Finite Group Databasebeta

← #34205088

Subgroups — #34205088

Order 20224.

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

normal subgroups only

This group has 416 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#23765065C211yesmaximal, characteristic
2#23763784C211yesmaximal, characteristic
3#23763685C212yesmaximal, complemented
4#23761521C212yesmaximal, complemented
5C79×#2980C211yesmaximal, characteristic
6#19099330C2211yescharacteristic
7#19098589—22no—
8#19098580—21no—
9#19098537C2211yescharacteristic
10#19098537C2212yes—
11#19098358C412yescomplemented
12#19098356C412yescomplemented
13#19098335—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—
24#14579120—22no—
25#14579120—21no—
26Dic79⋊C8—44no—
27#14579088—42no—
28#14579076—22no—
29#14579076—22no—
30#14579076C2×C414yes—
31C2×#41×C79D411yescharacteristic
32C2×#41×C79C2311yescharacteristic
33C2×M16×C79—21no—
34C2×M16×C79D412yes—
35C22×C8×C79—21no—
36C22×C8×C79D412yes—
37C4⋊Q8×C79Q811yescharacteristic
38#127×C79C2×C411yescharacteristic
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—42no—
52#11929608—41no—
53C2×C79⋊C8—24no—
54C2×C79⋊C8—41no—
55C4⋊Dic79—42no—
56C4⋊Dic79—41no—
57C22⋊Dic79—24no—
58C22⋊Dic79—41no—
59C4×Dic79—42no—
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×C79C2×Q811yescharacteristic
67C2×D4×C79—22no—
68C2×D4×C79—21no—
69C2×D4×C79C4⋊C411yescharacteristic
70C2×D4×C79C22×C411yescharacteristic
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×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×C79C2×C4⋊C411yescharacteristic
109C2×C4×C79#12011yescharacteristic
110C2×C4×C79C22⋊Q812yes—
111C2×Dic79—44no—
112C2×Dic79—81no—
113C8×C79—42no—
114C8×C79—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×C79B(4,𝔽2)11yescharacteristic
129C4×C79#45811yescharacteristic
130Dic79—84no—
131Dic79—161no—
132#12969—791nomaximal, Sylow
133C2×C79—21no—
134C2×C79—42no—
135C2×C79—41no—
136C2×C79—81no—
137C2×C79#186711yescharacteristic
138#2980D7911yescharacteristic
139#2126—791no—
140#1884—791no—
141#1868—792no—
142#1391—792no—
143C79#1296911yescharacteristic, complemented, Sylow
144#568—791no—
145#532C2×D7911yescharacteristic
146#435—22no—
147#435—22no—
148#432—21no—
149#424—1582no—
150#421—1581no—
151#410—792no—
152#410—791no—
153#408—21no—
154#408C2×D7912yes—
155#354—792no—
156#353—792no—
157#344—1582no—
158#339—1581no—
159C2×#41C22⋊D7911yescharacteristic
160C2×#41C22×D7911yescharacteristic
161#131—3162no—
162C2×M16—21no—
163C2×M16—1581no—
164C2×M16—1582no—
165C2×M16C22⋊D7912yes—
166C22×C8—21no—
167C22×C8C22⋊D7912yes—
168C4⋊Q8Dic15811yescharacteristic
169#127C4×D7911yescharacteristic
170#124Dic15811yescharacteristic
171#124C4×D7911yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—1581no—
175#117—1582no—
176C2×C42C22⋊D7911yescharacteristic
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—3164no—
180#104—3162no—
181#99—794no—
182#99—1582no—
183#99—1582no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187#97—22no—
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#1192962412yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4C2×C22⋊D7911yescharacteristic
202C22×C4C2×C22⋊D7912yes—
203M16—42no—
204M16—81no—
205M16—3162no—
206M16—3161no—
207M16—1582no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—42no—
211C2×C8—81no—
212C2×C8—3161no—
213C2×C8—1584no—
214C4⋊C4—22no—
215C4⋊C4—3162no—
216C4⋊C4—3161no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—3161no—
220C22⋊C4—1584no—
221C42—21no—
222C42—3162no—
223C42—3161no—
224C42Dic79⋊C411yescharacteristic
225C42Dic79⋊C411yescharacteristic
226C23—21no—
227C23C4×C22⋊D7912yes—
228C23#1457914411yescharacteristic
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—81no—
246C2×C4—3164no—
247C2×C4—6321no—
248C2×C4C2×Dic79⋊C411yescharacteristic
249C2×C4#1457913011yescharacteristic
250C2×C4#1457914012yes—
251C8—42no—
252C8—42no—
253C8—3162no—
254C22—22no—
255C22—21no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22#1909843611yescharacteristic
261C4—21no—
262C4—22no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—6324no—
267C4—12641no—
268C4#1909875011yescharacteristic
269C4#1909875311yescharacteristic
270C2—21no—
271C2—42no—
272C2—41no—
273C2—81no—
274C2#2376368111yescharacteristic
subgroupG/H
centerC4#19098750
commutatorC2×D4×C79C22×C4
FrattiniC2×#41C22×D79
FittingC79×#2980C2
socleC2×C79#1867
prime radical#532×C79C22
cosocleC2×#41×C79C23
hypercenter#2980D79
hypocenterC79#12969
LCSG ▷ C2×D4×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. #23763685C2semidirect
4. #23761521C2semidirect
11. #19098358C4semidirect
12. #19098356C4semidirect
143. C79#12969semidirect

Maximal subgroups

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

subgroupindex
#237650652
#237637842
#237636852
#237615212
C79×#29802
#1296979