Finite Group Databasebeta

← #34205087

Subgroups — #34205087

Order 20224.

416 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 91 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#23765065C211yesmaximal, characteristic
3#23763686C212yesmaximal, complemented
4#23761520C212yesmaximal, complemented
5C79×#1884C211yesmaximal, characteristic
6#19098635—22no—
7#19098635—22no—
8#19098620—21no—
9#19098591—22no—
10#19098537C2212yes—
11#19098530—21no—
12#19098530C2212yes—
13#19098358C412yescomplemented
14#19098356C412yescomplemented
15#19098334—22no—
16#568×C79C2211yescharacteristic
17#532×C79C2211yescharacteristic
18#421×C79—21no—
19#410×C79C2211yescharacteristic
20#339×C79—21no—
21C2×#11929608—21no—
22C2×#11929608D412yes—
23C22×C79⋊C8—21no—
24C22×C79⋊C8D412yes—
25#14579120—22no—
26C4⋊(C79⋊C8)—21no—
27C4⋊(C79⋊C8)—21no—
28Dic79⋊C8—44no—
29#14579076—22no—
30#14579076C2×C414yes—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—22no—
33C22⋊(C79⋊C8)—22no—
34#14579072—22no—
35C2×#41×C79D411yescharacteristic
36C2×#41×C79C2311yescharacteristic
37#131×C79—42no—
38C2×M16×C79—22no—
39C2×M16×C79—21no—
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—21no—
47C2×C42×C79D411yescharacteristic
48#104×C79—42no—
49#99×C79—22no—
50#11929608—42no—
51#11929608—81no—
52C2×C79⋊C8—22no—
53C2×C79⋊C8—22no—
54C2×C79⋊C8—42no—
55C2×C79⋊C8—81no—
56C4⋊Dic79—42no—
57C22⋊Dic79—24no—
58C4×Dic79—42no—
59#41×C79—41no—
60#41×C79—41no—
61#41×C79—41no—
62#41×C79—41no—
63C2×Q8×C79C4⋊C411yescharacteristic
64C2×Q8×C79C2×Q811yescharacteristic
65C2×D4×C79—22no—
66C2×D4×C79—21no—
67C2×D4×C79C4⋊C411yescharacteristic
68C2×D4×C79C22×C411yescharacteristic
69C2×D4×C79#4112yes—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79C2×D411yescharacteristic
73C22×C4×C79C2×D412yes—
74M16×C79—22no—
75M16×C79—42no—
76M16×C79—41no—
77C2×C8×C79—24no—
78C2×C8×C79—41no—
79C4⋊C4×C79—22no—
80C4⋊C4×C79—41no—
81C22⋊C4×C79—22no—
82C22⋊C4×C79—22no—
83C22⋊C4×C79—41no—
84C42×C79—21no—
85C42×C79—41no—
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—22no—
99C2×C4×C79—21no—
100C2×C4×C79—22no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79—81no—
108C2×C4×C79—81no—
109C2×C4×C79C2×C4⋊C411yescharacteristic
110C2×C4×C79#12011yescharacteristic
111C2×C4×C79C22⋊Q812yes—
112C2×Dic79—44no—
113C8×C79—42no—
114C79⋊C8—42no—
115C79⋊C8—42no—
116C22×C79—21no—
117C22×C79—22no—
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—161no—
129C4×C79B(4,𝔽2)11yescharacteristic
130C4×C79#45811yescharacteristic
131Dic79—84no—
132#12969—791nomaximal, Sylow
133C2×C79—21no—
134C2×C79—42no—
135C2×C79—41no—
136C2×C79—81no—
137C2×C79#186711yescharacteristic
138#2980—791no—
139#2126—791no—
140#1884D7911yescharacteristic
141#1868—792no—
142#1391—792no—
143C79#1296911yescharacteristic, complemented, Sylow
144#568C2×D7911yescharacteristic
145#532C2×D7911yescharacteristic
146#435—1582no—
147#435—1582no—
148#432—1581no—
149#424—1582no—
150#421—21no—
151#410—792no—
152#410C2×D7911yescharacteristic
153#408—792no—
154#408—1581no—
155#354—792no—
156#353—792no—
157#344—1582no—
158#339—21no—
159C2×#41C22⋊D7911yescharacteristic
160C2×#41C22×D7911yescharacteristic
161#131—42no—
162C2×M16—22no—
163C2×M16—21no—
164C2×M16—792no—
165C2×M16—1581no—
166C22×C8—792no—
167C22×C8—1581no—
168C4⋊Q8Dic15811yescharacteristic
169#127C4×D7911yescharacteristic
170#124Dic15811yescharacteristic
171#124C4×D7911yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—21no—
175#117—1582no—
176C2×C42C22⋊D7911yescharacteristic
177C4⋊C8—3164no—
178C4⋊C8—1581no—
179C4⋊C8—1581no—
180#104—42no—
181#99—22no—
182#99—794no—
183#99—1582no—
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×Q8Dic79⋊C411yescharacteristic
193C2×Q8C2×Dic15811yescharacteristic
194C2×D4—22no—
195C2×D4—21no—
196C2×D4Dic79⋊C411yescharacteristic
197C2×D4C2×C4×D7911yescharacteristic
198C2×D4#1192962312yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4D4×D7912yes—
202C22×C4C2×C22⋊D7911yescharacteristic
203M16—22no—
204M16—42no—
205M16—41no—
206M16—3162no—
207M16—6321no—
208C2×C8—24no—
209C2×C8—41no—
210C2×C8—3162no—
211C2×C8—6321no—
212C2×C8—1582no—
213C2×C8—1582no—
214C4⋊C4—22no—
215C4⋊C4—41no—
216C4⋊C4—3162no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—41no—
220C22⋊C4—1584no—
221C42—21no—
222C42—41no—
223C42—3162no—
224C42Dic79⋊C411yescharacteristic
225C42Dic79⋊C411yescharacteristic
226C23—21no—
227C23(C22⋊D79)⋊C412yes—
228C23#1457914211yescharacteristic
229Q8—41no—
230Q8—41no—
231D4—41no—
232D4—42no—
233D4—41no—
234D4—81no—
235D4—82no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—3164no—
248C2×C4C2×Dic79⋊C411yescharacteristic
249C2×C4#1457912911yescharacteristic
250C2×C4#1457913812yes—
251C8—42no—
252C8—3162no—
253C8—3162no—
254C22—21no—
255C22—22no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22#1909843511yescharacteristic
261C4—21no—
262C4—22no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—161no—
267C4—6324no—
268C4#1909874911yescharacteristic
269C4#1909875211yescharacteristic
270C2—21no—
271C2—41no—
272C2—42no—
273C2—81no—
274C2#2376368011yescharacteristic
subgroupG/H
centerC4#19098749
commutatorC2×D4×C79C22×C4
FrattiniC2×#41C22×D79
FittingC79×#1884C2
socleC2×C79#1867
prime radical#532×C79C22
cosocleC2×#41×C79C23
hypercenter#1884D79
hypocenterC79#12969
LCSG ▷ C2×D4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1884

Complements

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

NH ≅ G/Ntype
3. #23763686C2semidirect
4. #23761520C2semidirect
13. #19098358C4semidirect
14. #19098356C4semidirect
143. C79#12969semidirect

Maximal subgroups

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

subgroupindex
#237724712
#237650652
#237636862
#237615202
C79×#18842
#1296979