Finite Group Databasebeta

← #34205113

Subgroups — #34205113

Order 20224.

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

normal subgroups only

This group has 408 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#23765085C211yesmaximal, characteristic
2#23763784C211yesmaximal, characteristic
3#23763619C212yesmaximal, complemented
4#23761529C212yesmaximal, complemented
5C79×#2980C211yesmaximal, characteristic
6#19099330C2211yescharacteristic
7#19098611—22no—
8#19098580—21no—
9#19098547C2212yes—
10#19098537C2211yescharacteristic
11#19098362C412yescomplemented
12#19098360C412yescomplemented
13#19098328—22no—
14#19098323—21no—
15#532×C79C2211yescharacteristic
16#435×C79—22no—
17#435×C79—22no—
18#432×C79—21no—
19#408×C79—21no—
20#408×C79C2212yes—
21#14579168—42no—
22C2×#11929608—22no—
23C2×#11929608—21no—
24C2×#11929608—22no—
25#14579120—21no—
26#14579096—44no—
27#14579088—42no—
28#14579080C2×C414yes—
29#14579078—22no—
30#14579076—22no—
31C2×#41×C79D411yescharacteristic
32C2×#41×C79C2311yescharacteristic
33C2×M16×C79—21no—
34C2×M16×C79D412yes—
35C22×C8×C79—21no—
36C22×C8×C79D412yes—
37C4⋊Q8×C79C2×C411yescharacteristic
38#127×C79Q811yescharacteristic
39#124×C79C2×C411yescharacteristic
40#124×C79Q811yescharacteristic
41#121×C79—22no—
42C4×D4×C79—22no—
43C2×C42×C79D411yescharacteristic
44C4⋊C8×C79—21no—
45C4⋊C8×C79—21no—
46C22⋊C8×C79—22no—
47C22⋊C8×C79—22no—
48C22⋊C8×C79—22no—
49#97×C79—22no—
50#11929608—22no—
51#11929608—24no—
52#11929608—24no—
53#11929608—42no—
54#11929608—41no—
55C2×C79⋊C8—42no—
56C2×C79⋊C8—41no—
57C2×C79⋊C8—42no—
58C4⋊Dic79—41no—
59C22⋊Dic79—41no—
60C4×Dic79—41no—
61#41×C79—41no—
62#41×C79—41no—
63#41×C79—41no—
64#41×C79—41no—
65C2×Q8×C79C4⋊C411yescharacteristic
66C2×Q8×C79C22×C411yescharacteristic
67C2×D4×C79—22no—
68C2×D4×C79—21no—
69C2×D4×C79C4⋊C411yescharacteristic
70C2×D4×C79C2×Q811yescharacteristic
71C2×D4×C79#4112yes—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79C2×D411yescharacteristic
75C22×C4×C79C2×D412yes—
76M16×C79—42no—
77M16×C79—81no—
78C2×C8×C79—22no—
79C2×C8×C79—22no—
80C2×C8×C79—42no—
81C2×C8×C79—81no—
82C4⋊C4×C79—22no—
83C22⋊C4×C79—22no—
84C22⋊C4×C79—22no—
85C42×C79—21no—
86C42×C79C4⋊C411yescharacteristic
87C42×C79C4⋊C411yescharacteristic
88C23×C79—21no—
89C23×C79C22⋊Q812yes—
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—81no—
108C2×C4×C79C2×C4⋊C411yescharacteristic
109C2×C4×C79C4×D412yes—
110C2×C4×C79#12011yescharacteristic
111C2×Dic79—81no—
112C8×C79—42no—
113C8×C79—42no—
114C79⋊C8—44no—
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×C79B(4,𝔽2)11yescharacteristic
129C4×C79#45811yescharacteristic
130Dic79—161no—
131#12974—791nomaximal, Sylow
132C2×C79—21no—
133C2×C79—41no—
134C2×C79—42no—
135C2×C79—81no—
136C2×C79#186711yescharacteristic
137#2980D7911yescharacteristic
138#2132—791no—
139#1884—791no—
140#1858—792no—
141#1395—792no—
142C79#1297411yescharacteristic, complemented, Sylow
143#568—791no—
144#532C2×D7911yescharacteristic
145#435—22no—
146#435—22no—
147#432—21no—
148#430—1582no—
149#421—1581no—
150#413—792no—
151#410—791no—
152#408—21no—
153#408C2×D7912yes—
154#356—792no—
155#355—792no—
156#341—1582no—
157#339—1581no—
158C2×#41C22⋊D7911yescharacteristic
159C2×#41C22×D7911yescharacteristic
160#131—3162no—
161C2×M16—21no—
162C2×M16—1581no—
163C2×M16—1582no—
164C2×M16—1582no—
165C2×M16C22⋊D7912yes—
166C22×C8—21no—
167C22×C8C22⋊D7912yes—
168C4⋊Q8C4×D7911yescharacteristic
169#127Dic15811yescharacteristic
170#124Dic15811yescharacteristic
171#124C4×D7911yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—1581no—
175C2×C42C22⋊D7911yescharacteristic
176#108—3164no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179#104—3162no—
180#101—794no—
181#100—1582no—
182#99—1582no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186#97—22no—
187#41—41no—
188#41—41no—
189#41—41no—
190#41—41no—
191C2×Q8Dic79⋊C411yescharacteristic
192C2×Q8C2×C4×D7911yescharacteristic
193C2×D4—22no—
194C2×D4—21no—
195C2×D4Dic79⋊C411yescharacteristic
196C2×D4C2×Dic15811yescharacteristic
197C2×D4#1192962412yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4C2×C22⋊D7911yescharacteristic
201C22×C4C2×C22⋊D7912yes—
202M16—42no—
203M16—81no—
204M16—3162no—
205M16—3161no—
206M16—1582no—
207M16—1584no—
208M16—1584no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—42no—
212C2×C8—81no—
213C2×C8—3162no—
214C2×C8—3161no—
215C2×C8—3162no—
216C4⋊C4—22no—
217C4⋊C4—3161no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—3161no—
221C42—21no—
222C42—3161no—
223C42Dic79⋊C411yescharacteristic
224C42Dic79⋊C411yescharacteristic
225C23—21no—
226C23#1457914012yes—
227C23#1457914411yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—42no—
232D4—41no—
233D4—81no—
234D4—82no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—81no—
245C2×C4—6321no—
246C2×C4C2×Dic79⋊C411yescharacteristic
247C2×C4C4×C22⋊D7912yes—
248C2×C4#1457913011yescharacteristic
249C8—42no—
250C8—42no—
251C8—3164no—
252C8—3162no—
253C22—21no—
254C22—22no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22#1909843611yescharacteristic
260C4—22no—
261C4—21no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—12641no—
266C4#1909875011yescharacteristic
267C4#1909875311yescharacteristic
268C2—21no—
269C2—41no—
270C2—42no—
271C2—81no—
272C2#2376368111yescharacteristic
subgroupG/H
centerC4#19098750
commutatorC2×Q8×C79C22×C4
FrattiniC2×#41C22×D79
FittingC79×#2980C2
socleC2×C79#1867
prime radical#532×C79C22
cosocleC2×#41×C79C23
hypercenter#2980D79
hypocenterC79#12974
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2980

Complements

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

NH ≅ G/Ntype
3. #23763619C2semidirect
4. #23761529C2semidirect
11. #19098362C4semidirect
12. #19098360C4semidirect
142. C79#12974semidirect

Maximal subgroups

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

subgroupindex
#237650852
#237637842
#237636192
#237615292
C79×#29802
#1297479