Finite Group Databasebeta

← #34286227

Subgroups — #34286227

Order 20224.

1110 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098590C211yesmaximal, characteristic
2C2×#19098518C211yesmaximal, characteristic
3C22×#14579071C211yesmaximal, characteristic
4#23765884C211yesmaximal, characteristic
5#23763272C214yesmaximal, complemented
6#23763267C214yesmaximal, complemented
7#23763098C211yesmaximal, characteristic
8#23763093C211yesmaximal, characteristic
9C2×#405×C79C211yesmaximal, characteristic
10C22×#11929608C2211yescharacteristic
11C23×C79⋊C8C2211yescharacteristic
12C2×#14579120C2211yescharacteristic
13C22×C4×Dic79C2211yescharacteristic
14#19098590C418yescomplemented
15C2×C4⋊(C79⋊C8)C2214yes—
16#19098518—24no—
17C2×#14579071—24no—
18C2×#14579071C2212yes—
19C2×#14579071C2212yes—
20C2×#14579071C2212yes—
21C2×C8×Dic79C2212yes—
22#19098462C414yes—
23#19098438C414yes—
24C4×C4⋊Dic79C2214yes—
25#19098317C2214yes—
26#19098317C2214yes—
27C22×M16×C79C2211yescharacteristic
28C23×C8×C79C2211yescharacteristic
29C2×#117×C79C2211yescharacteristic
30#405×C79C418yescomplemented
31C2×#97×C79C2212yes—
32C2×C4×C8×C79C2212yes—
33C23×Dic79C2×C412yes—
34C2×#11929608—22no—
35C2×#11929608—22no—
36C2×#11929608C2×C414yes—
37C22×C79⋊C8—22no—
38C22×C79⋊C8—22no—
39C22×C79⋊C8C2×C414yes—
40C22×C79⋊C8C2312yes—
41C22×C79⋊C8C2312yes—
42#14579120—22no—
43#14579120—22no—
44C2×C4⋊Dic79C2×C412yes—
45C2×C4⋊Dic79C2×C418yes—
46C2×C22⋊Dic79C2×C412yes—
47C2×C4×Dic79—22no—
48C2×C4×Dic79—22no—
49C2×C4×Dic79C2×C412yes—
50C2×C4×Dic79C2312yes—
51C2×C4×Dic79C2312yes—
52C4⋊(C79⋊C8)C2×C4116yescomplemented
53#14579071—24no—
54#14579071—24no—
55#14579071—24no—
56C8×Dic79—24no—
57C79⋊#95C2×C418yes—
58C23×C4×C79C2311yescharacteristic
59C2×M16×C79—22no—
60C2×M16×C79—22no—
61C2×M16×C79C2×C414yes—
62C22×C8×C79C2×C414yes—
63C22×C8×C79D414yes—
64C22×C8×C79Q814yes—
65C22×C8×C79C2312yes—
66C22×C8×C79C2311yescharacteristic
67C22×C8×C79C2311yescharacteristic
68#117×C79C2×C414yes—
69#117×C79C2×C414yes—
70C2×C4⋊C4×C79C2×C412yes—
71C2×C22⋊C4×C79C2×C412yes—
72C2×C42×C79C2312yes—
73#97×C79C2×C418yescomplemented
74C4×C8×C79C2×C418yescomplemented
75C22×Dic79—24no—
76C22×Dic79—44no—
77C22×Dic79C22×C414yes—
78C22×Dic79C22×C414yes—
79#11929608—28no—
80C2×C79⋊C8—28no—
81C2×C79⋊C8—22no—
82C2×C79⋊C8—22no—
83C2×C79⋊C8—22no—
84C2×C79⋊C8—22no—
85C2×C79⋊C8C22×C414yes—
86C2×C79⋊C8C22×C414yes—
87C4⋊Dic79—24no—
88C4⋊Dic79—48no—
89C22⋊Dic79—24no—
90C4×Dic79—24no—
91C4×Dic79—22no—
92C4×Dic79—22no—
93C4×Dic79—22no—
94C4×Dic79—22no—
95C24×C79C22×C411yescharacteristic
96C22×C4×C79C22×C414yes—
97C22×C4×C79C22×C411yescharacteristic
98C22×C4×C79C22×C412yes—
99C22×C4×C79C22×C412yes—
100C22×C4×C79C2×D412yes—
101C22×C4×C79C2×Q812yes—
102C22×C4×C79#4112yes—
103C22×C4×C79#4112yes—
104C22×C4×C79C2411yescharacteristic
105M16×C79—28no—
106C2×C8×C79—22no—
107C2×C8×C79—22no—
108C2×C8×C79C4⋊C4116yescomplemented
109C2×C8×C79C22×C414yes—
110C2×C8×C79C22×C412yes—
111C2×C8×C79C22×C412yes—
112C2×C8×C79C2×D412yes—
113C2×C8×C79C2×D412yes—
114C2×C8×C79C2×Q812yes—
115C2×C8×C79C2×Q812yes—
116C4⋊C4×C79C4218yes—
117C22⋊C4×C79C4218yes—
118C42×C79C22×C414yes—
119C42×C79C22×C414yes—
120C23×C79—24no—
121C23×C79C2×C4212yes—
122C23×C79C4×D412yes—
123C23×C79C4×Q812yes—
124C23×C79C23×C411yescharacteristic
125C2×C4×C79—28no—
126C2×C4×C79C2×C4218yes—
127C2×C4×C79C2×C4212yes—
128C2×C4×C79C2×C4⋊C414yes—
129C2×C4×C79#11714yes—
130C2×C4×C79C4×D414yes—
131C2×C4×C79C4×D412yes—
132C2×C4×C79C4×Q814yes—
133C2×C4×C79C4×Q812yes—
134C2×C4×C79C23×C411yescharacteristic
135C2×C4×C79C23×C411yescharacteristic
136C2×C4×C79C22×D411yescharacteristic
137C2×C4×C79C22×Q811yescharacteristic
138C2×C4×C79C2×#4111yescharacteristic
139C2×C4×C79C2×#4111yescharacteristic
140C2×Dic79—24no—
141C2×Dic79—24no—
142C2×Dic79—24no—
143C2×Dic79—28no—
144C2×Dic79—24no—
145C2×Dic79—48no—
146C8×C79—24no—
147C8×C79C2×C4⋊C414yes—
148C8×C79C2×C4⋊C414yes—
149C79⋊C8—24no—
150C79⋊C8—24no—
151C22×C79—24no—
152C22×C79—24no—
153C22×C79—24no—
154C22×C79C4×C4⋊C414yes—
155C22×C79C22×C4211yescharacteristic
156C22×C79C2×C4×D411yescharacteristic
157C22×C79C2×C4×Q811yescharacteristic
158C22×C79#56814yes—
159C4×C79—28no—
160C4×C79C4×C4⋊C414yes—
161C4×C79C22×C4⋊C411yescharacteristic
162C4×C79C2×#11711yescharacteristic
163C4×C79C2×C4×D411yescharacteristic
164C4×C79C2×C4×Q811yescharacteristic
165Dic79—44no—
166Dic79—44no—
167#20436—791nomaximal, Sylow
168C2×C79—24no—
169C2×C79C2×C4×C4⋊C411yescharacteristic
170C2×C79#287512yes—
171C2×C79#293012yes—
172C2×C79#296112yes—
173C2×#424—791no—
174C2×#405—791no—
175C2×#405D7911yescharacteristic
176C22×#97—791no—
177#2296—791no—
178#1771—794no—
179#1770—794no—
180#1729—791no—
181#1728—791no—
182C79#2043611yescharacteristic, complemented, Sylow
183C22×M16—791no—
184C22×M16C2×D7911yescharacteristic
185C23×C8—791no—
186C23×C8C2×D7911yescharacteristic
187C2×#117—791no—
188C2×#117C2×D7911yescharacteristic
189C22×C42—791no—
190#424—798no—
191C2×C4⋊C8—794no—
192#405—1584no—
193#405Dic7918yescomplemented
194C2×#97—792no—
195C2×#97—792no—
196C2×#97—792no—
197C2×#97—1584no—
198C2×#97C2×D7912yes—
199C2×C4×C8—792no—
200C2×C4×C8C2×D7912yes—
201#389—794no—
202#385—794no—
203C4×C4⋊C4—794no—
204#336—794no—
205#336—794no—
206C23×C4—792no—
207C23×C4C22×D7911yescharacteristic
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—794no—
211C2×M16—1582no—
212C2×M16—1582no—
213C2×M16C2×Dic7914yes—
214C22×C8—794no—
215C22×C8—792no—
216C22×C8—792no—
217C22×C8—1582no—
218C22×C8—1582no—
219C22×C8Dic15814yes—
220C22×C8D31614yes—
221C22×C8C2×Dic7914yes—
222C22×C8C22×D7912yes—
223C22×C8C22×D7911yescharacteristic
224C22×C8C22×D7911yescharacteristic
225#117—1582no—
226#117—1582no—
227#117C4×D7914yes—
228#117C2×Dic7914yes—
229C2×C4⋊C4—798no—
230C2×C4⋊C4—792no—
231C2×C4⋊C4C4×D7912yes—
232C2×C22⋊C4—792no—
233C2×C22⋊C4C4×D7912yes—
234C2×C42—792no—
235C2×C42—792no—
236C2×C42—792no—
237C2×C42—1582no—
238C2×C42—1582no—
239C2×C42C22×D7912yes—
240C4⋊C8—7916no—
241#97—1584no—
242#97—1584no—
243#97—1584no—
244#97C2×Dic7918yescomplemented
245C4×C8—1584no—
246C4×C8C2×Dic7918yescomplemented
247#95—798no—
248C24C2×C4×D7911yescharacteristic
249C22×C4—3164no—
250C22×C4—794no—
251C22×C4—794no—
252C22×C4—1584no—
253C22×C4C2×C4×D7914yes—
254C22×C4C2×C4×D7911yescharacteristic
255C22×C4C2×C4×D7912yes—
256C22×C4C22×Dic7912yes—
257C22×C4C2×D31612yes—
258C22×C4C2×Dic15812yes—
259C22×C4#1192962412yes—
260C22×C4#1192962412yes—
261C22×C4C23×D7911yescharacteristic
262M16—28no—
263M16—1588no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—794no—
267C2×C8—794no—
268C2×C8—1582no—
269C2×C8—1582no—
270C2×C8—1588no—
271C2×C8—1582no—
272C2×C8—1582no—
273C2×C8C4⋊Dic79116yescomplemented
274C2×C8C22×Dic7914yes—
275C2×C8C22×Dic7912yes—
276C2×C8C22×Dic7912yes—
277C2×C8C2×D31612yes—
278C2×C8C2×D31612yes—
279C2×C8C2×Dic15812yes—
280C2×C8C2×Dic15812yes—
281C4⋊C4—3168no—
282C4⋊C4—1584no—
283C4⋊C4C4×Dic7918yes—
284C22⋊C4—1584no—
285C22⋊C4C4×Dic7918yes—
286C42—1582no—
287C42—1582no—
288C42—1584no—
289C42—1582no—
290C42—1582no—
291C42C2×C4×D7914yes—
292C42C22×Dic7914yes—
293C23—24no—
294C23C2×C4×Dic7912yes—
295C23C4×D31612yes—
296C23C4×Dic15812yes—
297C23C22×C4×D7911yescharacteristic
298C2×C4—28no—
299C2×C4—3168no—
300C2×C4—1584no—
301C2×C4—1584no—
302C2×C4—1584no—
303C2×C4—1588no—
304C2×C4—1584no—
305C2×C4C2×C4×Dic7918yes—
306C2×C4C2×C4×Dic7912yes—
307C2×C4C2×C4⋊Dic7914yes—
308C2×C4#1457912014yes—
309C2×C4C4×D31612yes—
310C2×C4C4×D31614yes—
311C2×C4C4×Dic15812yes—
312C2×C4C4×Dic15814yes—
313C2×C4C22×C4×D7911yescharacteristic
314C2×C4C23×Dic7911yescharacteristic
315C2×C4C22×D31611yescharacteristic
316C2×C4C22×Dic15811yescharacteristic
317C2×C4C2×#1192962411yescharacteristic
318C2×C4C2×#1192962411yescharacteristic
319C8—24no—
320C8—1584no—
321C8—1584no—
322C8C2×C4⋊Dic7914yes—
323C8C2×C4⋊Dic7914yes—
324C22—24no—
325C22—24no—
326C22—24no—
327C22C4×C4⋊Dic7914yes—
328C22C22×C4×Dic7911yescharacteristic
329C22C2×C4×D31611yescharacteristic
330C22C2×C4×Dic15811yescharacteristic
331C22#1909932814yes—
332C4—28no—
333C4—3164no—
334C4—3164no—
335C4C4×C4⋊Dic7914yes—
336C4C22×C4⋊Dic7911yescharacteristic
337C4C2×#1457912011yescharacteristic
338C4C2×C4×D31611yescharacteristic
339C4C2×C4×Dic15811yescharacteristic
340C2—24no—
341C2C2×C4×C4⋊Dic7911yescharacteristic
342C2#2377182612yes—
343C2#2377210512yes—
344C2#2377233812yes—
subgroupG/H
center, FrattiniC22×C4C23×D79
commutatorC22×C79C22×C42
FittingC2×#405×C79C2
socleC23×C79C23×C4
prime radicalC24×C79C22×C4
cosocleC22×C4×C79C24
hypercenterC2×#405D79
hypocenterC79#20436
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
5. #23763272C2semidirect
6. #23763267C2semidirect
14. #19098590C4semidirect
30. #405×C79C4semidirect
52. C4⋊(C79⋊C8)C2×C4semidirect
73. #97×C79C2×C4semidirect
74. C4×C8×C79C2×C4semidirect
108. C2×C8×C79C4⋊C4semidirect
182. C79#20436semidirect
193. #405Dic79semidirect
244. #97C2×Dic79semidirect
246. C4×C8C2×Dic79semidirect
273. C2×C8C4⋊Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190985902
C2×#190985182
C22×#145790712
#237658842
#237632722
#237632672
#237630982
#237630932
C2×#405×C792
#2043679