Finite Group Databasebeta

← #34183513

Subgroups — #34183513

Order 20224.

432 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772349C211yesmaximal, characteristic
2#23764345C211yesmaximal, characteristic
3#23761899C211yesmaximal, characteristic
4#23761803C211yesmaximal, characteristic
5#23761240C211yesmaximal, characteristic
6#23761237C211yesmaximal, characteristic
7C79×#3170C211yesmaximal, characteristic
8#19098694—21no—
9#19098694—22no—
10Q8×C79⋊C8—21no—
11#19098616C2211yescharacteristic
12#19098590—21no—
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14#19098519—21no—
15C4×#11929608C2211yescharacteristic
16C4×C4⋊Dic79C2211yescharacteristic
17#19098322—21no—
18#19098322—21no—
19#19098300—22no—
20Q8⋊(C79⋊C8)—21no—
21Q8⋊(C79⋊C8)—21no—
22#533×C79C2211yescharacteristic
23#533×C79C2211yescharacteristic
24#500×C79C411yescharacteristic, complemented
25#494×C79C411yescharacteristic, complemented
26#487×C79C411yescharacteristic, complemented
27#487×C79C411yescharacteristic, complemented
28#467×C79—22no—
29C4×Q16×C79—22no—
30C2×C4×C8×C79C2211yescharacteristic
31C2×#11929608D411yescharacteristic
32C2×#11929608D411yescharacteristic
33C22×C79⋊C8—21no—
34C2×C4⋊Dic79—21no—
35C2×C4×Dic79—21no—
36C4⋊(C79⋊C8)—22no—
37C4⋊(C79⋊C8)—22no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)D411yescharacteristic
41C4⋊(C79⋊C8)D411yescharacteristic
42C22⋊(C79⋊C8)—21no—
43#14579072—22no—
44#14579072—21no—
45C4×C79⋊C8—22no—
46C4×C79⋊C8—21no—
47C4×C79⋊C8—21no—
48C79⋊#95—21no—
49C2×Q16×C79—22no—
50C22×C8×C79—21no—
51C4⋊Q8×C79C2×C411yescharacteristic
52C4⋊Q8×C79C2×C411yescharacteristic
53C4⋊Q8×C79C2×C411yescharacteristic
54C4⋊Q8×C79C2×C411yescharacteristic
55#125×C79—21no—
56#125×C79—21no—
57C22⋊Q8×C79—21no—
58C22⋊Q8×C79—21no—
59C22⋊Q8×C79—21no—
60C22⋊Q8×C79D412yes—
61C4×Q8×C79—21no—
62C4×Q8×C79—21no—
63C4×Q8×C79—21no—
64C4×Q8×C79D412yes—
65#117×C79—21no—
66#117×C79—21no—
67C2×C42×C79C2311yescharacteristic
68#107×C79—22no—
69#103×C79—22no—
70#103×C79—22no—
71C4×C8×C79—21no—
72C4×C8×C79C2×C411yescharacteristic
73C4×C8×C79C2×C411yescharacteristic
74C22×Dic79—41no—
75#11929608—21no—
76#11929608—21no—
77#11929608—21no—
78#11929608Q1612yescomplemented
79C2×C79⋊C8—21no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—21no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8—41no—
85C2×C79⋊C8—41no—
86C4⋊Dic79—22no—
87C4⋊Dic79—41no—
88C4×Dic79—41no—
89C4×Dic79—41no—
90C2×Q8×C79—21no—
91C2×Q8×C79—21no—
92C2×Q8×C79—21no—
93C2×Q8×C79C22⋊C412yes—
94C22×C4×C79C2×D411yescharacteristic
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79#4111yescharacteristic
97Q16×C79—44no—
98C2×C8×C79—21no—
99C2×C8×C79—21no—
100C2×C8×C79—42no—
101C4⋊C4×C79—21no—
102C4⋊C4×C79—21no—
103C4⋊C4×C79—21no—
104C4⋊C4×C79—22no—
105C4⋊C4×C79—22no—
106C4⋊C4×C79—21no—
107C4⋊C4×C79—21no—
108C4⋊C4×C79—21no—
109C4⋊C4×C79—21no—
110C4⋊C4×C79—41no—
111C4⋊C4×C79—41no—
112C4⋊C4×C79C22⋊C412yes—
113C22⋊C4×C79—22no—
114C22⋊C4×C79—41no—
115C42×C79—22no—
116C42×C79—41no—
117C42×C79C22×C411yescharacteristic
118C42×C79C2×D411yescharacteristic
119C42×C79C2×D411yescharacteristic
120C42×C79#4111yescharacteristic
121C23×C79#12111yescharacteristic
122Q8×C79—42no—
123Q8×C79—42no—
124Q8×C79—41no—
125Q8×C79—41no—
126Q8×C79—41no—
127Q8×C79—41no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—22no—
131C2×C4×C79—22no—
132C2×C4×C79—41no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79C2×C22⋊C411yescharacteristic
136C2×C4×C79C4×D411yescharacteristic
137C2×C4×C79C4×D411yescharacteristic
138C2×C4×C79#12011yescharacteristic
139C2×C4×C79#12111yescharacteristic
140C2×C4×C79#12311yescharacteristic
141C2×C4×C79C2×Q1611yescharacteristic
142C2×C4×C79#13511yescharacteristic
143C2×C4×C79#13711yescharacteristic
144C2×C4×C79#13711yescharacteristic
145C2×Dic79—42no—
146C2×Dic79—81no—
147C2×Dic79—81no—
148C8×C79—22no—
149C8×C79—41no—
150C79⋊C8—22no—
151C79⋊C8—41no—
152C79⋊C8—41no—
153C79⋊C8—41no—
154C79⋊C8—81no—
155C22×C79—21no—
156C22×C79—21no—
157C22×C79#38611yescharacteristic
158C22×C79#46211yescharacteristic
159C22×C79#46411yescharacteristic
160C4×C79—21no—
161C4×C79—21no—
162C4×C79—21no—
163C4×C79—42no—
164C4×C79—42no—
165C4×C79—81no—
166C4×C79—81no—
167C4×C79C4×Q1611yescharacteristic
168C4×C79#44111yescharacteristic
169C4×C79#44311yescharacteristic
170C4×C79#44411yescharacteristic
171C4×C79#45111yescharacteristic
172C4×C79#45211yescharacteristic
173Dic79—82no—
174#9056—791nomaximal, Sylow
175C2×C79—21no—
176C2×C79—21no—
177C2×C79#187611yescharacteristic
178C2×C79#187811yescharacteristic
179C2×C79#188411yescharacteristic
180#3170D7911yescharacteristic
181#2962—791no—
182#1986—791no—
183#1503—791no—
184#1481—791no—
185#1279—791no—
186#1278—791no—
187C79#905611yescharacteristic, complemented, Sylow
188#533C2×D7911yescharacteristic
189#533C2×D7911yescharacteristic
190#500Dic7911yescharacteristic, complemented
191#494Dic7911yescharacteristic, complemented
192#487Dic7911yescharacteristic, complemented
193#487Dic7911yescharacteristic, complemented
194#467—22no—
195#446—1581no—
196#446—1582no—
197C8×Q8—1581no—
198C4×Q16—22no—
199#431—791no—
200#424—1581no—
201C2×C4⋊C8—791no—
202#405—1581no—
203C4×M16—791no—
204C2×C4×C8C2×D7911yescharacteristic
205C4×C4⋊C4—791no—
206#339—1581no—
207#339—1581no—
208#328—1582no—
209Q8⋊C8—1581no—
210Q8⋊C8—1581no—
211C2×Q16—22no—
212C2×M16—791no—
213C2×M16—791no—
214C22×C8—21no—
215C22×C8—1581no—
216C4⋊Q8C2×Dic7911yescharacteristic
217C4⋊Q8C2×Dic7911yescharacteristic
218C4⋊Q8C2×Dic7911yescharacteristic
219C4⋊Q8C2×Dic7911yescharacteristic
220#125—21no—
221#125—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8C22⋊D7912yes—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8C22⋊D7912yes—
230#117—21no—
231#117—21no—
232C2×C4⋊C4—1581no—
233C2×C42—1581no—
234C2×C42C22×D7911yescharacteristic
235#107—22no—
236C4⋊C8—791no—
237C4⋊C8—791no—
238C4⋊C8—1582no—
239C4⋊C8—1582no—
240C4⋊C8—1581no—
241C4⋊C8—1581no—
242#103—22no—
243#103—22no—
244C22⋊C8—1581no—
245#97—1582no—
246#97—1581no—
247C4×C8—21no—
248C4×C8—1582no—
249C4×C8—1581no—
250C4×C8—1581no—
251C4×C8C2×Dic7911yescharacteristic
252C4×C8C2×Dic7911yescharacteristic
253#95—1581no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8C22⋊Dic7912yes—
258C22×C4—3161no—
259C22×C4D4×D7911yescharacteristic
260C22×C4C2×C22⋊D7911yescharacteristic
261C22×C4#1192962311yescharacteristic
262Q16—44no—
263M16—792no—
264M16—1581no—
265M16—1581no—
266M16—1581no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—42no—
270C2×C8—3161no—
271C2×C8—3161no—
272C2×C8—3161no—
273C2×C8—1581no—
274C2×C8—1581no—
275C2×C8—1581no—
276C2×C8—1581no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—3161no—
289C4⋊C4—1582no—
290C4⋊C4C22⋊Dic7912yes—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C42—22no—
294C42—41no—
295C42—3161no—
296C42—3161no—
297C42C22×Dic7911yescharacteristic
298C42D4×D7911yescharacteristic
299C42C2×C22⋊D7911yescharacteristic
300C42#1192962311yescharacteristic
301C23#1457913311yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307Q8—41no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—3162no—
316C2×C4—6321no—
317C2×C4—6321no—
318C2×C4C2×C22⋊Dic7911yescharacteristic
319C2×C4D4×Dic7911yescharacteristic
320C2×C4D4×Dic7911yescharacteristic
321C2×C4#1457912911yescharacteristic
322C2×C4#1457913311yescharacteristic
323C2×C4#1457914211yescharacteristic
324C2×C4Q16×D7911yescharacteristic
325C2×C4#1457918411yescharacteristic
326C2×C4#1457919311yescharacteristic
327C2×C4#1457919311yescharacteristic
328C8—22no—
329C8—41no—
330C8—3161no—
331C8—3161no—
332C8—3161no—
333C8—6321no—
334C8—1582no—
335C22—21no—
336C22—21no—
337C22#1909844311yescharacteristic
338C22#1909877111yescharacteristic
339C22#1909878711yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—42no—
345C4—81no—
346C4—81no—
347C4—6322no—
348C4Q16×Dic7911yescharacteristic
349C4#1909867011yescharacteristic
350C4#1909868011yescharacteristic
351C4#1909868511yescharacteristic
352C4#1909871711yescharacteristic
353C4#1909872411yescharacteristic
354C2—21no—
355C2—21no—
356C2#2376372911yescharacteristic
357C2#2376374311yescharacteristic
358C2#2376378411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3170C2
socleC22×C79#386
prime radicalC23×C79#121
cosocleC2×C42×C79C23
hypercenter#3170D79
hypocenterC79#9056
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
24. #500×C79C4semidirect
25. #494×C79C4semidirect
26. #487×C79C4semidirect
27. #487×C79C4semidirect
78. #11929608Q16semidirect
187. C79#9056semidirect
190. #500Dic79semidirect
191. #494Dic79semidirect
192. #487Dic79semidirect
193. #487Dic79semidirect

Maximal subgroups

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

subgroupindex
#237723492
#237643452
#237618992
#237618032
#237612402
#237612372
C79×#31702
#905679