Finite Group Databasebeta

← #34205324

Subgroups — #34205324

Order 20224.

590 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579102C211yesmaximal, characteristic
2C2×D8×Dic79C211yesmaximal, characteristic, complemented
3C2×#19098365C211yesmaximal, characteristic, complemented
4#23764264C211yesmaximal, characteristic, complemented
5#23761491C211yesmaximal, characteristic, complemented
6#23761477C211yesmaximal, characteristic, complemented
7C2×#357×C79C211yesmaximal, characteristic, complemented
8C22×D632C2211yescharacteristic
9C2×D4×Dic79—21no—
10C2×#14579102—22no—
11C2×#14579102—22no—
12C2×#14579102C414yescomplemented
13C22×C79⋊C16C2211yescharacteristic
14D8×Dic79—22no—
15D8×Dic79—22no—
16C2×#14579095C2211yescharacteristic
17C2×#14579082—21no—
18C2×#14579081—21no—
19C2×C8×Dic79C2211yescharacteristic
20#19098464—21no—
21#19098365—22no—
22C22×D8×C79C2211yescharacteristic
23C22×C16×C79C2211yescharacteristic
24C2×#107×C79C2211yescharacteristic
25#357×C79—22no—
26C22×D316—21no—
27C23×Dic79—41no—
28C2×D632—21no—
29C2×D632—21no—
30C2×D632—21no—
31C2×D632—21no—
32C2×D632—21no—
33C2×D632C2×C412yescomplemented
34C22×C79⋊C8D411yescharacteristic
35D4×Dic79—24no—
36D4×Dic79—42no—
37C2×C4⋊Dic79—21no—
38C2×C22⋊Dic79—41no—
39C2×(C2×D79)⋊C4—41no—
40C2×(C2×D79)⋊C4—41no—
41C2×C4×Dic79D411yescharacteristic
42#14579102—28no—
43C2×C79⋊C16—21no—
44C2×C79⋊C16—21no—
45C2×C79⋊C16C2×C412yescomplemented
46#14579095—21no—
47#14579095D412yes—
48#14579082—24no—
49#14579081—42no—
50C8×Dic79—21no—
51C8×Dic79—21no—
52C22×D4×C79—21no—
53C2×D8×C79—21no—
54C2×D8×C79—21no—
55C2×D8×C79—21no—
56C2×D8×C79C2×C412yes—
57C2×D8×C79D412yes—
58C2×D8×C79D412yes—
59C22×C8×C79C2311yescharacteristic
60C2×C4⋊C4×C79—21no—
61C2×C16×C79—21no—
62C2×C16×C79—21no—
63C2×C16×C79—21no—
64#107×C79—21no—
65#107×C79—21no—
66C23×D79—41no—
67C2×D316—22no—
68C2×D316—41no—
69C2×D316—41no—
70C2×D316—41no—
71C2×D316—41no—
72C2×D316—41no—
73C22×Dic79—44no—
74C22×Dic79—41no—
75C22×Dic79—82no—
76C22×Dic79—82no—
77C22×Dic79D811yescharacteristic
78C22×Dic79D811yescharacteristic
79D632—22no—
80D632—22no—
81D632—41no—
82D632—41no—
83C2×C79⋊C8—21no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—21no—
86C4⋊Dic79—22no—
87C4⋊Dic79—41no—
88C22⋊Dic79—44no—
89(C2×D79)⋊C4—82no—
90(C2×D79)⋊C4—82no—
91C4×Dic79—21no—
92C4×Dic79—21no—
93C79⋊C16—22no—
94C24×C79—41no—
95C2×D4×C79—22no—
96C2×D4×C79—22no—
97C2×D4×C79—22no—
98C2×D4×C79—41no—
99C2×D4×C79—41no—
100C2×D4×C79—41no—
101C22×C4×C79—41no—
102C22×C4×C79C2×D411yescharacteristic
103D8×C79—22no—
104D8×C79—22no—
105D8×C79—22no—
106D8×C79C22⋊C414yescomplemented
107C2×C8×C79C22×C411yescharacteristic
108C2×C8×C79C2×D411yescharacteristic
109C2×C8×C79C2×D411yescharacteristic
110C2×C8×C79C2×D411yescharacteristic
111C2×C8×C79#4111yescharacteristic
112C2×C8×C79#4111yescharacteristic
113C4⋊C4×C79—41no—
114C4⋊C4×C79—41no—
115C16×C79—41no—
116C23×C79—42no—
117C23×C79—42no—
118C23×C79—42no—
119C23×C79—81no—
120C23×C79—81no—
121C23×C79—81no—
122C23×C79—81no—
123C23×C79C2×D811yescharacteristic
124C22×D79—42no—
125C22×D79—81no—
126C22×D79—81no—
127C22×D79—81no—
128C22×D79—81no—
129C22×D79—81no—
130C22×D79—81no—
131D4×C79—24no—
132D4×C79—42no—
133D4×C79—42no—
134D4×C79—42no—
135C2×C4×C79—81no—
136C2×C4×C79—81no—
137C2×C4×C79—81no—
138C2×C4×C79C4×D411yescharacteristic
139C2×C4×C79#12011yescharacteristic
140C2×C4×C79#12111yescharacteristic
141C2×C4×C79#13511yescharacteristic
142C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
144C2×Dic79—21no—
145C2×Dic79—21no—
146C2×Dic79—21no—
147C2×Dic79—21no—
148C2×Dic79—21no—
149C2×Dic79—42no—
150C2×Dic79—88no—
151C2×Dic79—81no—
152C2×Dic79—81no—
153C2×Dic79D1612yescomplemented
154D316—42no—
155D316—42no—
156D316—81no—
157D316—81no—
158C8×C79C2×C22⋊C411yescharacteristic
159C8×C79C4×D411yescharacteristic
160C8×C79#12111yescharacteristic
161C8×C79#12311yescharacteristic
162C79⋊C8—41no—
163C22×C79—44no—
164C22×C79—82no—
165C22×C79—82no—
166C22×C79—82no—
167C22×C79—82no—
168C22×C79—82no—
169C22×C79—82no—
170C22×C79C4×D811yescharacteristic
171C22×C79#44711yescharacteristic
172C22×C79#45911yescharacteristic
173C22×C79C2×D1611yescharacteristic
174C22×C79#50811yescharacteristic
175C22×C79#50911yescharacteristic
176C22×C79#50911yescharacteristic
177C2×D79—82no—
178C2×D79—82no—
179C2×D79—82no—
180C2×D79—161no—
181C2×D79—161no—
182C2×D79—161no—
183C2×D79—161no—
184C4×C79—161no—
185C4×C79#38611yescharacteristic
186C4×C79#44211yescharacteristic
187C4×C79#44911yescharacteristic
188C4×C79#46311yescharacteristic
189Dic79—22no—
190Dic79—41no—
191Dic79—82no—
192#13012—791nomaximal, Sylow
193C2×C79—84no—
194C2×C79#187911yescharacteristic
195C2×C79C4×D1611yescharacteristic
196C2×C79#217711yescharacteristic
197C2×C79#218411yescharacteristic
198C2×C79#218611yescharacteristic
199C2×C79#220611yescharacteristic
200C2×C79#221011yescharacteristic
201D79—162no—
202C22×D16—791no—
203C2×C4×D8—791no—
204C2×#357—791no—
205C2×#357D7911yescharacteristic, complemented
206#1970—791no—
207#1380—791no—
208#1374—791no—
209C79#1301211yescharacteristic, complemented, Sylow
210C22×D8—791no—
211C22×D8C2×D7911yescharacteristic
212C2×C4×D4—1581no—
213C2×D16—794no—
214C2×D16—1582no—
215C2×D16—1582no—
216C22×C16—791no—
217C22×C16C2×D7911yescharacteristic
218C4×D8—1582no—
219C4×D8—1582no—
220C2×#107—791no—
221C2×#107C2×D7911yescharacteristic
222C2×#102—1581no—
223C2×#102—1581no—
224C2×C4×C8—791no—
225#390—1581no—
226#357—22no—
227#357—1582no—
228C22×D4—21no—
229C22×D4—1581no—
230C23×C4—3161no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—792no—
235C2×D8—1581no—
236C2×D8—1581no—
237C2×D8—1581no—
238C2×D8—1581no—
239C2×D8—1581no—
240C2×D8C4×D7912yes—
241C2×D8D31612yes—
242C2×D8C22⋊D7912yes—
243C22×C8—791no—
244C22×C8C22×D7911yescharacteristic
245C4×D4—3162no—
246C4×D4—1584no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—1581no—
249C2×C22⋊C4—3161no—
250C2×C22⋊C4—3161no—
251C2×C22⋊C4—3161no—
252C2×C42—791no—
253D16—1588no—
254C2×C16—21no—
255C2×C16—21no—
256C2×C16—21no—
257C2×C16—792no—
258C2×C16—1581no—
259C2×C16—1581no—
260#107—21no—
261#107—21no—
262#107—792no—
263#107—1581no—
264#102—3162no—
265#102—1584no—
266C4×C8—1581no—
267C4×C8—1581no—
268C24—41no—
269C24—3161no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—3161no—
277C2×D4—3161no—
278C2×D4—3161no—
279C2×D4—3161no—
280C2×D4—3161no—
281C2×D4—1582no—
282C22×C4—41no—
283C22×C4—3161no—
284C22×C4—3164no—
285C22×C4—791no—
286C22×C4—791no—
287C22×C4—6322no—
288C22×C4—6322no—
289C22×C4D4×D7911yescharacteristic
290D8—22no—
291D8—22no—
292D8—22no—
293D8—3161no—
294D8—3161no—
295D8—1582no—
296D8—1582no—
297D8(C2×D79)⋊C414yescomplemented
298C2×C8—1581no—
299C2×C8—1581no—
300C2×C8—1581no—
301C2×C8C2×C4×D7911yescharacteristic
302C2×C8D4×D7911yescharacteristic
303C2×C8C2×C22⋊D7911yescharacteristic
304C2×C8C2×D31611yescharacteristic
305C2×C8#1192962311yescharacteristic
306C2×C8#1192962311yescharacteristic
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—3161no—
310C4⋊C4—1582no—
311C22⋊C4—3164no—
312C22⋊C4—6322no—
313C22⋊C4—6322no—
314C42—1581no—
315C42—1581no—
316C16—41no—
317C16—1582no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—3162no—
326C23—6321no—
327C23—6321no—
328C23—6321no—
329C23—6321no—
330C23—6321no—
331C23—6321no—
332C23D8×D7911yescharacteristic
333D4—24no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—3162no—
338D4—3162no—
339D4—6321no—
340D4—6321no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—3162no—
345C2×C4—792no—
346C2×C4—6321no—
347C2×C4—6321no—
348C2×C4—6328no—
349C2×C4—1581no—
350C2×C4—1581no—
351C2×C4—1581no—
352C2×C4—1581no—
353C2×C4—1581no—
354C2×C4(C22⋊D79)⋊C411yescharacteristic
355C2×C4#1457913111yescharacteristic
356C2×C4#1457913311yescharacteristic
357C2×C4#1457918211yescharacteristic
358C2×C4#1457918511yescharacteristic
359C2×C4#1457918811yescharacteristic
360C8—3161no—
361C8C2×(C2×D79)⋊C411yescharacteristic
362C8(C22⋊D79)⋊C411yescharacteristic
363C8#1457913311yescharacteristic
364C8#1457914511yescharacteristic
365C22—44no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—6322no—
373C22—6322no—
374C22—6322no—
375C22—12641no—
376C22—12641no—
377C22—12641no—
378C22—12641no—
379C22#1909864311yescharacteristic
380C22#1909869811yescharacteristic
381C22#1909875611yescharacteristic
382C22D16×D7911yescharacteristic
383C22#1909899411yescharacteristic
384C22#1909899711yescharacteristic
385C22#1909900011yescharacteristic
386C4—161no—
387C4—3161no—
388C4—6322no—
389C4—1582no—
390C4#1909844011yescharacteristic
391C4#1909867211yescharacteristic
392C4#1909870911yescharacteristic
393C4#1909878011yescharacteristic
394C2—84no—
395C2—12642no—
396C2#2376374611yescharacteristic
397C2#2376524411yescharacteristic
398C2#2376527311yescharacteristic
399C2#2376531311yescharacteristic
400C2#2376532411yescharacteristic
401C2#2376543311yescharacteristic
402C2#2376545711yescharacteristic
subgroupG/H
centerC23D8×D79
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC2×#357×C79C2
socleC23×C79C2×D8
prime radicalC22×#14579102C2
cosocleC22×C8×C79C23
hypercenterC2×#357D79
hypocenterC79#13012
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×D8×Dic79C2semidirect
3. C2×#19098365C2semidirect
4. #23764264C2semidirect
5. #23761491C2semidirect
6. #23761477C2semidirect
7. C2×#357×C79C2semidirect
12. C2×#14579102C4semidirect
33. C2×D632C2×C4semidirect
45. C2×C79⋊C16C2×C4semidirect
106. D8×C79C22⋊C4semidirect
153. C2×Dic79D16semidirect
205. C2×#357D79semidirect
209. C79#13012semidirect
297. D8(C2×D79)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#145791022
C2×D8×Dic792
C2×#190983652
#237642642
#237614912
#237614772
C2×#357×C792
#1301279