Finite Group Databasebeta

← #34204877

Subgroups — #34204877

Order 20224.

490 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×D8×Dic79C211yesmaximal, characteristic
2C2×#19098391C211yesmaximal, characteristic, complemented
3C2×#19098364C211yesmaximal, characteristic
4#23764134C211yesmaximal, characteristic, complemented
5#23761491C211yesmaximal, characteristic, complemented
6#23761477C211yesmaximal, characteristic, complemented
7C2×#357×C79C211yesmaximal, characteristic
8C2×D4×Dic79—21no—
9C22×C79⋊C16C2211yescharacteristic
10D8×Dic79—22no—
11D8×Dic79—22no—
12C2×#14579095C2211yescharacteristic
13C2×#14579095C2211yescharacteristic
14C2×#14579082—21no—
15C2×C8×Dic79C2211yescharacteristic
16#19098462—21no—
17#19098391—22no—
18#19098364—22no—
19#19098325—21no—
20C22×D8×C79C2211yescharacteristic
21C22×C16×C79C2211yescharacteristic
22C2×#107×C79C2211yescharacteristic
23#357×C79C414yescomplemented
24C23×Dic79—41no—
25C22×C79⋊C8D411yescharacteristic
26D4×Dic79—24no—
27D4×Dic79—42no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79—41no—
31C2×C22⋊Dic79—41no—
32C2×C4×Dic79D411yescharacteristic
33C2×C79⋊C16—21no—
34C2×C79⋊C16—21no—
35C2×C79⋊C16—21no—
36#14579095—21no—
37#14579095—21no—
38#14579095—21no—
39#14579095D412yes—
40#14579082—24no—
41C8×Dic79—21no—
42C8×Dic79—21no—
43C79⋊#95—41no—
44C22×D4×C79—21no—
45C2×D8×C79—21no—
46C2×D8×C79—21no—
47C2×D8×C79—21no—
48C2×D8×C79C2×C412yes—
49C2×D8×C79D412yes—
50C2×D8×C79Q812yes—
51C22×C8×C79C2311yescharacteristic
52C2×C4⋊C4×C79—21no—
53C2×C16×C79—21no—
54C2×C16×C79—21no—
55C2×C16×C79C2×C412yescomplemented
56#107×C79—21no—
57#107×C79C2×C412yescomplemented
58C22×Dic79—44no—
59C22×Dic79—41no—
60C22×Dic79—41no—
61C22×Dic79—82no—
62C22×Dic79—82no—
63C22×Dic79D811yescharacteristic
64C22×Dic79D811yescharacteristic
65C2×C79⋊C8—21no—
66C2×C79⋊C8—21no—
67C2×C79⋊C8—21no—
68C4⋊Dic79—22no—
69C4⋊Dic79—41no—
70C4⋊Dic79—41no—
71C4⋊Dic79—41no—
72C4⋊Dic79—82no—
73C22⋊Dic79—44no—
74C4×Dic79—21no—
75C4×Dic79—21no—
76C79⋊C16—41no—
77C24×C79—41no—
78C2×D4×C79—22no—
79C2×D4×C79—22no—
80C2×D4×C79—22no—
81C2×D4×C79—41no—
82C2×D4×C79—41no—
83C2×D4×C79—41no—
84C22×C4×C79—41no—
85C22×C4×C79C2×D411yescharacteristic
86D8×C79—22no—
87D8×C79—22no—
88D8×C79—22no—
89D8×C79C4⋊C414yescomplemented
90C2×C8×C79C22×C411yescharacteristic
91C2×C8×C79C2×D411yescharacteristic
92C2×C8×C79C2×D411yescharacteristic
93C2×C8×C79C2×Q811yescharacteristic
94C2×C8×C79#4111yescharacteristic
95C2×C8×C79#4111yescharacteristic
96C4⋊C4×C79—22no—
97C4⋊C4×C79—41no—
98C16×C79—22no—
99C23×C79—42no—
100C23×C79—42no—
101C23×C79—42no—
102C23×C79—81no—
103C23×C79—81no—
104C23×C79—81no—
105C23×C79—81no—
106C23×C79C2×D811yescharacteristic
107D4×C79—24no—
108D4×C79—42no—
109D4×C79—42no—
110D4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—81no—
113C2×C4×C79—81no—
114C2×C4×C79C4×D411yescharacteristic
115C2×C4×C79#12011yescharacteristic
116C2×C4×C79C22⋊Q811yescharacteristic
117C2×C4×C79#13511yescharacteristic
118C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C79#13711yescharacteristic
120C2×Dic79—21no—
121C2×Dic79—21no—
122C2×Dic79—21no—
123C2×Dic79—21no—
124C2×Dic79—21no—
125C2×Dic79—42no—
126C2×Dic79—88no—
127C2×Dic79—81no—
128C2×Dic79—81no—
129C2×Dic79—81no—
130C2×Dic79—81no—
131C2×Dic79—81no—
132C2×Dic79SD3212yes—
133C8×C79C2×C4⋊C411yescharacteristic
134C8×C79C4×D411yescharacteristic
135C8×C79C22⋊Q811yescharacteristic
136C8×C79#12311yescharacteristic
137C79⋊C8—41no—
138C22×C79—44no—
139C22×C79—82no—
140C22×C79—82no—
141C22×C79—82no—
142C22×C79—82no—
143C22×C79—82no—
144C22×C79—82no—
145C22×C79C4×D811yescharacteristic
146C22×C79#44711yescharacteristic
147C22×C79#47411yescharacteristic
148C22×C79C2×SD3211yescharacteristic
149C22×C79#50811yescharacteristic
150C22×C79#50911yescharacteristic
151C22×C79#50911yescharacteristic
152C4×C79—82no—
153C4×C79#38511yescharacteristic
154C4×C79#44011yescharacteristic
155C4×C79#45211yescharacteristic
156C4×C79#47811yescharacteristic
157Dic79—22no—
158Dic79—41no—
159Dic79—82no—
160Dic79—161no—
161#12937—791nomaximal, Sylow
162C2×C79—84no—
163C2×C79#186211yescharacteristic
164C2×C79C4×SD3211yescharacteristic
165C2×C79#217711yescharacteristic
166C2×C79#218611yescharacteristic
167C2×C79#218711yescharacteristic
168C2×C79#222611yescharacteristic
169C2×C79#222811yescharacteristic
170C2×C4×D8—791no—
171C2×#367—791no—
172C2×#357—791no—
173C2×#357D7911yescharacteristic
174#1942—791no—
175#1380—791no—
176#1374—791no—
177C79#1293711yescharacteristic, complemented, Sylow
178C22×D8C2×D7911yescharacteristic
179C2×C4×D4—1581no—
180C22×C16—791no—
181C22×C16C2×D7911yescharacteristic
182C4×D8—1582no—
183C4×D8—1582no—
184C2×#107—791no—
185C2×#107—791no—
186C2×#107C2×D7911yescharacteristic
187C2×#102—1581no—
188C2×C4×C8—791no—
189#389—1581no—
190#367—1582no—
191#357—1582no—
192#357Dic7914yescomplemented
193#340—1581no—
194C22×D4—21no—
195C23×C4—3161no—
196C2×D8—21no—
197C2×D8—21no—
198C2×D8—21no—
199C2×D8Dic15812yes—
200C2×D8D31612yes—
201C2×D8C2×Dic7912yes—
202C22×C8—791no—
203C22×C8C22×D7911yescharacteristic
204C4×D4—3162no—
205C4×D4—1584no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—3161no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4—1581no—
210C2×C22⋊C4—3161no—
211C2×C42—791no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—1581no—
215C2×C16—1581no—
216C2×C16—1581no—
217C2×C16C2×Dic7912yescomplemented
218#107—21no—
219#107—792no—
220#107—1581no—
221#107—1581no—
222#107—1581no—
223#107C2×Dic7912yescomplemented
224#102—1584no—
225C4×C8—1581no—
226C4×C8—1581no—
227#95—3161no—
228C24—41no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C22×C4—41no—
236C22×C4—3161no—
237C22×C4—3164no—
238C22×C4—3161no—
239C22×C4—791no—
240C22×C4—791no—
241C22×C4—6322no—
242C22×C4—6322no—
243C22×C4D4×D7911yescharacteristic
244D8—22no—
245D8—22no—
246D8—22no—
247D8C4⋊Dic7914yescomplemented
248C2×C8—1581no—
249C2×C8—1581no—
250C2×C8—1581no—
251C2×C8C22×Dic7911yescharacteristic
252C2×C8D4×D7911yescharacteristic
253C2×C8C2×D31611yescharacteristic
254C2×C8C2×Dic15811yescharacteristic
255C2×C8#1192962311yescharacteristic
256C2×C8#1192962311yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—3161no—
260C4⋊C4—3161no—
261C4⋊C4—3161no—
262C4⋊C4—6322no—
263C4⋊C4—1582no—
264C22⋊C4—3164no—
265C42—1581no—
266C42—1581no—
267C16—22no—
268C16—3161no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23D8×D7911yescharacteristic
277D4—24no—
278D4—42no—
279D4—42no—
280D4—42no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—3162no—
285C2×C4—792no—
286C2×C4—6321no—
287C2×C4—6321no—
288C2×C4—6328no—
289C2×C4—6321no—
290C2×C4—6321no—
291C2×C4—6321no—
292C2×C4—1581no—
293C2×C4—1581no—
294C2×C4—1581no—
295C2×C4—1581no—
296C2×C4—1581no—
297C2×C4D4×Dic7911yescharacteristic
298C2×C4#1457913111yescharacteristic
299C2×C4#1457913811yescharacteristic
300C2×C4#1457918011yescharacteristic
301C2×C4#1457918811yescharacteristic
302C2×C4#1457919011yescharacteristic
303C8—3161no—
304C8C2×C4⋊Dic7911yescharacteristic
305C8D4×Dic7911yescharacteristic
306C8#1457913811yescharacteristic
307C8#1457914511yescharacteristic
308C22—44no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22—82no—
313C22—82no—
314C22—82no—
315C22D8×Dic7911yescharacteristic
316C22#1909869811yescharacteristic
317C22#1909884111yescharacteristic
318C22SD32×D7911yescharacteristic
319C22#1909899411yescharacteristic
320C22#1909899711yescharacteristic
321C22#1909899711yescharacteristic
322C4—82no—
323C4—3161no—
324C4—6322no—
325C4—1582no—
326C4—12641no—
327C4#1909843811yescharacteristic
328C4#1909866411yescharacteristic
329C4#1909872611yescharacteristic
330C4#1909886111yescharacteristic
331C2—84no—
332C2#2376364911yescharacteristic
333C2SD32×Dic7911yescharacteristic
334C2#2376527611yescharacteristic
335C2#2376532411yescharacteristic
336C2#2376533011yescharacteristic
337C2#2376554511yescharacteristic
338C2#2376555911yescharacteristic
subgroupG/H
centerC23D8×D79
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC2×#357×C79C2
socleC23×C79C2×D8
prime radicalC22×D8×C79C22
cosocleC22×C8×C79C23
hypercenterC2×#357D79
hypocenterC79#12937
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#19098391C2semidirect
4. #23764134C2semidirect
5. #23761491C2semidirect
6. #23761477C2semidirect
23. #357×C79C4semidirect
55. C2×C16×C79C2×C4semidirect
57. #107×C79C2×C4semidirect
89. D8×C79C4⋊C4semidirect
177. C79#12937semidirect
192. #357Dic79semidirect
217. C2×C16C2×Dic79semidirect
223. #107C2×Dic79semidirect
247. D8C4⋊Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×D8×Dic792
C2×#190983912
C2×#190983642
#237641342
#237614912
#237614772
C2×#357×C792
#1293779