Finite Group Databasebeta

← #34296253

Subgroups — #34296253

Order 20224.

858 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098920C211yesmaximal, characteristic
2#125×Dic79C211yesmaximal, characteristic, complemented
3#23765939C211yesmaximal, characteristic
4#23764211C211yesmaximal, characteristic
5#23764084C212yesmaximal
6#23764077C212yesmaximal, complemented
7#23763530C211yesmaximal, characteristic, complemented
8#23763219C212yesmaximal, complemented
9#23763214C212yesmaximal
10C4×C8×Dic79C211yesmaximal, characteristic, complemented
11C79×#1991C211yesmaximal, characteristic, complemented
12C2×#14579149C2211yescharacteristic
13C2×#14579126C2211yescharacteristic
14C2×#14579123C2211yescharacteristic
15#19098920C418yescomplemented
16C2×#14579086C2212yes—
17C2×#14579084C2212yes—
18C2×#14579083C2212yes—
19C2×#14579081C2212yes—
20C2×C4×C79⋊C8C2211yescharacteristic
21C2×C8×Dic79C2212yes—
22C2×C8×Dic79C2212yes—
23#19098468C2211yescharacteristic
24#19098464C2211yescharacteristic
25#19098462C2212yes—
26#19098452—22no—
27#19098449—22no—
28#19098429C2211yescharacteristic
29C4⋊C4×Dic79—22no—
30C4⋊C4×Dic79C2212yes—
31#19098410—22no—
32C42×Dic79C2211yescharacteristic
33#19098325C2212yes—
34#19098325C2212yes—
35C2×#125×C79C2211yescharacteristic
36C2×C4×C8×C79C2211yescharacteristic
37#382×C79C2211yescharacteristic
38#340×C79C2212yes—
39#340×C79C2212yes—
40C22×Dic158C2311yescharacteristic
41C22×D316C2311yescharacteristic
42C22×C4×D79—21no—
43C22×C79⋊C8C2312yes—
44#14579149—22no—
45#14579149C2×C414yes—
46#14579126—24no—
47#14579123—24no—
48C2×C4⋊Dic79—22no—
49C2×C4⋊Dic79C2312yes—
50C2×Dic79⋊C4—21no—
51C2×(C2×D79)⋊C4—22no—
52C2×(C2×D79)⋊C4—22no—
53C2×(C2×D79)⋊C4—21no—
54C2×C4×Dic79—21no—
55C2×C4×Dic79—22no—
56C2×C4×Dic79D412yes—
57C2×C4×Dic79D412yes—
58C2×C4×Dic79C2311yescharacteristic
59C2×C4×Dic79C2311yescharacteristic
60#14579086—24no—
61#14579084C2×C418yescomplemented
62#14579083—24no—
63#14579081C2×C418yes—
64C4×C79⋊C8C2×C414yescomplemented
65C8×Dic79—24no—
66C8×Dic79—24no—
67C79⋊#95—22no—
68C79⋊#95—22no—
69C79⋊#95—22no—
70C79⋊#95—22no—
71C22×C8×C79C2312yes—
72#125×C79—22no—
73#125×C79C2×C414yescomplemented
74C2×C4⋊C4×C79—22no—
75C2×C4⋊C4×C79C2311yescharacteristic
76C2×C4⋊C4×C79C2311yescharacteristic
77C2×C4⋊C4×C79C2312yes—
78C2×C42×C79C2311yescharacteristic
79C4×C8×C79—22no—
80#95×C79—22no—
81C23×D79—21no—
82C2×Dic158—22no—
83C2×Dic158—22no—
84C2×Dic158—21no—
85C2×Dic158C22×C412yes—
86C2×D316—21no—
87C2×D316—22no—
88C2×D316—22no—
89C2×D316C22×C412yes—
90C22×Dic79—21no—
91C22×Dic79—22no—
92C22×Dic79C2×D411yescharacteristic
93C22×Dic79C2×D412yes—
94C22×Dic79C2×D411yescharacteristic
95C2×C4×D79—42no—
96C2×C4×D79—44no—
97C2×C79⋊C8—22no—
98C2×C79⋊C8—22no—
99C2×C79⋊C8C22×C414yes—
100C4⋊Dic79—22no—
101C4⋊Dic79—22no—
102C4⋊Dic79—44no—
103Dic79⋊C4—42no—
104(C2×D79)⋊C4—28no—
105(C2×D79)⋊C4—42no—
106(C2×D79)⋊C4—44no—
107C4×Dic79—24no—
108C4×Dic79—24no—
109C4×Dic79—22no—
110C4×Dic79—42no—
111C4×Dic79—44no—
112C4×Dic79#4114yes—
113C22×C4×C79—22no—
114C22×C4×C79—21no—
115C22×C4×C79—21no—
116C22×C4×C79C2×D412yes—
117C22×C4×C79C2411yescharacteristic
118C2×C8×C79—22no—
119C2×C8×C79—22no—
120C2×C8×C79#4114yes—
121C4⋊C4×C79—28no—
122C4⋊C4×C79—22no—
123C4⋊C4×C79—22no—
124C4⋊C4×C79—22no—
125C4⋊C4×C79C22×C414yes—
126C42×C79C22×C412yes—
127C42×C79#4112yes—
128C23×C79C22×D411yescharacteristic
129C22×D79—22no—
130C22×D79—41no—
131C22×D79—41no—
132C22×D79—42no—
133C22×D79—42no—
134C2×C4×C79—22no—
135C2×C4×C79—22no—
136C2×C4×C79—24no—
137C2×C4×C79—42no—
138C2×C4×C79—42no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—42no—
142C2×C4×C79—42no—
143C2×C4×C79C4×D414yes—
144C2×C4×C79C23×C411yescharacteristic
145C2×C4×C79C2×#4112yes—
146C2×C4×C79C2×#4111yescharacteristic
147C2×C4×C79C2×#4112yes—
148C2×Dic79—22no—
149C2×Dic79—22no—
150C2×Dic79—22no—
151C2×Dic79—22no—
152C2×Dic79—41no—
153C2×Dic79—41no—
154C2×Dic79—42no—
155C2×Dic79—42no—
156C2×Dic79—42no—
157C2×Dic79#12412yes—
158C2×Dic79#12412yes—
159C2×Dic79#13514yes—
160C2×Dic79#13514yes—
161D316—24no—
162D316—42no—
163C4×D79—84no—
164Dic158—24no—
165Dic158—42no—
166C8×C79—24no—
167C79⋊C8—24no—
168C22×C79C2×C4×D411yescharacteristic
169C22×C79C2×#12411yescharacteristic
170C22×C79#53211yescharacteristic
171C22×C79C2×#13512yes—
172C22×C79C2×#13512yes—
173C2×D79—44no—
174C2×D79—42no—
175C2×D79—81no—
176C2×D79—81no—
177C2×D79—82no—
178C4×C79—24no—
179C4×C79—44no—
180C4×C79—81no—
181C4×C79—81no—
182C4×C79C4×#4112yes—
183C4×C79#52912yes—
184Dic79—22no—
185Dic79—24no—
186Dic79—42no—
187Dic79—82no—
188Dic79#48814yescomplemented
189C4×#488—791nomaximal, Sylow
190C2×C79C4×#12411yescharacteristic
191C2×C79C4×#13512yes—
192C2×C79C2×#48811yescharacteristic
193C2×C79#313111yescharacteristic
194C2×C79#316812yes—
195D79—82no—
196C2×#488—791no—
197C4×#125—791no—
198C4×#124—791no—
199#1991D7911yescharacteristic, complemented
200#1958—791no—
201#1934—792no—
202#1933—792no—
203#1835—791no—
204C4×#103—792no—
205C4×#102—792no—
206C42×C8—791no—
207C79C4×#48811yescharacteristic, complemented, Sylow
208C2×#125C2×D7911yescharacteristic
209C2×#124—791no—
210C2×C4×Q8—791no—
211C2×C4×D4—791no—
212#488—798no—
213C2×#103—792no—
214C2×#103—792no—
215C2×#102—792no—
216C2×#102—792no—
217C2×C4×C8—791no—
218C2×C4×C8—792no—
219C2×C4×C8—792no—
220C2×C4×C8C2×D7911yescharacteristic
221#391—791no—
222#390—791no—
223#389—792no—
224#388—1582no—
225#387—1582no—
226#382—791no—
227#382C2×D7911yescharacteristic
228C4×C4⋊C4—792no—
229C4×C4⋊C4—1582no—
230C4×C22⋊C4—1582no—
231C43—791no—
232#340—792no—
233#340—792no—
234#340C2×D7912yes—
235#340C2×D7912yes—
236C22×Q8—791no—
237C22×D4—791no—
238C23×C4—1581no—
239C22×C8—792no—
240C22×C8C22×D7912yes—
241#125—22no—
242#125C4×D7914yescomplemented
243#124—794no—
244#124—1582no—
245C4×Q8—1584no—
246C4×D4—1584no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—792no—
249C2×C4⋊C4—1581no—
250C2×C4⋊C4—1582no—
251C2×C4⋊C4C22×D7911yescharacteristic
252C2×C4⋊C4C22×D7911yescharacteristic
253C2×C4⋊C4C22×D7912yes—
254C2×C22⋊C4—1581no—
255C2×C22⋊C4—1582no—
256C2×C22⋊C4—1582no—
257C2×C42—791no—
258C2×C42—792no—
259C2×C42—792no—
260C2×C42—791no—
261C2×C42—1581no—
262C2×C42—1582no—
263C2×C42C22×D7911yescharacteristic
264#103—798no—
265#103—1584no—
266#102—798no—
267#102—1584no—
268C4×C8—22no—
269C4×C8—794no—
270C4×C8—1584no—
271C4×C8—1584no—
272#95—22no—
273#95—1582no—
274#95—1582no—
275#95—1582no—
276#95—1582no—
277C24—1581no—
278C2×Q8—792no—
279C2×Q8—1582no—
280C2×Q8—1582no—
281C2×Q8—1581no—
282C2×D4—792no—
283C2×D4—1582no—
284C2×D4—1581no—
285C2×D4—1582no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—3162no—
290C22×C4—3164no—
291C22×C4—791no—
292C22×C4—791no—
293C22×C4—792no—
294C22×C4—1582no—
295C22×C4—1581no—
296C22×C4D4×D7912yes—
297C22×C4C23×D7911yescharacteristic
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—794no—
301C2×C8—1582no—
302C2×C8—1582no—
303C2×C8#1192962514yes—
304C4⋊C4—22no—
305C4⋊C4—28no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—3162no—
309C4⋊C4—3164no—
310C4⋊C4—1582no—
311C4⋊C4—1582no—
312C4⋊C4C2×C4×D7914yes—
313C22⋊C4—3164no—
314C22⋊C4—3162no—
315C22⋊C4—1588no—
316C42—3164no—
317C42—3162no—
318C42—794no—
319C42—1584no—
320C42—1584no—
321C42—1582no—
322C42C2×C4×D7912yes—
323C42#1192962312yes—
324C23—3161no—
325C23—3162no—
326C23—3162no—
327C23—3161no—
328C23—1582no—
329C23C2×D4×D7911yescharacteristic
330Q8—3162no—
331Q8—1584no—
332D4—3162no—
333D4—1584no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—3161no—
344C2×C4—3161no—
345C2×C4—3162no—
346C2×C4—3162no—
347C2×C4—3162no—
348C2×C4—792no—
349C2×C4—794no—
350C2×C4—792no—
351C2×C4—794no—
352C2×C4—6324no—
353C2×C4—1582no—
354C2×C4—1582no—
355C2×C4—1582no—
356C2×C4—1582no—
357C2×C4(C22⋊D79)⋊C414yes—
358C2×C4C22×C4×D7911yescharacteristic
359C2×C4#41×D7912yes—
360C2×C4C2×#1192962311yescharacteristic
361C2×C4C2×#1192962512yes—
362C8—24no—
363C8—1584no—
364C22—3164no—
365C22—3162no—
366C22—6321no—
367C22—6322no—
368C22—6321no—
369C22C2×(C22⋊D79)⋊C411yescharacteristic
370C22#124×D7911yescharacteristic
371C22#1909911411yescharacteristic
372C22#135×D7912yes—
373C22#135×D7912yes—
374C4—24no—
375C4—44no—
376C4—81no—
377C4—81no—
378C4—3162no—
379C4—794no—
380C4—6322no—
381C4—1584no—
382C4—1582no—
383C4C4×#1192962512yes—
384C4#1909909912yes—
385C2—6322no—
386C2#2376594311yescharacteristic
387C2#2377216812yes—
388C2#488×D7911yescharacteristic
389C2#2377372111yescharacteristic
390C2#2377397212yes—
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1991C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenter#1991D79
hypocenterC79C4×#488
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
2. #125×Dic79C2semidirect
6. #23764077C2semidirect
7. #23763530C2semidirect
8. #23763219C2semidirect
10. C4×C8×Dic79C2semidirect
11. C79×#1991C2semidirect
15. #19098920C4semidirect
61. #14579084C2×C4semidirect
64. C4×C79⋊C8C2×C4semidirect
73. #125×C79C2×C4semidirect
188. Dic79#488semidirect
199. #1991D79semidirect
207. C79C4×#488semidirect
242. #125C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190989202
#125×Dic792
#237659392
#237642112
#237640842
#237640772
#237635302
#237632192
#237632142
C4×C8×Dic792
C79×#19912
C4×#48879