Finite Group Databasebeta

← #34189228

Subgroups — #34189228

Order 20224.

468 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 85 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#23773526C212yesmaximal
2#23764339C211yesmaximal, characteristic
3#23762015C212yesmaximal
4#23761270C211yesmaximal, characteristic
5C79×#3249C211yesmaximal, characteristic
6C2×C4×Dic158C2211yescharacteristic
7#19098891—22no—
8#19098845—22no—
9#19098770—22no—
10#19098718—22no—
11#19098668—24no—
12#19098620C2212yes—
13C2×#14579072C2211yescharacteristic
14#19098317—22no—
15#19098305—24no—
16#533×C79C2212yes—
17#477×C79—22no—
18#467×C79—22no—
19#462×C79—22no—
20C2×C4⋊C8×C79C2211yescharacteristic
21C22×Dic158D411yescharacteristic
22C2×#11929614—24no—
23C22×C79⋊C8—21no—
24C4×Dic158—22no—
25C4×Dic158—22no—
26C2×C4⋊Dic79D411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×C4×Dic79—21no—
29#14579092—24no—
30C4⋊(C79⋊C8)—22no—
31#14579085—24no—
32#14579084—24no—
33C22⋊(C79⋊C8)—22no—
34#14579072—22no—
35#14579072D414yes—
36C2×Q16×C79—22no—
37C22×C8×C79D412yes—
38C4⋊Q8×C79D412yes—
39C4⋊Q8×C79D412yes—
40#125×C79—22no—
41C22⋊Q8×C79—22no—
42C22⋊Q8×C79—22no—
43C4×Q8×C79—22no—
44C4×Q8×C79—22no—
45#117×C79—22no—
46C2×C42×C79C2311yescharacteristic
47#107×C79—22no—
48C4⋊C8×C79—22no—
49#103×C79—24no—
50C2×Dic158—22no—
51C2×Dic158—42no—
52C22×Dic79—21no—
53C22×Dic79—21no—
54#11929614—48no—
55C2×C79⋊C8—22no—
56C2×C79⋊C8—24no—
57C2×C79⋊C8—41no—
58C2×C79⋊C8—41no—
59C4⋊Dic79—22no—
60Dic79⋊C4—42no—
61C4×Dic79—41no—
62C4×Dic79—41no—
63C2×Q8×C79—22no—
64C2×Q8×C79—22no—
65C22×C4×C79C2×D411yescharacteristic
66C22×C4×C79C2×D411yescharacteristic
67C22×C4×C79#4111yescharacteristic
68Q16×C79—82no—
69C2×C8×C79—22no—
70C2×C8×C79—42no—
71C4⋊C4×C79—22no—
72C4⋊C4×C79—22no—
73C4⋊C4×C79—22no—
74C4⋊C4×C79—22no—
75C4⋊C4×C79—42no—
76C4⋊C4×C79—42no—
77C22⋊C4×C79—42no—
78C42×C79—42no—
79C42×C79C2×D412yes—
80C42×C79C2×D412yes—
81C23×C79#12111yescharacteristic
82Q8×C79—44no—
83Q8×C79—42no—
84Q8×C79—42no—
85C2×C4×C79—22no—
86C2×C4×C79—21no—
87C2×C4×C79—21no—
88C2×C4×C79—41no—
89C2×C4×C79—42no—
90C2×C4×C79—42no—
91C2×C4×C79#12012yes—
92C2×C4×C79#12011yescharacteristic
93C2×C4×C79#12112yes—
94C2×C4×C79#12711yescharacteristic
95C2×Dic79—22no—
96C2×Dic79—41no—
97C2×Dic79—41no—
98C2×Dic79—41no—
99C2×Dic79—42no—
100Dic158—24no—
101Dic158—42no—
102C8×C79—42no—
103C79⋊C8—42no—
104C79⋊C8—44no—
105C22×C79—21no—
106C22×C79—21no—
107C22×C79#39211yescharacteristic
108C22×C79#47312yes—
109C4×C79—21no—
110C4×C79—22no—
111C4×C79—22no—
112C4×C79—82no—
113C4×C79—82no—
114C4×C79#45511yescharacteristic
115C4×C79#45611yescharacteristic
116Dic79—42no—
117Dic79—81no—
118#10031—791nomaximal, Sylow
119C2×C79—22no—
120C2×C79#200711yescharacteristic
121C2×C79#201011yescharacteristic
122C2×C79#202011yescharacteristic
123#3249D7911yescharacteristic
124#3113—792no—
125#1985—791no—
126#1531—792no—
127#1290—791no—
128C79#1003111yescharacteristic, complemented, Sylow
129#533C2×D7912yes—
130C2×C4×Q8—791no—
131#483—1582no—
132#477—22no—
133#475—1582no—
134#467—22no—
135#462—22no—
136#462—1582no—
137#451—1582no—
138#441—1584no—
139#432—792no—
140C2×C4⋊C8C2×D7911yescharacteristic
141C2×#97—791no—
142#336—1582no—
143#330—1584no—
144C22×Q8—791no—
145C2×Q16—22no—
146C2×Q16—1584no—
147C22×C8—1581no—
148C22×C8C22⋊D7912yes—
149C4⋊Q8C22⋊D7912yes—
150C4⋊Q8C22⋊D7912yes—
151#125—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—1582no—
157C4×Q8—1582no—
158#117—22no—
159C2×C4⋊C4—791no—
160C2×C4⋊C4—1581no—
161C2×C42—1581no—
162C2×C42C22×D7911yescharacteristic
163#107—22no—
164#106—1584no—
165C4⋊C8—22no—
166C4⋊C8—1582no—
167#103—24no—
168#103—1584no—
169#103—1584no—
170C22⋊C8—1582no—
171#97—794no—
172#97—1582no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—3162no—
176C2×Q8—1582no—
177C22×C4—1581no—
178C22×C4—1581no—
179C22×C4D4×D7911yescharacteristic
180C22×C4C2×C22⋊D7911yescharacteristic
181C22×C4#1192962311yescharacteristic
182Q16—82no—
183Q16—3168no—
184C2×C8—22no—
185C2×C8—42no—
186C2×C8—3161no—
187C2×C8—3161no—
188C2×C8—1582no—
189C2×C8—1584no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—3162no—
197C4⋊C4—1582no—
198C22⋊C4—42no—
199C42—42no—
200C42—3161no—
201C42—3161no—
202C42D4×D7912yes—
203C42C2×C22⋊D7912yes—
204C23#1457913211yescharacteristic
205Q8—44no—
206Q8—42no—
207Q8—42no—
208Q8—3162no—
209Q8—1584no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—3161no—
217C2×C4—3161no—
218C2×C4—3161no—
219C2×C4—3162no—
220C2×C4—1582no—
221C2×C4#1457912912yes—
222C2×C4#1457913011yescharacteristic
223C2×C4#1457913312yes—
224C2×C4#1457915711yescharacteristic
225C8—42no—
226C8—3162no—
227C8—3164no—
228C22—21no—
229C22—21no—
230C22#1909847111yescharacteristic
231C22#1909883512yes—
232C4—22no—
233C4—21no—
234C4—22no—
235C4—82no—
236C4—82no—
237C4—3162no—
238C4—6321no—
239C4#1909874111yescharacteristic
240C4#1909874611yescharacteristic
241C2—22no—
242C2#2376444711yescharacteristic
243C2#2376446411yescharacteristic
244C2#2376451911yescharacteristic
subgroupG/H
centerC22#19098471
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#3249C2
socleC22×C79#392
prime radicalC23×C79#121
hypercenter#3249D79
hypocenterC79#10031
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
128. C79#10031semidirect

Maximal subgroups

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

subgroupindex
#237735262
#237643392
#237620152
#237612702
C79×#32492
#1003179