Finite Group Databasebeta

← #34205110

Subgroups — #34205110

Order 20224.

460 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 103 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#23766490C211yesmaximal, characteristic
2#23763794C211yesmaximal, characteristic
3#23763617C212yesmaximal
4#23761526C212yesmaximal
5C2×#356×C79C211yesmaximal, characteristic
6#19099041C2211yescharacteristic
7#19098605—22no—
8#19098572—21no—
9C2×#14579080C2211yescharacteristic
10#19098470—21no—
11#19098470C2212yes—
12#19098462—22no—
13C4⋊C4×Dic79—22no—
14C4×C4⋊Dic79—21no—
15#19098324—22no—
16#19098323—21no—
17C2×C4⋊Q8×C79C2211yescharacteristic
18C2×#101×C79C2212yes—
19#356×C79C414yescomplemented
20C2×#11929608—21no—
21Q8×Dic79—41no—
22Q8×Dic79—41no—
23#14579120—21no—
24#14579120—22no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79D412yes—
29C2×C4×Dic79—22no—
30C2×C4×Dic79—21no—
31C2×C4×Dic79D412yes—
32#14579095—42no—
33#14579093—42no—
34#14579085—41no—
35#14579085—41no—
36#14579080—22no—
37C79⋊#95—22no—
38C79⋊#95—22no—
39C79⋊#95—22no—
40C22×Q8×C79D411yescharacteristic
41C22×Q8×C79C2311yescharacteristic
42C2×M16×C79—22no—
43C4⋊Q8×C79—22no—
44C4⋊Q8×C79C2×C412yes—
45C4⋊Q8×C79Q812yes—
46C2×C4⋊C4×C79—22no—
47C2×C42×C79D411yescharacteristic
48#101×C79—22no—
49#101×C79C2×C414yes—
50C22×Dic79—22no—
51C22×Dic79—22no—
52#11929608—41no—
53#11929608—41no—
54C2×C79⋊C8—41no—
55C4⋊Dic79—24no—
56C4⋊Dic79—21no—
57C4⋊Dic79—21no—
58C4⋊Dic79—22no—
59C4⋊Dic79—22no—
60C4⋊Dic79—42no—
61C4⋊Dic79—42no—
62C4⋊Dic79—81no—
63C22⋊Dic79—41no—
64C4×Dic79—42no—
65C4×Dic79—44no—
66C4×Dic79—42no—
67C4×Dic79—41no—
68C4×Dic79—81no—
69C2×Q8×C79—21no—
70C2×Q8×C79—41no—
71C2×Q8×C79—41no—
72C2×Q8×C79—41no—
73C2×Q8×C79—41no—
74C2×Q8×C79C4⋊C412yes—
75C2×Q8×C79C22×C411yescharacteristic
76C2×Q8×C79C2×Q811yescharacteristic
77C2×Q8×C79#4112yes—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79C2×D411yescharacteristic
81C22×C4×C79C2×D412yes—
82M16×C79—24no—
83M16×C79—42no—
84C2×C8×C79—42no—
85C4⋊C4×C79—24no—
86C4⋊C4×C79—24no—
87C42×C79—21no—
88C42×C79C4⋊C412yes—
89C23×C79#12011yescharacteristic
90Q8×C79—41no—
91Q8×C79—41no—
92Q8×C79—42no—
93Q8×C79—42no—
94Q8×C79—81no—
95C2×C4×C79—22no—
96C2×C4×C79—22no—
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—21no—
100C2×C4×C79—22no—
101C2×C4×C79—21no—
102C2×C4×C79—22no—
103C2×C4×C79—41no—
104C2×C4×C79—41no—
105C2×C4×C79C2×C4⋊C411yescharacteristic
106C2×C4×C79C4×D412yes—
107C2×C4×C79C22⋊Q812yes—
108C2×C4×C79#12311yescharacteristic
109C2×Dic79—24no—
110C2×Dic79—42no—
111C2×Dic79—42no—
112C2×Dic79—42no—
113C2×Dic79—42no—
114C2×Dic79—41no—
115C2×Dic79—41no—
116C2×Dic79—42no—
117C8×C79—44no—
118C79⋊C8—81no—
119C22×C79—21no—
120C22×C79—21no—
121C22×C79#38511yescharacteristic
122C22×C79B(4,𝔽2)11yescharacteristic
123C22×C79#45811yescharacteristic
124C4×C79—24no—
125C4×C79—21no—
126C4×C79—22no—
127C4×C79—22no—
128C4×C79—21no—
129C4×C79—42no—
130Dic79—44no—
131Dic79—81no—
132Dic79—82no—
133Dic79—81no—
134#12973—791nomaximal, Sylow
135C2×C79—21no—
136C2×C79—21no—
137C2×C79#186711yescharacteristic
138C2×C79#220411yescharacteristic
139C2×C79#220511yescharacteristic
140#2369—791no—
141C2×#356D7911yescharacteristic
142#1886—791no—
143#1857—792no—
144#1394—792no—
145C79#1297311yescharacteristic, complemented, Sylow
146C2×C4⋊Q8C2×D7911yescharacteristic
147#519—791no—
148#428—1582no—
149#419—1581no—
150C2×#101—791no—
151C2×#101C2×D7912yes—
152#391—792no—
153#391—1581no—
154#389—1582no—
155C4×C4⋊C4—1582no—
156C4×C4⋊C4—1581no—
157#356Dic7914yescomplemented
158#339—1582no—
159#339—1581no—
160C22×Q8D31611yescharacteristic
161C22×Q8C22×D7911yescharacteristic
162C2×M16—22no—
163C2×M16—1581no—
164C4⋊Q8—22no—
165C4⋊Q8Dic15812yes—
166C4⋊Q8C2×Dic7912yes—
167C4×Q8—3161no—
168C4×Q8—3161no—
169#117—1582no—
170#117—1581no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—792no—
173C2×C4⋊C4—1581no—
174C2×C4⋊C4—1581no—
175C2×C4⋊C4—1581no—
176C2×C42—792no—
177C2×C42—1582no—
178C2×C42—1581no—
179C2×C42D31611yescharacteristic
180#107—3162no—
181#106—3162no—
182#103—3161no—
183#103—3161no—
184#101—22no—
185#101—1582no—
186#101C2×Dic7914yes—
187#95—1582no—
188#95—1582no—
189#95—1582no—
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8C4⋊Dic7912yes—
196C2×Q8C22×Dic7911yescharacteristic
197C2×Q8C2×Dic15811yescharacteristic
198C2×Q8#1192962312yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—1582no—
202C22×C4—1582no—
203C22×C4D4×D7912yes—
204C22×C4C2×D31611yescharacteristic
205M16—24no—
206M16—42no—
207M16—3161no—
208M16—3161no—
209C2×C8—42no—
210C2×C8—3161no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—3162no—
214C4⋊C4—3162no—
215C4⋊C4—6321no—
216C4⋊C4—1581no—
217C4⋊C4—1581no—
218C4⋊C4—1582no—
219C4⋊C4—1582no—
220C4⋊C4—1584no—
221C22⋊C4—3161no—
222C42—21no—
223C42—3162no—
224C42—3164no—
225C42—3162no—
226C42—3161no—
227C42—6321no—
228C42C4⋊Dic7912yes—
229C23#1457913111yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—42no—
233Q8—42no—
234Q8—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—3162no—
246C2×C4—3162no—
247C2×C4—3162no—
248C2×C4—3162no—
249C2×C4—3161no—
250C2×C4—3161no—
251C2×C4—3162no—
252C2×C4—1584no—
253C2×C4C2×C4⋊Dic7911yescharacteristic
254C2×C4D4×Dic7912yes—
255C2×C4#1457913812yes—
256C2×C4#1457914511yescharacteristic
257C8—44no—
258C8—6321no—
259C22—21no—
260C22—21no—
261C22#1909843811yescharacteristic
262C22#1909875111yescharacteristic
263C22#1909875411yescharacteristic
264C4—24no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—22no—
269C4—42no—
270C4—3164no—
271C4—6322no—
272C4—6321no—
273C4—6321no—
274C2—21no—
275C2—21no—
276C2#2376368311yescharacteristic
277C2#2376542811yescharacteristic
278C2#2376543111yescharacteristic
subgroupG/H
centerC22#19098751
commutatorC2×Q8×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC2×#356×C79C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
cosocleC22×Q8×C79C23
hypercenterC2×#356D79
hypocenterC79#12973
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
19. #356×C79C4semidirect
145. C79#12973semidirect
157. #356Dic79semidirect

Maximal subgroups

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

subgroupindex
#237664902
#237637942
#237636172
#237615262
C2×#356×C792
#1297379