Finite Group Databasebeta

← #34205398

Subgroups — #34205398

Order 20224.

494 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
1#23772276C211yesmaximal, characteristic, complemented
2C2×#19098370C211yesmaximal, characteristic, complemented
3C2×#19098364C211yesmaximal, characteristic, complemented
4#23764311C211yesmaximal, characteristic, complemented
5#23761509C211yesmaximal, characteristic, complemented
6#23761490C211yesmaximal, characteristic, complemented
7C2×#509×C79C211yesmaximal, characteristic
8#19099046—21no—
9C22×C79⋊C16C2211yescharacteristic
10#19098664—22no—
11Q16×Dic79—21no—
12D8×Dic79—21no—
13C2×#14579097C2211yescharacteristic
14C2×#14579095C2211yescharacteristic
15#19098568—21no—
16#19098564—21no—
17C2×#14579078—21no—
18#19098518C2211yescharacteristic
19#19098370—22no—
20#19098364—22no—
21C2×#135×C79C2211yescharacteristic
22C22×D8×C79C2211yescharacteristic
23#509×C79—22no—
24#509×C79C414yescomplemented
25C2×SD32×C79—21no—
26C2×D16×C79—21no—
27C2×M32×C79C2211yescharacteristic
28C2×#11929608—21no—
29C2×#11929608D411yescharacteristic
30Q8×Dic79—22no—
31D4×Dic79—22no—
32D4×Dic79—42no—
33#14579120—41no—
34#14579120D411yescharacteristic
35C2×C4⋊Dic79—21no—
36C2×C4⋊Dic79—41no—
37C2×C79⋊C16—21no—
38C2×C79⋊C16—21no—
39C2×C79⋊C16—21no—
40#14579097—21no—
41#14579097—21no—
42#14579095—21no—
43#14579095D412yes—
44#14579085—22no—
45#14579082—22no—
46#14579082—41no—
47#14579082—41no—
48#14579078—42no—
49#14579071—21no—
50C8×Dic79—21no—
51C2×#41×C79—21no—
52C22×D4×C79—21no—
53#135×C79—21no—
54#135×C79—21no—
55#135×C79C2×C412yes—
56#135×C79D412yes—
57C2×Q16×C79D411yescharacteristic
58C2×SD16×C79—21no—
59C2×D8×C79—21no—
60C2×D8×C79—21no—
61C2×D8×C79—21no—
62C2×D8×C79—21no—
63C2×D8×C79—21no—
64C2×D8×C79C2×C412yescomplemented
65C2×D8×C79D411yescharacteristic
66C22×C8×C79C2311yescharacteristic
67SD32×C79—24no—
68D16×C79—24no—
69M32×C79—21no—
70M32×C79C2×C412yescomplemented
71C2×C16×C79—21no—
72C22×Dic79—42no—
73C22×Dic79—41no—
74#11929608—21no—
75#11929608—21no—
76#11929608—41no—
77#11929608—41no—
78C2×C79⋊C8—21no—
79C2×C79⋊C8—41no—
80C4⋊Dic79—22no—
81C4⋊Dic79—42no—
82C4⋊Dic79—41no—
83C4⋊Dic79D811yescharacteristic
84C4⋊Dic79D811yescharacteristic
85C22⋊Dic79—21no—
86C22⋊Dic79—42no—
87C4×Dic79—21no—
88C4×Dic79—42no—
89C79⋊C16—41no—
90C24×C79—41no—
91#41×C79—22no—
92#41×C79—22no—
93#41×C79—41no—
94#41×C79—41no—
95C2×Q8×C79—21no—
96C2×D4×C79—22no—
97C2×D4×C79—21no—
98C2×D4×C79—41no—
99C2×D4×C79—41no—
100C2×D4×C79—41no—
101C2×D4×C79—41no—
102C2×D4×C79—41no—
103C2×D4×C79—41no—
104C22×C4×C79—41no—
105C22×C4×C79C2×D411yescharacteristic
106Q16×C79—21no—
107Q16×C79C22⋊C412yes—
108SD16×C79—22no—
109SD16×C79—22no—
110D8×C79—21no—
111D8×C79—22no—
112D8×C79—22no—
113D8×C79—41no—
114D8×C79—41no—
115D8×C79C22⋊C412yes—
116C2×C8×C79C22×C411yescharacteristic
117C2×C8×C79C2×D411yescharacteristic
118C2×C8×C79C2×D411yescharacteristic
119C2×C8×C79C2×D411yescharacteristic
120C2×C8×C79#4111yescharacteristic
121C2×C8×C79#4111yescharacteristic
122C16×C79—22no—
123C23×C79—41no—
124C23×C79—42no—
125C23×C79—81no—
126C23×C79—81no—
127C23×C79—161no—
128C23×C79—161no—
129C23×C79AGL(1,ℤ/8)11yescharacteristic
130Q8×C79—22no—
131Q8×C79—41no—
132D4×C79—22no—
133D4×C79—42no—
134D4×C79—42no—
135D4×C79—42no—
136D4×C79—42no—
137D4×C79—41no—
138D4×C79—81no—
139D4×C79—81no—
140C2×C4×C79—41no—
141C2×C4×C79—42no—
142C2×C4×C79—42no—
143C2×C4×C79C4×D411yescharacteristic
144C2×C4×C79#12011yescharacteristic
145C2×C4×C79#12111yescharacteristic
146C2×C4×C79C2×D811yescharacteristic
147C2×C4×C79#13511yescharacteristic
148C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
149C2×Dic79—21no—
150C2×Dic79—21no—
151C2×Dic79—42no—
152C2×Dic79—162no—
153C2×Dic79—161no—
154C8×C79C2×C22⋊C411yescharacteristic
155C8×C79C4×D411yescharacteristic
156C8×C79#12111yescharacteristic
157C8×C79#12311yescharacteristic
158C79⋊C8—41no—
159C79⋊C8—81no—
160C22×C79—21no—
161C22×C79—21no—
162C22×C79—42no—
163C22×C79—81no—
164C22×C79—81no—
165C22×C79—82no—
166C22×C79—162no—
167C22×C79—162no—
168C22×C79—161no—
169C22×C79—161no—
170C22×C79#44211yescharacteristic
171C22×C79#44711yescharacteristic
172C22×C79#46311yescharacteristic
173C4×C79—42no—
174C4×C79#38611yescharacteristic
175C4×C79C4×D811yescharacteristic
176C4×C79#44911yescharacteristic
177C4×C79#45911yescharacteristic
178Dic79—41no—
179Dic79—41no—
180Dic79—82no—
181#13022—791nomaximal, Sylow
182C2×C79—21no—
183C2×C79—21no—
184C2×C79—82no—
185C2×C79—162no—
186C2×C79#187911yescharacteristic
187C2×C79#219211yescharacteristic
188C2×C79#219311yescharacteristic
189C2×#509D7911yescharacteristic
190#2951—791no—
191C2×#359—791no—
192C2×#357—791no—
193#1978—791no—
194#1387—791no—
195#1379—791no—
196C79#1302211yescharacteristic, complemented, Sylow
197C2×#135C2×D7911yescharacteristic
198C22×D8C2×D7911yescharacteristic
199#520—1581no—
200#509—22no—
201#509Dic7914yescomplemented
202C2×SD32—21no—
203C2×D16—21no—
204C2×M32C2×D7911yescharacteristic
205C22×C16—791no—
206#440—1582no—
207C4×Q16—1581no—
208C4×D8—1581no—
209C2×#108—791no—
210C2×#107—791no—
211#418—1581no—
212#417—1581no—
213C2×#100—1581no—
214#405—791no—
215#359—1582no—
216#357—1582no—
217C2×#41—21no—
218C22×D4—21no—
219#135—21no—
220#135—21no—
221#135C2×Dic7912yes—
222#135C22⋊D7912yes—
223C2×Q16C22⋊D7911yescharacteristic
224C2×SD16—21no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—21no—
229C2×D8—21no—
230C2×D8C2×Dic7912yescomplemented
231C2×D8C22⋊D7911yescharacteristic
232C2×M16—791no—
233C2×M16—1581no—
234C22×C8C22×D7911yescharacteristic
235C4×Q8—1582no—
236C4×D4—3162no—
237C4×D4—1582no—
238#117—3161no—
239#117—791no—
240C2×C4⋊C4—3161no—
241C2×C4⋊C4—1581no—
242SD32—24no—
243D16—24no—
244M32—21no—
245M32C2×Dic7912yescomplemented
246C2×C16—21no—
247C2×C16—1581no—
248C2×C16—1581no—
249C2×C16—1581no—
250#108—1581no—
251#108—1581no—
252#107—792no—
253#107—1581no—
254#103—1582no—
255#102—3161no—
256#102—3161no—
257#102—1582no—
258#100—3162no—
259#97—1581no—
260C4×C8—1581no—
261C24—41no—
262#41—22no—
263#41—22no—
264#41—41no—
265#41—41no—
266C2×Q8—21no—
267C2×D4—22no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C22×C4—41no—
276C22×C4—3161no—
277C22×C4—3162no—
278C22×C4D4×D7911yescharacteristic
279Q16—21no—
280Q16C22⋊Dic7912yes—
281SD16—22no—
282SD16—22no—
283D8—22no—
284D8—22no—
285D8—21no—
286D8—41no—
287D8—41no—
288D8C22⋊Dic7912yes—
289M16—3161no—
290M16—3161no—
291M16—1581no—
292M16—1581no—
293C2×C8—3161no—
294C2×C8—1581no—
295C2×C8C22×Dic7911yescharacteristic
296C2×C8D4×D7911yescharacteristic
297C2×C8C2×C22⋊D7911yescharacteristic
298C2×C8C2×C22⋊D7911yescharacteristic
299C2×C8#1192962311yescharacteristic
300C2×C8#1192962311yescharacteristic
301C4⋊C4—3162no—
302C4⋊C4—3161no—
303C4⋊C4—791no—
304C4⋊C4—791no—
305C4⋊C4—1582no—
306C22⋊C4—3162no—
307C22⋊C4—1581no—
308C42—3162no—
309C42—1581no—
310C16—22no—
311C16—3161no—
312C23—41no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—161no—
317C23—161no—
318C23#1457918511yescharacteristic
319Q8—22no—
320Q8—41no—
321D4—22no—
322D4—42no—
323D4—42no—
324D4—42no—
325D4—42no—
326D4—41no—
327D4—81no—
328D4—81no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—3162no—
333C2×C4—1581no—
334C2×C4—1581no—
335C2×C4—12642no—
336C2×C4—12641no—
337C2×C4D4×Dic7911yescharacteristic
338C2×C4#1457912911yescharacteristic
339C2×C4#1457913311yescharacteristic
340C2×C4D8×D7911yescharacteristic
341C2×C4#1457918011yescharacteristic
342C2×C4#1457918511yescharacteristic
343C8—3161no—
344C8—6321no—
345C8C2×C22⋊Dic7911yescharacteristic
346C8D4×Dic7911yescharacteristic
347C8#1457913311yescharacteristic
348C8#1457914211yescharacteristic
349C22—21no—
350C22—21no—
351C22—42no—
352C22—81no—
353C22—81no—
354C22—82no—
355C22—162no—
356C22—162no—
357C22—161no—
358C22—161no—
359C22#1909867511yescharacteristic
360C22#1909869511yescharacteristic
361C22#1909877611yescharacteristic
362C4—42no—
363C4—3161no—
364C4—3161no—
365C4—6322no—
366C4#1909844311yescharacteristic
367C4D8×Dic7911yescharacteristic
368C4#1909870511yescharacteristic
369C4#1909875611yescharacteristic
370C2—21no—
371C2—21no—
372C2—82no—
373C2—162no—
374C2#2376375011yescharacteristic
375C2#2376535411yescharacteristic
376C2#2376536111yescharacteristic
subgroupG/H
centerC22#19098695
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
Fitting, prime radicalC2×#509×C79C2
socleC22×C79#447
cosocleC22×C8×C79C23
hypercenterC2×#509D79
hypocenterC79#13022
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. #23772276C2semidirect
2. C2×#19098370C2semidirect
3. C2×#19098364C2semidirect
4. #23764311C2semidirect
5. #23761509C2semidirect
6. #23761490C2semidirect
24. #509×C79C4semidirect
64. C2×D8×C79C2×C4semidirect
70. M32×C79C2×C4semidirect
196. C79#13022semidirect
201. #509Dic79semidirect
230. C2×D8C2×Dic79semidirect
245. M32C2×Dic79semidirect

Maximal subgroups

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

subgroupindex
#237722762
C2×#190983702
C2×#190983642
#237643112
#237615092
#237614902
C2×#509×C792
#1302279