Finite Group Databasebeta

← #34205356

Subgroups — #34205356

Order 20224.

462 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772317C211yesmaximal, characteristic, complemented
2#23765136C211yesmaximal, characteristic, complemented
3#23765128C211yesmaximal, characteristic, complemented
4#23764299C211yesmaximal, characteristic, complemented
5#23761496C211yesmaximal, characteristic, complemented
6#23761488C211yesmaximal, characteristic, complemented
7C2×#508×C79C211yesmaximal, characteristic
8#19099330—21no—
9C2×#14579101C2211yescharacteristic
10#19098685—22no—
11#19098680—22no—
12C2×#14579097C2211yescharacteristic
13#19098601C2211yescharacteristic
14#19098580—21no—
15#19098580—21no—
16#19098537—21no—
17#19098518C2211yescharacteristic
18#19098373—22no—
19#19098370—22no—
20C2×#135×C79C2211yescharacteristic
21C2×#135×C79C2211yescharacteristic
22#508×C79—22no—
23#508×C79—22no—
24C2×Q32×C79C411yescharacteristic, complemented
25C2×SD32×C79C411yescharacteristic, complemented
26C2×SD32×C79C411yescharacteristic, complemented
27C2×D16×C79C411yescharacteristic, complemented
28C22×C16×C79C2211yescharacteristic
29#14579168—24no—
30#14579168—42no—
31C2×#11929608—21no—
32C2×#11929608—42no—
33C2×#11929608D411yescharacteristic
34#14579120—21no—
35#14579120D411yescharacteristic
36#14579101—21no—
37#14579101—21no—
38C2×C79⋊C16—21no—
39#14579097—21no—
40#14579097D412yes—
41#14579095—21no—
42#14579093—21no—
43#14579088—24no—
44#14579088—42no—
45#14579076—42no—
46#14579071—21no—
47C8×Dic79—21no—
48C2×#41×C79—21no—
49C2×#41×C79—21no—
50#135×C79—21no—
51#135×C79—21no—
52#135×C79—21no—
53#135×C79—21no—
54#135×C79—21no—
55#135×C79—21no—
56#135×C79D412yes—
57#135×C79D412yes—
58C2×Q16×C79C2×C411yescharacteristic
59C2×Q16×C79C2×C411yescharacteristic
60C2×SD16×C79—21no—
61C2×SD16×C79—21no—
62C2×D8×C79C2×C411yescharacteristic
63C2×D8×C79C2×C411yescharacteristic
64C22×C8×C79C2311yescharacteristic
65Q32×C79—22no—
66SD32×C79—22no—
67SD32×C79—22no—
68D16×C79—22no—
69C2×C16×C79—21no—
70C2×C16×C79—21no—
71C2×C16×C79C2×C411yescharacteristic
72C2×C16×C79C2×C411yescharacteristic
73#11929608—22no—
74#11929608—21no—
75#11929608—44no—
76#11929608—41no—
77#11929608D812yes—
78C2×C79⋊C8—21no—
79C2×C79⋊C8—44no—
80C2×C79⋊C8—41no—
81C4⋊Dic79—21no—
82C4⋊Dic79—41no—
83C22⋊Dic79—21no—
84C22⋊Dic79—41no—
85C4×Dic79—21no—
86C4×Dic79—41no—
87C79⋊C16—41no—
88#41×C79—22no—
89#41×C79—22no—
90#41×C79—41no—
91#41×C79—41no—
92#41×C79—41no—
93#41×C79—41no—
94#41×C79—41no—
95#41×C79—41no—
96C2×Q8×C79—21no—
97C2×Q8×C79—21no—
98C2×D4×C79—21no—
99C2×D4×C79—21no—
100C2×D4×C79—41no—
101C2×D4×C79—41no—
102C22×C4×C79—41no—
103C22×C4×C79—41no—
104C22×C4×C79C2×D411yescharacteristic
105Q16×C79—21no—
106Q16×C79—21no—
107Q16×C79—21no—
108Q16×C79C22⋊C412yes—
109SD16×C79—22no—
110SD16×C79—22no—
111SD16×C79—41no—
112SD16×C79—41no—
113D8×C79—21no—
114D8×C79—21no—
115D8×C79—21no—
116D8×C79C22⋊C412yes—
117C2×C8×C79C22×C411yescharacteristic
118C2×C8×C79C2×D411yescharacteristic
119C2×C8×C79C2×D411yescharacteristic
120C2×C8×C79C2×D411yescharacteristic
121C2×C8×C79#4111yescharacteristic
122C2×C8×C79#4111yescharacteristic
123C16×C79—21no—
124C16×C79—21no—
125C23×C79—41no—
126C23×C79—41no—
127C23×C79AGL(1,ℤ/8)11yescharacteristic
128Q8×C79—22no—
129Q8×C79—41no—
130Q8×C79—41no—
131Q8×C79—41no—
132D4×C79—22no—
133D4×C79—42no—
134D4×C79—42no—
135D4×C79—41no—
136D4×C79—41no—
137D4×C79—41no—
138D4×C79—81no—
139D4×C79—81no—
140C2×C4×C79—41no—
141C2×C4×C79—42no—
142C2×C4×C79—42no—
143C2×C4×C79—41no—
144C2×C4×C79—81no—
145C2×C4×C79—81no—
146C2×C4×C79C4×D411yescharacteristic
147C2×C4×C79#12011yescharacteristic
148C2×C4×C79#12111yescharacteristic
149C2×C4×C79C2×D811yescharacteristic
150C2×C4×C79#13511yescharacteristic
151C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
152C2×Dic79—41no—
153C2×Dic79—81no—
154C8×C79C2×C22⋊C411yescharacteristic
155C8×C79C4×D411yescharacteristic
156C8×C79#12111yescharacteristic
157C8×C79#12311yescharacteristic
158C79⋊C8—22no—
159C79⋊C8—42no—
160C22×C79—41no—
161C22×C79—42no—
162C22×C79—81no—
163C22×C79—161no—
164C22×C79—161no—
165C22×C79#44211yescharacteristic
166C22×C79#44911yescharacteristic
167C22×C79#45911yescharacteristic
168C4×C79—42no—
169C4×C79—81no—
170C4×C79#38611yescharacteristic
171C4×C79C4×D811yescharacteristic
172C4×C79#44711yescharacteristic
173C4×C79#46311yescharacteristic
174Dic79—81no—
175Dic79—161no—
176#13016—791nomaximal, Sylow
177C2×C79—21no—
178C2×C79—21no—
179C2×C79—21no—
180C2×C79—82no—
181C2×C79—161no—
182C2×C79#187911yescharacteristic
183C2×#508D7911yescharacteristic
184#2957—791no—
185#2145—791no—
186#2143—791no—
187#1976—791no—
188#1382—791no—
189#1379—791no—
190C79#1301611yescharacteristic, complemented, Sylow
191C2×#135C2×D7911yescharacteristic
192C2×#135C2×D7911yescharacteristic
193#568—1581no—
194#508—22no—
195#508—22no—
196C2×Q32Dic7911yescharacteristic, complemented
197C2×SD32Dic7911yescharacteristic, complemented
198C2×SD32Dic7911yescharacteristic, complemented
199C2×D16Dic7911yescharacteristic, complemented
200C2×M32—791no—
201C22×C16C2×D7911yescharacteristic
202#444—1582no—
203#443—1582no—
204C2×#108—791no—
205#427—791no—
206#421—1581no—
207#421—1581no—
208#410—1581no—
209#405—791no—
210#360—1582no—
211#359—1582no—
212C2×#41—21no—
213C2×#41—21no—
214#135—21no—
215#135—21no—
216#135—21no—
217#135—21no—
218#135—21no—
219#135—21no—
220#135C22⋊D7912yes—
221#135C22⋊D7912yes—
222C2×Q16C2×Dic7911yescharacteristic
223C2×Q16C2×Dic7911yescharacteristic
224C2×SD16—21no—
225C2×SD16—21no—
226C2×D8C2×Dic7911yescharacteristic
227C2×D8C2×Dic7911yescharacteristic
228#131—3162no—
229#131—1584no—
230C2×M16—3162no—
231C2×M16—791no—
232C2×M16—1581no—
233C22×C8C22×D7911yescharacteristic
234#117—791no—
235#117—1581no—
236Q32—22no—
237SD32—22no—
238SD32—22no—
239D16—22no—
240M32—1581no—
241M32—1581no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—1581no—
245C2×C16C2×Dic7911yescharacteristic
246C2×C16C2×Dic7911yescharacteristic
247#108—792no—
248#108—1581no—
249#107—1581no—
250#106—1581no—
251#104—3162no—
252#104—1584no—
253#99—3162no—
254#97—1581no—
255C4×C8—1581no—
256#41—22no—
257#41—22no—
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4D4×D7911yescharacteristic
273Q16—21no—
274Q16—21no—
275Q16—21no—
276Q16C22⋊Dic7912yes—
277SD16—22no—
278SD16—22no—
279SD16—41no—
280SD16—41no—
281D8—21no—
282D8—21no—
283D8—21no—
284D8C22⋊Dic7912yes—
285M16—3161no—
286M16—3164no—
287M16—792no—
288M16—1582no—
289M16—1581no—
290C2×C8—3161no—
291C2×C8—3164no—
292C2×C8—1581no—
293C2×C8C22×Dic7911yescharacteristic
294C2×C8D4×D7911yescharacteristic
295C2×C8C2×C22⋊D7911yescharacteristic
296C2×C8C2×C22⋊D7911yescharacteristic
297C2×C8#1192962311yescharacteristic
298C2×C8#1192962311yescharacteristic
299C4⋊C4—3161no—
300C4⋊C4—1581no—
301C22⋊C4—3161no—
302C22⋊C4—1581no—
303C42—3161no—
304C42—1581no—
305C16—21no—
306C16—21no—
307C16—3161no—
308C23—41no—
309C23—41no—
310C23#1457918511yescharacteristic
311Q8—22no—
312Q8—41no—
313Q8—41no—
314Q8—41no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—81no—
322D4—81no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—3161no—
330C2×C4—6321no—
331C2×C4D4×Dic7911yescharacteristic
332C2×C4#1457912911yescharacteristic
333C2×C4#1457913311yescharacteristic
334C2×C4D8×D7911yescharacteristic
335C2×C4#1457918011yescharacteristic
336C2×C4#1457918511yescharacteristic
337C8—3162no—
338C8—1582no—
339C8C2×C22⋊Dic7911yescharacteristic
340C8D4×Dic7911yescharacteristic
341C8#1457913311yescharacteristic
342C8#1457914211yescharacteristic
343C22—41no—
344C22—42no—
345C22—81no—
346C22—161no—
347C22—161no—
348C22#1909867511yescharacteristic
349C22#1909870511yescharacteristic
350C22#1909875611yescharacteristic
351C4—42no—
352C4—81no—
353C4—6321no—
354C4—12641no—
355C4#1909844311yescharacteristic
356C4D8×Dic7911yescharacteristic
357C4#1909869511yescharacteristic
358C4#1909877611yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—82no—
363C2—161no—
364C2#2376375011yescharacteristic
subgroupG/H
centerC4#19098695
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
Fitting, prime radicalC2×#508×C79C2
socleC2×C79#1879
cosocleC22×C8×C79C23
hypercenterC2×#508D79
hypocenterC79#13016
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#508

Complements

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

NH ≅ G/Ntype
1. #23772317C2semidirect
2. #23765136C2semidirect
3. #23765128C2semidirect
4. #23764299C2semidirect
5. #23761496C2semidirect
6. #23761488C2semidirect
24. C2×Q32×C79C4semidirect
25. C2×SD32×C79C4semidirect
26. C2×SD32×C79C4semidirect
27. C2×D16×C79C4semidirect
190. C79#13016semidirect
196. C2×Q32Dic79semidirect
197. C2×SD32Dic79semidirect
198. C2×SD32Dic79semidirect
199. C2×D16Dic79semidirect

Maximal subgroups

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

subgroupindex
#237723172
#237651362
#237651282
#237642992
#237614962
#237614882
C2×#508×C792
#1301679