Finite Group Databasebeta

← #34177113

Subgroups — #34177113

Order 20224.

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

normal subgroups only

This group has 424 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#23764734C212yesmaximal
3#23764499C211yesmaximal, characteristic
4#23761224C212yesmaximal
5C79×#1272C211yesmaximal, characteristic
6C22×#11929614—21no—
7C2×#14579161C2211yescharacteristic
8C2×#14579093—22no—
9C2×#14579084—21no—
10#19098502—21no—
11#19098495—22no—
12#19098470—21no—
13#19098461C2212yes—
14#19098450—22no—
15#19098326—22no—
16#19098318C2211yescharacteristic
17#391×C79C2211yescharacteristic
18#336×C79C2212yes—
19C22×Dic158—21no—
20C2×#11929614—42no—
21C2×#11929614—42no—
22C2×#11929614—42no—
23C22×C79⋊C8—21no—
24#14579161—42no—
25C2×C4⋊Dic79—22no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79D412yes—
29C2×C4⋊Dic79D412yes—
30C2×Dic79⋊C4—22no—
31C2×C4×Dic79—21no—
32#14579093—44no—
33#14579084—42no—
34C79⋊#95—21no—
35C79⋊#95—22no—
36C79⋊#95—22no—
37C79⋊#95—22no—
38C22×Q8×C79D411yescharacteristic
39C22×C8×C79—22no—
40C2×C4⋊C4×C79D411yescharacteristic
41C2×C42×C79C2311yescharacteristic
42#95×C79—22no—
43C2×Dic158—22no—
44C2×Dic158—41no—
45C2×Dic158—41no—
46C2×Dic158—41no—
47C2×Dic158—41no—
48C2×Dic158—41no—
49C22×Dic79—22no—
50C22×Dic79—22no—
51C22×Dic79—41no—
52#11929614—48no—
53C2×C79⋊C8—22no—
54C2×C79⋊C8—41no—
55C2×C79⋊C8—41no—
56C4⋊Dic79—24no—
57C4⋊Dic79—22no—
58C4⋊Dic79—22no—
59C4⋊Dic79—42no—
60C4⋊Dic79—42no—
61C4⋊Dic79—81no—
62C4⋊Dic79—81no—
63Dic79⋊C4—42no—
64Dic79⋊C4—42no—
65C4×Dic79—81no—
66C2×Q8×C79—21no—
67C2×Q8×C79—41no—
68C2×Q8×C79—41no—
69C2×Q8×C79SD1612yes—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79C2×D411yescharacteristic
73C22×C4×C79C2×D412yes—
74C2×C8×C79—24no—
75C2×C8×C79—42no—
76C2×C8×C79—42no—
77C4⋊C4×C79—41no—
78C42×C79—41no—
79C23×C79#12011yescharacteristic
80Q8×C79—42no—
81Q8×C79—81no—
82C2×C4×C79—21no—
83C2×C4×C79—22no—
84C2×C4×C79—22no—
85C2×C4×C79—21no—
86C2×C4×C79—22no—
87C2×C4×C79—41no—
88C2×C4×C79—41no—
89C2×C4×C79—41no—
90C2×C4×C79—42no—
91C2×C4×C79C2×SD1611yescharacteristic
92C2×C4×C79#13512yes—
93C2×C4×C79#13712yes—
94C2×C4×C79#13711yescharacteristic
95C2×Dic79—42no—
96C2×Dic79—42no—
97C2×Dic79—42no—
98C2×Dic79—42no—
99C2×Dic79—42no—
100C2×Dic79—42no—
101C2×Dic79—42no—
102C2×Dic79—81no—
103C2×Dic79—81no—
104Dic158—42no—
105Dic158—42no—
106Dic158—81no—
107Dic158—81no—
108C8×C79—44no—
109C79⋊C8—42no—
110C22×C79#44811yescharacteristic
111C22×C79#45212yes—
112C22×C79#45311yescharacteristic
113C22×C79#45612yes—
114C22×C79B(4,𝔽2)11yescharacteristic
115C4×C79—21no—
116C4×C79—21no—
117C4×C79—22no—
118C4×C79—22no—
119C4×C79—42no—
120C4×C79—81no—
121Dic79—82no—
122Dic79—82no—
123Dic79—82no—
124#7760—791nomaximal, Sylow
125C2×C79#159611yescharacteristic
126C2×C79#160312yes—
127C2×C79#163411yescharacteristic
128C2×C79#165312yes—
129C2×C79#166011yescharacteristic
130#2809—791no—
131#2058—792no—
132#2016—791no—
133#1272D7911yescharacteristic
134#1271—792no—
135C79#776011yescharacteristic, complemented, Sylow
136C22×Q16—1581no—
137C2×C4⋊Q8—791no—
138C2×#106—1582no—
139C2×#103—1581no—
140#400—1581no—
141#398—1582no—
142#391—1581no—
143#391C2×D7911yescharacteristic
144#389—792no—
145#387—1582no—
146#340—1582no—
147#336—791no—
148#336C2×D7912yes—
149C22×Q8—1581no—
150C22×Q8D31611yescharacteristic
151C2×Q16—3162no—
152C2×Q16—3162no—
153C2×Q16—3162no—
154C22×C8—22no—
155C22×C8—1581no—
156C4⋊Q8—3162no—
157C2×C4⋊C4—792no—
158C2×C4⋊C4—792no—
159C2×C4⋊C4—1582no—
160C2×C4⋊C4—1582no—
161C2×C4⋊C4—1581no—
162C2×C4⋊C4—1581no—
163C2×C4⋊C4D31611yescharacteristic
164C2×C42—1581no—
165C2×C42C22×D7911yescharacteristic
166#106—3164no—
167#103—3162no—
168#95—22no—
169#95—1581no—
170#95—1582no—
171#95—1582no—
172#95—1582no—
173C2×Q8—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—3161no—
177C2×Q8—3161no—
178C2×Q8—3161no—
179C2×Q8—3161no—
180C2×Q8—3161no—
181C2×Q8—1582no—
182C2×Q8#1192961312yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—3161no—
186C22×C4—1582no—
187C22×C4—1582no—
188C22×C4D4×D7912yes—
189C22×C4C2×D31611yescharacteristic
190Q16—3168no—
191C2×C8—24no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—3161no—
195C2×C8—3161no—
196C2×C8—1582no—
197C4⋊C4—41no—
198C4⋊C4—3162no—
199C4⋊C4—3162no—
200C4⋊C4—3162no—
201C4⋊C4—3162no—
202C4⋊C4—6321no—
203C4⋊C4—6321no—
204C4⋊C4—1584no—
205C4⋊C4—1582no—
206C4⋊C4—1582no—
207C42—41no—
208C42—6321no—
209C23#1457913111yescharacteristic
210Q8—42no—
211Q8—81no—
212Q8—3162no—
213Q8—3162no—
214Q8—6321no—
215Q8—6321no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—3162no—
226C2×C4—3162no—
227C2×C4—3162no—
228C2×C4—3162no—
229C2×C4—3162no—
230C2×C4—3162no—
231C2×C4—3162no—
232C2×C4—6321no—
233C2×C4—6321no—
234C2×C4C2×#1192961311yescharacteristic
235C2×C4#1457918212yes—
236C2×C4#1457919312yes—
237C2×C4#1457919411yescharacteristic
238C8—44no—
239C8—3162no—
240C22#1909870411yescharacteristic
241C22#1909872812yes—
242C22#1909873311yescharacteristic
243C22#1909874712yes—
244C22#1909875111yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—42no—
250C4—81no—
251C4—6322no—
252C4—6322no—
253C4—6322no—
254C2#2376232511yescharacteristic
255C2#2376236612yes—
256C2#2376254611yescharacteristic
257C2#2376267912yes—
258C2#2376271211yescharacteristic
subgroupG/H
centerC23#14579131
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1272C2
socle, prime radicalC23×C79#120
hypercenter#1272D79
hypocenterC79#7760
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
135. C79#7760semidirect

Maximal subgroups

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

subgroupindex
#237713882
#237647342
#237644992
#237612242
C79×#12722
#776079