Finite Group Databasebeta

← #34205103

Subgroups — #34205103

Order 20224.

492 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#19098360C211yesmaximal, characteristic
2#23763804C211yesmaximal, characteristic
3#23763614C212yesmaximal, complemented
4#23761527C212yesmaximal, complemented
5C79×#2372C211yesmaximal, characteristic
6#19099041C2211yescharacteristic
7#19098603—22no—
8#19098568—21no—
9C2×#14579080C2211yescharacteristic
10C2×#14579080C2212yes—
11#19098360C414yescomplemented
12#19098323—21no—
13#19098323—22no—
14C2×#124×C79C2211yescharacteristic
15#391×C79C2212yes—
16#388×C79—22no—
17#386×C79—21no—
18C4×C22⋊C4×C79—22no—
19#376×C79—21no—
20C2×#11929608—22no—
21C2×#11929608—21no—
22Q8×Dic79—42no—
23#14579120—21no—
24#14579120—22no—
25#14579094—42no—
26#14579092—42no—
27#14579082—42no—
28#14579080—22no—
29#14579080—22no—
30#14579080C2×C414yescomplemented
31C22×Q8×C79C2311yescharacteristic
32C22×D4×C79D411yescharacteristic
33C23×C4×C79—21no—
34#124×C79—22no—
35#124×C79C2×C412yes—
36#124×C79Q812yes—
37C2×C4⋊C4×C79D412yes—
38C2×C22⋊C4×C79—22no—
39C2×C22⋊C4×C79—22no—
40C2×C22⋊C4×C79—21no—
41C2×C42×C79—22no—
42C2×C42×C79D411yescharacteristic
43C2×C42×C79D412yes—
44#95×C79—21no—
45#95×C79—21no—
46#95×C79—22no—
47#95×C79—22no—
48#11929608—24no—
49#11929608—42no—
50#11929608—41no—
51#11929608—41no—
52C2×C79⋊C8—42no—
53C2×C79⋊C8—41no—
54C4⋊Dic79—22no—
55C4⋊Dic79—22no—
56C4⋊Dic79—22no—
57C22⋊Dic79—41no—
58C4×Dic79—42no—
59C4×Dic79—41no—
60C24×C79—21no—
61C2×Q8×C79—41no—
62C2×Q8×C79—41no—
63C2×Q8×C79C22×C411yescharacteristic
64C2×Q8×C79C2×Q811yescharacteristic
65C2×Q8×C79#4112yes—
66C2×D4×C79—21no—
67C2×D4×C79—41no—
68C2×D4×C79—41no—
69C2×D4×C79C4⋊C412yes—
70C22×C4×C79—22no—
71C22×C4×C79—21no—
72C22×C4×C79—22no—
73C22×C4×C79—44no—
74C22×C4×C79—81no—
75C22×C4×C79C2×D411yescharacteristic
76C22×C4×C79C2×D412yes—
77C4⋊C4×C79—42no—
78C22⋊C4×C79—24no—
79C22⋊C4×C79—24no—
80C22⋊C4×C79—44no—
81C22⋊C4×C79—81no—
82C42×C79—21no—
83C42×C79—44no—
84C42×C79—42no—
85C42×C79C4⋊C412yes—
86C23×C79—22no—
87C23×C79—42no—
88C23×C79—42no—
89C23×C79—41no—
90C23×C79—41no—
91C23×C79#12011yescharacteristic
92Q8×C79—41no—
93Q8×C79—41no—
94Q8×C79—42no—
95D4×C79—42no—
96D4×C79—81no—
97C2×C4×C79—22no—
98C2×C4×C79—21no—
99C2×C4×C79—22no—
100C2×C4×C79—21no—
101C2×C4×C79—24no—
102C2×C4×C79—22no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—42no—
108C2×C4×C79—42no—
109C2×C4×C79—84no—
110C2×C4×C79C2×C4⋊C411yescharacteristic
111C2×C4×C79C4×D412yes—
112C2×C4×C79C22⋊Q812yes—
113C2×C4×C79#12311yescharacteristic
114C2×Dic79—42no—
115C79⋊C8—44no—
116C79⋊C8—81no—
117C22×C79—22no—
118C22×C79—44no—
119C22×C79—42no—
120C22×C79—81no—
121C22×C79—81no—
122C22×C79—82no—
123C22×C79#38511yescharacteristic
124C22×C79B(4,𝔽2)11yescharacteristic
125C22×C79#45811yescharacteristic
126C4×C79—21no—
127C4×C79—22no—
128C4×C79—22no—
129C4×C79—21no—
130C4×C79—42no—
131C4×C79—44no—
132C4×C79—82no—
133Dic79—82no—
134#12972—791nomaximal, Sylow
135C2×C79—21no—
136C2×C79—21no—
137C2×C79—82no—
138C2×C79#186711yescharacteristic
139C2×C79#220412yes—
140#2372D7911yescharacteristic
141C2×#355—791no—
142#1888—791no—
143#1857—792no—
144#1394—792no—
145C79#1297211yescharacteristic, complemented, Sylow
146C2×#124C2×D7911yescharacteristic
147#519—791no—
148#428—1582no—
149#418—1581no—
150C2×#101—792no—
151C2×#101—791no—
152#391C2×D7912yes—
153#388—22no—
154#386—21no—
155C4×C22⋊C4—22no—
156#376—21no—
157#355—794no—
158#339—1582no—
159#339—1581no—
160C22×Q8C22×D7911yescharacteristic
161C22×D4C22⋊D7911yescharacteristic
162C23×C4—21no—
163C2×M16—1581no—
164C2×M16—1582no—
165#124—22no—
166#124Dic15812yes—
167#124C4×D7912yes—
168C4×Q8—3162no—
169#117—1581no—
170#117—1582no—
171C2×C4⋊C4C22⋊D7912yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C42—22no—
176C2×C42C22⋊D7911yescharacteristic
177C2×C42C22⋊D7912yes—
178#107—3162no—
179#106—3162no—
180#102—3162no—
181#101—794no—
182#101—1582no—
183#101—1582no—
184#95—21no—
185#95—21no—
186#95—22no—
187#95—22no—
188C24—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C2×C4×D7911yescharacteristic
192C2×Q8C2×Dic15811yescharacteristic
193C2×Q8#1192962412yes—
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4Dic79⋊C412yes—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—44no—
202C22×C4—81no—
203C22×C4C2×C22⋊D7911yescharacteristic
204C22×C4C2×C22⋊D7912yes—
205M16—3162no—
206M16—3161no—
207M16—3161no—
208M16—1584no—
209C2×C8—3162no—
210C2×C8—3161no—
211C4⋊C4—42no—
212C4⋊C4—1582no—
213C4⋊C4—1582no—
214C4⋊C4—1582no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—44no—
218C22⋊C4—81no—
219C22⋊C4—3161no—
220C42—21no—
221C42—42no—
222C42—44no—
223C42—3162no—
224C42—3161no—
225C42Dic79⋊C412yes—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23#1457913011yescharacteristic
232Q8—41no—
233Q8—41no—
234Q8—42no—
235D4—42no—
236D4—81no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—84no—
250C2×C4—3162no—
251C2×C4C2×Dic79⋊C411yescharacteristic
252C2×C4C4×C22⋊D7912yes—
253C2×C4#1457914012yes—
254C2×C4#1457914411yescharacteristic
255C8—3164no—
256C8—6321no—
257C22—22no—
258C22—44no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—82no—
263C22#1909843611yescharacteristic
264C22#1909875011yescharacteristic
265C22#1909875311yescharacteristic
266C4—22no—
267C4—21no—
268C4—21no—
269C4—22no—
270C4—44no—
271C4—42no—
272C4—82no—
273C4—6322no—
274C2—21no—
275C2—21no—
276C2—82no—
277C2#2376368111yescharacteristic
278C2#2376542612yes—
subgroupG/H
centerC22#19098750
commutatorC2×Q8×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC79×#2372C2
socleC22×C79B(4,𝔽2)
prime radicalC22×D4×C79D4
cosocleC22×Q8×C79C23
hypercenter#2372D79
hypocenterC79#12972
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2372

Complements

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

NH ≅ G/Ntype
3. #23763614C2semidirect
4. #23761527C2semidirect
11. #19098360C4semidirect
30. #14579080C2×C4semidirect
145. C79#12972semidirect

Maximal subgroups

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

subgroupindex
C2×#190983602
#237638042
#237636142
#237615272
C79×#23722
#1297279