Finite Group Databasebeta

← #34205221

Subgroups — #34205221

Order 20224.

530 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098805C211yesmaximal, characteristic
2C2×#19098600C211yesmaximal, characteristic
3C2×#19098368C211yesmaximal, characteristic, complemented
4C2×#19098366C211yesmaximal, characteristic, complemented
5C2×(C2×D79)⋊C16C211yesmaximal, characteristic
6#23761491C211yesmaximal, characteristic, complemented
7C79×#1380C211yesmaximal, characteristic, complemented
8C22×Dic316C2211yescharacteristic
9C22×C8×D79C2211yescharacteristic
10C2×#14579141—21no—
11C2×#14579119—21no—
12C22×C79⋊C16C2211yescharacteristic
13#19098805—22no—
14#19098805C414yescomplemented
15#19098600—22no—
16#19098600—22no—
17C2×#14579095C2211yescharacteristic
18C2×#14579092—21no—
19C2×#14579084—21no—
20#19098368—22no—
21#19098366—22no—
22(C2×D79)⋊C16—22no—
23C22×C16×C79C2211yescharacteristic
24C2×#107×C79C2211yescharacteristic
25C2×#107×C79C2211yescharacteristic
26C22×Dic158—21no—
27C22×C4×D79D411yescharacteristic
28C2×Dic316—21no—
29C2×Dic316—21no—
30C2×Dic316—21no—
31C2×Dic316—21no—
32C2×Dic316—21no—
33C2×Dic316C2×C412yes—
34C22×C79⋊C8D411yescharacteristic
35C2×C8×D79—21no—
36C2×C8×D79—21no—
37C2×C8×D79—21no—
38C2×C8×D79—21no—
39C2×C8×D79C2×C412yes—
40C2×C8×D79D412yes—
41#14579141—24no—
42#14579141—42no—
43#14579119—24no—
44#14579119—42no—
45C2×C4⋊Dic79—21no—
46C2×C4⋊Dic79—41no—
47C2×(C2×D79)⋊C4—41no—
48C2×(C2×D79)⋊C4—41no—
49C2×C4×Dic79—41no—
50C2×C79⋊C16—21no—
51C2×C79⋊C16—21no—
52C2×C79⋊C16—21no—
53#14579095—21no—
54#14579095D412yes—
55#14579092—24no—
56#14579084—24no—
57C22×C8×C79C2311yescharacteristic
58C2×C4⋊C4×C79—21no—
59C2×C4⋊C4×C79—21no—
60C2×C16×C79—21no—
61C2×C16×C79—21no—
62C2×C16×C79—21no—
63#107×C79—21no—
64#107×C79—21no—
65#107×C79C2×C412yes—
66#107×C79D412yes—
67C23×D79D811yescharacteristic
68C2×Dic158—22no—
69C2×Dic158—41no—
70C2×Dic158—41no—
71C2×Dic158—41no—
72C2×Dic158—41no—
73C2×Dic158—41no—
74C22×Dic79—41no—
75C22×Dic79—41no—
76C22×Dic79D811yescharacteristic
77C2×C4×D79—21no—
78C2×C4×D79—21no—
79C2×C4×D79—21no—
80C2×C4×D79—21no—
81C2×C4×D79C22⋊C412yes—
82C2×C4×D79SD1612yes—
83Dic316—22no—
84Dic316—22no—
85Dic316—41no—
86Dic316—41no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8C22⋊C412yes—
90C8×D79—22no—
91C8×D79—22no—
92C8×D79—22no—
93C8×D79—22no—
94C4⋊Dic79—22no—
95C4⋊Dic79—44no—
96C4⋊Dic79—41no—
97(C2×D79)⋊C4—44no—
98(C2×D79)⋊C4—44no—
99C4×Dic79—44no—
100C79⋊C16—41no—
101C22×C4×C79—41no—
102C22×C4×C79—41no—
103C22×C4×C79C2×D411yescharacteristic
104C2×C8×C79C22×C411yescharacteristic
105C2×C8×C79C2×D411yescharacteristic
106C2×C8×C79C2×D411yescharacteristic
107C2×C8×C79C2×D411yescharacteristic
108C2×C8×C79#4111yescharacteristic
109C2×C8×C79#4111yescharacteristic
110C4⋊C4×C79—22no—
111C4⋊C4×C79—22no—
112C4⋊C4×C79—41no—
113C4⋊C4×C79—41no—
114C16×C79—41no—
115C23×C79C2×D811yescharacteristic
116C22×D79—21no—
117C22×D79—21no—
118C22×D79—21no—
119C22×D79—21no—
120C22×D79—21no—
121C22×D79—21no—
122C22×D79#10212yes—
123C2×C4×C79—42no—
124C2×C4×C79—42no—
125C2×C4×C79—81no—
126C2×C4×C79—81no—
127C2×C4×C79—81no—
128C2×C4×C79—81no—
129C2×C4×C79C2×C22⋊C411yescharacteristic
130C2×C4×C79#12011yescharacteristic
131C2×C4×C79#12311yescharacteristic
132C2×C4×C79C2×SD1611yescharacteristic
133C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C79#13711yescharacteristic
135C2×Dic79—21no—
136C2×Dic79—21no—
137C2×Dic79—42no—
138C2×Dic79—42no—
139C2×Dic79—81no—
140C2×Dic79—81no—
141C2×Dic79—81no—
142C2×Dic79—81no—
143C2×Dic79#10212yes—
144C4×D79—22no—
145C4×D79—22no—
146C4×D79—22no—
147C4×D79—22no—
148Dic158—42no—
149Dic158—42no—
150Dic158—81no—
151Dic158—81no—
152C8×C79C4×D411yescharacteristic
153C8×C79C4×D411yescharacteristic
154C8×C79#12111yescharacteristic
155C8×C79#12111yescharacteristic
156C79⋊C8—22no—
157C22×C79C2×#10211yescharacteristic
158C22×C79#44711yescharacteristic
159C22×C79#48011yescharacteristic
160C22×C79#50811yescharacteristic
161C22×C79#50811yescharacteristic
162C22×C79#51011yescharacteristic
163C22×C79#51011yescharacteristic
164C2×D79—22no—
165C2×D79—22no—
166C2×D79—22no—
167C2×D79—41no—
168C2×D79—41no—
169C2×D79—41no—
170C2×D79—41no—
171C4×C79—82no—
172C4×C79—82no—
173C4×C79#38611yescharacteristic
174C4×C79#41711yescharacteristic
175C4×C79#44811yescharacteristic
176C4×C79#48111yescharacteristic
177Dic79—22no—
178Dic79—82no—
179Dic79—82no—
180#12994—791nomaximal, Sylow
181C2×C79#187411yescharacteristic
182C2×C79#213811yescharacteristic
183C2×C79#214111yescharacteristic
184C2×C79#218911yescharacteristic
185C2×C79#218911yescharacteristic
186C2×C79#223711yescharacteristic
187C2×C79#223711yescharacteristic
188D79—42no—
189C2×#467—791no—
190C2×#427—791no—
191C2×#358—791no—
192C2×#358—791no—
193C2×C22⋊C16—791no—
194#1380—791no—
195#1380D7911yescharacteristic, complemented
196C79#1299411yescharacteristic, complemented, Sylow
197C22×Q16—791no—
198C23×C8—791no—
199C2×C22⋊Q8—1581no—
200C2×#117—1581no—
201C22×C16—791no—
202C22×C16C2×D7911yescharacteristic
203#467—794no—
204#467—1582no—
205#427—1582no—
206#427—1582no—
207C2×#107—791no—
208C2×#107C2×D7911yescharacteristic
209C2×#107C2×D7911yescharacteristic
210C2×#106—1581no—
211C2×#103—1581no—
212#358—1582no—
213#358—1582no—
214C22⋊C16—1582no—
215C22×Q8—1581no—
216C23×C4—791no—
217C2×Q16—792no—
218C2×Q16—1581no—
219C2×Q16—1581no—
220C2×Q16—1581no—
221C2×Q16—1581no—
222C2×Q16—1581no—
223C22×C8—791no—
224C22×C8—792no—
225C22×C8—792no—
226C22×C8—1581no—
227C22×C8—1581no—
228C22×C8—1581no—
229C22×C8—1581no—
230C22×C8C22×D7911yescharacteristic
231C22⋊Q8—3162no—
232C22⋊Q8—1584no—
233#117—3162no—
234#117—1584no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—3161no—
238C2×C4⋊C4—1581no—
239C2×C22⋊C4—3161no—
240C2×C22⋊C4—3161no—
241C2×C42—3161no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—1581no—
246C2×C16—1581no—
247C2×C16—1581no—
248#107—21no—
249#107—21no—
250#107—792no—
251#107—1581no—
252#107C4×D7912yes—
253#107D31612yes—
254#106—1584no—
255#103—1584no—
256C24—791no—
257C2×Q8—3161no—
258C2×Q8—3161no—
259C2×Q8—3161no—
260C2×Q8—3161no—
261C2×Q8—3161no—
262C2×Q8—1582no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—3161no—
266C22×C4—3161no—
267C22×C4—791no—
268C22×C4—792no—
269C22×C4—792no—
270C22×C4—1581no—
271C22×C4—1581no—
272C22×C4—1581no—
273C22×C4—1581no—
274C22×C4D4×D7911yescharacteristic
275Q16—3161no—
276Q16—3161no—
277Q16—1582no—
278Q16—1582no—
279C2×C8—792no—
280C2×C8—1581no—
281C2×C8—1581no—
282C2×C8—1582no—
283C2×C8—1582no—
284C2×C8—1582no—
285C2×C8—1582no—
286C2×C8C2×C4×D7911yescharacteristic
287C2×C8D4×D7911yescharacteristic
288C2×C8D4×D7911yescharacteristic
289C2×C8C2×D31611yescharacteristic
290C2×C8#1192962411yescharacteristic
291C2×C8#1192962511yescharacteristic
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—3164no—
297C4⋊C4—3161no—
298C4⋊C4—1582no—
299C22⋊C4—3164no—
300C22⋊C4—3164no—
301C42—3164no—
302C16—41no—
303C16—3161no—
304C23—792no—
305C23—1581no—
306C23—1581no—
307C23—1581no—
308C23—1581no—
309C23—1581no—
310C23—1581no—
311C23D8×D7911yescharacteristic
312Q8—3162no—
313Q8—3162no—
314Q8—6321no—
315Q8—6321no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—3162no—
323C2×C4—3162no—
324C2×C4—792no—
325C2×C4—6321no—
326C2×C4—6321no—
327C2×C4—6321no—
328C2×C4—6321no—
329C2×C4—1581no—
330C2×C4—1581no—
331C2×C4—1582no—
332C2×C4—1582no—
333C2×C4—1582no—
334C2×C4—1582no—
335C2×C4C22⋊C4×D7911yescharacteristic
336C2×C4#1457913111yescharacteristic
337C2×C4#1457914611yescharacteristic
338C2×C4SD16×D7911yescharacteristic
339C2×C4#1457918811yescharacteristic
340C2×C4#1457919311yescharacteristic
341C8—1582no—
342C8#1457912311yescharacteristic
343C8C4×D31611yescharacteristic
344C8#1457913411yescharacteristic
345C8#1457913611yescharacteristic
346C22—3161no—
347C22—3161no—
348C22—3161no—
349C22—3161no—
350C22—1582no—
351C22—1582no—
352C22—1582no—
353C22#102×D7911yescharacteristic
354C22#1909869811yescharacteristic
355C22#1909887811yescharacteristic
356C22#1909899211yescharacteristic
357C22#1909899411yescharacteristic
358C22#1909900211yescharacteristic
359C22#1909900511yescharacteristic
360C4—82no—
361C4—82no—
362C4—6322no—
363C4—6322no—
364C4—1582no—
365C4#1909844111yescharacteristic
366C4#1909856311yescharacteristic
367C4#1909870311yescharacteristic
368C4#1909888311yescharacteristic
369C2—3162no—
370C2#2376371611yescharacteristic
371C2#2376510411yescharacteristic
372C2#2376511811yescharacteristic
373C2#2376534111yescharacteristic
374C2#2376534311yescharacteristic
375C2#2376561011yescharacteristic
376C2#2376561111yescharacteristic
subgroupG/H
centerC23D8×D79
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC79×#1380C2
socleC23×C79C2×D8
prime radicalC23×D79D8
cosocleC22×C8×C79C23
hypercenter#1380D79
hypocenterC79#12994
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
3. C2×#19098368C2semidirect
4. C2×#19098366C2semidirect
6. #23761491C2semidirect
7. C79×#1380C2semidirect
14. #19098805C4semidirect
195. #1380D79semidirect
196. C79#12994semidirect

Maximal subgroups

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

subgroupindex
C2×#190988052
C2×#190986002
C2×#190983682
C2×#190983662
C2×(C2×D79)⋊C162
#237614912
C79×#13802
#1299479