Finite Group Databasebeta

← #34186133

Subgroups — #34186133

Order 20224.

454 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 115 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#23768762C211yesmaximal, characteristic
2#23763396C211yesmaximal, characteristic
3#23761830C212yesmaximal
4#23761259C212yesmaximal
5C79×#2983C211yesmaximal, characteristic
6C2×C4×Dic158C2211yescharacteristic
7#19098587C2212yes—
8C2×#14579080—21no—
9#19098502C412yes—
10#19098483C412yes—
11#19098470—21no—
12#19098459—21no—
13#19098447—22no—
14#19098429C2211yescharacteristic
15#19098332—21no—
16#19098295—24no—
17C2×C4×Q8×C79C2211yescharacteristic
18#446×C79—24no—
19#432×C79—22no—
20#431×C79C2212yes—
21C22×Dic158C2×C411yescharacteristic
22C2×#11929608—22no—
23C4×Dic158—22no—
24C4×Dic158—22no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—22no—
28C2×Dic79⋊C4—21no—
29C2×C4×Dic79—21no—
30C4⋊(C79⋊C8)—22no—
31C4⋊(C79⋊C8)—22no—
32#14579080—42no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95—21no—
37C79⋊#95C2×C412yes—
38C22×Q8×C79D411yescharacteristic
39C4×Q8×C79—22no—
40C4×Q8×C79—22no—
41C2×C4⋊C4×C79C2×C412yes—
42C2×C4⋊C4×C79D411yescharacteristic
43C2×C42×C79D412yes—
44C2×C42×C79C2311yescharacteristic
45C4⋊C8×C79—24no—
46C4⋊C8×C79—22no—
47C22⋊C8×C79—22no—
48#97×C79—24no—
49C4×C8×C79D414yes—
50C2×Dic158—22no—
51C2×Dic158—42no—
52C22×Dic79—21no—
53C22×Dic79—22no—
54C22×Dic79—21no—
55#11929608—42no—
56#11929608—42no—
57C2×C79⋊C8—42no—
58C4⋊Dic79—22no—
59C4⋊Dic79—41no—
60C4⋊Dic79—41no—
61Dic79⋊C4—44no—
62Dic79⋊C4—41no—
63Dic79⋊C4—41no—
64C4×Dic79—42no—
65C2×Q8×C79—22no—
66C2×Q8×C79—21no—
67C2×Q8×C79—21no—
68C2×Q8×C79—22no—
69C22×C4×C79C22⋊C412yes—
70C22×C4×C79C22⋊C412yes—
71C22×C4×C79C22×C411yescharacteristic
72C22×C4×C79C2×D411yescharacteristic
73C22×C4×C79C2×D411yescharacteristic
74C2×C8×C79—24no—
75C4⋊C4×C79—24no—
76C4⋊C4×C79—22no—
77C42×C79—22no—
78C42×C79—22no—
79C42×C79C2×D412yes—
80C42×C79#4112yes—
81C23×C79C2×C22⋊C411yescharacteristic
82Q8×C79—42no—
83Q8×C79—42no—
84C2×C4×C79—22no—
85C2×C4×C79—22no—
86C2×C4×C79—22no—
87C2×C4×C79—22no—
88C2×C4×C79—41no—
89C2×C4×C79—41no—
90C2×C4×C79—42no—
91C2×C4×C79C4×D412yes—
92C2×C4×C79#12011yescharacteristic
93C2×C4×C79#12112yes—
94C2×C4×C79#12311yescharacteristic
95C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C79#13712yes—
97C2×C4×C79#13711yescharacteristic
98C2×Dic79—22no—
99C2×Dic79—41no—
100C2×Dic79—41no—
101C2×Dic79—42no—
102C2×Dic79—42no—
103C2×Dic79—42no—
104C2×Dic79—41no—
105C2×Dic79—42no—
106Dic158—24no—
107Dic158—42no—
108C8×C79—44no—
109C79⋊C8—82no—
110C22×C79—21no—
111C22×C79—21no—
112C22×C79#38611yescharacteristic
113C22×C79#41711yescharacteristic
114C22×C79#41911yescharacteristic
115C4×C79—22no—
116C4×C79—21no—
117C4×C79—21no—
118C4×C79—42no—
119C4×C79—42no—
120C4×C79#46812yes—
121C4×C79#47012yes—
122Dic79—42no—
123Dic79—82no—
124Dic79—81no—
125#9488—791nomaximal, Sylow
126C2×C79—21no—
127C2×C79—21no—
128C2×C79#189111yescharacteristic
129C2×C79#189411yescharacteristic
130C2×C79#191211yescharacteristic
131#2983D7911yescharacteristic
132#2581—791no—
133#1801—791no—
134#1487—792no—
135#1286—792no—
136C79#948811yescharacteristic, complemented, Sylow
137C2×C4×Q8—791no—
138C2×C4×Q8C2×D7911yescharacteristic
139#446—24no—
140#432—22no—
141#431C2×D7912yes—
142#423—792no—
143C2×#101—1581no—
144#400—792no—
145#395—792no—
146#391—1581no—
147#389—1581no—
148#387—1582no—
149#382—791no—
150#343—1581no—
151Q8⋊C8—1584no—
152C22×Q8—791no—
153C22×Q8D31611yescharacteristic
154C2×M16—1582no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—1582no—
158C4×Q8—1582no—
159C2×C4⋊C4—791no—
160C2×C4⋊C4—1582no—
161C2×C4⋊C4—1581no—
162C2×C4⋊C4—1581no—
163C2×C4⋊C4C4×D7912yes—
164C2×C4⋊C4D31611yescharacteristic
165C2×C42—1581no—
166C2×C42C22⋊D7912yes—
167C2×C42C22×D7911yescharacteristic
168C4⋊C8—24no—
169C4⋊C8—22no—
170C4⋊C8—1582no—
171C4⋊C8—1582no—
172#101—3162no—
173C22⋊C8—22no—
174#97—24no—
175C4×C8C22⋊D7914yes—
176#95—792no—
177#95—1582no—
178#95—1582no—
179#95—1582no—
180#95—1581no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—22no—
185C2×Q8—3162no—
186C2×Q8—1582no—
187C22×C4—1582no—
188C22×C4—1581no—
189C22×C4—1581no—
190C22×C4(C2×D79)⋊C412yes—
191C22×C4(C2×D79)⋊C412yes—
192C22×C4C2×C4×D7911yescharacteristic
193C22×C4C2×C22⋊D7911yescharacteristic
194C22×C4C2×D31611yescharacteristic
195M16—3162no—
196M16—3162no—
197C2×C8—24no—
198C2×C8—3162no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—3164no—
202C4⋊C4—3161no—
203C4⋊C4—3161no—
204C4⋊C4—3161no—
205C4⋊C4—3161no—
206C4⋊C4—1582no—
207C42—22no—
208C42—22no—
209C42—3162no—
210C42C2×C22⋊D7912yes—
211C42#1192962412yes—
212C23C2×(C2×D79)⋊C411yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—3162no—
216Q8—1584no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—3161no—
225C2×C4—3161no—
226C2×C4—3162no—
227C2×C4—3162no—
228C2×C4—3162no—
229C2×C4—3161no—
230C2×C4—3162no—
231C2×C4—1582no—
232C2×C4C4×C22⋊D7912yes—
233C2×C4#1457913011yescharacteristic
234C2×C4#1457913512yes—
235C2×C4#1457914411yescharacteristic
236C2×C4#1457918911yescharacteristic
237C2×C4#1457919412yes—
238C2×C4#1457919411yescharacteristic
239C8—44no—
240C8—6322no—
241C22—21no—
242C22—21no—
243C22#1909844211yescharacteristic
244C22#1909856511yescharacteristic
245C22#1909857311yescharacteristic
246C4—22no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—3162no—
252C4—6322no—
253C4—6321no—
254C4#1909880912yes—
255C4#1909882112yes—
256C2—21no—
257C2—21no—
258C2#2376381811yescharacteristic
259C2#2376383311yescharacteristic
260C2#2376393711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2983C2
socleC22×C79#386
prime radicalC23×C79C2×C22⋊C4
cosocleC2×C42×C79C23
hypercenter#2983D79
hypocenterC79#9488
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
136. C79#9488semidirect

Maximal subgroups

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

subgroupindex
#237687622
#237633962
#237618302
#237612592
C79×#29832
#948879