Finite Group Databasebeta

← #34192887

Subgroups — #34192887

Order 20224.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764362C211yesmaximal, characteristic
2#23764324C211yesmaximal, characteristic
3#23764164C211yesmaximal, characteristic
4#23764116C211yesmaximal, characteristic
5C2×#19098315C211yesmaximal, characteristic
6#23761284C211yesmaximal, characteristic
7C2×#332×C79C211yesmaximal, characteristic
8C2×#14579093—21no—
9C2×C4⋊(C79⋊C8)C2211yescharacteristic
10C2×Dic79⋊C8—21no—
11C2×#14579085—21no—
12C2×#14579085—21no—
13C2×C4×C79⋊C8C2211yescharacteristic
14#19098433C2211yescharacteristic
15C4×C4⋊Dic79C2211yescharacteristic
16#19098326—21no—
17#19098325—21no—
18#19098325—21no—
19#19098317—21no—
20#19098315—22no—
21C2×C4⋊Q8×C79C2211yescharacteristic
22C2×C4⋊C8×C79C2211yescharacteristic
23C2×C4⋊C8×C79C2211yescharacteristic
24#332×C79—22no—
25C22×C79⋊C8—21no—
26C22×C79⋊C8D411yescharacteristic
27C22×C79⋊C8D411yescharacteristic
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79—21no—
31C2×C4⋊Dic79D411yescharacteristic
32C2×C4⋊Dic79D411yescharacteristic
33C2×C4×Dic79—21no—
34#14579093—42no—
35C4⋊(C79⋊C8)—21no—
36C4⋊(C79⋊C8)—21no—
37Dic79⋊C8—42no—
38#14579085—42no—
39#14579085—42no—
40C4×C79⋊C8—21no—
41C4×C79⋊C8—21no—
42C79⋊#95—21no—
43C22×Q8×C79—21no—
44C22×C8×C79—21no—
45C22×C8×C79—21no—
46C4⋊Q8×C79—21no—
47C4⋊Q8×C79—21no—
48C4⋊Q8×C79—21no—
49C4⋊Q8×C79Q812yes—
50C2×C4⋊C4×C79—21no—
51C2×C4⋊C4×C79—21no—
52C2×C42×C79C2311yescharacteristic
53C4⋊C8×C79—21no—
54C4⋊C8×C79—21no—
55C4⋊C8×C79—21no—
56C4⋊C8×C79D412yes—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—41no—
60C2×C79⋊C8—21no—
61C2×C79⋊C8—21no—
62C2×C79⋊C8—21no—
63C2×C79⋊C8—21no—
64C2×C79⋊C8—21no—
65C2×C79⋊C8—41no—
66C2×C79⋊C8—41no—
67C2×C79⋊C8—41no—
68C2×C79⋊C8SD1612yes—
69C4⋊Dic79—21no—
70C4⋊Dic79—21no—
71C4⋊Dic79—21no—
72C4⋊Dic79—41no—
73C4⋊Dic79—41no—
74C4⋊Dic79—41no—
75C4⋊Dic79—41no—
76C4⋊Dic79—81no—
77C4⋊Dic79D812yes—
78C4×Dic79—41no—
79C4×Dic79—41no—
80C2×Q8×C79—22no—
81C2×Q8×C79—41no—
82C2×Q8×C79—41no—
83C2×Q8×C79—41no—
84C2×Q8×C79—41no—
85C2×Q8×C79—41no—
86C22×C4×C79—41no—
87C22×C4×C79C2×D411yescharacteristic
88C22×C4×C79C2×D411yescharacteristic
89C22×C4×C79#4111yescharacteristic
90C2×C8×C79—22no—
91C2×C8×C79—41no—
92C2×C8×C79—41no—
93C2×C8×C79—41no—
94C2×C8×C79—41no—
95C2×C8×C79—41no—
96C4⋊C4×C79—22no—
97C4⋊C4×C79—22no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—41no—
100C42×C79C2×D411yescharacteristic
101C42×C79C2×Q811yescharacteristic
102C42×C79#4111yescharacteristic
103C42×C79#4111yescharacteristic
104C23×C79#12111yescharacteristic
105Q8×C79—42no—
106Q8×C79—42no—
107Q8×C79—81no—
108Q8×C79—81no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—42no—
112C2×C4×C79—81no—
113C2×C4×C79—81no—
114C2×C4×C79#12111yescharacteristic
115C2×C4×C79#12111yescharacteristic
116C2×C4×C79C22⋊Q811yescharacteristic
117C2×C4×C79C22⋊Q811yescharacteristic
118C2×C4×C79C22⋊Q811yescharacteristic
119C2×C4×C79#12611yescharacteristic
120C2×C4×C79C2×D811yescharacteristic
121C2×C4×C79C2×SD1611yescharacteristic
122C2×C4×C79#13511yescharacteristic
123C2×C4×C79#13511yescharacteristic
124C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C79#13711yescharacteristic
126C2×C4×C79#13711yescharacteristic
127C2×C4×C79#13711yescharacteristic
128C2×Dic79—22no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—81no—
135C2×Dic79—81no—
136C2×Dic79—81no—
137C8×C79—42no—
138C8×C79—81no—
139C79⋊C8—22no—
140C79⋊C8—41no—
141C79⋊C8—81no—
142C22×C79#39611yescharacteristic
143C22×C79#46111yescharacteristic
144C22×C79#46411yescharacteristic
145C22×C79#46611yescharacteristic
146C22×C79#46811yescharacteristic
147C22×C79#47111yescharacteristic
148C22×C79#47311yescharacteristic
149C4×C79—21no—
150C4×C79—21no—
151C4×C79—82no—
152C4×C79#45911yescharacteristic
153C4×C79#46411yescharacteristic
154C4×C79#46511yescharacteristic
155C4×C79#47011yescharacteristic
156C4×C79#47411yescharacteristic
157C4×C79#47511yescharacteristic
158C4×C79#47811yescharacteristic
159C4×C79#47911yescharacteristic
160Dic79—42no—
161Dic79—81no—
162Dic79—161no—
163#10602—791nomaximal, Sylow
164C2×C79#167311yescharacteristic
165C2×C79#168211yescharacteristic
166C2×C79#169111yescharacteristic
167C2×C79#170011yescharacteristic
168C2×C79#206411yescharacteristic
169C2×C79#206611yescharacteristic
170C2×C79#206711yescharacteristic
171#1990—791no—
172#1981—791no—
173#1948—791no—
174#1939—791no—
175C2×#335—791no—
176C2×#332D7911yescharacteristic
177#1296—791no—
178C79#1060211yescharacteristic, complemented, Sylow
179C2×C4⋊Q8C2×D7911yescharacteristic
180C2×#106—1581no—
181C2×C4⋊C8—791no—
182C2×C4⋊C8—1581no—
183C2×C4⋊C8C2×D7911yescharacteristic
184C2×C4⋊C8C2×D7911yescharacteristic
185C2×#103—1581no—
186C2×#103—1581no—
187C2×C4×C8—791no—
188#384—791no—
189C4×C4⋊C4—791no—
190#340—1581no—
191#340—1581no—
192#340—1581no—
193#336—1581no—
194#335—1582no—
195#332—22no—
196C22×Q8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—791no—
200C22×C8—791no—
201C22×C8—1581no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8Dic15812yes—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—791no—
209C2×C4⋊C4—791no—
210C2×C4⋊C4—1581no—
211C2×C4⋊C4—1581no—
212C2×C4⋊C4—1581no—
213C2×C42—1581no—
214C2×C42C22×D7911yescharacteristic
215#106—3162no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—3162no—
220C4⋊C8—1581no—
221C4⋊C8—1581no—
222C4⋊C8C22⋊D7912yes—
223#103—3162no—
224#103—3162no—
225C4×C8—1581no—
226C4×C8—1581no—
227#95—1581no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C22×C4—41no—
235C22×C4—3161no—
236C22×C4—1581no—
237C22×C4—1581no—
238C22×C4D4×D7911yescharacteristic
239C22×C4D4×D7911yescharacteristic
240C22×C4#1192962411yescharacteristic
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—3161no—
248C2×C8—3161no—
249C2×C8—3161no—
250C2×C8—792no—
251C2×C8—1581no—
252C2×C8—1581no—
253C2×C8—1581no—
254C2×C8—1581no—
255C2×C8—1581no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—3161no—
261C4⋊C4—3161no—
262C4⋊C4—3161no—
263C4⋊C4—3161no—
264C4⋊C4—792no—
265C4⋊C4—6321no—
266C4⋊C4—1581no—
267C4⋊C4—1581no—
268C4⋊C4—1581no—
269C42—3161no—
270C42—3161no—
271C42C2×C22⋊D7911yescharacteristic
272C42C2×Dic15811yescharacteristic
273C42#1192962311yescharacteristic
274C42#1192962311yescharacteristic
275C23#1457913411yescharacteristic
276Q8—42no—
277Q8—42no—
278Q8—81no—
279Q8—81no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4—3161no—
289C2×C4—3161no—
290C2×C4—6321no—
291C2×C4—6321no—
292C2×C4—6321no—
293C2×C4—1582no—
294C2×C4#1457913211yescharacteristic
295C2×C4#1457913311yescharacteristic
296C2×C4#1457913811yescharacteristic
297C2×C4#1457913811yescharacteristic
298C2×C4#1457914011yescharacteristic
299C2×C4#1457915411yescharacteristic
300C2×C4D8×D7911yescharacteristic
301C2×C4SD16×D7911yescharacteristic
302C2×C4#1457918211yescharacteristic
303C2×C4#1457918211yescharacteristic
304C2×C4#1457918511yescharacteristic
305C2×C4#1457919011yescharacteristic
306C2×C4#1457919311yescharacteristic
307C2×C4#1457919311yescharacteristic
308C8—42no—
309C8—81no—
310C8—3161no—
311C8—6321no—
312C8—1582no—
313C22#1909848611yescharacteristic
314C22#1909876711yescharacteristic
315C22#1909878811yescharacteristic
316C22#1909879811yescharacteristic
317C22#1909880811yescharacteristic
318C22#1909882911yescharacteristic
319C22#1909883711yescharacteristic
320C4—21no—
321C4—21no—
322C4—82no—
323C4—3162no—
324C4—6321no—
325C4—12641no—
326C4#1909875611yescharacteristic
327C4#1909878311yescharacteristic
328C4#1909878911yescharacteristic
329C4#1909881811yescharacteristic
330C4#1909884111yescharacteristic
331C4#1909884611yescharacteristic
332C4#1909886511yescharacteristic
333C4#1909886811yescharacteristic
334C2#2376278811yescharacteristic
335C2#2376284711yescharacteristic
336C2#2376289811yescharacteristic
337C2#2376295711yescharacteristic
338C2#2376476311yescharacteristic
339C2#2376477911yescharacteristic
340C2#2376478411yescharacteristic
subgroupG/H
centerC23#14579134
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC2×#332×C79C2
socle, prime radicalC23×C79#121
hypercenterC2×#332D79
hypocenterC79#10602
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
178. C79#10602semidirect

Maximal subgroups

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

subgroupindex
#237643622
#237643242
#237641642
#237641162
C2×#190983152
#237612842
C2×#332×C792
#1060279