Finite Group Databasebeta

← #34201352

Subgroups — #34201352

Order 20224.

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

normal subgroups only

This group has 330 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#23762442C212yesmaximal
3#23761426C212yesmaximal
4#23761344C211yesmaximal, characteristic
5C79×#2061C211yesmaximal, characteristic
6#19098896—22no—
7#19098505—22no—
8#19098502C2212yes—
9#19098484—22no—
10#19098483—21no—
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×Dic158—21no—
20C2×C4⋊Dic79—21no—
21C2×C4⋊Dic79—21no—
22C2×C4⋊Dic79D412yes—
23C2×Dic79⋊C4—22no—
24#14579094—42no—
25#14579085—42no—
26C22⋊(C79⋊C8)—22no—
27C79⋊#95—22no—
28C79⋊#95—21no—
29C79⋊#95—22no—
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—82no—
44C2×Dic158—81no—
45C22×Dic79—22no—
46C22×Dic79—22no—
47C2×C79⋊C8—42no—
48C4⋊Dic79—42no—
49C4⋊Dic79—81no—
50C4⋊Dic79—81no—
51Dic79⋊C4—82no—
52C2×Q8×C79—22no—
53C22×C4×C79—21no—
54C22×C4×C79—21no—
55C22×C4×C79C2×D411yescharacteristic
56C22×C4×C79C2×D412yes—
57M16×C79—22no—
58M16×C79—41no—
59C2×C8×C79—41no—
60C4⋊C4×C79—21no—
61C4⋊C4×C79—21no—
62C4⋊C4×C79—22no—
63C4⋊C4×C79—41no—
64C4⋊C4×C79—42no—
65C4⋊C4×C79—41no—
66C22⋊C4×C79—41no—
67C42×C79—21no—
68C42×C79—41no—
69C42×C79Q1612yes—
70C23×C79#12011yescharacteristic
71Q8×C79—82no—
72C2×C4×C79—21no—
73C2×C4×C79—21no—
74C2×C4×C79—22no—
75C2×C4×C79—22no—
76C2×C4×C79—41no—
77C2×C4×C79—41no—
78C2×C4×C79—42no—
79C2×C4×C79—42no—
80C2×C4×C79—41no—
81C2×C4×C79—41no—
82C2×C4×C79—41no—
83C2×C4×C79C2×Q1611yescharacteristic
84C2×C4×C79#13711yescharacteristic
85C2×Dic79—42no—
86C2×Dic79—42no—
87C2×Dic79—42no—
88C2×Dic79—42no—
89C2×Dic79—42no—
90C2×Dic79—42no—
91Dic158—84no—
92C8×C79—42no—
93C79⋊C8—82no—
94C22×C79—21no—
95C22×C79—21no—
96C22×C79#45111yescharacteristic
97C22×C79#45611yescharacteristic
98C22×C79B(4,𝔽2)11yescharacteristic
99C4×C79—21no—
100C4×C79—21no—
101C4×C79—42no—
102C4×C79—42no—
103C4×C79—81no—
104C4×C79—81no—
105C4×C79—81no—
106Dic79—82no—
107Dic79—82no—
108#12261—791nomaximal, Sylow
109C2×C79—21no—
110C2×C79—21no—
111C2×C79#160511yescharacteristic
112C2×C79#220111yescharacteristic
113C2×C79#220211yescharacteristic
114#2844—791no—
115#2061D7911yescharacteristic
116#1617—792no—
117#1353—792no—
118#1324—791no—
119C79#1226111yescharacteristic, complemented, Sylow
120#544C2×D7911yescharacteristic
121#484—1582no—
122#428—21no—
123#400—792no—
124#400—1582no—
125#395—1582no—
126#395—1581no—
127#382—1581no—
128#382C2×D7911yescharacteristic
129#343C2×D7911yescharacteristic
130#339—21no—
131#328—1582no—
132#324—792no—
133C22×Q8—1581no—
134C2×M16—21no—
135C4⋊Q8—41no—
136#125—41no—
137C22⋊Q8—22no—
138#117—21no—
139C2×C4⋊C4—792no—
140C2×C4⋊C4—1581no—
141C2×C4⋊C4—1581no—
142C2×C4⋊C4—1582no—
143C2×C4⋊C4C22⋊D7911yescharacteristic
144C2×C4⋊C4C22×D7911yescharacteristic
145C2×C42C22⋊D7911yescharacteristic
146#107—41no—
147#107—3162no—
148#106—41no—
149#103—3162no—
150C22⋊C8—1582no—
151#95—22no—
152#95—792no—
153#95—1582no—
154#95—1581no—
155#95—1582no—
156#95—1582no—
157C2×Q8—22no—
158C2×Q8—6322no—
159C2×Q8—6321no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—1582no—
163C22×C4—1582no—
164C22×C4D4×D7912yes—
165C22×C4C2×C22⋊D7911yescharacteristic
166M16—22no—
167M16—41no—
168C2×C8—41no—
169C2×C8—3162no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—3162no—
177C4⋊C4—6322no—
178C4⋊C4—6321no—
179C4⋊C4—6321no—
180C22⋊C4—41no—
181C42—21no—
182C42—41no—
183C42#1192961412yes—
184C23#1457912911yescharacteristic
185Q8—82no—
186Q8—6324no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—3162no—
199C2×C4—3162no—
200C2×C4—3162no—
201C2×C4—3162no—
202C2×C4—3162no—
203C2×C4—3162no—
204C2×C4C2×#1192961411yescharacteristic
205C2×C4#1457919111yescharacteristic
206C8—42no—
207C8—6322no—
208C22—21no—
209C22—21no—
210C22#1909871811yescharacteristic
211C22#1909874511yescharacteristic
212C22#1909874911yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—81no—
218C4—81no—
219C4—81no—
220C4—6322no—
221C4—6322no—
222C2—21no—
223C2—21no—
224C2#2376237311yescharacteristic
225C2#2376541211yescharacteristic
226C2#2376541611yescharacteristic
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#12261
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
119. C79#12261semidirect

Maximal subgroups

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

subgroupindex
#237716722
#237624422
#237614262
#237613442
C79×#20612
#1226179