Finite Group Databasebeta

← #34184317

Subgroups — #34184317

Order 20224.

560 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 131 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#23769213C211yesmaximal, characteristic
2#23761971C212yesmaximal
3C2×#19098289C211yesmaximal, characteristic
4#23761267C212yesmaximal
5C79×#2316C211yesmaximal, characteristic
6C2×#14579140C2211yescharacteristic
7C2×C22⋊(C79⋊C8)C2212yes—
8#19098502C412yescomplemented
9#19098496—22no—
10#19098483C412yescomplemented
11#19098436—21no—
12#19098426—21no—
13#19098423C2211yescharacteristic
14#19098299—24no—
15#19098289—22no—
16C2×#117×C79—21no—
17C22×C4⋊C4×C79C2211yescharacteristic
18#389×C79—24no—
19#380×C79—21no—
20#380×C79C2212yes—
21C22×Dic158C2×C411yescharacteristic
22C22×C79⋊C8—22no—
23#14579140—22no—
24#14579140—22no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—22no—
29C2×C22⋊Dic79—21no—
30C2×C22⋊Dic79—21no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—22no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95C2×C412yes—
37C23×C4×C79D411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C23×C4×C79C2311yescharacteristic
40#117×C79—42no—
41#117×C79—42no—
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79—22no—
44C2×C4⋊C4×C79—22no—
45C2×C4⋊C4×C79—22no—
46C2×C4⋊C4×C79—22no—
47C2×C4⋊C4×C79C2×C412yes—
48C2×C4⋊C4×C79D412yes—
49C2×C22⋊C4×C79—22no—
50C2×C42×C79—22no—
51#95×C79—22no—
52#95×C79D414yes—
53C2×Dic158—22no—
54C2×Dic158—81no—
55C22×Dic79—21no—
56C22×Dic79—21no—
57C22×Dic79—22no—
58C2×C79⋊C8—42no—
59C2×C79⋊C8—82no—
60C4⋊Dic79—22no—
61C4⋊Dic79—81no—
62Dic79⋊C4—42no—
63Dic79⋊C4—82no—
64C22⋊Dic79—41no—
65C22⋊Dic79—41no—
66C22⋊Dic79—41no—
67C22⋊Dic79—41no—
68C24×C79C2×D411yescharacteristic
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—24no—
76C22×C4×C79—81no—
77C22×C4×C79—81no—
78C22×C4×C79C22⋊C412yes—
79C22×C4×C79C22⋊C412yes—
80C22×C4×C79SD1612yes—
81C22×C4×C79Q1612yes—
82C22×C4×C79C22×C411yescharacteristic
83C22×C4×C79C2×D412yes—
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79#4112yes—
86C4⋊C4×C79—24no—
87C4⋊C4×C79—24no—
88C4⋊C4×C79—44no—
89C4⋊C4×C79—82no—
90C4⋊C4×C79—82no—
91C22⋊C4×C79—42no—
92C22⋊C4×C79—42no—
93C42×C79—44no—
94C23×C79—41no—
95C23×C79—41no—
96C23×C79C2×C22⋊C411yescharacteristic
97C23×C79#12011yescharacteristic
98C23×C79#12311yescharacteristic
99C23×C79C2×SD1611yescharacteristic
100C23×C79C2×Q1611yescharacteristic
101C23×C79AGL(1,ℤ/8)11yescharacteristic
102C23×C79#13711yescharacteristic
103C2×C4×C79—22no—
104C2×C4×C79—22no—
105C2×C4×C79—22no—
106C2×C4×C79—22no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—41no—
111C2×C4×C79—42no—
112C2×C4×C79—41no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—44no—
117C2×C4×C79—44no—
118C2×C4×C79—44no—
119C2×C4×C79—81no—
120C2×C4×C79#10314yes—
121C2×C4×C79C4×D412yes—
122C2×C4×C79#12112yes—
123C2×Dic79—42no—
124C2×Dic79—42no—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—42no—
130Dic158—82no—
131C79⋊C8—82no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—41no—
139C22×C79—41no—
140C22×C79—41no—
141C22×C79—41no—
142C22×C79#38611yescharacteristic
143C22×C79C2×#10311yescharacteristic
144C22×C79#41711yescharacteristic
145C22×C79#45011yescharacteristic
146C22×C79#45111yescharacteristic
147C22×C79#48311yescharacteristic
148C22×C79#48411yescharacteristic
149C4×C79—24no—
150C4×C79—22no—
151C4×C79—22no—
152C4×C79—42no—
153C4×C79—84no—
154Dic79—82no—
155Dic79—81no—
156Dic79—81no—
157#9200—791nomaximal, Sylow
158C2×C79—21no—
159C2×C79—21no—
160C2×C79—21no—
161C2×C79—21no—
162C2×C79—21no—
163C2×C79—21no—
164C2×C79#187211yescharacteristic
165C2×C79#191211yescharacteristic
166C2×C79#191311yescharacteristic
167#2627—791no—
168#2316D7911yescharacteristic
169#1520—792no—
170C2×#324—791no—
171#1289—792no—
172C79#920011yescharacteristic, complemented, Sylow
173C2×C22⋊Q8—791no—
174C2×#117—21no—
175C22×C4⋊C4C2×D7911yescharacteristic
176C2×C22⋊C8—792no—
177#400—792no—
178#398—1582no—
179#395—792no—
180#389—24no—
181#385—1581no—
182#381—1581no—
183#380—21no—
184#380—791no—
185#380C2×D7912yes—
186#328—1584no—
187#324—1582no—
188C22×Q8—791no—
189C23×C4C22⋊D7911yescharacteristic
190C23×C4C22⋊D7911yescharacteristic
191C23×C4C22×D7911yescharacteristic
192C22×C8—1582no—
193C22⋊Q8—1582no—
194C22⋊Q8—1582no—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—791no—
203C2×C4⋊C4—1581no—
204C2×C4⋊C4—1582no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4C4×D7912yes—
207C2×C4⋊C4D31612yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—1581no—
210C2×C22⋊C4—1581no—
211C2×C42—22no—
212C22⋊C8—1582no—
213C22⋊C8—1582no—
214#95—22no—
215#95—792no—
216#95—1582no—
217#95—1582no—
218#95—1582no—
219#95C22⋊D7914yes—
220C24C2×C22⋊D7911yescharacteristic
221C2×Q8—6321no—
222C2×Q8—1582no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—1582no—
233C22×C4—1581no—
234C22×C4—1581no—
235C22×C4(C2×D79)⋊C412yes—
236C22×C4(C2×D79)⋊C412yes—
237C22×C4#1192961212yes—
238C22×C4#1192961412yes—
239C22×C4C2×C4×D7911yescharacteristic
240C22×C4C2×C22⋊D7912yes—
241C22×C4C2×D31611yescharacteristic
242C22×C4#1192962412yes—
243C2×C8—3162no—
244C2×C8—6322no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—44no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—3162no—
251C4⋊C4—6322no—
252C4⋊C4—6321no—
253C4⋊C4—1582no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—3161no—
257C22⋊C4—3161no—
258C22⋊C4—3161no—
259C22⋊C4—3161no—
260C42—44no—
261C23—41no—
262C23—41no—
263C23C2×(C2×D79)⋊C411yescharacteristic
264C23#1457913011yescharacteristic
265C23#1457914411yescharacteristic
266C23C2×#1192961211yescharacteristic
267C23C2×#1192961411yescharacteristic
268C23#1457918711yescharacteristic
269C23#1457919211yescharacteristic
270Q8—6322no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—81no—
288C2×C4—3162no—
289C2×C4—3162no—
290C2×C4—3162no—
291C2×C4—3162no—
292C2×C4—3161no—
293C2×C4—3161no—
294C2×C4—3162no—
295C2×C4#1457908414yes—
296C2×C4C4×C22⋊D7912yes—
297C2×C4#1457913512yes—
298C8—6322no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#1909844211yescharacteristic
310C22C2×#1457908411yescharacteristic
311C22#1909856211yescharacteristic
312C22#1909871411yescharacteristic
313C22#1909871911yescharacteristic
314C22#1909889311yescharacteristic
315C22#1909889811yescharacteristic
316C4—22no—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—84no—
321C4—6322no—
322C4—6321no—
323C4—6321no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2#2376370711yescharacteristic
331C2#2376393711yescharacteristic
332C2#2376394211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2316C2
socleC22×C79#386
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#2316D79
hypocenterC79#9200
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
8. #19098502C4semidirect
10. #19098483C4semidirect
172. C79#9200semidirect

Maximal subgroups

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

subgroupindex
#237692132
#237619712
C2×#190982892
#237612672
C79×#23162
#920079