Finite Group Databasebeta

← #34201116

Subgroups — #34201116

Order 20224.

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

normal subgroups only

This group has 336 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#23771388C211yesmaximal, characteristic
2#23762442C212yesmaximal
3#23761426C212yesmaximal
4#23761334C211yesmaximal, characteristic
5C79×#2028C211yesmaximal, characteristic
6#19098896—22no—
7#19098502C2212yes—
8#19098495—22no—
9#19098470—21no—
10#19098461—21no—
11#19098450—22no—
12#19098300—22no—
13#19098289C2212yes—
14#544×C79C2211yescharacteristic
15#419×C79—21no—
16#391×C79C2211yescharacteristic
17#343×C79C2211yescharacteristic
18#339×C79—21no—
19C2×C4⋊Dic79—21no—
20C2×C4⋊Dic79—21no—
21C2×C4⋊Dic79—21no—
22C2×C4⋊Dic79D412yes—
23C2×Dic79⋊C4—22no—
24C2×C4×Dic79—21no—
25#14579094—42no—
26#14579085—42no—
27C22⋊(C79⋊C8)—22no—
28C79⋊#95—22no—
29C79⋊#95—22no—
30C79⋊#95—22no—
31C79⋊#95D412yes—
32C22×Q8×C79D411yescharacteristic
33C2×M16×C79—21no—
34C4⋊Q8×C79—41no—
35#125×C79—41no—
36C22⋊Q8×C79—22no—
37#117×C79—21no—
38C2×C4⋊C4×C79C2311yescharacteristic
39C2×C42×C79D411yescharacteristic
40#103×C79—41no—
41#103×C79—41no—
42#95×C79—22no—
43C22×Dic79—22no—
44C22×Dic79—22no—
45C2×C79⋊C8—42no—
46C4⋊Dic79—42no—
47C4⋊Dic79—42no—
48C4⋊Dic79—81no—
49C4⋊Dic79—81no—
50Dic79⋊C4—44no—
51C4×Dic79—81no—
52C2×Q8×C79—21no—
53C2×Q8×C79—22no—
54C2×Q8×C79—41no—
55C2×Q8×C79—41no—
56C2×Q8×C79Q1612yes—
57C22×C4×C79—21no—
58C22×C4×C79—21no—
59C22×C4×C79C2×D411yescharacteristic
60C22×C4×C79C2×D412yes—
61M16×C79—41no—
62M16×C79—41no—
63C2×C8×C79—41no—
64C4⋊C4×C79—21no—
65C4⋊C4×C79—21no—
66C4⋊C4×C79—22no—
67C4⋊C4×C79—41no—
68C4⋊C4×C79—42no—
69C22⋊C4×C79—41no—
70C42×C79—41no—
71C42×C79—41no—
72C23×C79#12011yescharacteristic
73Q8×C79—42no—
74Q8×C79—81no—
75Q8×C79—82no—
76C2×C4×C79—21no—
77C2×C4×C79—21no—
78C2×C4×C79—22no—
79C2×C4×C79—22no—
80C2×C4×C79—22no—
81C2×C4×C79—41no—
82C2×C4×C79—41no—
83C2×C4×C79—41no—
84C2×C4×C79—41no—
85C2×C4×C79—42no—
86C2×C4×C79—41no—
87C2×C4×C79—41no—
88C2×C4×C79C2×Q1611yescharacteristic
89C2×C4×C79#13711yescharacteristic
90C2×Dic79—42no—
91C2×Dic79—42no—
92C2×Dic79—42no—
93C2×Dic79—42no—
94C2×Dic79—42no—
95C2×Dic79—42no—
96C8×C79—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—81no—
109C4×C79—81no—
110Dic79—82no—
111Dic79—82no—
112#12225—791nomaximal, Sylow
113C2×C79—21no—
114C2×C79—21no—
115C2×C79#160211yescharacteristic
116C2×C79#220111yescharacteristic
117C2×C79#220411yescharacteristic
118#2809—791no—
119#2028D7911yescharacteristic
120#1617—792no—
121#1353—792no—
122#1319—791no—
123C79#1222511yescharacteristic, complemented, Sylow
124#544C2×D7911yescharacteristic
125#484—1582no—
126#419—21no—
127#400—792no—
128#398—1582no—
129#391—1581no—
130#391C2×D7911yescharacteristic
131#389—1581no—
132#387—1582no—
133#343C2×D7911yescharacteristic
134#339—21no—
135#328—1582no—
136#324—792no—
137C22×Q8C22⋊D7911yescharacteristic
138C2×M16—21no—
139C4⋊Q8—41no—
140#125—41no—
141C22⋊Q8—22no—
142#117—21no—
143C2×C4⋊C4—792no—
144C2×C4⋊C4—1582no—
145C2×C4⋊C4—1581no—
146C2×C4⋊C4—1581no—
147C2×C4⋊C4—1581no—
148C2×C4⋊C4C22×D7911yescharacteristic
149C2×C42—1581no—
150C2×C42C22⋊D7911yescharacteristic
151#107—3162no—
152#103—41no—
153#103—41no—
154#103—3162no—
155C22⋊C8—1582no—
156#95—22no—
157#95—792no—
158#95—1582no—
159#95—1582no—
160#95—1582no—
161C2×Q8—21no—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#1192961412yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—1582no—
169C22×C4—1582no—
170C22×C4D4×D7912yes—
171C22×C4C2×C22⋊D7911yescharacteristic
172M16—41no—
173M16—41no—
174C2×C8—41no—
175C2×C8—3162no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—3162no—
182C4⋊C4—3164no—
183C4⋊C4—3162no—
184C4⋊C4—6321no—
185C4⋊C4—6321no—
186C22⋊C4—41no—
187C42—41no—
188C42—41no—
189C42—6321no—
190C23#1457912911yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—82no—
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—41no—
205C2×C4—41no—
206C2×C4—3162no—
207C2×C4—3162no—
208C2×C4—3162no—
209C2×C4—3162no—
210C2×C4—3162no—
211C2×C4—3162no—
212C2×C4C2×#1192961411yescharacteristic
213C2×C4#1457919111yescharacteristic
214C8—81no—
215C8—6322no—
216C22—21no—
217C22—21no—
218C22#1909871811yescharacteristic
219C22#1909874011yescharacteristic
220C22#1909874911yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—81no—
227C4—81no—
228C4—6322no—
229C4—6322no—
230C2—21no—
231C2—21no—
232C2#2376235611yescharacteristic
233C2#2376541211yescharacteristic
234C2#2376542511yescharacteristic
subgroupG/H
centerC22#19098749
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2028C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2028D79
hypocenterC79#12225
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2028

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
#237713882
#237624422
#237614262
#237613342
C79×#20282
#1222579