Finite Group Databasebeta

← #34205396

Subgroups — #34205396

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
1C2×#19098998C211yesmaximal, characteristic
2C2×#19098370C211yesmaximal, characteristic, complemented
3C2×#19098363C211yesmaximal, characteristic, complemented
4#23764310C211yesmaximal, characteristic, complemented
5#23761508C211yesmaximal, characteristic, complemented
6#23761489C211yesmaximal, characteristic, complemented
7C79×#2951C211yesmaximal, characteristic, complemented
8C22×D632C2211yescharacteristic
9#19098998—22no—
10#19098998C414yescomplemented
11C2×#14579104—21no—
12C2×#14579102—21no—
13C2×#14579101C2211yescharacteristic
14C22×C79⋊C16C2211yescharacteristic
15C2×#14579097C2211yescharacteristic
16#19098567—21no—
17C2×#14579077—21no—
18#19098370—22no—
19#19098363—22no—
20C2×#135×C79C2211yescharacteristic
21#520×C79—21no—
22#440×C79—22no—
23C4×Q16×C79—21no—
24C4×D8×C79—21no—
25C2×#107×C79C2211yescharacteristic
26#417×C79—21no—
27#405×C79C2211yescharacteristic
28C22×D316—21no—
29C2×D632—21no—
30C2×D632—21no—
31C2×D632—21no—
32C2×D632—21no—
33C2×D632—21no—
34C2×D632C2×C412yescomplemented
35C2×#11929608—21no—
36#14579104—24no—
37#14579102—24no—
38#14579101—21no—
39#14579101C2×C412yescomplemented
40C2×C79⋊C16—21no—
41C2×C79⋊C16—21no—
42C2×C79⋊C16—21no—
43C2×C79⋊C16—21no—
44#14579097—21no—
45#14579097—21no—
46#14579081—41no—
47#14579081—41no—
48#14579077—42no—
49C2×#41×C79—21no—
50#135×C79—21no—
51#135×C79—21no—
52#135×C79C2×C412yes—
53#135×C79D412yes—
54C2×Q16×C79D411yescharacteristic
55C2×SD16×C79—21no—
56C2×D8×C79D411yescharacteristic
57C2×M16×C79D411yescharacteristic
58C22×C8×C79C2311yescharacteristic
59C4×Q8×C79—22no—
60C4×D4×C79—22no—
61C4×D4×C79—42no—
62#117×C79—41no—
63#117×C79D411yescharacteristic
64C2×C4⋊C4×C79—21no—
65C2×C4⋊C4×C79—41no—
66#107×C79—21no—
67#107×C79D412yes—
68#103×C79—22no—
69#102×C79—22no—
70#97×C79—21no—
71C4×C8×C79—21no—
72C23×D79—41no—
73C2×D316—22no—
74C2×D316—41no—
75C2×D316—41no—
76C2×D316—41no—
77C2×D316—41no—
78C2×D316—41no—
79D632—22no—
80D632—22no—
81D632—41no—
82D632—41no—
83#11929608—41no—
84#11929608—41no—
85C2×C79⋊C8—41no—
86C79⋊C16—22no—
87C79⋊C16—41no—
88#41×C79—22no—
89#41×C79—22no—
90#41×C79—41no—
91#41×C79—41no—
92C2×Q8×C79—21no—
93C2×D4×C79—21no—
94C2×D4×C79—41no—
95C22×C4×C79—42no—
96C22×C4×C79—41no—
97C22×C4×C79—41no—
98C22×C4×C79C2×D411yescharacteristic
99Q16×C79—21no—
100Q16×C79C22⋊C412yes—
101SD16×C79—22no—
102SD16×C79—22no—
103D8×C79—21no—
104D8×C79C22⋊C412yes—
105M16×C79—21no—
106M16×C79—21no—
107C2×C8×C79—21no—
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—42no—
116C4⋊C4×C79—41no—
117C4⋊C4×C79D811yescharacteristic
118C4⋊C4×C79D811yescharacteristic
119C22⋊C4×C79—21no—
120C22⋊C4×C79—42no—
121C42×C79—21no—
122C42×C79—42no—
123C23×C79—41no—
124C23×C79AGL(1,ℤ/8)11yescharacteristic
125C22×D79—42no—
126C22×D79—81no—
127C22×D79—81no—
128C22×D79—161no—
129C22×D79—161no—
130Q8×C79—22no—
131Q8×C79—41no—
132D4×C79—22no—
133D4×C79—42no—
134D4×C79—42no—
135D4×C79—41no—
136C2×C4×C79—21no—
137C2×C4×C79—21no—
138C2×C4×C79—41no—
139C2×C4×C79—42no—
140C2×C4×C79—42no—
141C2×C4×C79—42no—
142C2×C4×C79—162no—
143C2×C4×C79—161no—
144C2×C4×C79C4×D411yescharacteristic
145C2×C4×C79#12011yescharacteristic
146C2×C4×C79#12111yescharacteristic
147C2×C4×C79C2×D811yescharacteristic
148C2×C4×C79#13511yescharacteristic
149C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
150D316—42no—
151D316—42no—
152D316—81no—
153D316—81no—
154C8×C79—41no—
155C8×C79C2×C22⋊C411yescharacteristic
156C8×C79C4×D411yescharacteristic
157C8×C79#12111yescharacteristic
158C8×C79#12311yescharacteristic
159C79⋊C8—81no—
160C22×C79—21no—
161C22×C79—21no—
162C22×C79—42no—
163C22×C79—81no—
164C22×C79—81no—
165C22×C79#44211yescharacteristic
166C22×C79#44711yescharacteristic
167C22×C79#46311yescharacteristic
168C2×D79—82no—
169C2×D79—162no—
170C2×D79—161no—
171C2×D79—161no—
172C2×D79—162no—
173C4×C79—42no—
174C4×C79—41no—
175C4×C79—41no—
176C4×C79—82no—
177C4×C79#38611yescharacteristic
178C4×C79C4×D811yescharacteristic
179C4×C79#44911yescharacteristic
180C4×C79#45911yescharacteristic
181#13022—791nomaximal, Sylow
182C2×C79—21no—
183C2×C79—21no—
184C2×C79—82no—
185C2×C79#187911yescharacteristic
186C2×C79#219211yescharacteristic
187C2×C79#219311yescharacteristic
188D79—162no—
189C2×#509—791no—
190#2951D7911yescharacteristic, complemented
191C2×#359—791no—
192C2×#357—791no—
193#1978—791no—
194#1387—791no—
195#1379—791no—
196C79#1302211yescharacteristic, complemented, Sylow
197C2×#135C2×D7911yescharacteristic
198C22×D8—791no—
199#520—21no—
200#509—794no—
201#509—1582no—
202C2×SD32—1581no—
203C2×D16—1581no—
204C2×M32—791no—
205C22×C16—791no—
206#440—22no—
207C4×Q16—21no—
208C4×D8—21no—
209C2×#108—791no—
210C2×#107C2×D7911yescharacteristic
211#418—1581no—
212#417—21no—
213C2×#100—1581no—
214#405C2×D7911yescharacteristic
215#359—1582no—
216#357—1582no—
217C2×#41—21no—
218C22×D4—1581no—
219#135—21no—
220#135—21no—
221#135C4×D7912yes—
222#135D31612yes—
223C2×Q16C22⋊D7911yescharacteristic
224C2×SD16—21no—
225C2×D8—792no—
226C2×D8—1581no—
227C2×D8—1581no—
228C2×D8—1581no—
229C2×D8—1581no—
230C2×D8—1581no—
231C2×D8C22⋊D7911yescharacteristic
232C2×M16—1581no—
233C2×M16C22⋊D7911yescharacteristic
234C22×C8C22×D7911yescharacteristic
235C4×Q8—22no—
236C4×D4—22no—
237C4×D4—42no—
238#117—41no—
239#117C22⋊D7911yescharacteristic
240C2×C4⋊C4—21no—
241C2×C4⋊C4—41no—
242SD32—1584no—
243D16—1584no—
244M32—792no—
245M32—1581no—
246C2×C16—1581no—
247C2×C16—1581no—
248C2×C16—1581no—
249C2×C16—1581no—
250#108—1581no—
251#108—1581no—
252#107—21no—
253#107C22⋊D7912yes—
254#103—22no—
255#102—22no—
256#102—3161no—
257#102—3161no—
258#100—3162no—
259#97—21no—
260C4×C8—21no—
261C24—3161no—
262#41—22no—
263#41—22no—
264#41—41no—
265#41—41no—
266C2×Q8—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—3161no—
270C2×D4—3161no—
271C2×D4—3161no—
272C2×D4—3161no—
273C2×D4—3161no—
274C2×D4—1582no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C2×C22⋊D7911yescharacteristic
279Q16—21no—
280Q16(C2×D79)⋊C412yes—
281SD16—22no—
282SD16—22no—
283D8—21no—
284D8—3161no—
285D8—3161no—
286D8—1582no—
287D8—1582no—
288D8(C2×D79)⋊C412yes—
289M16—21no—
290M16—21no—
291M16—3161no—
292M16—3161no—
293C2×C8—21no—
294C2×C8—3161no—
295C2×C8C2×C4×D7911yescharacteristic
296C2×C8C2×C22⋊D7911yescharacteristic
297C2×C8C2×C22⋊D7911yescharacteristic
298C2×C8C2×D31611yescharacteristic
299C2×C8#1192962411yescharacteristic
300C2×C8#1192962411yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4#1192960911yescharacteristic
305C4⋊C4#1192960911yescharacteristic
306C22⋊C4—21no—
307C22⋊C4—42no—
308C42—21no—
309C42—42no—
310C16—3161no—
311C16—1582no—
312C23—41no—
313C23—3162no—
314C23—6321no—
315C23—6321no—
316C23—12641no—
317C23—12641no—
318C23#1457918711yescharacteristic
319Q8—22no—
320Q8—41no—
321D4—22no—
322D4—42no—
323D4—42no—
324D4—41no—
325D4—3162no—
326D4—3162no—
327D4—6321no—
328D4—6321no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—162no—
336C2×C4—161no—
337C2×C4C4×C22⋊D7911yescharacteristic
338C2×C4#1457913011yescharacteristic
339C2×C4#1457913511yescharacteristic
340C2×C4C2×#1192960911yescharacteristic
341C2×C4#1457918111yescharacteristic
342C2×C4#1457918611yescharacteristic
343C8—41no—
344C8—6321no—
345C8C2×(C2×D79)⋊C411yescharacteristic
346C8C4×C22⋊D7911yescharacteristic
347C8#1457913511yescharacteristic
348C8#1457914411yescharacteristic
349C22—21no—
350C22—21no—
351C22—42no—
352C22—81no—
353C22—81no—
354C22—6322no—
355C22—12642no—
356C22—12642no—
357C22—12641no—
358C22—12641no—
359C22#1909867311yescharacteristic
360C22#1909869711yescharacteristic
361C22#1909877811yescharacteristic
362C4—42no—
363C4—41no—
364C4—41no—
365C4—82no—
366C4#1909844211yescharacteristic
367C4C4×#1192960911yescharacteristic
368C4#1909870811yescharacteristic
369C4#1909875811yescharacteristic
370C2—21no—
371C2—21no—
372C2—82no—
373C2—12642no—
374C2#2376374811yescharacteristic
375C2#2376535711yescharacteristic
376C2#2376536411yescharacteristic
subgroupG/H
centerC22#19098697
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC79×#2951C2
socleC22×C79#447
prime radicalC2×#19098998C2
cosocleC22×C8×C79C23
hypercenter#2951D79
hypocenterC79#13022
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
2. C2×#19098370C2semidirect
3. C2×#19098363C2semidirect
4. #23764310C2semidirect
5. #23761508C2semidirect
6. #23761489C2semidirect
7. C79×#2951C2semidirect
10. #19098998C4semidirect
34. C2×D632C2×C4semidirect
39. #14579101C2×C4semidirect
190. #2951D79semidirect
196. C79#13022semidirect

Maximal subgroups

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

subgroupindex
C2×#190989982
C2×#190983702
C2×#190983632
#237643102
#237615082
#237614892
C79×#29512
#1302279