Finite Group Databasebeta

← #34177080

Subgroups — #34177080

Order 20224.

536 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 95 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#23764733C212yesmaximal, complemented
2#23764482C211yesmaximal, characteristic
3#23761223C212yesmaximal, complemented
4#23761221C211yesmaximal, characteristic
5C79×#2683C211yesmaximal, characteristic
6C22×#11929612—21no—
7C2×#14579161C2211yescharacteristic
8C2×#14579092—22no—
9C2×#14579084—21no—
10#19098325—22no—
11#19098318C2211yescharacteristic
12#19098318C2212yes—
13C2×C22⋊Q8×C79—22no—
14#396×C79—22no—
15#395×C79—21no—
16#390×C79C2211yescharacteristic
17#389×C79C2212yes—
18#386×C79—21no—
19C22×Dic158—21no—
20C2×#11929612—42no—
21C2×#11929612—42no—
22C2×#11929612—42no—
23C22×C79⋊C8—21no—
24C22×C79⋊C8—22no—
25#14579161—42no—
26C2×C4⋊Dic79—22no—
27#14579092—44no—
28#14579084—42no—
29C22×Q8×C79—21no—
30C22×D4×C79D411yescharacteristic
31C23×C4×C79—21no—
32C22⋊Q8×C79—44no—
33C22⋊Q8×C79—44no—
34C2×C4⋊C4×C79—22no—
35C2×C4⋊C4×C79—22no—
36C2×C4⋊C4×C79D412yes—
37C2×C4⋊C4×C79D411yescharacteristic
38C2×C4⋊C4×C79D412yes—
39C2×C22⋊C4×C79—22no—
40C2×C22⋊C4×C79—22no—
41C2×C22⋊C4×C79—21no—
42C2×C42×C79C2311yescharacteristic
43#95×C79—21no—
44#95×C79—22no—
45C2×Dic158—22no—
46C2×Dic158—41no—
47C2×Dic158—41no—
48C2×Dic158—41no—
49C2×Dic158—41no—
50C2×Dic158—41no—
51C22×Dic79—41no—
52#11929612—48no—
53C2×C79⋊C8—22no—
54C2×C79⋊C8—24no—
55C2×C79⋊C8—42no—
56C2×C79⋊C8—41no—
57C2×C79⋊C8—41no—
58C2×C79⋊C8—42no—
59C4⋊Dic79—24no—
60C4⋊Dic79—42no—
61C24×C79—21no—
62C2×Q8×C79—44no—
63C2×Q8×C79—81no—
64C2×D4×C79—21no—
65C2×D4×C79—41no—
66C2×D4×C79—41no—
67C2×D4×C79SD1612yes—
68C22×C4×C79—22no—
69C22×C4×C79—22no—
70C22×C4×C79—21no—
71C22×C4×C79—44no—
72C22×C4×C79—81no—
73C22×C4×C79C2×D412yes—
74C22×C4×C79C2×D411yescharacteristic
75C4⋊C4×C79—22no—
76C4⋊C4×C79—44no—
77C4⋊C4×C79—41no—
78C4⋊C4×C79—42no—
79C4⋊C4×C79—42no—
80C4⋊C4×C79—42no—
81C4⋊C4×C79SD1614yescomplemented
82C22⋊C4×C79—44no—
83C22⋊C4×C79—42no—
84C22⋊C4×C79—42no—
85C22⋊C4×C79—81no—
86C42×C79—41no—
87C23×C79—22no—
88C23×C79—42no—
89C23×C79—41no—
90C23×C79—42no—
91C23×C79—41no—
92C23×C79#12011yescharacteristic
93Q8×C79—84no—
94D4×C79—42no—
95D4×C79—81no—
96C2×C4×C79—21no—
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—21no—
100C2×C4×C79—24no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—41no—
106C2×C4×C79—42no—
107C2×C4×C79—42no—
108C2×C4×C79—84no—
109C2×C4×C79C2×SD1611yescharacteristic
110C2×C4×C79C2×SD1612yes—
111C2×C4×C79AGL(1,ℤ/8)12yes—
112C2×C4×C79#13711yescharacteristic
113C2×Dic79—42no—
114C2×Dic79—81no—
115C2×Dic79—81no—
116Dic158—42no—
117Dic158—42no—
118Dic158—81no—
119Dic158—81no—
120C79⋊C8—44no—
121C79⋊C8—42no—
122C22×C79—42no—
123C22×C79—44no—
124C22×C79—82no—
125C22×C79—81no—
126C22×C79—81no—
127C22×C79#44811yescharacteristic
128C22×C79#45012yes—
129C22×C79#45311yescharacteristic
130C22×C79#45612yes—
131C22×C79B(4,𝔽2)11yescharacteristic
132C4×C79—21no—
133C4×C79—21no—
134C4×C79—22no—
135C4×C79—22no—
136C4×C79—44no—
137C4×C79—82no—
138C4×C79—81no—
139Dic79—82no—
140#7754—791nomaximal, Sylow
141C2×C79—82no—
142C2×C79#159611yescharacteristic
143C2×C79#160412yes—
144C2×C79#162911yescharacteristic
145C2×C79#165112yes—
146C2×C79#166011yescharacteristic
147#2683D7911yescharacteristic
148#2058—792no—
149#2013—791no—
150#1271—792no—
151#1270—791no—
152C79#775411yescharacteristic, complemented, Sylow
153C22×SD16—1581no—
154C2×C4⋊Q8—791no—
155C2×C22⋊Q8—22no—
156C2×#106—1582no—
157C2×#103—1581no—
158#396—22no—
159#395—21no—
160#390C2×D7911yescharacteristic
161#389C2×D7912yes—
162#386—21no—
163#340—1582no—
164#336—792no—
165#336—791no—
166C22×Q8—21no—
167C22×Q8—1581no—
168C22×D4C22⋊D7911yescharacteristic
169C23×C4—21no—
170C2×SD16—3162no—
171C2×SD16—3162no—
172C2×SD16—3162no—
173C22×C8—1581no—
174C22×C8—1582no—
175C4⋊Q8—3162no—
176C22⋊Q8—44no—
177C22⋊Q8—44no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—1582no—
181C2×C4⋊C4C22⋊D7912yes—
182C2×C4⋊C4C22⋊D7912yes—
183C2×C4⋊C4C22⋊D7911yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C42C22×D7911yescharacteristic
188#106—3164no—
189#103—3162no—
190#95—21no—
191#95—22no—
192C24—21no—
193C2×Q8—44no—
194C2×Q8—81no—
195C2×Q8—3161no—
196C2×Q8—3161no—
197C2×Q8—3161no—
198C2×Q8—3161no—
199C2×Q8—3161no—
200C2×Q8—1582no—
201C2×D4—21no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4#1192961112yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—44no—
209C22×C4—81no—
210C22×C4—3161no—
211C22×C4C2×C22⋊D7911yescharacteristic
212C22×C4C2×C22⋊D7912yes—
213SD16—3168no—
214C2×C8—3162no—
215C2×C8—3161no—
216C2×C8—3161no—
217C2×C8—3162no—
218C2×C8—1582no—
219C2×C8—1584no—
220C4⋊C4—22no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—44no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—3162no—
227C4⋊C4—1584no—
228C4⋊C4#1192961214yescomplemented
229C22⋊C4—44no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—81no—
233C42—41no—
234C23—22no—
235C23—42no—
236C23—41no—
237C23—42no—
238C23—41no—
239C23#1457913011yescharacteristic
240Q8—84no—
241Q8—3162no—
242Q8—3162no—
243Q8—6321no—
244Q8—6321no—
245D4—42no—
246D4—81no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—84no—
260C2×C4—3162no—
261C2×C4—6321no—
262C2×C4—6321no—
263C2×C4C2×#1192961111yescharacteristic
264C2×C4C2×#1192961212yes—
265C2×C4#1457918712yes—
266C2×C4#1457919211yescharacteristic
267C8—3162no—
268C8—3164no—
269C22—42no—
270C22—44no—
271C22—81no—
272C22—81no—
273C22—82no—
274C22#1909870211yescharacteristic
275C22#1909871412yes—
276C22#1909873111yescharacteristic
277C22#1909874612yes—
278C22#1909875011yescharacteristic
279C4—21no—
280C4—21no—
281C4—22no—
282C4—22no—
283C4—44no—
284C4—81no—
285C4—82no—
286C4—6322no—
287C2—82no—
288C2#2376232311yescharacteristic
289C2#2376236912yes—
290C2#2376251611yescharacteristic
291C2#2376266312yes—
292C2#2376271111yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2683C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#2683D79
hypocenterC79#7754
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2683

Complements

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

NH ≅ G/Ntype
1. #23764733C2semidirect
3. #23761223C2semidirect
81. C4⋊C4×C79SD16semidirect
152. C79#7754semidirect
228. C4⋊C4#11929612semidirect

Maximal subgroups

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

subgroupindex
#237647332
#237644822
#237612232
#237612212
C79×#26832
#775479