Finite Group Databasebeta

← #34184322

Subgroups — #34184322

Order 20224.

612 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 115 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#23769205C211yesmaximal, characteristic
2#23761971C212yesmaximal, complemented
3C2×#19098287C211yesmaximal, characteristic
4#23761267C212yesmaximal, complemented
5C79×#2312C211yesmaximal, characteristic
6C2×#14579140C2211yescharacteristic
7C2×C22⋊(C79⋊C8)C2212yes—
8#19098499C412yescomplemented
9#19098486—22no—
10#19098475C412yescomplemented
11#19098436—21no—
12#19098423C2211yescharacteristic
13#19098419—21no—
14#19098299—24no—
15#19098287—22no—
16C22×C22⋊C4×C79C2211yescharacteristic
17#388×C79—24no—
18#380×C79C2212yes—
19#379×C79—21no—
20C4×C22⋊C4×C79—21no—
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—
31C2×C22⋊Dic79—22no—
32C2×C22⋊Dic79—22no—
33C22⋊(C79⋊C8)—22no—
34C22⋊(C79⋊C8)—22no—
35C79⋊#95—22no—
36C79⋊#95C2×C412yes—
37C25×C79D411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C23×C4×C79C2311yescharacteristic
40C2×C4⋊C4×C79—22no—
41C2×C22⋊C4×C79—22no—
42C2×C22⋊C4×C79—22no—
43C2×C22⋊C4×C79—22no—
44C2×C22⋊C4×C79—22no—
45C2×C22⋊C4×C79—22no—
46C2×C22⋊C4×C79C2×C412yes—
47C2×C22⋊C4×C79D412yes—
48C2×C42×C79—22no—
49#95×C79—22no—
50#95×C79D414yes—
51C2×Dic158—22no—
52C2×Dic158—81no—
53C22×Dic79—21no—
54C22×Dic79—22no—
55C22×Dic79—21no—
56C2×C79⋊C8—42no—
57C2×C79⋊C8—82no—
58C4⋊Dic79—22no—
59C4⋊Dic79—81no—
60Dic79⋊C4—42no—
61Dic79⋊C4—82no—
62C22⋊Dic79—41no—
63C22⋊Dic79—41no—
64C22⋊Dic79—42no—
65C22⋊Dic79—82no—
66C22⋊Dic79—82no—
67C24×C79—21no—
68C24×C79—21no—
69C24×C79—22no—
70C24×C79—22no—
71C24×C79—81no—
72C24×C79—81no—
73C24×C79C22⋊C412yes—
74C24×C79C2×D411yescharacteristic
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—24no—
78C22×C4×C79—81no—
79C22×C4×C79—81no—
80C22×C4×C79C22⋊C412yes—
81C22×C4×C79C22×C411yescharacteristic
82C22×C4×C79C2×D412yes—
83C22×C4×C79C2×D411yescharacteristic
84C22×C4×C79#4112yes—
85C4⋊C4×C79—82no—
86C22⋊C4×C79—24no—
87C22⋊C4×C79—24no—
88C22⋊C4×C79—42no—
89C22⋊C4×C79—42no—
90C22⋊C4×C79—82no—
91C22⋊C4×C79—82no—
92C42×C79—82no—
93C23×C79—22no—
94C23×C79—22no—
95C23×C79—22no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79—42no—
99C23×C79—42no—
100C23×C79—42no—
101C23×C79—42no—
102C23×C79—42no—
103C23×C79—42no—
104C23×C79—81no—
105C23×C79—81no—
106C23×C79—84no—
107C23×C79—84no—
108C23×C79C2×C22⋊C411yescharacteristic
109C23×C79#12011yescharacteristic
110C23×C79#12311yescharacteristic
111C23×C79#13512yes—
112C23×C79#13712yes—
113C2×C4×C79—22no—
114C2×C4×C79—41no—
115C2×C4×C79—42no—
116C2×C4×C79—41no—
117C2×C4×C79—42no—
118C2×C4×C79—42no—
119C2×C4×C79—42no—
120C2×C4×C79—44no—
121C2×C4×C79—44no—
122C2×C4×C79—44no—
123C2×C4×C79—81no—
124C2×C4×C79C4×D412yes—
125C2×C4×C79#12112yes—
126C2×Dic79—42no—
127C2×Dic79—42no—
128C2×Dic79—42no—
129C2×Dic79—42no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—42no—
133Dic158—82no—
134C79⋊C8—82no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—21no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79—42no—
142C22×C79—42no—
143C22×C79—44no—
144C22×C79—42no—
145C22×C79—42no—
146C22×C79—42no—
147C22×C79—82no—
148C22×C79—84no—
149C22×C79—84no—
150C22×C79#38611yescharacteristic
151C22×C79#41611yescharacteristic
152C22×C79#41911yescharacteristic
153C22×C79#45212yes—
154C22×C79#48512yes—
155C4×C79—22no—
156C4×C79—22no—
157C4×C79—42no—
158C4×C79—84no—
159Dic79—82no—
160Dic79—81no—
161Dic79—81no—
162#9201—791nomaximal, Sylow
163C2×C79—21no—
164C2×C79—22no—
165C2×C79—21no—
166C2×C79—22no—
167C2×C79—82no—
168C2×C79#187311yescharacteristic
169C2×C79#191111yescharacteristic
170C2×C79#191311yescharacteristic
171#2626—791no—
172#2312D7911yescharacteristic
173#1520—792no—
174C2×#323—791no—
175#1289—792no—
176C79#920111yescharacteristic, complemented, Sylow
177C2×C22⋊Q8—791no—
178C22×C22⋊C4C2×D7911yescharacteristic
179C2×C22⋊C8—792no—
180#399—792no—
181#396—1582no—
182#393—792no—
183#388—24no—
184#385—1581no—
185#380—791no—
186#380C2×D7912yes—
187#379—21no—
188#379—1581no—
189C4×C22⋊C4—21no—
190#328—1584no—
191#323—1582no—
192C25C22⋊D7911yescharacteristic
193C22×Q8—791no—
194C23×C4C22⋊D7911yescharacteristic
195C23×C4C22×D7911yescharacteristic
196C22×C8—1582no—
197C22⋊Q8—1582no—
198C22⋊Q8—1582no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—791no—
201C2×C4⋊C4—1581no—
202C2×C4⋊C4—1582no—
203C2×C4⋊C4—1581no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—1581no—
210C2×C22⋊C4—1581no—
211C2×C22⋊C4—1582no—
212C2×C22⋊C4—1582no—
213C2×C22⋊C4C4×D7912yes—
214C2×C22⋊C4D31612yes—
215C2×C42—22no—
216C22⋊C8—1582no—
217C22⋊C8—1582no—
218#95—22no—
219#95—792no—
220#95—1582no—
221#95C22⋊D7914yes—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—81no—
227C24—81no—
228C24(C2×D79)⋊C412yes—
229C24C2×C22⋊D7911yescharacteristic
230C2×Q8—6321no—
231C2×Q8—1582no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—1582no—
238C22×C4—1581no—
239C22×C4—1581no—
240C22×C4(C2×D79)⋊C412yes—
241C22×C4C2×C4×D7911yescharacteristic
242C22×C4C2×C22⋊D7912yes—
243C22×C4C2×D31611yescharacteristic
244C22×C4#1192962412yes—
245C2×C8—3162no—
246C2×C8—6322no—
247C4⋊C4—82no—
248C4⋊C4—3162no—
249C4⋊C4—6322no—
250C4⋊C4—6321no—
251C4⋊C4—1582no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C22⋊C4—3161no—
259C22⋊C4—3161no—
260C22⋊C4—3162no—
261C22⋊C4—6322no—
262C22⋊C4—6322no—
263C42—82no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—84no—
278C23—84no—
279C23C2×(C2×D79)⋊C411yescharacteristic
280C23#1457913011yescharacteristic
281C23#1457914411yescharacteristic
282C23#1457918112yes—
283C23#1457919112yes—
284Q8—6322no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—81no—
296C2×C4—3162no—
297C2×C4—3162no—
298C2×C4—3162no—
299C2×C4—3162no—
300C2×C4—3161no—
301C2×C4—3161no—
302C2×C4—3162no—
303C2×C4C4×C22⋊D7912yes—
304C2×C4#1457913512yes—
305C8—6322no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22#1909844211yescharacteristic
322C22#1909855711yescharacteristic
323C22#1909857111yescharacteristic
324C22#1909872512yes—
325C22#1909890412yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—84no—
330C4—6322no—
331C4—6321no—
332C4—6321no—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2#2376371211yescharacteristic
339C2#2376393211yescharacteristic
340C2#2376394211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2312C2
socleC22×C79#386
prime radicalC25×C79D4
cosocleC23×C4×C79C23
hypercenter#2312D79
hypocenterC79#9201
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #23761971C2semidirect
4. #23761267C2semidirect
8. #19098499C4semidirect
10. #19098475C4semidirect
176. C79#9201semidirect

Maximal subgroups

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

subgroupindex
#237692052
#237619712
C2×#190982872
#237612672
C79×#23122
#920179