Finite Group Databasebeta

← #34207331

Subgroups — #34207331

Order 20224.

450 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
1C2×#19098967C211yesmaximal, characteristic, complemented
2C2×#19098364C211yesmaximal, characteristic
3#23764267C211yesmaximal, characteristic
4#23761492C211yesmaximal, characteristic, complemented
5#23761491C211yesmaximal, characteristic, complemented
6#23761477C211yesmaximal, characteristic, complemented
7C2×#357×C79C211yesmaximal, characteristic
8C2×#14579160—21no—
9C22×C79⋊C16C2211yescharacteristic
10#19098967—22no—
11#19098967—22no—
12C2×#14579095C2211yescharacteristic
13C2×#14579095C2211yescharacteristic
14C2×#14579094—21no—
15C2×#14579082—21no—
16C2×C8×Dic79C2211yescharacteristic
17#19098466—21no—
18#19098364—22no—
19C22×D8×C79C2211yescharacteristic
20C22×C16×C79C2211yescharacteristic
21C2×#107×C79C2211yescharacteristic
22#357×C79—22no—
23C22×Dic158—41no—
24C22×C79⋊C8D411yescharacteristic
25#14579160—24no—
26#14579160—42no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79—21no—
29C2×Dic79⋊C4—41no—
30C2×C22⋊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#14579094—24no—
41#14579082—42no—
42C8×Dic79—21no—
43C8×Dic79—21no—
44C22×D4×C79—21no—
45C2×D8×C79—21no—
46C2×D8×C79—21no—
47C2×D8×C79—21no—
48C2×D8×C79—21no—
49C2×D8×C79—21no—
50C2×D8×C79Q812yescomplemented
51C22×C8×C79C2311yescharacteristic
52C2×C4⋊C4×C79—21no—
53C2×C16×C79—21no—
54C2×C16×C79—21no—
55C2×C16×C79—21no—
56#107×C79—21no—
57#107×C79D412yes—
58C2×Dic158—44no—
59C2×Dic158—82no—
60C2×Dic158—82no—
61C22×Dic79—41no—
62C22×Dic79—41no—
63C22×Dic79D811yescharacteristic
64C22×Dic79D811yescharacteristic
65C2×C79⋊C8—21no—
66C2×C79⋊C8—21no—
67C2×C79⋊C8SD1612yes—
68C4⋊Dic79—22no—
69C4⋊Dic79—41no—
70C4⋊Dic79—41no—
71C4⋊Dic79—41no—
72Dic79⋊C4—44no—
73C22⋊Dic79—82no—
74C22⋊Dic79—82no—
75C4×Dic79—21no—
76C4×Dic79—21no—
77C79⋊C16—41no—
78C24×C79—41no—
79C2×D4×C79—22no—
80C2×D4×C79—41no—
81C2×D4×C79—41no—
82C2×D4×C79—41no—
83C2×D4×C79—41no—
84C2×D4×C79—41no—
85C22×C4×C79—41no—
86C22×C4×C79C2×D411yescharacteristic
87D8×C79—22no—
88D8×C79—22no—
89D8×C79—41no—
90D8×C79—41no—
91C2×C8×C79C2×D411yescharacteristic
92C2×C8×C79C2×D411yescharacteristic
93C2×C8×C79C2×Q811yescharacteristic
94C2×C8×C79#4111yescharacteristic
95C2×C8×C79#4111yescharacteristic
96C2×C8×C79#4111yescharacteristic
97C4⋊C4×C79—22no—
98C4⋊C4×C79—41no—
99C16×C79—41no—
100C23×C79—42no—
101C23×C79—81no—
102C23×C79—81no—
103C23×C79—81no—
104C23×C79—81no—
105C23×C79—81no—
106C23×C79—81no—
107C23×C79C2×D811yescharacteristic
108D4×C79—42no—
109D4×C79—42no—
110D4×C79—81no—
111D4×C79—81no—
112C2×C4×C79—42no—
113C2×C4×C79—81no—
114C2×C4×C79—81no—
115C2×C4×C79#12011yescharacteristic
116C2×C4×C79C22⋊Q811yescharacteristic
117C2×C4×C79#12411yescharacteristic
118C2×C4×C79C2×SD1611yescharacteristic
119C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C79#13711yescharacteristic
121C2×Dic79—21no—
122C2×Dic79—21no—
123C2×Dic79—21no—
124C2×Dic79—21no—
125C2×Dic79—42no—
126C2×Dic79—81no—
127C2×Dic79—81no—
128C2×Dic79—81no—
129C2×Dic79—81no—
130C2×Dic79—81no—
131C2×Dic79D1612yes—
132C2×Dic79SD3212yes—
133Dic158—88no—
134C8×C79C22⋊Q811yescharacteristic
135C8×C79C22⋊Q811yescharacteristic
136C8×C79#12411yescharacteristic
137C8×C79#12411yescharacteristic
138C79⋊C8—22no—
139C22×C79—82no—
140C22×C79—82no—
141C22×C79—82no—
142C22×C79—161no—
143C22×C79—161no—
144C22×C79—161no—
145C22×C79—161no—
146C22×C79#44711yescharacteristic
147C22×C79#47411yescharacteristic
148C22×C79#48611yescharacteristic
149C22×C79C2×D1611yescharacteristic
150C22×C79C2×SD3211yescharacteristic
151C22×C79#50911yescharacteristic
152C22×C79#51011yescharacteristic
153C4×C79—82no—
154C4×C79#39311yescharacteristic
155C4×C79#44811yescharacteristic
156C4×C79#47611yescharacteristic
157C4×C79#48911yescharacteristic
158Dic79—22no—
159Dic79—22no—
160Dic79—82no—
161Dic79—161no—
162#13363—791nomaximal, Sylow
163C2×C79—162no—
164C2×C79#202311yescharacteristic
165C2×C79#218411yescharacteristic
166C2×C79#218511yescharacteristic
167C2×C79#222411yescharacteristic
168C2×C79#222611yescharacteristic
169C2×C79#224011yescharacteristic
170C2×C79#224311yescharacteristic
171C2×#500—791no—
172C2×#357—791no—
173C2×#357D7911yescharacteristic
174#1970—791no—
175#1380—791no—
176#1380—791no—
177#1374—791no—
178C79#1336311yescharacteristic, complemented, Sylow
179C22×D8C2×D7911yescharacteristic
180C2×C4⋊Q8—1581no—
181C22×C16—791no—
182C22×C16C2×D7911yescharacteristic
183#500—1582no—
184#500—1582no—
185C2×#107—791no—
186C2×#107—791no—
187C2×#107—1581no—
188C2×#107C2×D7911yescharacteristic
189C2×#102—1581no—
190C2×C4×C8—791no—
191#390—1581no—
192#357—22no—
193#357—1582no—
194C22×Q8—3161no—
195C22×D4—21no—
196C2×D8—21no—
197C2×D8—21no—
198C2×D8—21no—
199C2×D8—21no—
200C2×D8—21no—
201C2×D8Dic15812yescomplemented
202C22×C8—791no—
203C22×C8C22×D7911yescharacteristic
204C4⋊Q8—3162no—
205C4⋊Q8—1584no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—3161no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4—1581no—
210C2×C22⋊C4—3161no—
211C2×C22⋊C4—3161no—
212C2×C42—791no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—1581no—
217C2×C16—1581no—
218C2×C16—1581no—
219#107—21no—
220#107—792no—
221#107—1584no—
222#107—1581no—
223#107—1581no—
224#107—1581no—
225#107C22⋊D7912yes—
226#102—3162no—
227C4×C8—1581no—
228C4×C8—1581no—
229C24—41no—
230C2×Q8—3164no—
231C2×Q8—6322no—
232C2×Q8—6322no—
233C2×D4—22no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C22×C4—41no—
240C22×C4—3161no—
241C22×C4—3161no—
242C22×C4—791no—
243C22×C4—791no—
244C22×C4D4×D7911yescharacteristic
245D8—22no—
246D8—22no—
247D8—41no—
248D8—41no—
249C2×C8—792no—
250C2×C8—1581no—
251C2×C8—1581no—
252C2×C8D4×D7911yescharacteristic
253C2×C8C2×C22⋊D7911yescharacteristic
254C2×C8C2×Dic15811yescharacteristic
255C2×C8#1192962311yescharacteristic
256C2×C8#1192962311yescharacteristic
257C2×C8#1192962411yescharacteristic
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—3164no—
261C4⋊C4—3161no—
262C4⋊C4—3161no—
263C4⋊C4—3161no—
264C4⋊C4—1582no—
265C22⋊C4—6322no—
266C22⋊C4—6322no—
267C42—1581no—
268C42—1581no—
269C16—41no—
270C16—3161no—
271C23—42no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23D8×D7911yescharacteristic
279Q8—6328no—
280D4—42no—
281D4—42no—
282D4—81no—
283D4—81no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—3162no—
288C2×C4—792no—
289C2×C4—792no—
290C2×C4—6321no—
291C2×C4—6321no—
292C2×C4—6321no—
293C2×C4—6321no—
294C2×C4—6321no—
295C2×C4—1581no—
296C2×C4—1581no—
297C2×C4—1581no—
298C2×C4—1581no—
299C2×C4#1457912911yescharacteristic
300C2×C4#1457913811yescharacteristic
301C2×C4#1457914711yescharacteristic
302C2×C4SD16×D7911yescharacteristic
303C2×C4#1457918511yescharacteristic
304C2×C4#1457919011yescharacteristic
305C8—1582no—
306C8#1457913811yescharacteristic
307C8#1457914011yescharacteristic
308C8#1457914711yescharacteristic
309C8#1457914811yescharacteristic
310C22—82no—
311C22—82no—
312C22—82no—
313C22—161no—
314C22—161no—
315C22—161no—
316C22—161no—
317C22#1909869511yescharacteristic
318C22#1909884111yescharacteristic
319C22#1909890811yescharacteristic
320C22D16×D7911yescharacteristic
321C22SD32×D7911yescharacteristic
322C22#1909899711yescharacteristic
323C22#1909900211yescharacteristic
324C4—82no—
325C4—6322no—
326C4—1582no—
327C4—1582no—
328C4—12641no—
329C4#1909847511yescharacteristic
330C4#1909870011yescharacteristic
331C4#1909885111yescharacteristic
332C4#1909892111yescharacteristic
333C2—162no—
334C2#2376453411yescharacteristic
335C2#2376531011yescharacteristic
336C2#2376531511yescharacteristic
337C2#2376553511yescharacteristic
338C2#2376554511yescharacteristic
339C2#2376562211yescharacteristic
340C2#2376563511yescharacteristic
subgroupG/H
centerC23D8×D79
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
FittingC2×#357×C79C2
socleC23×C79C2×D8
prime radicalC22×D8×C79C22
hypercenterC2×#357D79
hypocenterC79#13363
LCSG ▷ C22×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
1. C2×#19098967C2semidirect
4. #23761492C2semidirect
5. #23761491C2semidirect
6. #23761477C2semidirect
50. C2×D8×C79Q8semidirect
178. C79#13363semidirect
201. C2×D8Dic158semidirect

Maximal subgroups

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

subgroupindex
C2×#190989672
C2×#190983642
#237642672
#237614922
#237614912
#237614772
C2×#357×C792
#1336379