Finite Group Databasebeta

← #34185418

Subgroups — #34185418

Order 20224.

540 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772525C211yesmaximal, characteristic, complemented
2#23764241C211yesmaximal, characteristic, complemented
3#23762068C211yesmaximal, characteristic, complemented
4#23761839C211yesmaximal, characteristic, complemented
5C79⋊#1309C211yesmaximal, characteristic, complemented
6#23761289C211yesmaximal, characteristic, complemented
7C79×#3038C211yesmaximal, characteristic
8#19098694—21no—
9#19098635—22no—
10D4×C79⋊C8—21no—
11#19098620—21no—
12#19098616C2211yescharacteristic
13#19098587C2211yescharacteristic
14#19098587C2211yescharacteristic
15C2×C4⋊(C79⋊C8)—21no—
16C2×#14579078—22no—
17C4×#11929608C2211yescharacteristic
18#19098307—21no—
19#19098307—21no—
20Q8⋊(C79⋊C8)—21no—
21D4⋊(C79⋊C8)—21no—
22C2×AGL(1,ℤ/8)×C79C412yes—
23C2×#127×C79C2211yescharacteristic
24C4×#41×C79C2211yescharacteristic
25#460×C79—21no—
26#460×C79—21no—
27#459×C79—21no—
28#459×C79—21no—
29#423×C79C2211yescharacteristic
30#418×C79C412yes—
31C2×#11929608—21no—
32C2×#11929608—21no—
33C2×#11929608D412yes—
34C22×C79⋊C8—22no—
35C4⋊(C79⋊C8)—22no—
36C4⋊(C79⋊C8)—21no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)D411yescharacteristic
41C4⋊(C79⋊C8)D411yescharacteristic
42#14579078—44no—
43C22⋊(C79⋊C8)—22no—
44C22⋊(C79⋊C8)—21no—
45#14579072—22no—
46#14579072—21no—
47C4×C79⋊C8—21no—
48C4×C79⋊C8—21no—
49C2×#41×C79C2×C411yescharacteristic
50C22×D4×C79—21no—
51C22×D4×C79—21no—
52C22×D4×C79C2×C412yes—
53AGL(1,ℤ/8)×C79—44no—
54C2×SD16×C79—22no—
55C2×D8×C79—22no—
56C2×M16×C79C2×C412yes—
57#127×C79—21no—
58#127×C79—21no—
59#127×C79—21no—
60#127×C79—21no—
61C4×Q8×C79D411yescharacteristic
62C4×Q8×C79D411yescharacteristic
63C4×D4×C79—21no—
64C4×D4×C79—21no—
65C4×D4×C79D411yescharacteristic
66C4×D4×C79D411yescharacteristic
67#117×C79—21no—
68#117×C79C2×C411yescharacteristic
69C2×C42×C79—21no—
70C2×C42×C79C2311yescharacteristic
71C4⋊C8×C79—21no—
72C4⋊C8×C79—21no—
73#102×C79—22no—
74#102×C79—22no—
75#11929608—22no—
76#11929608—42no—
77#11929608—41no—
78#11929608—81no—
79C2×C79⋊C8—21no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—22no—
82C2×C79⋊C8—41no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8—82no—
85C24×C79—22no—
86#41×C79—22no—
87#41×C79—41no—
88C2×Q8×C79C22⋊C411yescharacteristic
89C2×D4×C79—21no—
90C2×D4×C79—22no—
91C2×D4×C79—22no—
92C2×D4×C79—42no—
93C2×D4×C79—42no—
94C2×D4×C79—42no—
95C2×D4×C79—42no—
96C2×D4×C79—41no—
97C2×D4×C79—41no—
98C2×D4×C79—41no—
99C2×D4×C79—41no—
100C2×D4×C79—81no—
101C2×D4×C79—81no—
102C2×D4×C79C22⋊C411yescharacteristic
103C22×C4×C79—21no—
104C22×C4×C79—21no—
105C22×C4×C79—21no—
106C22×C4×C79C22×C411yescharacteristic
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79#4111yescharacteristic
109SD16×C79—44no—
110D8×C79—44no—
111M16×C79—42no—
112C2×C8×C79—22no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79—21no—
115C4⋊C4×C79C22⋊C411yescharacteristic
116C4⋊C4×C79C22⋊C411yescharacteristic
117C22⋊C4×C79—21no—
118C22⋊C4×C79—21no—
119C22⋊C4×C79—21no—
120C42×C79—21no—
121C42×C79—21no—
122C42×C79—21no—
123C42×C79C2×D411yescharacteristic
124C42×C79C2×D411yescharacteristic
125C42×C79C2×D411yescharacteristic
126C42×C79#4111yescharacteristic
127C23×C79—21no—
128C23×C79—44no—
129C23×C79—42no—
130C23×C79—82no—
131C23×C79—82no—
132C23×C79C4×D411yescharacteristic
133Q8×C79—21no—
134Q8×C79—21no—
135D4×C79—21no—
136D4×C79—42no—
137D4×C79—42no—
138D4×C79—42no—
139D4×C79—82no—
140D4×C79—82no—
141D4×C79—82no—
142D4×C79—81no—
143D4×C79—81no—
144D4×C79—81no—
145D4×C79—81no—
146D4×C79#10012yescomplemented
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—21no—
154C2×C4×C79—21no—
155C2×C4×C79—21no—
156C2×C4×C79—42no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—41no—
161C2×C4×C79C2×C22⋊C411yescharacteristic
162C2×C4×C79C4×D411yescharacteristic
163C2×C4×C79#12011yescharacteristic
164C2×C4×C79#12111yescharacteristic
165C2×C4×C79#12111yescharacteristic
166C2×C4×C79#12311yescharacteristic
167C8×C79—42no—
168C79⋊C8—42no—
169C79⋊C8—41no—
170C79⋊C8—42no—
171C79⋊C8—41no—
172C79⋊C8—81no—
173C22×C79—21no—
174C22×C79—21no—
175C22×C79—22no—
176C22×C79—21no—
177C22×C79—41no—
178C22×C79—41no—
179C22×C79—82no—
180C22×C79—82no—
181C22×C79—82no—
182C22×C79—164no—
183C22×C79#38611yescharacteristic
184C4×C79—21no—
185C4×C79—21no—
186C4×C79—21no—
187C4×C79—41no—
188C4×C79—41no—
189C4×C79—41no—
190C4×C79C2×#10011yescharacteristic
191C4×C79#41311yescharacteristic
192C4×C79#45311yescharacteristic
193C4×C79#45411yescharacteristic
194C4×C79#47111yescharacteristic
195C4×C79#47211yescharacteristic
196#9359—791nomaximal, Sylow
197C2×C79—41no—
198C2×C79—42no—
199C2×C79—162no—
200C2×C79#188311yescharacteristic
201C2×C79#190011yescharacteristic
202C2×C79#190711yescharacteristic
203#3038D7911yescharacteristic
204#2990—791no—
205#1965—791no—
206#1545—791no—
207#1489—791no—
208#1309—791no—
209#1299—791no—
210C79#935911yescharacteristic, complemented, Sylow
211C2×AGL(1,ℤ/8)Dic7912yes—
212C2×#127C2×D7911yescharacteristic
213C4×#41C2×D7911yescharacteristic
214#460—21no—
215#460—21no—
216#459—21no—
217#459—21no—
218#446—1581no—
219#435—1582no—
220C8×D4—1581no—
221#432—1581no—
222#431—791no—
223#423—791no—
224#423—791no—
225#423C2×D7911yescharacteristic
226C2×C4⋊C8—1581no—
227#418Dic7912yes—
228C2×#100—1582no—
229C4×M16—791no—
230#331—1581no—
231#331—1581no—
232Q8⋊C8—1581no—
233D4⋊C8—1581no—
234C2×#41C2×Dic7911yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2×Dic7912yes—
238AGL(1,ℤ/8)—44no—
239C2×SD16—22no—
240C2×D8—22no—
241C2×M16—792no—
242C2×M16—1581no—
243C2×M16—1581no—
244C2×M16C2×Dic7912yes—
245C22×C8—1582no—
246#127—21no—
247#127—21no—
248#127—21no—
249#127—21no—
250C4×Q8C22⋊D7911yescharacteristic
251C4×Q8C22⋊D7911yescharacteristic
252C4×D4—21no—
253C4×D4—21no—
254C4×D4C22⋊D7911yescharacteristic
255C4×D4C22⋊D7911yescharacteristic
256#117—21no—
257#117C2×Dic7911yescharacteristic
258C2×C42—21no—
259C2×C42C22×D7911yescharacteristic
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—791no—
263C4⋊C8—791no—
264C4⋊C8—1582no—
265C4⋊C8—1581no—
266C4⋊C8—1581no—
267C4⋊C8—1581no—
268C4⋊C8—1581no—
269#102—22no—
270#102—22no—
271#100—3164no—
272C22⋊C8—1582no—
273C22⋊C8—1581no—
274#97—1582no—
275#97—1581no—
276C4×C8—1581no—
277C4×C8—1581no—
278C24—22no—
279#41—22no—
280#41—41no—
281C2×Q8C22⋊Dic7911yescharacteristic
282C2×D4—21no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—81no—
294C2×D4—81no—
295C2×D4C22⋊Dic7911yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4C22×Dic7911yescharacteristic
300C22×C4D4×D7911yescharacteristic
301C22×C4#1192962311yescharacteristic
302SD16—44no—
303D8—44no—
304M16—42no—
305M16—3161no—
306M16—3162no—
307M16—6321no—
308M16—1582no—
309C2×C8—22no—
310C2×C8—3161no—
311C2×C8—3161no—
312C2×C8—6322no—
313C2×C8—1581no—
314C2×C8—1581no—
315C2×C8—1582no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4C22⋊Dic7911yescharacteristic
319C4⋊C4C22⋊Dic7911yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42D4×D7911yescharacteristic
327C42C2×C22⋊D7911yescharacteristic
328C42C2×C22⋊D7911yescharacteristic
329C42#1192962311yescharacteristic
330C23—21no—
331C23—44no—
332C23—42no—
333C23—82no—
334C23—82no—
335C23D4×Dic7911yescharacteristic
336Q8—21no—
337Q8—21no—
338D4—21no—
339D4—42no—
340D4—42no—
341D4—42no—
342D4—82no—
343D4—82no—
344D4—82no—
345D4—81no—
346D4—81no—
347D4—81no—
348D4—81no—
349D4#1457907812yescomplemented
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4C2×C22⋊Dic7911yescharacteristic
365C2×C4D4×Dic7911yescharacteristic
366C2×C4#1457912911yescharacteristic
367C2×C4#1457913311yescharacteristic
368C2×C4#1457913311yescharacteristic
369C2×C4#1457914211yescharacteristic
370C8—42no—
371C8—3162no—
372C8—3161no—
373C8—3162no—
374C8—3161no—
375C8—6321no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22—164no—
386C22#1909844311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4C2×#1457907811yescharacteristic
394C4#1909854711yescharacteristic
395C4#1909872911yescharacteristic
396C4#1909873411yescharacteristic
397C4#1909882411yescharacteristic
398C4#1909883111yescharacteristic
399C2—41no—
400C2—42no—
401C2—162no—
402C2#2376377811yescharacteristic
403C2#2376386411yescharacteristic
404C2#2376391011yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3038C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3038D79
hypocenterC79#9359
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #23772525C2semidirect
2. #23764241C2semidirect
3. #23762068C2semidirect
4. #23761839C2semidirect
5. C79⋊#1309C2semidirect
6. #23761289C2semidirect
146. D4×C79#100semidirect
210. C79#9359semidirect
349. D4#14579078semidirect

Maximal subgroups

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

subgroupindex
#237725252
#237642412
#237620682
#237618392
C79⋊#13092
#237612892
C79×#30382
#935979