21no—
28#14579095—44no—
29#14579092—42no—
30#14579085—42no—
31C79⋊#95—21no—
32C79⋊#95—22no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C22×Q8×C79D411yescharacteristic
36C22×C8×C79—22no—
37C2×C4⋊C4×C79D411yescharacteristic
38C2×C42×C79C2311yescharacteristic
39#95×C79—22no—
40C22×Dic79—22no—
41C22×Dic79—22no—
42C22×Dic79—41no—
43C2×C79⋊C8—22no—
44C2×C79⋊C8—41no—
45C2×C79⋊C8—41no—
46C4⋊Dic79—22no—
47C4⋊Dic79—22no—
48C4⋊Dic79—42no—
49C4⋊Dic79—41no—
50C4⋊Dic79—41no—
51C4⋊Dic79—42no—
52C4⋊Dic79—42no—
53C4⋊Dic79—81no—
54C4⋊Dic79—81no—
55Dic79⋊C4—42no—
56Dic79⋊C4—42no—
57C4×Dic79—81no—
58C2×Q8×C79—21no—
59C2×Q8×C79—41no—
60C2×Q8×C79—41no—
61C2×Q8×C79Q1612yes—
62C22×C4×C79—21no—
63C22×C4×C79—21no—
64C22×C4×C79C2×D411yescharacteristic
65C22×C4×C79C2×D412yes—
66C2×C8×C79—24no—
67C2×C8×C79—42no—
68C2×C8×C79—42no—
69C4⋊C4×C79—41no—
70C42×C79—41no—
71C23×C79#12011yescharacteristic
72Q8×C79—42no—
73Q8×C79—81no—
74C2×C4×C79—21no—
75C2×C4×C79—22no—
76C2×C4×C79—22no—
77C2×C4×C79—21no—
78C2×C4×C79—22no—
79C2×C4×C79—41no—
80C2×C4×C79—41no—
81C2×C4×C79—41no—
82C2×C4×C79—42no—
83C2×C4×C79C2×Q1611yescharacteristic
84C2×C4×C79#13512yes—
85C2×C4×C79AGL(1,ℤ/8)12yes—
86C2×C4×C79#13711yescharacteristic
87C2×Dic79—42no—
88C2×Dic79—42no—
89C2×Dic79—42no—
90C2×Dic79—42no—
91C2×Dic79—42no—
92C2×Dic79—42no—
93C2×Dic79—82no—
94C2×Dic79—81no—
95C8×C79—44no—
96C79⋊C8—42no—
97C22×C79#44912yes—
98C22×C79#45111yescharacteristic
99C22×C79#45311yescharacteristic
100C22×C79#45612yes—
101C22×C79#45811yescharacteristic
102C4×C79—22no—
103C4×C79—22no—
104C4×C79—21no—
105C4×C79—21no—
106C4×C79—42no—
107C4×C79—81no—
108Dic79—82no—
109Dic79—82no—
110Dic79—161no—
111#7802—791nomaximal, Sylow
112C2×C79#160912yes—
113C2×C79#161411yescharacteristic
114C2×C79#162411yescharacteristic
115C2×C79#164812yes—
116C2×C79#166011yescharacteristic
117#2809—791no—
118#2096—791no—
119#2083—792no—
120#1272D7911yescharacteristic
121#1271—792no—
122C79#780211yescharacteristic, complemented, Sylow
123C2×#107—1582no—
124C2×#106—1581no—
125C2×#103—1581no—
126#400—1581no—
127#398—1582no—
128#391—1581no—
129#391C2×D7911yescharacteristic
130#389—792no—
131#387—1582no—
132#384—791no—
133#340—1582no—
134#336—791no—
135#336C2×D7912yes—
136C22×Q8D31611yescharacteristic
137C22×C8—22no—
138C22×C8—1581no—
139C2×C4⋊C4—792no—
140C2×C4⋊C4—792no—
141C2×C4⋊C4—1582no—
142C2×C4⋊C4—1581no—
143C2×C4⋊C4—1582no—
144C2×C4⋊C4—1581no—
145C2×C4⋊C4—1581no—
146C2×C4⋊C4D31611yescharacteristic
147C2×C42—1581no—
148C2×C42C22×D7911yescharacteristic
149#107—3164no—
150#106—3162no—
151#103—3162no—
152#95—22no—
153#95—1581no—
154#95—1582no—
155#95—1582no—
156#95—1582no—
157C2×Q8—21no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8Dic31612yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—3161no—
164C22×C4—1582no—
165C22×C4—1582no—
166C22×C4D4×D7912yes—
167C22×C4C2×D31611yescharacteristic
168C2×C8—24no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—3161no—
172C2×C8—3161no—
173C2×C8—1582no—
174C4⋊C4—41no—
175C4⋊C4—3162no—
176C4⋊C4—3161no—
177C4⋊C4—3161no—
178C4⋊C4—3162no—
179C4⋊C4—3162no—
180C4⋊C4—3162no—
181C4⋊C4—3162no—
182C4⋊C4—6321no—
183C4⋊C4—6321no—
184C4⋊C4—1582no—
185C4⋊C4—1582no—
186C42—41no—
187C42—6321no—
188C23#1457913111yescharacteristic
189Q8—42no—
190Q8—81no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—3162no—
201C2×C4—3162no—
202C2×C4—3162no—
203C2×C4—3162no—
204C2×C4—3162no—
205C2×C4—3162no—
206C2×C4—6322no—
207C2×C4—6321no—
208C2×C4C2×Dic31611yescharacteristic
209C2×C4#1457918412yes—
210C2×C4#1457918812yes—
211C2×C4#1457919411yescharacteristic
212C8—44no—
213C8—3162no—
214C22#1909871112yes—
215C22#1909872111yescharacteristic
216C22#1909873311yescharacteristic
217C22#1909874712yes—
218C22#1909875411yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—81no—
225C4—6322no—
226C4—6322no—
227C4—12641no—
228C2#2376240012yes—
229C2#2376242711yescharacteristic
230C2#2376248411yescharacteristic
231C2#2376264512yes—
232C2#2376271211yescharacteristic
subgroupG/H
centerC23#14579131
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1272C2
socle, prime radicalC23×C79#120
hypercenter#1272D79
hypocenterC79#7802
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
122. C79#7802semidirect

Maximal subgroups

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

subgroupindex
#237713882
#237649432
#237648732
#237612242
C79×#12722
#780279