Finite Group Databasebeta

← #34181320

Subgroups — #34181320

Order 20224.

542 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C8×D316C211yesmaximal, characteristic
2#23763397C212yesmaximal
3C2×#19098316C211yesmaximal, characteristic, complemented
4#23761283C212yesmaximal, complemented
5C79×#1845C211yesmaximal, characteristic, complemented
6C22×C8×D79—21no—
7C2×C4×D316C2211yescharacteristic
8C8×D316—22no—
9C8×D316C414yescomplemented
10C2×C4⋊(C79⋊C8)C2212yes—
11C2×C4⋊(C79⋊C8)C2211yescharacteristic
12C2×#14579081—22no—
13C2×(C2×D79)⋊C8—21no—
14#19098325—22no—
15#19098316—22no—
16C2×#107×C79—21no—
17C2×#107×C79—21no—
18C2×C4×C8×C79C2211yescharacteristic
19#384×C79C2212yes—
20C22×D316D411yescharacteristic
21C22×C4×D79—21no—
22C22×C79⋊C8—21no—
23C22×C79⋊C8—22no—
24C2×C8×D79—24no—
25C2×C8×D79—42no—
26C2×C8×D79—42no—
27C4×D316—21no—
28C4×D316—21no—
29C4×D316C2×C412yes—
30C4×D316Q812yes—
31C2×C4⋊Dic79D411yescharacteristic
32C2×(C2×D79)⋊C4—21no—
33C4⋊(C79⋊C8)—21no—
34C4⋊(C79⋊C8)—22no—
35C4⋊(C79⋊C8)—22no—
36C4⋊(C79⋊C8)C2×C412yes—
37#14579081—44no—
38(C2×D79)⋊C8—24no—
39C22×C8×C79D411yescharacteristic
40C22×C8×C79D411yescharacteristic
41C2×C4⋊C4×C79—22no—
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79—22no—
44C2×C42×C79C2311yescharacteristic
45#107×C79—42no—
46#107×C79—42no—
47C4×C8×C79C2×C412yes—
48C4×C8×C79D412yes—
49C23×D79—21no—
50C2×D316—21no—
51C2×D316—21no—
52C2×D316—21no—
53C2×D316C4⋊C412yes—
54C2×D316D812yes—
55C2×D316Q1612yes—
56C22×Dic79—21no—
57C2×C4×D79—22no—
58C2×C4×D79—22no—
59C2×C4×D79—41no—
60C2×C4×D79—41no—
61C2×C4×D79—41no—
62C2×C4×D79—41no—
63C2×C79⋊C8—22no—
64C2×C79⋊C8—42no—
65C2×C79⋊C8—41no—
66C2×C79⋊C8—42no—
67C2×C79⋊C8—41no—
68C2×C79⋊C8—42no—
69C8×D79—48no—
70C4⋊Dic79—21no—
71C4⋊Dic79C4⋊C412yes—
72(C2×D79)⋊C4—22no—
73(C2×D79)⋊C4—22no—
74C22×C4×C79—42no—
75C22×C4×C79C2×D411yescharacteristic
76C22×C4×C79C2×D411yescharacteristic
77C22×C4×C79#4111yescharacteristic
78C2×C8×C79—21no—
79C2×C8×C79—21no—
80C2×C8×C79C22⋊C412yes—
81C2×C8×C79C22⋊C412yes—
82C2×C8×C79D812yes—
83C2×C8×C79SD1612yes—
84C4⋊C4×C79—42no—
85C4⋊C4×C79—42no—
86C4⋊C4×C79—42no—
87C4⋊C4×C79—42no—
88C4⋊C4×C79—82no—
89C42×C79C22×C411yescharacteristic
90C42×C79C2×D411yescharacteristic
91C42×C79C2×Q811yescharacteristic
92C42×C79#4111yescharacteristic
93C23×C79#12111yescharacteristic
94C22×D79—22no—
95C22×D79—22no—
96C22×D79—22no—
97C22×D79—41no—
98C22×D79—41no—
99C22×D79—41no—
100C22×D79—41no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
103C2×C4×C79—82no—
104C2×C4×C79—82no—
105C2×C4×C79—82no—
106C2×C4×C79C2×C22⋊C411yescharacteristic
107C2×C4×C79C2×C4⋊C411yescharacteristic
108C2×C4×C79#11711yescharacteristic
109C2×C4×C79#12111yescharacteristic
110C2×C4×C79C22⋊Q811yescharacteristic
111C2×C4×C79C22⋊Q811yescharacteristic
112C2×C4×C79C2×D811yescharacteristic
113C2×C4×C79C2×D811yescharacteristic
114C2×C4×C79C2×SD1611yescharacteristic
115C2×C4×C79C2×Q1611yescharacteristic
116C2×C4×C79#13511yescharacteristic
117C2×C4×C79#13511yescharacteristic
118C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C79#13711yescharacteristic
120C2×Dic79—22no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123D316—22no—
124D316—22no—
125D316—22no—
126D316#10714yescomplemented
127C4×D79—42no—
128C4×D79—42no—
129C4×D79—42no—
130C4×D79—42no—
131C8×C79—22no—
132C8×C79#10214yescomplemented
133C79⋊C8—42no—
134C79⋊C8—82no—
135C22×C79#38011yescharacteristic
136C22×C79#45911yescharacteristic
137C22×C79#46111yescharacteristic
138C22×C79#46611yescharacteristic
139C22×C79#46711yescharacteristic
140C22×C79#47211yescharacteristic
141C22×C79#47311yescharacteristic
142C2×D79—24no—
143C2×D79—42no—
144C2×D79—42no—
145C2×D79—42no—
146C2×D79—42no—
147C2×D79—42no—
148C2×D79—42no—
149C4×C79—21no—
150C4×C79—21no—
151C4×C79—162no—
152C4×C79C2×#10211yescharacteristic
153C4×C79#41711yescharacteristic
154C4×C79C2×#10711yescharacteristic
155C4×C79#42711yescharacteristic
156C4×C79#46611yescharacteristic
157C4×C79#46711yescharacteristic
158C4×C79#47411yescharacteristic
159C4×C79#47611yescharacteristic
160Dic79—42no—
161#8644—791nomaximal, Sylow
162C2×C79#166711yescharacteristic
163C2×C79#166911yescharacteristic
164C2×C79#167311yescharacteristic
165C2×C79#167511yescharacteristic
166C2×C79#180211yescharacteristic
167C2×C79#180811yescharacteristic
168C2×C79#181611yescharacteristic
169D79—44no—
170C2×C8×D4—791no—
171#1845D7911yescharacteristic, complemented
172#1802—792no—
173C2×#335—791no—
174#1296—792no—
175C79#864411yescharacteristic, complemented, Sylow
176C23×C8—1581no—
177C2×C4×D4—791no—
178C8×D4—794no—
179C8×D4—1582no—
180C2×#107—21no—
181C2×#107—21no—
182C2×C4⋊C8—791no—
183C2×C4⋊C8—792no—
184C2×#102—1582no—
185C2×C22⋊C8—1581no—
186C2×C4×C8C2×D7911yescharacteristic
187#384C2×D7912yes—
188#340—1582no—
189#335—1582no—
190C22×D4—791no—
191C23×C4—1581no—
192C22×C8—3162no—
193C22×C8—3162no—
194C22×C8—1582no—
195C22×C8—1581no—
196C22×C8—1584no—
197C22×C8C22⋊D7911yescharacteristic
198C22×C8C22⋊D7911yescharacteristic
199C4×D4—792no—
200C4×D4—792no—
201C4×D4—1581no—
202C4×D4—1581no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—791no—
207C2×C22⋊C4—1581no—
208C2×C42C22×D7911yescharacteristic
209#107—42no—
210#107—42no—
211C4⋊C8—792no—
212C4⋊C8—1582no—
213C4⋊C8—1581no—
214C4⋊C8—1582no—
215#102—3164no—
216C22⋊C8—1584no—
217C4×C8C4×D7912yes—
218C4×C8D31612yes—
219C24—1581no—
220C2×D4—792no—
221C2×D4—792no—
222C2×D4—792no—
223C2×D4—1581no—
224C2×D4—1581no—
225C2×D4—1581no—
226C22×C4—42no—
227C22×C4—3161no—
228C22×C4—3161no—
229C22×C4—3161no—
230C22×C4—3161no—
231C22×C4—1581no—
232C22×C4—1582no—
233C22×C4—1582no—
234C22×C4D4×D7911yescharacteristic
235C22×C4C2×C22⋊D7911yescharacteristic
236C22×C4#1192962311yescharacteristic
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—3161no—
240C2×C8—3161no—
241C2×C8—3168no—
242C2×C8—3162no—
243C2×C8—3162no—
244C2×C8—3162no—
245C2×C8—1582no—
246C2×C8(C2×D79)⋊C412yes—
247C2×C8(C2×D79)⋊C412yes—
248C2×C8#1192960912yes—
249C2×C8#1192961112yes—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—82no—
255C4⋊C4—792no—
256C4⋊C4—1581no—
257C22⋊C4—1582no—
258C22⋊C4—1582no—
259C42C2×C4×D7911yescharacteristic
260C42C2×D31611yescharacteristic
261C42Q8×D7911yescharacteristic
262C42#1192962511yescharacteristic
263C23—3161no—
264C23—3161no—
265C23—3161no—
266C23—3161no—
267C23—1582no—
268C23—1582no—
269C23—1582no—
270C23#1457913211yescharacteristic
271D4—794no—
272D4—1582no—
273D4—1582no—
274D4—1582no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—3161no—
281C2×C4—3161no—
282C2×C4—3162no—
283C2×C4—3162no—
284C2×C4—3162no—
285C2×C4—3162no—
286C2×C4—1582no—
287C2×C4C2×(C2×D79)⋊C411yescharacteristic
288C2×C4C4⋊C4×D7911yescharacteristic
289C2×C4#1457911911yescharacteristic
290C2×C4#1457913611yescharacteristic
291C2×C4#1457913711yescharacteristic
292C2×C4#1457914111yescharacteristic
293C2×C4D8×D7911yescharacteristic
294C2×C4C2×#1192960911yescharacteristic
295C2×C4C2×#1192961111yescharacteristic
296C2×C4Q16×D7911yescharacteristic
297C2×C4#1457918011yescharacteristic
298C2×C4#1457918411yescharacteristic
299C2×C4#1457918711yescharacteristic
300C2×C4#1457919211yescharacteristic
301C8—22no—
302C8—3162no—
303C8—6322no—
304C8#1457908114yescomplemented
305C22—3162no—
306C22—3162no—
307C22—3162no—
308C22—3162no—
309C22—3162no—
310C22—3162no—
311C22—1584no—
312C22#1909842011yescharacteristic
313C22#1909875511yescharacteristic
314C22#1909876511yescharacteristic
315C22#1909879611yescharacteristic
316C22#1909880111yescharacteristic
317C22#1909883011yescharacteristic
318C22#1909883511yescharacteristic
319C4—21no—
320C4—21no—
321C4—162no—
322C4—3162no—
323C4C2×#1457908111yescharacteristic
324C4#1909856211yescharacteristic
325C4#107×D7911yescharacteristic
326C4#1909860011yescharacteristic
327C4#1909880011yescharacteristic
328C4#1909880511yescharacteristic
329C4#1909884011yescharacteristic
330C4#1909885011yescharacteristic
331C2—3164no—
332C2#2376275211yescharacteristic
333C2#2376276211yescharacteristic
334C2#2376278611yescharacteristic
335C2#2376279611yescharacteristic
336C2#2376339711yescharacteristic
337C2#2376342611yescharacteristic
338C2#2376346511yescharacteristic
subgroupG/H
centerC23#14579132
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1845C2
socleC23×C79#121
prime radicalC22×D316D4
cosocleC2×C42×C79C23
hypercenter#1845D79
hypocenterC79#8644
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1845

Complements

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

NH ≅ G/Ntype
3. C2×#19098316C2semidirect
4. #23761283C2semidirect
5. C79×#1845C2semidirect
9. C8×D316C4semidirect
126. D316#107semidirect
132. C8×C79#102semidirect
171. #1845D79semidirect
175. C79#8644semidirect
304. C8#14579081semidirect

Maximal subgroups

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

subgroupindex
C2×C8×D3162
#237633972
C2×#190983162
#237612832
C79×#18452
#864479