Finite Group Databasebeta

← #34288679

Subgroups — #34288679

Order 20224.

1126 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098600C212yesmaximal
2C2×#19098598C212yesmaximal
3C2×C4×C8×D79C211yesmaximal, characteristic
4#23765806C212yesmaximal
5#23763564C211yesmaximal, characteristic, complemented
6#23763562C212yesmaximal, complemented
7#23763558C212yesmaximal, complemented
8#23763555C212yesmaximal, complemented
9C79×#1846C211yesmaximal, characteristic, complemented
10C22×C8×D79C2211yescharacteristic
11C22×C8×D79C2211yescharacteristic
12C2×#14579119C2214yes—
13C2×C42×D79C2211yescharacteristic
14#19098600—28no—
15#19098598—28no—
16C2×#14579095C2212yes—
17C2×#14579094C2214yes—
18C2×#14579093C2212yes—
19C2×#14579092C2214yes—
20C2×C4×C79⋊C8C2211yescharacteristic
21C2×C8×Dic79C2211yescharacteristic
22C2×C8×Dic79C2211yescharacteristic
23C4×C8×D79C418yescomplemented
24#19098432C2214yes—
25#19098429C2212yes—
26#19098422—24no—
27#19098414—24no—
28C2×#107×C79C2212yes—
29C2×#106×C79C2212yes—
30C2×C4×C8×C79C2211yescharacteristic
31#382×C79C2212yes—
32C22×C4×D79D412yes—
33C22×C4×D79C2311yescharacteristic
34C22×C79⋊C8C2311yescharacteristic
35C22×C79⋊C8C2311yescharacteristic
36C2×C8×D79C2×C414yes—
37C2×C8×D79C2×C414yes—
38C2×C8×D79D414yes—
39C2×C8×D79D414yes—
40C2×C8×D79Q814yes—
41C2×C8×D79Q814yes—
42#14579119—28no—
43#14579119—28no—
44C2×C4⋊Dic79C2314yes—
45C2×Dic79⋊C4—28no—
46C2×(C2×D79)⋊C4—24no—
47C2×C4×Dic79—24no—
48C2×C4×Dic79D412yes—
49C2×C4×Dic79C2311yescharacteristic
50C42×D79C2×C414yes—
51C42×D79Q814yes—
52#14579095—24no—
53#14579094—28no—
54#14579093—24no—
55#14579092—28no—
56C4×C79⋊C8C2×C414yes—
57C8×Dic79C2×C414yes—
58C8×Dic79C2×C414yes—
59C79⋊#95—24no—
60C22×C8×C79C2311yescharacteristic
61C22×C8×C79C2311yescharacteristic
62C2×C4⋊C4×C79C2314yes—
63C2×C42×C79C2311yescharacteristic
64#107×C79—24no—
65#106×C79—24no—
66C4×C8×C79C2×C414yes—
67#95×C79—24no—
68C23×D79C2×D411yescharacteristic
69C22×Dic79—24no—
70C22×Dic79C2×D411yescharacteristic
71C22×Dic79C2×D412yes—
72C2×C4×D79—28no—
73C2×C4×D79C4⋊C418yes—
74C2×C4×D79C22×C412yes—
75C2×C4×D79C2×D412yes—
76C2×C4×D79C2×D412yes—
77C2×C4×D79C2×Q812yes—
78C2×C4×D79C2×Q812yes—
79C2×C4×D79C2×Q812yes—
80C2×C79⋊C8C22×C412yes—
81C2×C79⋊C8C22×C412yes—
82C2×C79⋊C8C2×D412yes—
83C2×C79⋊C8C2×D412yes—
84C2×C79⋊C8C2×Q812yes—
85C2×C79⋊C8C2×Q812yes—
86C8×D79C4⋊C4116yescomplemented
87C8×D79C4⋊C4116yescomplemented
88C4⋊Dic79—24no—
89C4⋊Dic79—24no—
90Dic79⋊C4—416no—
91(C2×D79)⋊C4—48no—
92C4×Dic79—48no—
93C4×Dic79C4⋊C418yes—
94C4×Dic79C22×C412yes—
95C4×Dic79C2×Q812yes—
96C22×C4×C79—24no—
97C22×C4×C79C2×D412yes—
98C22×C4×C79C2411yescharacteristic
99C2×C8×C79C22×C412yes—
100C2×C8×C79C22×C412yes—
101C2×C8×C79C2×D412yes—
102C2×C8×C79C2×D412yes—
103C2×C8×C79C2×Q812yes—
104C2×C8×C79C2×Q812yes—
105C4⋊C4×C79—24no—
106C4⋊C4×C79—24no—
107C42×C79C22×C412yes—
108C42×C79C2×Q812yes—
109C23×C79C22×D411yescharacteristic
110C22×D79—24no—
111C22×D79C2×C4⋊C412yes—
112C22×D79#12712yes—
113C22×D79C4⋊Q812yes—
114C2×C4×C79—24no—
115C2×C4×C79—44no—
116C2×C4×C79—48no—
117C2×C4×C79C2×C4⋊C414yes—
118C2×C4×C79C23×C411yescharacteristic
119C2×C4×C79C22×D411yescharacteristic
120C2×C4×C79C22×D411yescharacteristic
121C2×C4×C79C22×Q811yescharacteristic
122C2×C4×C79C22×Q811yescharacteristic
123C2×C4×C79C22×Q811yescharacteristic
124C2×Dic79—44no—
125C2×Dic79—48no—
126C2×Dic79C2×C4⋊C412yes—
127C2×Dic79C2×C4⋊C414yes—
128C2×Dic79#12712yes—
129C2×Dic79C4⋊Q812yes—
130C2×Dic79#13518yes—
131C4×D79—216no—
132C4×D79C2×C4⋊C414yes—
133C4×D79C2×C4⋊C414yes—
134C4×D79C4⋊Q814yes—
135C4×D79C4⋊Q814yes—
136C8×C79C2×C4⋊C414yes—
137C8×C79C2×C4⋊C414yes—
138C79⋊C8C2×C4⋊C414yes—
139C79⋊C8C2×C4⋊C414yes—
140C22×C79C22×C4⋊C411yescharacteristic
141C22×C79C2×#12711yescharacteristic
142C22×C79C2×C4⋊Q811yescharacteristic
143C22×C79C2×#13514yes—
144C2×D79—24no—
145C2×D79—24no—
146C2×D79—24no—
147C2×D79#38414yes—
148C4×C79—24no—
149C4×C79—84no—
150C4×C79C22×C4⋊C411yescharacteristic
151C4×C79C22×C4⋊C411yescharacteristic
152C4×C79C2×C4⋊Q811yescharacteristic
153C4×C79C2×C4⋊Q811yescharacteristic
154Dic79—84no—
155Dic79#38414yes—
156Dic79#42718yescomplemented
157#20797—791nomaximal, Sylow
158C2×C79C2×#38411yescharacteristic
159C2×C79C2×#42712yes—
160C2×C79#314712yes—
161C2×C79#316012yes—
162D79—24no—
163C2×#427—792no—
164C2×#427—792no—
165C22×C4×C8—791no—
166#2285—792no—
167#1846—791no—
168#1846—792no—
169#1846D7911yescharacteristic, complemented
170#1845—792no—
171#1844—792no—
172C79#2079711yescharacteristic, complemented, Sylow
173C23×C8—791no—
174C23×C8—791no—
175C2×#117—794no—
176C22×C42—791no—
177#427—1588no—
178#427—1588no—
179C2×#107—792no—
180C2×#107—794no—
181C2×#107C2×D7912yes—
182C2×#106—792no—
183C2×#106—794no—
184C2×#106C2×D7912yes—
185C2×C4×C8—791no—
186C2×C4×C8—791no—
187C2×C4×C8—791no—
188C2×C4×C8—798no—
189C2×C4×C8C2×D7911yescharacteristic
190#384—794no—
191#382—792no—
192#382C2×D7912yes—
193#380—1584no—
194C4×C4⋊C4—1584no—
195C23×C4—791no—
196C23×C4—792no—
197C22×C8—791no—
198C22×C8—791no—
199C22×C8—794no—
200C22×C8—794no—
201C22×C8—794no—
202C22×C8—794no—
203C22×C8—794no—
204C22×C8—794no—
205C22×C8C22×D7911yescharacteristic
206C22×C8C22×D7911yescharacteristic
207#117—1588no—
208#117—1588no—
209C2×C4⋊C4—794no—
210C2×C4⋊C4—1588no—
211C2×C4⋊C4C22×D7914yes—
212C2×C22⋊C4—1584no—
213C2×C42—792no—
214C2×C42—791no—
215C2×C42—794no—
216C2×C42—794no—
217C2×C42—1584no—
218C2×C42C22×D7911yescharacteristic
219#107—24no—
220#107—1588no—
221#107—1584no—
222#106—24no—
223#106—1588no—
224#106—1584no—
225C4×C8—794no—
226C4×C8—794no—
227C4×C8—794no—
228C4×C8C4×D7914yes—
229#95—24no—
230#95—1584no—
231C24—791no—
232C22×C4—24no—
233C22×C4—791no—
234C22×C4—792no—
235C22×C4—792no—
236C22×C4—792no—
237C22×C4—792no—
238C22×C4—792no—
239C22×C4—792no—
240C22×C4—792no—
241C22×C4—798no—
242C22×C4—1588no—
243C22×C4—1584no—
244C22×C4D4×D7912yes—
245C22×C4C23×D7911yescharacteristic
246C2×C8—792no—
247C2×C8—792no—
248C2×C8—792no—
249C2×C8—792no—
250C2×C8—792no—
251C2×C8—792no—
252C2×C8—7916no—
253C2×C8—7916no—
254C2×C8C2×C4×D7912yes—
255C2×C8C2×C4×D7912yes—
256C2×C8D4×D7912yes—
257C2×C8D4×D7912yes—
258C2×C8Q8×D7912yes—
259C2×C8Q8×D7912yes—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—31616no—
263C4⋊C4—1584no—
264C4⋊C4—1584no—
265C22⋊C4—3168no—
266C42—3168no—
267C42—792no—
268C42—798no—
269C42—792no—
270C42C2×C4×D7912yes—
271C42Q8×D7912yes—
272C23—792no—
273C23—792no—
274C23—792no—
275C23—1584no—
276C23C2×D4×D7911yescharacteristic
277C2×C4—24no—
278C2×C4—48no—
279C2×C4—44no—
280C2×C4—3168no—
281C2×C4—3164no—
282C2×C4—792no—
283C2×C4—794no—
284C2×C4—798no—
285C2×C4—792no—
286C2×C4—792no—
287C2×C4—794no—
288C2×C4—794no—
289C2×C4—794no—
290C2×C4—794no—
291C2×C4—15816no—
292C2×C4C4⋊C4×D7914yes—
293C2×C4C22×C4×D7911yescharacteristic
294C2×C4C2×D4×D7911yescharacteristic
295C2×C4C2×D4×D7911yescharacteristic
296C2×C4C2×Q8×D7911yescharacteristic
297C2×C4C2×Q8×D7911yescharacteristic
298C2×C4C2×Q8×D7911yescharacteristic
299C8—794no—
300C8—794no—
301C8C4⋊C4×D7914yes—
302C8C4⋊C4×D7914yes—
303C22—794no—
304C22—1584no—
305C22—1584no—
306C22—1584no—
307C22C2×C4⋊C4×D7911yescharacteristic
308C22#127×D7911yescharacteristic
309C22C4⋊Q8×D7911yescharacteristic
310C22#135×D7914yes—
311C4—24no—
312C4—84no—
313C4—794no—
314C4—798no—
315C4—6324no—
316C4C2×C4⋊C4×D7911yescharacteristic
317C4C2×C4⋊C4×D7911yescharacteristic
318C4C4⋊Q8×D7911yescharacteristic
319C4C4⋊Q8×D7911yescharacteristic
320C2—1584no—
321C2#384×D7911yescharacteristic
322C2#427×D7912yes—
323C2#2377382512yes—
324C2#2377391312yes—
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1846C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
cosocleC22×C4×C79C24
hypercenter#1846D79
hypocenterC79#20797
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1846

Complements

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

NH ≅ G/Ntype
5. #23763564C2semidirect
6. #23763562C2semidirect
7. #23763558C2semidirect
8. #23763555C2semidirect
9. C79×#1846C2semidirect
23. C4×C8×D79C4semidirect
86. C8×D79C4⋊C4semidirect
87. C8×D79C4⋊C4semidirect
156. Dic79#427semidirect
169. #1846D79semidirect
172. C79#20797semidirect

Maximal subgroups

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

subgroupindex
C2×#190986002
C2×#190985982
C2×C4×C8×D792
#237658062
#237635642
#237635622
#237635582
#237635552
C79×#18462
#2079779