Finite Group Databasebeta

← #34201290

Subgroups — #34201290

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#23771468C211yesmaximal, characteristic
2#23762408C212yesmaximal
3#23761405C212yesmaximal
4#23761344C211yesmaximal, characteristic
5C79×#2061C211yesmaximal, characteristic
6C2×#14579153—21no—
7#19098891—22no—
8#19098503—22no—
9#19098495—22no—
10#19098493—21no—
11#19098493C2212yes—
12#19098300—22no—
13#19098289C2212yes—
14#544×C79C2211yescharacteristic
15#428×C79—21no—
16#382×C79C2211yescharacteristic
17#343×C79C2211yescharacteristic
18#339×C79—21no—
19#14579153—42no—
20#14579153—42no—
21C2×C4⋊Dic79—21no—
22C2×C4⋊Dic79—21no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79D412yes—
25C2×Dic79⋊C4—22no—
26C2×Dic79⋊C4—22no—
27#14579092—42no—
28#14579085—42no—
29C22⋊(C79⋊C8)—22no—
30C79⋊#95—22no—
31C79⋊#95—21no—
32C79⋊#95—22no—
33C79⋊#95D412yes—
34C2×M16×C79—21no—
35C4⋊Q8×C79—41no—
36#125×C79—41no—
37C22⋊Q8×C79—22no—
38#117×C79—21no—
39C2×C4⋊C4×C79D411yescharacteristic
40C2×C4⋊C4×C79C2311yescharacteristic
41C2×C42×C79D411yescharacteristic
42#107×C79—41no—
43#106×C79—41no—
44#95×C79—22no—
45C22×Dic79—22no—
46C22×Dic79—22no—
47C2×C79⋊C8—42no—
48C4⋊Dic79—42no—
49C4⋊Dic79—42no—
50C4⋊Dic79—42no—
51C4⋊Dic79—81no—
52Dic79⋊C4—44no—
53Dic79⋊C4—42no—
54Dic79⋊C4—42no—
55C2×Q8×C79—22no—
56C22×C4×C79—21no—
57C22×C4×C79—21no—
58C22×C4×C79C2×D411yescharacteristic
59C22×C4×C79C2×D412yes—
60M16×C79—22no—
61M16×C79—41no—
62C2×C8×C79—41no—
63C4⋊C4×C79—21no—
64C4⋊C4×C79—21no—
65C4⋊C4×C79—22no—
66C4⋊C4×C79—41no—
67C4⋊C4×C79—42no—
68C4⋊C4×C79—41no—
69C22⋊C4×C79—41no—
70C42×C79—21no—
71C42×C79—41no—
72C42×C79SD1612yes—
73C23×C79#12011yescharacteristic
74Q8×C79—82no—
75C2×C4×C79—21no—
76C2×C4×C79—21no—
77C2×C4×C79—22no—
78C2×C4×C79—22no—
79C2×C4×C79—41no—
80C2×C4×C79—41no—
81C2×C4×C79—42no—
82C2×C4×C79—42no—
83C2×C4×C79—41no—
84C2×C4×C79—41no—
85C2×C4×C79—41no—
86C2×C4×C79C2×SD1611yescharacteristic
87C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
88C2×Dic79—42no—
89C2×Dic79—42no—
90C2×Dic79—42no—
91C2×Dic79—42no—
92C2×Dic79—42no—
93C2×Dic79—42no—
94C8×C79—42no—
95C79⋊C8—82no—
96C22×C79—21no—
97C22×C79—21no—
98C22×C79#45011yescharacteristic
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#12251—791nomaximal, Sylow
111C2×C79—21no—
112C2×C79—21no—
113C2×C79#160411yescharacteristic
114C2×C79#220111yescharacteristic
115C2×C79#220311yescharacteristic
116#2822—791no—
117#2061D7911yescharacteristic
118#1611—792no—
119#1346—792no—
120#1324—791no—
121C79#1225111yescharacteristic, complemented, Sylow
122#544C2×D7911yescharacteristic
123C2×#125—1581no—
124#483—1582no—
125#428—21no—
126#400—1582no—
127#398—792no—
128#398—1582no—
129#398—1581no—
130#382C2×D7911yescharacteristic
131#343C2×D7911yescharacteristic
132#339—21no—
133#328—1582no—
134#324—792no—
135C2×M16—21no—
136C4⋊Q8—41no—
137#125—41no—
138#125—3162no—
139#125—3162no—
140C22⋊Q8—22no—
141#117—21no—
142C2×C4⋊C4—792no—
143C2×C4⋊C4—1581no—
144C2×C4⋊C4—1582no—
145C2×C4⋊C4—1581no—
146C2×C4⋊C4—1582no—
147C2×C4⋊C4—1581no—
148C2×C4⋊C4C22⋊D7911yescharacteristic
149C2×C4⋊C4C22×D7911yescharacteristic
150C2×C42C22⋊D7911yescharacteristic
151#107—41no—
152#106—41no—
153#106—3162no—
154#103—3162no—
155C22⋊C8—1582no—
156#95—22no—
157#95—792no—
158#95—1582no—
159#95—1581no—
160#95—1582no—
161C2×Q8—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—1582no—
165C22×C4—1582no—
166C22×C4D4×D7912yes—
167C22×C4C2×C22⋊D7911yescharacteristic
168M16—22no—
169M16—41no—
170C2×C8—41no—
171C2×C8—3162no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—3162no—
179C4⋊C4—3164no—
180C4⋊C4—3162no—
181C4⋊C4—3162no—
182C4⋊C4—3162no—
183C4⋊C4—3162no—
184C4⋊C4—6321no—
185C22⋊C4—41no—
186C42—21no—
187C42—41no—
188C42#1192961112yes—
189C23#1457912911yescharacteristic
190Q8—82no—
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×#1192961111yescharacteristic
209C2×C4#1457918611yescharacteristic
210C8—42no—
211C8—6322no—
212C22—21no—
213C22—21no—
214C22#1909871311yescharacteristic
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#2376236811yescharacteristic
229C2#2376541211yescharacteristic
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#12251
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
121. C79#12251semidirect

Maximal subgroups

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

subgroupindex
#237714682
#237624082
#237614052
#237613442
C79×#20612
#1225179