Finite Group Databasebeta

← #34205090

Subgroups — #34205090

Order 20224.

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

normal subgroups only

This group has 416 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
1#23772471C211yesmaximal, characteristic
2#23763784C211yesmaximal, characteristic
3#23763688C212yesmaximal, complemented
4#23761520C212yesmaximal, complemented
5C79×#2126C211yesmaximal, characteristic
6#19099330C2211yescharacteristic
7#19098635—22no—
8#19098635—22no—
9#19098620—21no—
10#19098590—22no—
11#19098580—21no—
12#19098537C2211yescharacteristic
13#19098530—21no—
14#19098530C2212yes—
15#19098333—22no—
16#19098323—21no—
17#532×C79C2211yescharacteristic
18#410×C79C2212yes—
19#354×C79C412yescomplemented
20#353×C79C412yescomplemented
21#14579168—42no—
22C2×#11929608—22no—
23C2×#11929608—21no—
24C2×#11929608—21no—
25C2×#11929608D412yes—
26C22×C79⋊C8—21no—
27C22×C79⋊C8D412yes—
28#14579120—21no—
29C4⋊(C79⋊C8)—21no—
30C4⋊(C79⋊C8)—21no—
31C4⋊(C79⋊C8)—44no—
32#14579088—42no—
33#14579076—22no—
34C22⋊(C79⋊C8)—22no—
35C22⋊(C79⋊C8)—22no—
36C22⋊(C79⋊C8)—22no—
37#14579072—22no—
38C2×#41×C79D411yescharacteristic
39C2×#41×C79C2311yescharacteristic
40C4⋊Q8×C79Q811yescharacteristic
41#127×C79C2×C411yescharacteristic
42#124×C79C2×C411yescharacteristic
43#124×C79Q811yescharacteristic
44#121×C79—22no—
45C4×D4×C79—22no—
46#117×C79—22no—
47C2×C42×C79D411yescharacteristic
48#99×C79—22no—
49#99×C79C2×C414yes—
50#11929608—22no—
51#11929608—42no—
52#11929608—42no—
53#11929608—41no—
54#11929608—81no—
55C2×C79⋊C8—24no—
56C2×C79⋊C8—22no—
57C2×C79⋊C8—22no—
58C2×C79⋊C8—42no—
59C2×C79⋊C8—41no—
60C2×C79⋊C8—81no—
61C4⋊Dic79—41no—
62C22⋊Dic79—41no—
63C4×Dic79—41no—
64#41×C79—41no—
65#41×C79—41no—
66#41×C79—41no—
67#41×C79—41no—
68C2×Q8×C79C4⋊C411yescharacteristic
69C2×Q8×C79C2×Q811yescharacteristic
70C2×D4×C79—22no—
71C2×D4×C79—21no—
72C2×D4×C79C4⋊C411yescharacteristic
73C2×D4×C79C22×C411yescharacteristic
74C2×D4×C79#4112yes—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79C2×D411yescharacteristic
78C22×C4×C79C2×D412yes—
79C4⋊C4×C79—22no—
80C4⋊C4×C79—42no—
81C22⋊C4×C79—22no—
82C22⋊C4×C79—22no—
83C22⋊C4×C79—24no—
84C42×C79—21no—
85C42×C79—42no—
86C42×C79C4⋊C411yescharacteristic
87C42×C79C4⋊C411yescharacteristic
88C23×C79—21no—
89C23×C79C4×D412yes—
90C23×C79#12311yescharacteristic
91Q8×C79—41no—
92Q8×C79—41no—
93D4×C79—41no—
94D4×C79—42no—
95D4×C79—41no—
96D4×C79—81no—
97D4×C79—82no—
98C2×C4×C79—21no—
99C2×C4×C79—22no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79—44no—
108C2×C4×C79—81no—
109C2×C4×C79C2×C4⋊C411yescharacteristic
110C2×C4×C79#12011yescharacteristic
111C2×C4×C79C22⋊Q812yes—
112C2×Dic79—81no—
113C79⋊C8—42no—
114C79⋊C8—42no—
115C79⋊C8—42no—
116C22×C79—22no—
117C22×C79—21no—
118C22×C79—41no—
119C22×C79—42no—
120C22×C79—42no—
121C22×C79—41no—
122C22×C79#38511yescharacteristic
123C4×C79—21no—
124C4×C79—22no—
125C4×C79—41no—
126C4×C79—41no—
127C4×C79—41no—
128C4×C79—84no—
129C4×C79B(4,𝔽2)11yescharacteristic
130C4×C79#45811yescharacteristic
131Dic79—161no—
132#12969—791nomaximal, Sylow
133C2×C79—21no—
134C2×C79—41no—
135C2×C79—42no—
136C2×C79—81no—
137C2×C79#186711yescharacteristic
138#2980—791no—
139#2126D7911yescharacteristic
140#1884—791no—
141#1868—792no—
142#1391—792no—
143C79#1296911yescharacteristic, complemented, Sylow
144#568—791no—
145#532C2×D7911yescharacteristic
146#435—1582no—
147#435—1582no—
148#432—1581no—
149#424—1582no—
150#421—1581no—
151#410—791no—
152#410C2×D7912yes—
153#408—792no—
154#408—1581no—
155#354Dic7912yescomplemented
156#353Dic7912yescomplemented
157#344—1582no—
158#339—1581no—
159C2×#41D31611yescharacteristic
160C2×#41C22×D7911yescharacteristic
161#131—3162no—
162C2×M16—792no—
163C2×M16—1581no—
164C2×M16—1582no—
165C2×M16—1581no—
166C22×C8—792no—
167C22×C8—1581no—
168C4⋊Q8Dic15811yescharacteristic
169#127C2×Dic7911yescharacteristic
170#124Dic15811yescharacteristic
171#124C2×Dic7911yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—22no—
175#117—1581no—
176C2×C42D31611yescharacteristic
177C4⋊C8—3164no—
178C4⋊C8—1581no—
179C4⋊C8—1581no—
180#104—3162no—
181#99—22no—
182#99—1582no—
183#99C2×Dic7914yes—
184C22⋊C8—1582no—
185C22⋊C8—1582no—
186C22⋊C8—1582no—
187#97—1582no—
188#41—41no—
189#41—41no—
190#41—41no—
191#41—41no—
192C2×Q8C4⋊Dic7911yescharacteristic
193C2×Q8C2×Dic15811yescharacteristic
194C2×D4—22no—
195C2×D4—21no—
196C2×D4C4⋊Dic7911yescharacteristic
197C2×D4C22×Dic7911yescharacteristic
198C2×D4#1192962312yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4D4×D7912yes—
202C22×C4C2×D31611yescharacteristic
203M16—3162no—
204M16—3161no—
205M16—3162no—
206M16—6321no—
207M16—1582no—
208C2×C8—3161no—
209C2×C8—3162no—
210C2×C8—6321no—
211C2×C8—1584no—
212C2×C8—1582no—
213C2×C8—1582no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—3161no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—24no—
220C22⋊C4—3161no—
221C42—21no—
222C42—42no—
223C42—3161no—
224C42C4⋊Dic7911yescharacteristic
225C42C4⋊Dic7911yescharacteristic
226C23—21no—
227C23D4×Dic7912yes—
228C23#1457914511yescharacteristic
229Q8—41no—
230Q8—41no—
231D4—41no—
232D4—42no—
233D4—41no—
234D4—81no—
235D4—82no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—44no—
246C2×C4—81no—
247C2×C4—6321no—
248C2×C4C2×C4⋊Dic7911yescharacteristic
249C2×C4#1457913111yescharacteristic
250C2×C4#1457913812yes—
251C8—3162no—
252C8—3162no—
253C8—3162no—
254C22—22no—
255C22—21no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22#1909843811yescharacteristic
261C4—21no—
262C4—22no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—84no—
267C4—12641no—
268C4#1909875111yescharacteristic
269C4#1909875411yescharacteristic
270C2—21no—
271C2—42no—
272C2—41no—
273C2—81no—
274C2#2376368311yescharacteristic
subgroupG/H
centerC4#19098751
commutatorC2×D4×C79C22×C4
FrattiniC2×#41C22×D79
FittingC79×#2126C2
socleC2×C79#1867
prime radical#532×C79C22
cosocleC2×#41×C79C23
hypercenter#2126D79
hypocenterC79#12969
LCSG ▷ C2×D4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2126

Complements

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

NH ≅ G/Ntype
3. #23763688C2semidirect
4. #23761520C2semidirect
19. #354×C79C4semidirect
20. #353×C79C4semidirect
143. C79#12969semidirect
155. #354Dic79semidirect
156. #353Dic79semidirect

Maximal subgroups

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

subgroupindex
#237724712
#237637842
#237636882
#237615202
C79×#21262
#1296979