Finite Group Databasebeta

← #34197939

Subgroups — #34197939

Order 20224.

374 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 87 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#23768605C211yesmaximal, characteristic
2#23761724C211yesmaximal, characteristic
3#23761345C214yesmaximal
4C79×#1467C211yesmaximal, characteristic
5C2×#14579161C411yescharacteristic
6#19098616—21no—
7#19098483—21no—
8#19098483—21no—
9#19098470C412yes—
10#19098461—21no—
11#19098461—21no—
12#19098432C2212yes—
13C4×C4⋊Dic79C411yescharacteristic
14#19098289—21no—
15#19098289C2212yes—
16C2×C4×Q8×C79C2211yescharacteristic
17#432×C79—21no—
18#324×C79—21no—
19#324×C79C2212yes—
20C22×Dic158—21no—
21#14579161—41no—
22#14579161—41no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79C2×C412yes—
26C2×Dic79⋊C4—22no—
27C2×Dic79⋊C4—22no—
28C2×C4×Dic79C2×C412yes—
29C4⋊(C79⋊C8)—41no—
30C22⋊(C79⋊C8)—22no—
31C4×C79⋊C8—41no—
32C79⋊#95—21no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C22×Q8×C79C2×C411yescharacteristic
36C4×Q8×C79—41no—
37C4×Q8×C79—41no—
38C2×C4⋊C4×C79D411yescharacteristic
39C2×C4⋊C4×C79D411yescharacteristic
40C2×C4⋊C4×C79C2311yescharacteristic
41C2×C42×C79C2×C411yescharacteristic
42C2×C42×C79D411yescharacteristic
43C2×C42×C79D411yescharacteristic
44C4⋊C8×C79—41no—
45C22⋊C8×C79—22no—
46#97×C79—41no—
47C2×Dic158—42no—
48C2×Dic158—82no—
49C22×Dic79—22no—
50C22×Dic79—22no—
51C2×C79⋊C8—42no—
52C4⋊Dic79—24no—
53C4⋊Dic79—42no—
54C4⋊Dic79—41no—
55C4⋊Dic79—41no—
56Dic79⋊C4—82no—
57Dic79⋊C4—82no—
58C4×Dic79—24no—
59C2×Q8×C79—22no—
60C2×Q8×C79—22no—
61C2×Q8×C79—41no—
62C22×C4×C79C22⋊C411yescharacteristic
63C22×C4×C79C22⋊C411yescharacteristic
64C22×C4×C79C22⋊C411yescharacteristic
65C22×C4×C79C22⋊C411yescharacteristic
66C22×C4×C79C22×C411yescharacteristic
67C22×C4×C79C2×D411yescharacteristic
68C22×C4×C79C2×D411yescharacteristic
69C2×C8×C79—42no—
70C4⋊C4×C79—41no—
71C4⋊C4×C79—41no—
72C4⋊C4×C79—41no—
73C42×C79—21no—
74C42×C79—21no—
75C42×C79—21no—
76C42×C79—21no—
77C42×C79—21no—
78C42×C79—21no—
79C23×C79C2×C22⋊C411yescharacteristic
80Q8×C79—44no—
81C2×C4×C79—21no—
82C2×C4×C79—21no—
83C2×C4×C79—21no—
84C2×C4×C79—21no—
85C2×C4×C79—22no—
86C2×C4×C79—21no—
87C2×C4×C79—21no—
88C2×C4×C79—21no—
89C2×C4×C79—21no—
90C2×C4×C79—21no—
91C2×C4×C79—21no—
92C2×C4×C79—41no—
93C2×C4×C79—41no—
94C2×C4×C79—41no—
95C2×C4×C79#9911yescharacteristic
96C2×C4×C79#9911yescharacteristic
97C2×C4×C79#10112yes—
98C2×C4×C79#13112yes—
99C2×Dic79—24no—
100C2×Dic79—42no—
101C2×Dic79—42no—
102C2×Dic79—42no—
103C2×Dic79—42no—
104C2×Dic79—42no—
105Dic158—84no—
106C8×C79—82no—
107C79⋊C8—82no—
108C22×C79—21no—
109C22×C79—21no—
110C22×C79#40811yescharacteristic
111C22×C79C2×#9911yescharacteristic
112C22×C79C2×#10111yescharacteristic
113C4×C79—21no—
114C4×C79—22no—
115C4×C79—22no—
116C4×C79—41no—
117C4×C79—41no—
118C4×C79—41no—
119C4×C79—41no—
120C4×C79#35612yes—
121Dic79—44no—
122Dic79—82no—
123#11574—791nomaximal, Sylow
124C2×C79—21no—
125C2×C79—21no—
126C2×C79#147211yescharacteristic
127C2×C79#212811yescharacteristic
128C2×C79C2×#35611yescharacteristic
129#2567—791no—
130#1467D7911yescharacteristic
131#1461—791no—
132#1325—794no—
133C79#1157411yescharacteristic, complemented, Sylow
134C2×C4⋊Q8—791no—
135C2×C4×Q8C2×D7911yescharacteristic
136#432—21no—
137#431—1581no—
138#395—1581no—
139#395—1581no—
140#391—792no—
141#389—1581no—
142#389—1581no—
143#384—792no—
144C4×C4⋊C4—791no—
145#324—21no—
146#324—792no—
147#324—1581no—
148#324C2×D7912yes—
149C22×Q8—1581no—
150C22×Q8C4×D7911yescharacteristic
151C4⋊Q8—3161no—
152C4⋊Q8—3161no—
153C4×Q8—41no—
154C4×Q8—41no—
155C2×C4⋊C4—792no—
156C2×C4⋊C4—1581no—
157C2×C4⋊C4—1582no—
158C2×C4⋊C4—1582no—
159C2×C4⋊C4—1581no—
160C2×C4⋊C4D31611yescharacteristic
161C2×C4⋊C4C22⋊D7911yescharacteristic
162C2×C4⋊C4C22×D7911yescharacteristic
163C2×C42—792no—
164C2×C42C4×D7911yescharacteristic
165C2×C42D31611yescharacteristic
166C2×C42C22⋊D7911yescharacteristic
167C4⋊C8—41no—
168C4⋊C8—3161no—
169C22⋊C8—22no—
170C22⋊C8—1582no—
171#97—41no—
172C4×C8—3161no—
173#95—1581no—
174#95—1582no—
175#95—1582no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—41no—
179C2×Q8—3162no—
180C2×Q8—6322no—
181C22×C4—1582no—
182C22×C4—1582no—
183C22×C4(C2×D79)⋊C411yescharacteristic
184C22×C4(C2×D79)⋊C411yescharacteristic
185C22×C4(C2×D79)⋊C411yescharacteristic
186C22×C4(C2×D79)⋊C411yescharacteristic
187C22×C4C2×C4×D7911yescharacteristic
188C22×C4C2×C22⋊D7911yescharacteristic
189C22×C4C2×D31611yescharacteristic
190C2×C8—42no—
191C2×C8—3162no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—3162no—
196C4⋊C4—3161no—
197C4⋊C4—3161no—
198C4⋊C4—6322no—
199C4⋊C4—6322no—
200C4⋊C4—1584no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—1584no—
208C23C2×(C2×D79)⋊C411yescharacteristic
209Q8—44no—
210Q8—6324no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—3162no—
226C2×C4—3162no—
227C2×C4—3162no—
228C2×C4—3162no—
229C2×C4—3162no—
230C2×C4—1584no—
231C2×C4#1457907511yescharacteristic
232C2×C4#1457907511yescharacteristic
233C2×C4#1457907912yes—
234C2×C4#1457916912yes—
235C8—82no—
236C8—6322no—
237C22—21no—
238C22—21no—
239C22#1909853111yescharacteristic
240C22C2×#1457907511yescharacteristic
241C22C2×#1457907911yescharacteristic
242C4—21no—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—3164no—
250C4—6322no—
251C4#1909836112yes—
252C2—21no—
253C2—21no—
254C2#2376176511yescharacteristic
255C2#2376507111yescharacteristic
256C2C2×#1909836111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1467C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1467D79
hypocenterC79#11574
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
133. C79#11574semidirect

Maximal subgroups

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

subgroupindex
#237686052
#237617242
#237613452
C79×#14672
#1157479