Finite Group Databasebeta

← #34186643

Subgroups — #34186643

Order 20224.

386 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
1#23764351C211yesmaximal, characteristic
2#23764201C211yesmaximal, characteristic
3#23764161C211yesmaximal, characteristic
4#23763270C211yesmaximal, characteristic
5C2×#19098310C211yesmaximal, characteristic
6#23761234C211yesmaximal, characteristic
7C79×#1296C211yesmaximal, characteristic
8C2×#14579161C2211yescharacteristic
9C2×#14579092—21no—
10C2×C4⋊(C79⋊C8)C2211yescharacteristic
11C2×#14579084—21no—
12C2×#14579084—21no—
13C2×#14579071—21no—
14C2×C4×C79⋊C8C2211yescharacteristic
15C4×C4⋊Dic79C2211yescharacteristic
16#19098326—21no—
17#19098326—21no—
18#19098325—21no—
19#19098317—21no—
20#19098310C414yescomplemented
21C2×C4⋊C8×C79C2211yescharacteristic
22C2×C4⋊C8×C79C2211yescharacteristic
23#384×C79C2211yescharacteristic
24C22×Dic158—21no—
25C22×C79⋊C8—21no—
26C22×C79⋊C8D411yescharacteristic
27C22×C79⋊C8D411yescharacteristic
28#14579161—21no—
29#14579161—21no—
30#14579161—21no—
31#14579161C2×C412yes—
32C2×C4⋊Dic79—21no—
33C2×C4⋊Dic79—21no—
34C2×C4⋊Dic79D411yescharacteristic
35C2×C4⋊Dic79D411yescharacteristic
36C2×C4×Dic79—21no—
37#14579092—42no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)C2×C412yes—
40#14579084—42no—
41#14579084—42no—
42#14579071—42no—
43C4×C79⋊C8—21no—
44C4×C79⋊C8—21no—
45C79⋊#95—21no—
46C22×C8×C79—21no—
47C22×C8×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79—21no—
50C2×C4⋊C4×C79—21no—
51C2×C42×C79C2311yescharacteristic
52C4⋊C8×C79—21no—
53C4⋊C8×C79—21no—
54C4⋊C8×C79—21no—
55C4⋊C8×C79C2×C412yes—
56C2×Dic158—22no—
57C2×Dic158—41no—
58C2×Dic158—41no—
59C2×Dic158—41no—
60C2×Dic158—41no—
61C2×Dic158—41no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C22×Dic79—41no—
65C2×C79⋊C8—22no—
66C2×C79⋊C8—21no—
67C2×C79⋊C8—21no—
68C2×C79⋊C8—21no—
69C2×C79⋊C8—21no—
70C2×C79⋊C8—21no—
71C2×C79⋊C8—41no—
72C2×C79⋊C8—41no—
73C2×C79⋊C8SD1612yes—
74C4⋊Dic79—22no—
75C4⋊Dic79—22no—
76C4⋊Dic79—21no—
77C4⋊Dic79—21no—
78C4⋊Dic79—21no—
79C4⋊Dic79—21no—
80C4⋊Dic79—41no—
81C4⋊Dic79—41no—
82C4×Dic79—41no—
83C4×Dic79—41no—
84C22×C4×C79—41no—
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C22×C4×C79#4111yescharacteristic
88C2×C8×C79—22no—
89C2×C8×C79—41no—
90C2×C8×C79—41no—
91C2×C8×C79—41no—
92C2×C8×C79—41no—
93C2×C8×C79—41no—
94C4⋊C4×C79—41no—
95C4⋊C4×C79—41no—
96C4⋊C4×C79—41no—
97C4⋊C4×C79—41no—
98C4⋊C4×C79—81no—
99C42×C79C22×C411yescharacteristic
100C42×C79#4111yescharacteristic
101C42×C79#4111yescharacteristic
102C42×C79#4111yescharacteristic
103C23×C79#12111yescharacteristic
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—81no—
107C2×C4×C79—81no—
108C2×C4×C79—81no—
109C2×C4×C79#11711yescharacteristic
110C2×C4×C79C4×D411yescharacteristic
111C2×C4×C79C4×D411yescharacteristic
112C2×C4×C79#12311yescharacteristic
113C2×C4×C79#12411yescharacteristic
114C2×C4×C79#12611yescharacteristic
115C2×C4×C79C2×SD1611yescharacteristic
116C2×C4×C79#13511yescharacteristic
117C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C79#13711yescharacteristic
121C2×C4×C79#13711yescharacteristic
122C2×C4×C79#13711yescharacteristic
123C2×Dic79—42no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—81no—
131C2×Dic79—81no—
132Dic158—42no—
133Dic158—42no—
134Dic158—81no—
135Dic158—81no—
136C8×C79—42no—
137C8×C79—81no—
138C79⋊C8—22no—
139C79⋊C8—42no—
140C79⋊C8—41no—
141C22×C79#38811yescharacteristic
142C22×C79#46011yescharacteristic
143C22×C79#46311yescharacteristic
144C22×C79#46811yescharacteristic
145C22×C79#47011yescharacteristic
146C22×C79#47111yescharacteristic
147C22×C79#47311yescharacteristic
148C4×C79—21no—
149C4×C79—21no—
150C4×C79—161no—
151C4×C79C4×SD1611yescharacteristic
152C4×C79#44011yescharacteristic
153C4×C79#44111yescharacteristic
154C4×C79#44211yescharacteristic
155C4×C79#48111yescharacteristic
156C4×C79#48211yescharacteristic
157C4×C79#48911yescharacteristic
158C4×C79#49111yescharacteristic
159Dic79—82no—
160Dic79—81no—
161Dic79—81no—
162#9570—791nomaximal, Sylow
163C2×C79#168611yescharacteristic
164C2×C79#169011yescharacteristic
165C2×C79#169911yescharacteristic
166C2×C79#170311yescharacteristic
167C2×C79#192611yescharacteristic
168C2×C79#193011yescharacteristic
169C2×C79#193511yescharacteristic
170#1988—791no—
171#1956—791no—
172#1948—791no—
173#1770—791no—
174C2×#332—791no—
175#1296D7911yescharacteristic
176#1277—791no—
177C79#957011yescharacteristic, complemented, Sylow
178C2×C4⋊Q8—791no—
179C2×#106—1581no—
180C2×C4⋊C8—791no—
181C2×C4⋊C8C2×D7911yescharacteristic
182C2×C4⋊C8C2×D7911yescharacteristic
183C2×#103—1581no—
184C2×#103—1581no—
185C2×#97—1581no—
186C2×C4×C8—791no—
187#384C2×D7911yescharacteristic
188C4×C4⋊C4—791no—
189#340—1581no—
190#340—1581no—
191#340—1581no—
192#336—1581no—
193#332—794no—
194C22×Q8—1581no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—791no—
198C22×C8—791no—
199C22×C8—1581no—
200C4⋊Q8—792no—
201C4⋊Q8—1581no—
202C4⋊Q8—1581no—
203C4⋊Q8—1581no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—791no—
208C2×C4⋊C4—791no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4—1581no—
211C2×C42—1581no—
212C2×C42C22×D7911yescharacteristic
213#106—3162no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—792no—
218C4⋊C8—1581no—
219C4⋊C8C4×D7912yes—
220#103—3162no—
221#103—3162no—
222#97—3162no—
223C4×C8—1581no—
224C4×C8—1581no—
225#95—1581no—
226C2×Q8—3161no—
227C2×Q8—3161no—
228C2×Q8—3161no—
229C2×Q8—3161no—
230C2×Q8—3161no—
231C2×Q8—1582no—
232C22×C4—41no—
233C22×C4—3161no—
234C22×C4—1581no—
235C22×C4—1581no—
236C22×C4D4×D7911yescharacteristic
237C22×C4D4×D7911yescharacteristic
238C22×C4#1192962411yescharacteristic
239C2×C8—22no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—3161no—
246C2×C8—3161no—
247C2×C8—792no—
248C2×C8—1582no—
249C2×C8—1581no—
250C2×C8—1581no—
251C2×C8—1581no—
252C2×C8—1581no—
253C2×C8—1581no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—81no—
259C4⋊C4—3161no—
260C4⋊C4—3161no—
261C4⋊C4—1582no—
262C4⋊C4—1582no—
263C4⋊C4—1581no—
264C4⋊C4—1581no—
265C4⋊C4—1581no—
266C4⋊C4—1581no—
267C42—3161no—
268C42—3161no—
269C42C2×C4×D7911yescharacteristic
270C42#1192962311yescharacteristic
271C42#1192962411yescharacteristic
272C42#1192962511yescharacteristic
273C23#1457913411yescharacteristic
274Q8—3162no—
275Q8—3162no—
276Q8—6321no—
277Q8—6321no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—3162no—
284C2×C4—3161no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4—3161no—
289C2×C4—3161no—
290C2×C4—6321no—
291C2×C4—6321no—
292C2×C4#1457911811yescharacteristic
293C2×C4(C22⋊D79)⋊C411yescharacteristic
294C2×C4#1457912311yescharacteristic
295C2×C4#1457914611yescharacteristic
296C2×C4#1457914711yescharacteristic
297C2×C4#1457915611yescharacteristic
298C2×C4SD16×D7911yescharacteristic
299C2×C4#1457918411yescharacteristic
300C2×C4#1457918511yescharacteristic
301C2×C4#1457918511yescharacteristic
302C2×C4#1457918811yescharacteristic
303C2×C4#1457919011yescharacteristic
304C2×C4#1457919011yescharacteristic
305C2×C4#1457919311yescharacteristic
306C8—42no—
307C8—81no—
308C8—3161no—
309C8—3162no—
310C8—1582no—
311C22#1909845211yescharacteristic
312C22#1909876211yescharacteristic
313C22#1909878111yescharacteristic
314C22#1909880811yescharacteristic
315C22#1909882011yescharacteristic
316C22#1909882511yescharacteristic
317C22#1909883711yescharacteristic
318C4—21no—
319C4—21no—
320C4—161no—
321C4—6321no—
322C4—6321no—
323C4—6322no—
324C4#1909864811yescharacteristic
325C4#1909866211yescharacteristic
326C4#1909866911yescharacteristic
327C4#1909867211yescharacteristic
328C4#1909888311yescharacteristic
329C4#1909889011yescharacteristic
330C4#1909892111yescharacteristic
331C4#1909893011yescharacteristic
332C2#2376287111yescharacteristic
333C2#2376289311yescharacteristic
334C2#2376295011yescharacteristic
335C2#2376297811yescharacteristic
336C2#2376402511yescharacteristic
337C2#2376405311yescharacteristic
338C2#2376408811yescharacteristic
subgroupG/H
centerC23#14579134
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1296C2
socle, prime radicalC23×C79#121
cosocleC2×C42×C79C23
hypercenter#1296D79
hypocenterC79#9570
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
20. #19098310C4semidirect
177. C79#9570semidirect

Maximal subgroups

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

subgroupindex
#237643512
#237642012
#237641612
#237632702
C2×#190983102
#237612342
C79×#12962
#957079