Finite Group Databasebeta

← #34183787

Subgroups — #34183787

Order 20224.

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

normal subgroups only

This group has 474 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#23772485C211yesmaximal, characteristic
2#23766116C211yesmaximal, characteristic
3#23761794C212yesmaximal
4#23761228C212yesmaximal
5C79×#1461C211yesmaximal, characteristic
6C2×#14579159C411yescharacteristic
7C2×C4×Dic158C2211yescharacteristic
8#19098694—24no—
9#19098620—22no—
10#19098616C2212yes—
11#19098470C411yescharacteristic
12#19098468C411yescharacteristic
13#19098447—22no—
14#19098444—22no—
15#19098428C411yescharacteristic
16#19098409C2211yescharacteristic
17#19098293—24no—
18C2×C4×Q8×C79C2211yescharacteristic
19#431×C79C2212yes—
20#324×C79—22no—
21C22×Dic158C2×C411yescharacteristic
22#14579159—22no—
23#14579159—22no—
24C4×Dic158—22no—
25C4×Dic158—22no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—22no—
28C2×Dic79⋊C4—21no—
29C2×C4×Dic79—21no—
30C2×C4×Dic79C2×C411yescharacteristic
31C2×C4×Dic79C2×C411yescharacteristic
32C4⋊(C79⋊C8)—24no—
33C4⋊(C79⋊C8)—22no—
34C22⋊(C79⋊C8)—22no—
35#14579072—24no—
36C4×C79⋊C8D414yes—
37C79⋊#95—22no—
38C79⋊#95—22no—
39C79⋊#95—22no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C22×Q8×C79C2×C411yescharacteristic
43C4×Q8×C79—22no—
44C4×Q8×C79—22no—
45C2×C4⋊C4×C79C2×C411yescharacteristic
46C2×C4⋊C4×C79D412yes—
47C2×C42×C79D412yes—
48C2×C42×C79C2311yescharacteristic
49C4⋊C8×C79—22no—
50C22⋊C8×C79—22no—
51C4×C8×C79—22no—
52C2×Dic158—22no—
53C2×Dic158—21no—
54C2×Dic158—22no—
55C2×Dic158M1612yes—
56C22×Dic79—21no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C2×C79⋊C8—24no—
61C4⋊Dic79—22no—
62Dic79⋊C4—28no—
63Dic79⋊C4—41no—
64Dic79⋊C4—41no—
65C4×Dic79—21no—
66C4×Dic79—22no—
67C4×Dic79—42no—
68C4×Dic79M1612yes—
69C2×Q8×C79—22no—
70C2×Q8×C79—21no—
71C2×Q8×C79—21no—
72C2×Q8×C79—22no—
73C22×C4×C79C22⋊C412yes—
74C22×C4×C79C22⋊C412yes—
75C22×C4×C79C22×C411yescharacteristic
76C22×C4×C79C2×D411yescharacteristic
77C22×C4×C79C2×D411yescharacteristic
78C2×C8×C79—42no—
79C4⋊C4×C79—22no—
80C4⋊C4×C79—24no—
81C42×C79—22no—
82C42×C79—22no—
83C42×C79C2×D412yes—
84C42×C79#4112yes—
85C23×C79C2×C22⋊C411yescharacteristic
86Q8×C79—42no—
87Q8×C79—42no—
88C2×C4×C79—22no—
89C2×C4×C79—22no—
90C2×C4×C79—22no—
91C2×C4×C79—22no—
92C2×C4×C79—41no—
93C2×C4×C79—41no—
94C2×C4×C79—42no—
95C2×C4×C79C4×D412yes—
96C2×C4×C79#12011yescharacteristic
97C2×C4×C79#12112yes—
98C2×C4×C79#12311yescharacteristic
99C2×C4×C79C2×M1611yescharacteristic
100C2×C4×C79C2×M1611yescharacteristic
101C2×C4×C79#13112yes—
102C2×Dic79—22no—
103C2×Dic79—22no—
104C2×Dic79—41no—
105C2×Dic79—41no—
106C2×Dic79—41no—
107C2×Dic79—41no—
108C2×Dic79—41no—
109C2×Dic79—41no—
110C2×Dic79—42no—
111C2×Dic79—41no—
112C2×Dic79—41no—
113Dic158—24no—
114Dic158—42no—
115C8×C79—82no—
116C79⋊C8—44no—
117C22×C79—21no—
118C22×C79—21no—
119C22×C79#38611yescharacteristic
120C22×C79#40711yescharacteristic
121C22×C79#40811yescharacteristic
122C4×C79—22no—
123C4×C79—21no—
124C4×C79—21no—
125C4×C79—42no—
126C4×C79—42no—
127C4×C79#43512yes—
128C4×C79#43612yes—
129Dic79—42no—
130Dic79—42no—
131Dic79—81no—
132Dic79—81no—
133#9104—791nomaximal, Sylow
134C2×C79—21no—
135C2×C79—21no—
136C2×C79#187011yescharacteristic
137C2×C79#190911yescharacteristic
138C2×C79#191211yescharacteristic
139#2983—791no—
140#2327—791no—
141#1479—792no—
142#1461D7911yescharacteristic
143#1274—792no—
144C79#910411yescharacteristic, complemented, Sylow
145C2×C4⋊Q8—791no—
146C2×C4×Q8—791no—
147C2×C4×Q8C2×D7911yescharacteristic
148#446—1584no—
149#432—1582no—
150#431—792no—
151#431C2×D7912yes—
152#391—791no—
153#391—791no—
154#387—1582no—
155#387—1582no—
156#382—791no—
157#376—791no—
158Q8⋊C8—1584no—
159#324—22no—
160C22×Q8—791no—
161C22×Q8C4×D7911yescharacteristic
162C4⋊Q8—1582no—
163C4⋊Q8—1582no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—1582no—
167C4×Q8—1582no—
168C2×C4⋊C4—791no—
169C2×C4⋊C4—1581no—
170C2×C4⋊C4—1582no—
171C2×C4⋊C4C4×D7911yescharacteristic
172C2×C4⋊C4C22⋊D7912yes—
173C2×C42—791no—
174C2×C42—791no—
175C2×C42—1581no—
176C2×C42D31612yes—
177C2×C42C22×D7911yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—1584no—
180C4⋊C8—1582no—
181C22⋊C8—22no—
182C22⋊C8—1582no—
183#97—1584no—
184C4×C8—22no—
185C4×C8—794no—
186#95—1582no—
187#95—1582no—
188#95—1582no—
189#95—1581no—
190#95—1581no—
191C2×Q8—22no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—792no—
196C2×Q8—1582no—
197C2×Q8—1582no—
198C2×Q8—1581no—
199C22×C4—1581no—
200C22×C4—1581no—
201C22×C4—1581no—
202C22×C4—1581no—
203C22×C4(C2×D79)⋊C412yes—
204C22×C4(C2×D79)⋊C412yes—
205C22×C4C2×C4×D7911yescharacteristic
206C22×C4C2×C22⋊D7911yescharacteristic
207C22×C4C2×D31611yescharacteristic
208C2×C8—42no—
209C2×C8—1584no—
210C4⋊C4—22no—
211C4⋊C4—24no—
212C4⋊C4—3161no—
213C4⋊C4—3161no—
214C4⋊C4—1588no—
215C4⋊C4—1582no—
216C42—22no—
217C42—22no—
218C42—3162no—
219C42—792no—
220C42—1582no—
221C42—1581no—
222C42D4×D7912yes—
223C42#1192962312yes—
224C23C2×(C2×D79)⋊C411yescharacteristic
225Q8—42no—
226Q8—42no—
227Q8—3162no—
228Q8—1584no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—3161no—
237C2×C4—3161no—
238C2×C4—3161no—
239C2×C4—3161no—
240C2×C4—3161no—
241C2×C4—3161no—
242C2×C4—3162no—
243C2×C4—3161no—
244C2×C4—3161no—
245C2×C4—1582no—
246C2×C4—1582no—
247C2×C4(C22⋊D79)⋊C412yes—
248C2×C4#1457913111yescharacteristic
249C2×C4#1457913312yes—
250C2×C4#1457914511yescharacteristic
251C2×C4M16×D7911yescharacteristic
252C2×C4M16×D7911yescharacteristic
253C2×C4#1457916712yes—
254C8—82no—
255C8—3164no—
256C22—21no—
257C22—21no—
258C22#1909844011yescharacteristic
259C22#1909852611yescharacteristic
260C22#1909852911yescharacteristic
261C4—22no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—42no—
266C4—3162no—
267C4—3162no—
268C4—6321no—
269C4—6321no—
270C4#1909863712yes—
271C4#1909863812yes—
272C2—21no—
273C2—21no—
274C2#2376369511yescharacteristic
275C2#2376392011yescharacteristic
276C2#2376393511yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1461C2
socleC22×C79#386
prime radicalC23×C79C2×C22⋊C4
cosocleC2×C42×C79C23
hypercenter#1461D79
hypocenterC79#9104
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
144. C79#9104semidirect

Maximal subgroups

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

subgroupindex
#237724852
#237661162
#237617942
#237612282
C79×#14612
#910479