Finite Group Databasebeta

← #34201117

Subgroups — #34201117

Order 20224.

336 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764561C211yesmaximal, characteristic
2#23762441C212yesmaximal
3#23761427C212yesmaximal
4#23761334C211yesmaximal, characteristic
5C79×#2809C211yesmaximal, characteristic
6#19099194C2211yescharacteristic
7#19098898—22no—
8#19098572—21no—
9#19098332C2211yescharacteristic
10#19098323—21no—
11#19098299—22no—
12#19098289C2212yes—
13#400×C79C2212yes—
14#398×C79—22no—
15#391×C79—21no—
16#391×C79C2211yescharacteristic
17#389×C79—21no—
18#387×C79—22no—
19C2×#11929608—21no—
20#14579160—41no—
21#14579152—41no—
22#14579140—22no—
23#14579120—21no—
24#14579094—42no—
25#14579085—41no—
26#14579085—41no—
27#14579084—42no—
28C22⋊(C79⋊C8)—22no—
29C22×Q8×C79D411yescharacteristic
30C2×C4⋊C4×C79—22no—
31C2×C4⋊C4×C79—21no—
32C2×C4⋊C4×C79—21no—
33C2×C4⋊C4×C79—21no—
34C2×C4⋊C4×C79D412yes—
35C2×C4⋊C4×C79C2311yescharacteristic
36C2×C42×C79—21no—
37C2×C42×C79D411yescharacteristic
38#95×C79—22no—
39#95×C79—22no—
40#95×C79—22no—
41#95×C79—22no—
42#95×C79D412yes—
43C2×Dic158—22no—
44#11929608—41no—
45#11929608—41no—
46C2×C79⋊C8—42no—
47C2×C79⋊C8—41no—
48C4⋊Dic79—21no—
49C4⋊Dic79—21no—
50C4⋊Dic79—22no—
51Dic79⋊C4—42no—
52C22⋊Dic79—41no—
53C4×Dic79—41no—
54C2×Q8×C79—21no—
55C2×Q8×C79—41no—
56C2×Q8×C79—41no—
57C2×Q8×C79Q1612yes—
58C22×C4×C79—21no—
59C22×C4×C79—21no—
60C22×C4×C79—22no—
61C22×C4×C79—22no—
62C22×C4×C79C2×D411yescharacteristic
63C22×C4×C79C2×D412yes—
64C4⋊C4×C79—41no—
65C4⋊C4×C79—44no—
66C4⋊C4×C79—42no—
67C4⋊C4×C79—42no—
68C4⋊C4×C79—81no—
69C4⋊C4×C79—81no—
70C42×C79—41no—
71C42×C79—81no—
72C23×C79#12011yescharacteristic
73Q8×C79—42no—
74Q8×C79—81no—
75C2×C4×C79—21no—
76C2×C4×C79—21no—
77C2×C4×C79—22no—
78C2×C4×C79—22no—
79C2×C4×C79—22no—
80C2×C4×C79—41no—
81C2×C4×C79—41no—
82C2×C4×C79—41no—
83C2×C4×C79—41no—
84C2×C4×C79—42no—
85C2×C4×C79—42no—
86C2×C4×C79—42no—
87C2×C4×C79—42no—
88C2×C4×C79—42no—
89C2×C4×C79—42no—
90C2×C4×C79—42no—
91C2×C4×C79C2×Q1611yescharacteristic
92C2×C4×C79#13711yescharacteristic
93C2×Dic79—41no—
94C2×Dic79—41no—
95Dic158—82no—
96C79⋊C8—81no—
97C79⋊C8—82no—
98C22×C79—21no—
99C22×C79—21no—
100C22×C79#45111yescharacteristic
101C22×C79#45511yescharacteristic
102C22×C79B(4,𝔽2)11yescharacteristic
103C4×C79—21no—
104C4×C79—21no—
105C4×C79—42no—
106C4×C79—42no—
107C4×C79—42no—
108C4×C79—82no—
109C4×C79—82no—
110Dic79—81no—
111Dic79—81no—
112#12225—791nomaximal, Sylow
113C2×C79—21no—
114C2×C79—21no—
115C2×C79#160211yescharacteristic
116C2×C79#220111yescharacteristic
117C2×C79#220411yescharacteristic
118#2809D7911yescharacteristic
119#2028—791no—
120#1617—792no—
121#1353—792no—
122#1319—791no—
123C79#1222511yescharacteristic, complemented, Sylow
124#544—791no—
125#484—1582no—
126#419—1581no—
127#400C2×D7912yes—
128#398—22no—
129#391—21no—
130#391C2×D7911yescharacteristic
131#389—21no—
132#387—22no—
133#343—791no—
134#339—1581no—
135#328—1582no—
136#324—792no—
137C22×Q8C22⋊D7911yescharacteristic
138C2×M16—1581no—
139C4⋊Q8—3161no—
140#125—3161no—
141C22⋊Q8—1582no—
142#117—1581no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C22⋊D7912yes—
148C2×C4⋊C4C22×D7911yescharacteristic
149C2×C42—21no—
150C2×C42C22⋊D7911yescharacteristic
151#107—3162no—
152#103—3162no—
153#103—3161no—
154#103—3161no—
155C22⋊C8—1582no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95C22⋊D7912yes—
161C2×Q8—21no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—1582no—
165C2×Q8#1192961412yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4C2×C22⋊D7911yescharacteristic
171C22×C4C2×C22⋊D7912yes—
172M16—3161no—
173M16—3161no—
174C2×C8—3162no—
175C2×C8—3161no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—3162no—
183C4⋊C4—1581no—
184C4⋊C4—1581no—
185C4⋊C4—1582no—
186C22⋊C4—3161no—
187C42—41no—
188C42—81no—
189C42—3161no—
190C23#1457913011yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—6322no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—3161no—
211C2×C4—3161no—
212C2×C4C2×#1192961411yescharacteristic
213C2×C4#1457919211yescharacteristic
214C8—6321no—
215C8—6322no—
216C22—21no—
217C22—21no—
218C22#1909871911yescharacteristic
219C22#1909874111yescharacteristic
220C22#1909875011yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—82no—
228C4—6321no—
229C4—6321no—
230C2—21no—
231C2—21no—
232C2#2376235711yescharacteristic
233C2#2376541311yescharacteristic
234C2#2376542611yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2809C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2809D79
hypocenterC79#12225
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2809

Complements

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

NH ≅ G/Ntype
123. C79#12225semidirect

Maximal subgroups

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

subgroupindex
#237645612
#237624412
#237614272
#237613342
C79×#28092
#1222579