Finite Group Databasebeta

← #34205214

Subgroups — #34205214

Order 20224.

574 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098801C211yesmaximal, characteristic
2C2×#107×D79C211yesmaximal, characteristic
3C2×#19098367C211yesmaximal, characteristic, complemented
4C2×(C2×D79)⋊C16C211yesmaximal, characteristic
5#23761492C211yesmaximal, characteristic, complemented
6#23761491C211yesmaximal, characteristic, complemented
7C2×#358×C79C211yesmaximal, characteristic, complemented
8C22×C8×D79C2211yescharacteristic
9C2×#14579137—21no—
10C2×C4⋊C4×D79—21no—
11C22×C79⋊C16C2211yescharacteristic
12#19098801—22no—
13#19098801C414yescomplemented
14C2×#14579095C2211yescharacteristic
15C2×#14579095C2211yescharacteristic
16C2×#14579094—21no—
17#107×D79—22no—
18#107×D79—22no—
19C2×#14579085—21no—
20#19098367—22no—
21(C2×D79)⋊C16—22no—
22C22×Q16×C79C2211yescharacteristic
23C22×C16×C79C2211yescharacteristic
24C2×#107×C79C2211yescharacteristic
25#358×C79—22no—
26C22×C4×D79—41no—
27C22×C4×D79D411yescharacteristic
28C22×C79⋊C8D411yescharacteristic
29C2×C8×D79—21no—
30C2×C8×D79—21no—
31C2×C8×D79—21no—
32C2×C8×D79—21no—
33C2×C8×D79C2×C412yes—
34C2×C8×D79D412yes—
35#14579137—24no—
36#14579137—42no—
37C2×C4⋊Dic79—21no—
38C2×C4⋊Dic79—21no—
39C2×Dic79⋊C4—41no—
40C2×Dic79⋊C4—41no—
41C4⋊C4×D79—24no—
42C4⋊C4×D79—42no—
43C2×(C2×D79)⋊C4—41no—
44C2×C79⋊C16—21no—
45C2×C79⋊C16—21no—
46C2×C79⋊C16—21no—
47#14579095—21no—
48#14579095—21no—
49#14579095C2×C412yescomplemented
50#14579095D412yes—
51#14579094—24no—
52#14579085—24no—
53C22×Q8×C79—21no—
54C2×Q16×C79—21no—
55C2×Q16×C79—21no—
56C2×Q16×C79—21no—
57C2×Q16×C79—21no—
58C2×Q16×C79—21no—
59C2×Q16×C79C2×C412yescomplemented
60C22×C8×C79C2311yescharacteristic
61C2×C4⋊C4×C79—21no—
62C2×C16×C79—21no—
63C2×C16×C79—21no—
64C2×C16×C79—21no—
65#107×C79—21no—
66#107×C79D412yes—
67C23×D79D811yescharacteristic
68C22×Dic79—41no—
69C22×Dic79—41no—
70C22×Dic79D811yescharacteristic
71C2×C4×D79—21no—
72C2×C4×D79—21no—
73C2×C4×D79—21no—
74C2×C4×D79—21no—
75C2×C4×D79—44no—
76C2×C4×D79—82no—
77C2×C4×D79—82no—
78C2×C4×D79C22⋊C412yes—
79C2×C4×D79SD1612yes—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8C22⋊C412yes—
83C8×D79—22no—
84C8×D79—22no—
85C8×D79—22no—
86C8×D79—22no—
87C4⋊Dic79—22no—
88C4⋊Dic79—22no—
89C4⋊Dic79—41no—
90C4⋊Dic79—41no—
91Dic79⋊C4—44no—
92Dic79⋊C4—44no—
93(C2×D79)⋊C4—44no—
94C79⋊C16—41no—
95C2×Q8×C79—22no—
96C2×Q8×C79—41no—
97C2×Q8×C79—41no—
98C2×Q8×C79—41no—
99C2×Q8×C79—41no—
100C2×Q8×C79—41no—
101C22×C4×C79—41no—
102C22×C4×C79—41no—
103C22×C4×C79C2×D411yescharacteristic
104Q16×C79—22no—
105Q16×C79—22no—
106Q16×C79—41no—
107Q16×C79—41no—
108C2×C8×C79C22×C411yescharacteristic
109C2×C8×C79C2×D411yescharacteristic
110C2×C8×C79C2×D411yescharacteristic
111C2×C8×C79C2×D411yescharacteristic
112C2×C8×C79#4111yescharacteristic
113C2×C8×C79#4111yescharacteristic
114C4⋊C4×C79—22no—
115C4⋊C4×C79—41no—
116C16×C79—41no—
117C23×C79C2×D811yescharacteristic
118C22×D79—21no—
119C22×D79—21no—
120C22×D79—21no—
121C22×D79—21no—
122C22×D79#10212yes—
123C22×D79SD3212yes—
124C22×D79Q3212yes—
125Q8×C79—42no—
126Q8×C79—42no—
127Q8×C79—81no—
128Q8×C79—81no—
129C2×C4×C79—42no—
130C2×C4×C79—42no—
131C2×C4×C79—81no—
132C2×C4×C79—81no—
133C2×C4×C79—81no—
134C2×C4×C79—81no—
135C2×C4×C79C2×C22⋊C411yescharacteristic
136C2×C4×C79#12011yescharacteristic
137C2×C4×C79#12311yescharacteristic
138C2×C4×C79C2×SD1611yescharacteristic
139C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C79#13711yescharacteristic
141C2×Dic79—21no—
142C2×Dic79—21no—
143C2×Dic79—42no—
144C2×Dic79—42no—
145C2×Dic79—81no—
146C2×Dic79—81no—
147C2×Dic79—81no—
148C2×Dic79—81no—
149C2×Dic79#10212yes—
150C4×D79—22no—
151C4×D79—22no—
152C4×D79—22no—
153C4×D79—22no—
154C4×D79—88no—
155C8×C79C4×D411yescharacteristic
156C8×C79C4×D411yescharacteristic
157C8×C79#12111yescharacteristic
158C8×C79#12111yescharacteristic
159C79⋊C8—22no—
160C22×C79C2×#10211yescharacteristic
161C22×C79#44711yescharacteristic
162C22×C79#48011yescharacteristic
163C22×C79C2×SD3211yescharacteristic
164C22×C79C2×Q3211yescharacteristic
165C22×C79#50911yescharacteristic
166C22×C79#51011yescharacteristic
167C2×D79—22no—
168C2×D79—22no—
169C2×D79—22no—
170C2×D79—22no—
171C2×D79—22no—
172C2×D79—22no—
173C2×D79#35814yescomplemented
174C4×C79—82no—
175C4×C79—82no—
176C4×C79#38611yescharacteristic
177C4×C79#41711yescharacteristic
178C4×C79#44811yescharacteristic
179C4×C79#48111yescharacteristic
180Dic79—22no—
181Dic79—82no—
182Dic79—82no—
183#12993—791nomaximal, Sylow
184C2×C79#187411yescharacteristic
185C2×C79C2×#35811yescharacteristic
186C2×C79#213911yescharacteristic
187C2×C79#218711yescharacteristic
188C2×C79#218811yescharacteristic
189C2×C79#223511yescharacteristic
190C2×C79#223611yescharacteristic
191D79—24no—
192C2×#467—791no—
193C22×#107—791no—
194C2×#358—791no—
195C2×#358D7911yescharacteristic, complemented
196C2×C22⋊C16—791no—
197#1380—791no—
198#1380—791no—
199C79#1299311yescharacteristic, complemented, Sylow
200C22×Q16C2×D7911yescharacteristic
201C23×C8—791no—
202C2×C22⋊Q8—1581no—
203C22×C4⋊C4—1581no—
204C22×C16—791no—
205C22×C16C2×D7911yescharacteristic
206#467—794no—
207#467—1582no—
208C2×#107—791no—
209C2×#107—791no—
210C2×#107—1582no—
211C2×#107—1581no—
212C2×#107—1582no—
213C2×#107C2×D7911yescharacteristic
214C2×#103—1581no—
215#358—22no—
216#358—1582no—
217C22⋊C16—1582no—
218C22×Q8—21no—
219C23×C4—3161no—
220C23×C4—791no—
221C2×Q16—21no—
222C2×Q16—21no—
223C2×Q16—21no—
224C2×Q16—21no—
225C2×Q16—21no—
226C2×Q16C4×D7912yescomplemented
227C22×C8—791no—
228C22×C8—792no—
229C22×C8—792no—
230C22×C8—1581no—
231C22×C8—1581no—
232C22×C8—1581no—
233C22×C8—1581no—
234C22×C8C22×D7911yescharacteristic
235C22⋊Q8—3162no—
236C22⋊Q8—1584no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—3162no—
239C2×C4⋊C4—3161no—
240C2×C4⋊C4—3161no—
241C2×C4⋊C4—1584no—
242C2×C4⋊C4—1581no—
243C2×C4⋊C4—1581no—
244C2×C22⋊C4—3161no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—1581no—
249C2×C16—1581no—
250C2×C16—1581no—
251#107—21no—
252#107—792no—
253#107—792no—
254#107—1584no—
255#107—1581no—
256#107—1581no—
257#107C22⋊D7912yes—
258#103—1584no—
259C24—791no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—3164no—
269C22×C4—3161no—
270C22×C4—3161no—
271C22×C4—791no—
272C22×C4—792no—
273C22×C4—792no—
274C22×C4—6322no—
275C22×C4—6322no—
276C22×C4—1581no—
277C22×C4—1581no—
278C22×C4—1581no—
279C22×C4—1581no—
280C22×C4D4×D7911yescharacteristic
281Q16—22no—
282Q16—22no—
283Q16—41no—
284Q16—41no—
285C2×C8—792no—
286C2×C8—1581no—
287C2×C8—1581no—
288C2×C8—1582no—
289C2×C8—1582no—
290C2×C8—1582no—
291C2×C8—1582no—
292C2×C8C2×C4×D7911yescharacteristic
293C2×C8D4×D7911yescharacteristic
294C2×C8D4×D7911yescharacteristic
295C2×C8C2×C22⋊D7911yescharacteristic
296C2×C8#1192962311yescharacteristic
297C2×C8#1192962411yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—3164no—
301C4⋊C4—3164no—
302C4⋊C4—3161no—
303C4⋊C4—3161no—
304C4⋊C4—1582no—
305C4⋊C4—1582no—
306C22⋊C4—3164no—
307C16—41no—
308C16—3161no—
309C23—792no—
310C23—792no—
311C23—792no—
312C23—1581no—
313C23—1581no—
314C23—1581no—
315C23—1581no—
316C23D8×D7911yescharacteristic
317Q8—42no—
318Q8—42no—
319Q8—81no—
320Q8—81no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—3162no—
328C2×C4—3162no—
329C2×C4—792no—
330C2×C4—6328no—
331C2×C4—6321no—
332C2×C4—6321no—
333C2×C4—6321no—
334C2×C4—6321no—
335C2×C4—1581no—
336C2×C4—1581no—
337C2×C4—1582no—
338C2×C4—1582no—
339C2×C4—1582no—
340C2×C4—1582no—
341C2×C4C22⋊C4×D7911yescharacteristic
342C2×C4#1457912911yescharacteristic
343C2×C4#1457914311yescharacteristic
344C2×C4SD16×D7911yescharacteristic
345C2×C4#1457918511yescharacteristic
346C2×C4#1457919011yescharacteristic
347C8—1582no—
348C8(C22⋊D79)⋊C411yescharacteristic
349C8C4×C22⋊D7911yescharacteristic
350C8#1457913211yescharacteristic
351C8#1457913411yescharacteristic
352C22—794no—
353C22—1582no—
354C22—1582no—
355C22—1582no—
356C22—1582no—
357C22—1582no—
358C22—1582no—
359C22#102×D7911yescharacteristic
360C22#1909869511yescharacteristic
361C22#1909887511yescharacteristic
362C22SD32×D7911yescharacteristic
363C22Q32×D7911yescharacteristic
364C22#1909900011yescharacteristic
365C22#1909900511yescharacteristic
366C4—82no—
367C4—82no—
368C4—6322no—
369C4—6322no—
370C4—1582no—
371C4#1909843911yescharacteristic
372C4#1909856111yescharacteristic
373C4#1909870011yescharacteristic
374C4#1909888011yescharacteristic
375C2—1584no—
376C2#2376371411yescharacteristic
377C2#358×D7911yescharacteristic
378C2#2376511111yescharacteristic
379C2#2376532711yescharacteristic
380C2#2376533211yescharacteristic
381C2#2376559611yescharacteristic
382C2#2376560111yescharacteristic
subgroupG/H
centerC23D8×D79
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC2×#358×C79C2
socleC23×C79C2×D8
prime radicalC23×D79D8
cosocleC22×C8×C79C23
hypercenterC2×#358D79
hypocenterC79#12993
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
3. C2×#19098367C2semidirect
5. #23761492C2semidirect
6. #23761491C2semidirect
7. C2×#358×C79C2semidirect
13. #19098801C4semidirect
49. #14579095C2×C4semidirect
59. C2×Q16×C79C2×C4semidirect
173. C2×D79#358semidirect
195. C2×#358D79semidirect
199. C79#12993semidirect
226. C2×Q16C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190988012
C2×#107×D792
C2×#190983672
C2×(C2×D79)⋊C162
#237614922
#237614912
C2×#358×C792
#1299379