Finite Group Databasebeta

← #34186141

Subgroups — #34186141

Order 20224.

522 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 111 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, complemented
2#23768305C211yesmaximal, characteristic
3#23761824C212yesmaximal
4#23761260C212yesmaximal, complemented
5C79×#1801C211yesmaximal, characteristic, complemented
6C2×#14579137—21no—
7C2×C4×D316C2211yescharacteristic
8#19098694—24no—
9#19098620—22no—
10#19098616C2212yes—
11#19098493C412yes—
12#19098480C412yes—
13#19098454—22no—
14#19098429C2211yescharacteristic
15#19098420—21no—
16#19098290—24no—
17C2×C4×Q8×C79C2211yescharacteristic
18#423×C79C2212yes—
19C2×#101×C79—21no—
20#343×C79—21no—
21C22×D316C2×C411yescharacteristic
22C22×C4×D79—21no—
23#14579137—44no—
24C4×D316—22no—
25C4×D316—22no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—22no—
29C2×(C2×D79)⋊C4—22no—
30C2×(C2×D79)⋊C4—22no—
31C2×(C2×D79)⋊C4—21no—
32C4⋊(C79⋊C8)—24no—
33C4⋊(C79⋊C8)—22no—
34C22⋊(C79⋊C8)—22no—
35#14579072—24no—
36C4×C79⋊C8D414yes—
37C79⋊#95—22no—
38C79⋊#95—21no—
39C79⋊#95C2×C412yes—
40C22×Q8×C79D411yescharacteristic
41C2×M16×C79—22no—
42C4×Q8×C79—22no—
43C4×Q8×C79—22no—
44C2×C4⋊C4×C79C2×C412yes—
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C42×C79D412yes—
47C2×C42×C79C2311yescharacteristic
48C4⋊C8×C79—22no—
49C4⋊C8×C79—22no—
50#101×C79—42no—
51C23×D79—21no—
52C2×D316—22no—
53C2×D316—42no—
54C22×Dic79—22no—
55C22×Dic79—21no—
56C2×C4×D79—42no—
57C2×C4×D79—42no—
58C2×C4×D79—41no—
59C2×C4×D79—41no—
60C2×C79⋊C8—24no—
61C4⋊Dic79—22no—
62C4⋊Dic79—42no—
63Dic79⋊C4—44no—
64(C2×D79)⋊C4—44no—
65(C2×D79)⋊C4—44no—
66(C2×D79)⋊C4—41no—
67(C2×D79)⋊C4—41no—
68C2×Q8×C79—22no—
69C2×Q8×C79—21no—
70C2×Q8×C79—22no—
71C2×Q8×C79—21no—
72C22×C4×C79C22⋊C412yes—
73C22×C4×C79C22⋊C412yes—
74C22×C4×C79C22×C411yescharacteristic
75C22×C4×C79C2×D411yescharacteristic
76C22×C4×C79C2×D411yescharacteristic
77M16×C79—42no—
78M16×C79—42no—
79C2×C8×C79—42no—
80C4⋊C4×C79—24no—
81C4⋊C4×C79—22no—
82C42×C79—22no—
83C42×C79—22no—
84C42×C79C2×D412yes—
85C42×C79#4112yes—
86C23×C79C2×C22⋊C411yescharacteristic
87C22×D79—22no—
88C22×D79—41no—
89C22×D79—41no—
90C22×D79—41no—
91C22×D79—41no—
92C22×D79—41no—
93C22×D79—41no—
94Q8×C79—42no—
95Q8×C79—42no—
96C2×C4×C79—22no—
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—22no—
100C2×C4×C79—41no—
101C2×C4×C79—41no—
102C2×C4×C79—42no—
103C2×C4×C79C4×D412yes—
104C2×C4×C79#12011yescharacteristic
105C2×C4×C79#12112yes—
106C2×C4×C79#12311yescharacteristic
107C2×C4×C79AGL(1,ℤ/8)12yes—
108C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
109C2×C4×C79#13711yescharacteristic
110C2×Dic79—42no—
111C2×Dic79—42no—
112C2×Dic79—41no—
113C2×Dic79—41no—
114C2×Dic79—42no—
115C2×Dic79—41no—
116D316—24no—
117D316—42no—
118C4×D79—82no—
119C4×D79—82no—
120C8×C79—82no—
121C79⋊C8—44no—
122C22×C79—21no—
123C22×C79—21no—
124C22×C79#38611yescharacteristic
125C22×C79#41711yescharacteristic
126C22×C79#41811yescharacteristic
127C2×D79—42no—
128C2×D79—42no—
129C2×D79—42no—
130C2×D79—81no—
131C2×D79—81no—
132C2×D79—81no—
133C2×D79—81no—
134C4×C79—22no—
135C4×C79—21no—
136C4×C79—21no—
137C4×C79—42no—
138C4×C79—42no—
139C4×C79#46812yes—
140C4×C79#46912yes—
141Dic79—82no—
142Dic79—81no—
143#9490—791nomaximal, Sylow
144C2×C79—21no—
145C2×C79—21no—
146C2×C79#189111yescharacteristic
147C2×C79#189511yescharacteristic
148C2×C79#191211yescharacteristic
149D79—82no—
150#2983—791no—
151#2536—791no—
152#1801D7911yescharacteristic, complemented
153#1486—792no—
154#1286—792no—
155C79#949011yescharacteristic, complemented, Sylow
156C2×C22⋊Q8—1581no—
157C2×C4×Q8C2×D7911yescharacteristic
158C2×C4×D4—791no—
159#446—1584no—
160#432—1582no—
161#431—792no—
162#423C2×D7912yes—
163C2×#101—21no—
164#398—792no—
165#394—792no—
166#388—1582no—
167#382—791no—
168#380—1581no—
169#343—21no—
170D4⋊C8—1584no—
171C22×Q8C22⋊D7911yescharacteristic
172C22×D4—791no—
173C23×C4—1581no—
174C2×M16—22no—
175C22⋊Q8—3164no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—1582no—
179C4×D4—1582no—
180C2×C4⋊C4—791no—
181C2×C4⋊C4—1582no—
182C2×C4⋊C4—1581no—
183C2×C4⋊C4C4×D7912yes—
184C2×C4⋊C4C22⋊D7911yescharacteristic
185C2×C22⋊C4—1582no—
186C2×C22⋊C4—1582no—
187C2×C22⋊C4—1581no—
188C2×C42D31612yes—
189C2×C42C22×D7911yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—1584no—
193C4⋊C8—1582no—
194#101—42no—
195C22⋊C8—1582no—
196#97—1584no—
197C4×C8—794no—
198#95—792no—
199#95—1582no—
200#95—1581no—
201C24—1581no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×D4—3162no—
207C2×D4—1582no—
208C22×C4—3162no—
209C22×C4—3162no—
210C22×C4—3161no—
211C22×C4—3161no—
212C22×C4—1582no—
213C22×C4—1581no—
214C22×C4(C2×D79)⋊C412yes—
215C22×C4(C2×D79)⋊C412yes—
216C22×C4C2×C4×D7911yescharacteristic
217C22×C4C2×C22⋊D7911yescharacteristic
218C22×C4C2×D31611yescharacteristic
219M16—42no—
220M16—42no—
221C2×C8—42no—
222C2×C8—1584no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—3164no—
226C4⋊C4—3162no—
227C4⋊C4—1582no—
228C22⋊C4—3164no—
229C22⋊C4—3164no—
230C22⋊C4—3161no—
231C22⋊C4—3161no—
232C42—22no—
233C42—22no—
234C42D4×D7912yes—
235C42#1192962312yes—
236C23—3161no—
237C23—3161no—
238C23—3161no—
239C23—3161no—
240C23—3161no—
241C23—3161no—
242C23—1582no—
243C23C2×(C2×D79)⋊C411yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—3162no—
247D4—1584no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—3162no—
256C2×C4—3162no—
257C2×C4—3161no—
258C2×C4—3161no—
259C2×C4—3162no—
260C2×C4—3161no—
261C2×C4—6322no—
262C2×C4—6322no—
263C2×C4(C22⋊D79)⋊C412yes—
264C2×C4#1457913111yescharacteristic
265C2×C4#1457913312yes—
266C2×C4#1457914511yescharacteristic
267C2×C4#1457918612yes—
268C2×C4#1457918711yescharacteristic
269C2×C4#1457919211yescharacteristic
270C8—82no—
271C8—3164no—
272C22—21no—
273C22—21no—
274C22—3162no—
275C22—3162no—
276C22—3162no—
277C22—6321no—
278C22—6321no—
279C22—6321no—
280C22—6321no—
281C22#1909844011yescharacteristic
282C22#1909856211yescharacteristic
283C22#1909856711yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—42no—
289C4—6322no—
290C4—6321no—
291C4#1909881112yes—
292C4#1909881412yes—
293C2—21no—
294C2—21no—
295C2—6322no—
296C2#2376381611yescharacteristic
297C2#2376383611yescharacteristic
298C2#2376393511yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1801C2
socleC22×C79#386
prime radicalC22×D316C2×C4
cosocleC2×C42×C79C23
hypercenter#1801D79
hypocenterC79#9490
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #23772485C2semidirect
4. #23761260C2semidirect
5. C79×#1801C2semidirect
152. #1801D79semidirect
155. C79#9490semidirect

Maximal subgroups

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

subgroupindex
#237724852
#237683052
#237618242
#237612602
C79×#18012
#949079