Finite Group Databasebeta

← #34189264

Subgroups — #34189264

Order 20224.

476 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 103 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#23773924C211yesmaximal, characteristic
2#23773439C212yesmaximal
3#23761802C212yesmaximal
4#23761231C211yesmaximal, characteristic
5C79×#1985C211yesmaximal, characteristic
6#19099120C2212yes—
7#19098972—22no—
8#19098972—22no—
9#19098898—22no—
10#19098858—22no—
11#19098773—22no—
12#19098719—22no—
13C4×#11929614—24no—
14#19098616C2212yes—
15#19098601—22no—
16#19098293—24no—
17C2×C4×Q8×C79C2211yescharacteristic
18C2×C4⋊C8×C79C2211yescharacteristic
19C2×#97×C79C2211yescharacteristic
20#336×C79—22no—
21C2×#11929614—24no—
22#14579161D412yes—
23#14579161D412yes—
24#14579153—22no—
25#14579140—22no—
26#14579140—22no—
27C4×Dic158—22no—
28C4×Dic158—22no—
29#14579120—22no—
30#14579095—22no—
31#14579095—22no—
32#14579094—24no—
33#14579093—24no—
34C4⋊(C79⋊C8)—22no—
35#14579085—24no—
36#14579084—24no—
37C22⋊(C79⋊C8)—22no—
38C4×C79⋊C8D414yes—
39C22×Q8×C79D411yescharacteristic
40C22×C8×C79—21no—
41C22×C8×C79D412yes—
42C4×Q8×C79—22no—
43C4×Q8×C79—22no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C42×C79—21no—
47C2×C42×C79C2311yescharacteristic
48C4⋊C8×C79—22no—
49#97×C79—22no—
50C2×Dic158—22no—
51C2×Dic158—22no—
52#11929614—48no—
53C2×C79⋊C8—24no—
54C4⋊Dic79—22no—
55C4⋊Dic79—22no—
56C4⋊Dic79—22no—
57C4⋊Dic79—22no—
58Dic79⋊C4—42no—
59Dic79⋊C4—42no—
60C22⋊Dic79—42no—
61C4×Dic79—42no—
62C2×Q8×C79—22no—
63C2×Q8×C79—42no—
64C22×C4×C79—21no—
65C22×C4×C79—21no—
66C22×C4×C79C2×D411yescharacteristic
67C22×C4×C79C2×D411yescharacteristic
68C22×C4×C79#4111yescharacteristic
69C2×C8×C79—22no—
70C2×C8×C79—22no—
71C2×C8×C79—41no—
72C2×C8×C79—42no—
73C2×C8×C79Q1614yescomplemented
74C4⋊C4×C79—22no—
75C4⋊C4×C79—42no—
76C42×C79—41no—
77C42×C79—41no—
78C42×C79C2×D412yes—
79C42×C79C2×D412yes—
80C23×C79#12111yescharacteristic
81Q8×C79—24no—
82Q8×C79—42no—
83C2×C4×C79—22no—
84C2×C4×C79—41no—
85C2×C4×C79—41no—
86C2×C4×C79—41no—
87C2×C4×C79—41no—
88C2×C4×C79—42no—
89C2×C4×C79—41no—
90C2×C4×C79#12011yescharacteristic
91C2×C4×C79#12012yes—
92C2×C4×C79#12112yes—
93C2×C4×C79#12711yescharacteristic
94C2×C4×C79C2×Q1612yes—
95C2×C4×C79#13712yes—
96C2×Dic79—42no—
97C2×Dic79—42no—
98Dic158—44no—
99Dic158—44no—
100C8×C79—24no—
101C8×C79—81no—
102C79⋊C8—44no—
103C22×C79—21no—
104C22×C79—21no—
105C22×C79#39211yescharacteristic
106C22×C79#46212yes—
107C4×C79—22no—
108C4×C79—22no—
109C4×C79—42no—
110C4×C79—81no—
111C4×C79#45112yes—
112C4×C79#49411yescharacteristic
113C4×C79#49711yescharacteristic
114Dic79—82no—
115Dic79—82no—
116#10037—791nomaximal, Sylow
117C2×C79—22no—
118C2×C79#201011yescharacteristic
119C2×C79#201611yescharacteristic
120C2×C79#201711yescharacteristic
121#3162—791no—
122#3104—792no—
123#1985D7911yescharacteristic
124#1481—792no—
125#1275—791no—
126C79#1003711yescharacteristic, complemented, Sylow
127#533—792no—
128C2×C4×Q8C2×D7911yescharacteristic
129#501—1582no—
130#501—1582no—
131#484—1582no—
132#477—1582no—
133#462—1582no—
134#451—1582no—
135C4×Q16—1584no—
136#431—792no—
137#427—1582no—
138C2×C4⋊C8C2×D7911yescharacteristic
139C2×#97C2×D7911yescharacteristic
140#336—22no—
141Q8⋊C8—1584no—
142C22×Q8D31611yescharacteristic
143C2×Q16—1584no—
144C22×C8—21no—
145C22×C8C22⋊D7912yes—
146C4⋊Q8—792no—
147C4⋊Q8—792no—
148#125—1582no—
149C22⋊Q8—1582no—
150C22⋊Q8—1582no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—1582no—
154C4×Q8—1582no—
155#117—1582no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4D31611yescharacteristic
158C2×C42—21no—
159C2×C42C22×D7911yescharacteristic
160#107—1584no—
161#107—1582no—
162#107—1582no—
163#106—1584no—
164C4⋊C8—22no—
165C4⋊C8—1582no—
166#103—1584no—
167#103—1584no—
168C22⋊C8—1582no—
169#97—22no—
170C4×C8—794no—
171C2×Q8—22no—
172C2×Q8—42no—
173C2×Q8—1582no—
174C2×Q8—1582no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4C2×C22⋊D7911yescharacteristic
178C22×C4C2×D31611yescharacteristic
179C22×C4#1192962411yescharacteristic
180Q16—3168no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—1584no—
186C2×C8#1192961414yescomplemented
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—3162no—
190C4⋊C4—3162no—
191C4⋊C4—1582no—
192C4⋊C4—1582no—
193C4⋊C4—1582no—
194C4⋊C4—1582no—
195C22⋊C4—3162no—
196C42—41no—
197C42—41no—
198C42—3162no—
199C42D4×D7912yes—
200C42D4×D7912yes—
201C23#1457913511yescharacteristic
202Q8—24no—
203Q8—42no—
204Q8—3164no—
205Q8—3164no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—3162no—
214C2×C4—3162no—
215C2×C4#1457912911yescharacteristic
216C2×C4#1457913112yes—
217C2×C4#1457913412yes—
218C2×C4#1457915711yescharacteristic
219C2×C4C2×#1192961412yes—
220C2×C4#1457919112yes—
221C8—24no—
222C8—81no—
223C8—3164no—
224C22—21no—
225C22—21no—
226C22#1909847211yescharacteristic
227C22#1909877312yes—
228C4—22no—
229C4—22no—
230C4—42no—
231C4—81no—
232C4—6322no—
233C4—6322no—
234C4#1909871812yes—
235C4#1909894211yescharacteristic
236C4#1909895811yescharacteristic
237C2—22no—
238C2#2376446711yescharacteristic
239C2#2376449911yescharacteristic
240C2#2376450411yescharacteristic
subgroupG/H
centerC2×C4#14579157
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1985C2
socleC22×C79#392
prime radicalC23×C79#121
hypercenter#1985D79
hypocenterC79#10037
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
73. C2×C8×C79Q16semidirect
126. C79#10037semidirect
186. C2×C8#11929614semidirect

Maximal subgroups

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

subgroupindex
#237739242
#237734392
#237618022
#237612312
C79×#19852
#1003779