Finite Group Databasebeta

← #34201347

Subgroups — #34201347

Order 20224.

342 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 59 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#23771672C211yesmaximal, characteristic
2#23762374C212yesmaximal
3#23761429C212yesmaximal
4#23761344C211yesmaximal, characteristic
5C79×#2061C211yesmaximal, characteristic
6#19098718—22no—
7#19098505—22no—
8#19098502—21no—
9#19098484—22no—
10#19098483C2212yes—
11#19098429—21no—
12#19098300—22no—
13#19098289C2212yes—
14#544×C79C2211yescharacteristic
15#428×C79—21no—
16#382×C79C2211yescharacteristic
17#343×C79C2211yescharacteristic
18#339×C79—21no—
19C22×Dic158D412yes—
20C2×#11929614—42no—
21C2×C4⋊Dic79—21no—
22C2×C4⋊Dic79—21no—
23C2×C4⋊Dic79—21no—
24C2×Dic79⋊C4—22no—
25#14579084—42no—
26C22⋊(C79⋊C8)—22no—
27C79⋊#95—22no—
28C79⋊#95—22no—
29C79⋊#95—21no—
30C79⋊#95—22no—
31C79⋊#95D412yes—
32C2×M16×C79—21no—
33C4⋊Q8×C79—41no—
34#125×C79—41no—
35C22⋊Q8×C79—22no—
36#117×C79—21no—
37C2×C4⋊C4×C79D411yescharacteristic
38C2×C4⋊C4×C79C2311yescharacteristic
39C2×C42×C79D411yescharacteristic
40#107×C79—41no—
41#106×C79—41no—
42#95×C79—22no—
43C2×Dic158—42no—
44C2×Dic158—82no—
45C22×Dic79—22no—
46C22×Dic79—22no—
47#11929614—84no—
48C2×C79⋊C8—42no—
49C4⋊Dic79—81no—
50C4⋊Dic79—81no—
51C4⋊Dic79—81no—
52Dic79⋊C4—82no—
53C2×Q8×C79—22no—
54C22×C4×C79—21no—
55C22×C4×C79—21no—
56C22×C4×C79C2×D411yescharacteristic
57C22×C4×C79C2×D412yes—
58M16×C79—22no—
59M16×C79—41no—
60C2×C8×C79—41no—
61C4⋊C4×C79—21no—
62C4⋊C4×C79—21no—
63C4⋊C4×C79—22no—
64C4⋊C4×C79—41no—
65C4⋊C4×C79—42no—
66C4⋊C4×C79—41no—
67C22⋊C4×C79—41no—
68C42×C79—21no—
69C42×C79—41no—
70C42×C79Q1612yescomplemented
71C23×C79#12011yescharacteristic
72Q8×C79—82no—
73C2×C4×C79—21no—
74C2×C4×C79—21no—
75C2×C4×C79—22no—
76C2×C4×C79—22no—
77C2×C4×C79—41no—
78C2×C4×C79—41no—
79C2×C4×C79—42no—
80C2×C4×C79—42no—
81C2×C4×C79—41no—
82C2×C4×C79—41no—
83C2×C4×C79—41no—
84C2×C4×C79C2×Q1611yescharacteristic
85C2×C4×C79#13711yescharacteristic
86C2×Dic79—42no—
87C2×Dic79—42no—
88C2×Dic79—42no—
89C2×Dic79—42no—
90C2×Dic79—42no—
91C2×Dic79—42no—
92Dic158—44no—
93Dic158—82no—
94C8×C79—42no—
95C79⋊C8—82no—
96C22×C79—21no—
97C22×C79—21no—
98C22×C79#45111yescharacteristic
99C22×C79#45611yescharacteristic
100C22×C79B(4,𝔽2)11yescharacteristic
101C4×C79—21no—
102C4×C79—21no—
103C4×C79—42no—
104C4×C79—42no—
105C4×C79—81no—
106C4×C79—81no—
107C4×C79—81no—
108Dic79—82no—
109Dic79—82no—
110#12260—791nomaximal, Sylow
111C2×C79—21no—
112C2×C79—21no—
113C2×C79#160511yescharacteristic
114C2×C79#219911yescharacteristic
115C2×C79#220311yescharacteristic
116#2844—791no—
117#2061D7911yescharacteristic
118#1605—792no—
119#1354—792no—
120#1324—791no—
121C79#1226011yescharacteristic, complemented, Sylow
122#544C2×D7911yescharacteristic
123#451—1582no—
124#428—21no—
125#400—1582no—
126#400—1581no—
127#395—792no—
128#395—1582no—
129#382—1581no—
130#382C2×D7911yescharacteristic
131#343C2×D7911yescharacteristic
132#339—21no—
133#328—1582no—
134#324—792no—
135C22×Q8—792no—
136C2×Q16—3162no—
137C2×M16—21no—
138C4⋊Q8—41no—
139#125—41no—
140C22⋊Q8—22no—
141#117—21no—
142C2×C4⋊C4—1581no—
143C2×C4⋊C4—1581no—
144C2×C4⋊C4—1581no—
145C2×C4⋊C4—1582no—
146C2×C4⋊C4C22⋊D7911yescharacteristic
147C2×C4⋊C4C22×D7911yescharacteristic
148C2×C42C22⋊D7911yescharacteristic
149#107—41no—
150#106—41no—
151#103—3162no—
152C22⋊C8—1582no—
153#95—22no—
154#95—792no—
155#95—1582no—
156#95—1582no—
157#95—1581no—
158#95—1582no—
159C2×Q8—22no—
160C2×Q8—3162no—
161C2×Q8—6322no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—1582no—
165C22×C4—1582no—
166C22×C4D4×D7912yes—
167C22×C4C2×C22⋊D7911yescharacteristic
168Q16—6324no—
169M16—22no—
170M16—41no—
171C2×C8—41no—
172C2×C8—3162no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—6322no—
180C4⋊C4—6321no—
181C4⋊C4—6321no—
182C4⋊C4—6321no—
183C22⋊C4—41no—
184C42—21no—
185C42—41no—
186C42#1192961412yescomplemented
187C23#1457912911yescharacteristic
188Q8—82no—
189Q8—3164no—
190Q8—6322no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—3162no—
203C2×C4—3162no—
204C2×C4—3162no—
205C2×C4—3162no—
206C2×C4—3162no—
207C2×C4—3162no—
208C2×C4C2×#1192961411yescharacteristic
209C2×C4#1457919111yescharacteristic
210C8—42no—
211C8—6322no—
212C22—21no—
213C22—21no—
214C22#1909871811yescharacteristic
215C22#1909874511yescharacteristic
216C22#1909874911yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—81no—
222C4—81no—
223C4—81no—
224C4—6322no—
225C4—6322no—
226C2—21no—
227C2—21no—
228C2#2376237311yescharacteristic
229C2#2376540011yescharacteristic
230C2#2376542011yescharacteristic
subgroupG/H
centerC22#19098749
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2061C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2061D79
hypocenterC79#12260
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
70. C42×C79Q16semidirect
121. C79#12260semidirect
186. C42#11929614semidirect

Maximal subgroups

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

subgroupindex
#237716722
#237623742
#237614292
#237613442
C79×#20612
#1226079