Finite Group Databasebeta

← #34207951

Subgroups — #34207951

Order 20224.

484 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 79 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#23771346C211yesmaximal, characteristic
2#23763681C212yesmaximal
3#23761525C211yesmaximal, characteristic
4#23761519C212yesmaximal
5C79×#1856C211yesmaximal, characteristic
6C2×#14579076C2212yes—
7#19098502—21no—
8#19098475—22no—
9#19098469—21no—
10#19098443C2212yes—
11#19098438—22no—
12#19098435—22no—
13#19098330—22no—
14#518×C79C2211yescharacteristic
15#428×C79—21no—
16C2×#100×C79C2211yescharacteristic
17#390×C79C2211yescharacteristic
18#339×C79—21no—
19C22×Dic158—21no—
20C23×Dic79D412yes—
21C2×C4⋊Dic79—21no—
22C2×C4⋊Dic79—21no—
23C2×Dic79⋊C4—42no—
24C2×C22⋊Dic79—22no—
25C2×C22⋊Dic79—22no—
26C2×C22⋊Dic79D412yes—
27#14579076—22no—
28#14579076Q814yescomplemented
29C79⋊#95—22no—
30C79⋊#95—22no—
31C79⋊#95—21no—
32C79⋊#95—22no—
33C79⋊#95—42no—
34C22×D4×C79C2311yescharacteristic
35C2×M16×C79—21no—
36C4×D4×C79—41no—
37C4×D4×C79—41no—
38#117×C79—21no—
39C2×C4⋊C4×C79D411yescharacteristic
40C2×C22⋊C4×C79—22no—
41C2×C22⋊C4×C79—22no—
42C2×C42×C79D411yescharacteristic
43#107×C79—41no—
44#106×C79—41no—
45#100×C79—21no—
46#100×C79Q812yescomplemented
47C2×Dic158—42no—
48C2×Dic158—81no—
49C2×Dic158—81no—
50C22×Dic79—22no—
51C22×Dic79—22no—
52C22×Dic79—42no—
53C22×Dic79—42no—
54C22×Dic79—82no—
55C4⋊Dic79—81no—
56C4⋊Dic79—81no—
57Dic79⋊C4—84no—
58C22⋊Dic79—24no—
59C22⋊Dic79—42no—
60C22⋊Dic79—42no—
61C22⋊Dic79—82no—
62C24×C79C2×D412yes—
63C2×D4×C79—41no—
64C2×D4×C79—41no—
65C2×D4×C79C2×Q811yescharacteristic
66C2×D4×C79C2×Q812yes—
67C2×D4×C79#4111yescharacteristic
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—22no—
71C22×C4×C79—22no—
72C22×C4×C79C2×D411yescharacteristic
73M16×C79—22no—
74M16×C79—41no—
75C2×C8×C79—41no—
76C4⋊C4×C79—21no—
77C4⋊C4×C79—21no—
78C4⋊C4×C79—41no—
79C22⋊C4×C79—44no—
80C22⋊C4×C79—42no—
81C22⋊C4×C79—41no—
82C42×C79—41no—
83C42×C79—41no—
84C23×C79—22no—
85C23×C79—22no—
86C23×C79—42no—
87C23×C79—42no—
88C23×C79#12011yescharacteristic
89C23×C79C22⋊Q812yes—
90C23×C79C4⋊Q812yes—
91D4×C79—42no—
92D4×C79—41no—
93D4×C79—41no—
94C2×C4×C79—21no—
95C2×C4×C79—21no—
96C2×C4×C79—22no—
97C2×C4×C79—41no—
98C2×C4×C79—41no—
99C2×C4×C79—42no—
100C2×C4×C79—41no—
101C2×C4×C79—41no—
102C2×C4×C79—41no—
103C2×C4×C79—82no—
104C2×C4×C79—82no—
105C2×C4×C79C22⋊Q811yescharacteristic
106C2×C4×C79C4⋊Q811yescharacteristic
107C2×Dic79—44no—
108C2×Dic79—42no—
109C2×Dic79—42no—
110C2×Dic79—42no—
111C2×Dic79—42no—
112C2×Dic79—44no—
113C2×Dic79—42no—
114C2×Dic79—42no—
115C2×Dic79—82no—
116C2×Dic79—82no—
117C2×Dic79—82no—
118Dic158—84no—
119C8×C79—42no—
120C22×C79—21no—
121C22×C79—21no—
122C22×C79—22no—
123C22×C79—22no—
124C22×C79—42no—
125C22×C79—42no—
126C22×C79—42no—
127C22×C79—42no—
128C22×C79—82no—
129C22×C79#39511yescharacteristic
130C22×C79B(4,𝔽2)11yescharacteristic
131C22×C79#45811yescharacteristic
132C4×C79—21no—
133C4×C79—21no—
134C4×C79—42no—
135C4×C79—81no—
136C4×C79—81no—
137C4×C79—81no—
138Dic79—82no—
139Dic79—84no—
140Dic79—82no—
141#13463—791nomaximal, Sylow
142C2×C79—21no—
143C2×C79—21no—
144C2×C79—42no—
145C2×C79—42no—
146C2×C79#206211yescharacteristic
147C2×C79#219811yescharacteristic
148C2×C79#219911yescharacteristic
149#2804—791no—
150#1867—792no—
151#1856D7911yescharacteristic
152#1393—791no—
153#1390—792no—
154C79#1346311yescharacteristic, complemented, Sylow
155#518C2×D7911yescharacteristic
156#428—21no—
157C2×#100C2×D7911yescharacteristic
158C2×#99—792no—
159#400—1581no—
160#393—1582no—
161#391—1581no—
162#390C2×D7911yescharacteristic
163#386—792no—
164#385—1582no—
165#385—1582no—
166#342—1582no—
167#339—21no—
168C22×Q8—1581no—
169C22×D4C22×D7911yescharacteristic
170C23×C4—792no—
171C2×M16—21no—
172C4×D4—41no—
173C4×D4—41no—
174#117—21no—
175C2×C4⋊C4—3162no—
176C2×C4⋊C4—1581no—
177C2×C4⋊C4—1581no—
178C2×C4⋊C4C22⋊D7911yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—792no—
182C2×C22⋊C4—1582no—
183C2×C22⋊C4—1582no—
184C2×C42C22⋊D7911yescharacteristic
185#107—41no—
186#106—41no—
187#100—21no—
188#100Dic15812yescomplemented
189#99—794no—
190#99—1582no—
191#95—3162no—
192#95—1582no—
193#95—1582no—
194#95—1581no—
195#95—1582no—
196C24D4×D7912yes—
197C2×Q8—3162no—
198C2×Q8—6321no—
199C2×Q8—6321no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4Q8×D7912yes—
203C2×D4C2×Dic15811yescharacteristic
204C2×D4#1192962411yescharacteristic
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—3162no—
210C22×C4—3162no—
211C22×C4—6322no—
212C22×C4—1582no—
213C22×C4—1582no—
214C22×C4C2×C22⋊D7911yescharacteristic
215M16—22no—
216M16—41no—
217C2×C8—41no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—41no—
221C4⋊C4—6321no—
222C4⋊C4—6321no—
223C4⋊C4—6324no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C22⋊C4—41no—
227C22⋊C4—3162no—
228C22⋊C4—3162no—
229C22⋊C4—6322no—
230C22⋊C4—1584no—
231C42—41no—
232C42—41no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23#1457912911yescharacteristic
238C23#1457913912yes—
239C23#1457916012yes—
240Q8—6324no—
241D4—42no—
242D4—41no—
243D4—41no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—3164no—
256C2×C4—3162no—
257C2×C4—3162no—
258C2×C4—3162no—
259C2×C4—3162no—
260C2×C4—3164no—
261C2×C4—3162no—
262C2×C4—3162no—
263C2×C4—6322no—
264C2×C4—6322no—
265C2×C4—6322no—
266C2×C4#1457914011yescharacteristic
267C2×C4#1457915911yescharacteristic
268C8—42no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—82no—
278C22#1909848311yescharacteristic
279C22#1909874911yescharacteristic
280C22#1909875211yescharacteristic
281C4—21no—
282C4—21no—
283C4—42no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—6322no—
288C4—6324no—
289C4—6322no—
290C2—21no—
291C2—21no—
292C2—42no—
293C2—42no—
294C2#2376475311yescharacteristic
295C2#2376539511yescharacteristic
296C2#2376540011yescharacteristic
subgroupG/H
centerC22#19098749
commutator, prime radical, cosocleC22×D4×C79C23
FrattiniC22×D4C22×D79
FittingC79×#1856C2
socleC22×C79B(4,𝔽2)
hypercenter#1856D79
hypocenterC79#13463
LCSG ▷ C22×D4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1856

Complements

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

NH ≅ G/Ntype
28. #14579076Q8semidirect
46. #100×C79Q8semidirect
154. C79#13463semidirect
188. #100Dic158semidirect

Maximal subgroups

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

subgroupindex
#237713462
#237636812
#237615252
#237615192
C79×#18562
#1346379