Finite Group Databasebeta

← #28311596

Subgroups — #28311596

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291682C211yesmaximal, characteristic, complemented
2#20290957C211yesmaximal, characteristic, complemented
3#20290957C211yesmaximal, characteristic, complemented
4#20290951C211yesmaximal, characteristic, complemented
5#20289763C211yesmaximal, characteristic, complemented
6#20289757C211yesmaximal, characteristic, complemented
7C2×#16810745C211yesmaximal, characteristic
8#16811681—22no—
9#16811681—22no—
10C2×#13133291—22no—
11C2×#13133291C2211yescharacteristic
12C2×#13133291C2211yescharacteristic
13C2×#13133291C2211yescharacteristic
14C2×#13133291C2211yescharacteristic
15#16811521—22no—
16#16811521—22no—
17#16811519—22no—
18C2×#13133224C2211yescharacteristic
19C2×#13133224C2211yescharacteristic
20#16810745—22no—
21#16810745—22no—
22C2×#13133153C411yescharacteristic
23C2×#13133144C411yescharacteristic
24C2×#13133144C411yescharacteristic
25C2×#13133127C2211yescharacteristic
26C2×#135×C53C411yescharacteristic
27#13133291—24no—
28#13133291—21no—
29#13133291—21no—
30#13133291—21no—
31#13133291—21no—
32#13133291—21no—
33#13133291—21no—
34#13133291—21no—
35#13133291D412yes—
36C2×#11719910—24no—
37C2×#11719910—21no—
38C2×#11719910—21no—
39C2×#11719910—21no—
40C2×#11719910D412yes—
41#13133224—21no—
42#13133224—21no—
43#13133224—21no—
44#13133224—21no—
45#13133224—21no—
46#13133224—21no—
47#13133224D412yes—
48#13133224D412yes—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917C2×C411yescharacteristic
52C2×#11719917C2×C411yescharacteristic
53C22×#88322—21no—
54C22×#88322—21no—
55C22×#88322C2311yescharacteristic
56C2×C4×Dic53C2×C411yescharacteristic
57#13133153—22no—
58#13133144—22no—
59#13133144—22no—
60#13133127—21no—
61#13133127—21no—
62C2×#41×C53C2×C411yescharacteristic
63C2×#41×C53C2×C411yescharacteristic
64#135×C53—22no—
65#135×C53—22no—
66C2×Q16×C53C811yescharacteristic
67C2×SD16×C53C811yescharacteristic
68C2×SD16×C53C811yescharacteristic
69C2×D8×C53C811yescharacteristic
70C22×C8×C53C2×C411yescharacteristic
71#41×C53—21no—
72#41×C53—21no—
73#41×C53—21no—
74#41×C53—21no—
75#41×C53—21no—
76#41×C53—21no—
77#41×C53C22⋊C412yes—
78#41×C53M1612yes—
79C2×Q8×C53C2×C811yescharacteristic
80C2×Q8×C53C2×C811yescharacteristic
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53C2×C811yescharacteristic
84C2×D4×C53C2×C811yescharacteristic
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53C22×C411yescharacteristic
88C22×Dic53—21no—
89Q16×C53—22no—
90SD16×C53—22no—
91SD16×C53—22no—
92D8×C53—22no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53C2×C811yescharacteristic
96C2×C8×C53C2×C811yescharacteristic
97C4×Dic53—21no—
98C4×Dic53—21no—
99#11719917—21no—
100#11719917—22no—
101#11719917—22no—
102#11719917—21no—
103#11719917—21no—
104#11719917—41no—
105#11719917—41no—
106#11719917C22⋊C412yes—
107C2×#88322—21no—
108C2×#88322—22no—
109C2×#88322—22no—
110C2×#88322—21no—
111C2×#88322—41no—
112C2×#88322—41no—
113C2×#88322C2×D411yescharacteristic
114C2×#88322C2×D411yescharacteristic
115C2×#88322C2×D411yescharacteristic
116C2×#88322C2×D411yescharacteristic
117C2×#88322#4111yescharacteristic
118C2×#88322#4111yescharacteristic
119#11719910—22no—
120#11719910—22no—
121#11719910—41no—
122#11719910—41no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53#13111yescharacteristic
126Q8×C53—21no—
127Q8×C53—21no—
128Q8×C53—21no—
129Q8×C53C22⋊C812yes—
130D4×C53—22no—
131D4×C53—22no—
132D4×C53—21no—
133D4×C53—21no—
134D4×C53—21no—
135D4×C53—41no—
136D4×C53—41no—
137D4×C53C22⋊C812yes—
138C2×C4×C53—21no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—21no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53C2×C22⋊C411yescharacteristic
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53C22×C811yescharacteristic
148C2×C4×C53C2×M1611yescharacteristic
149C2×C4×C53#13111yescharacteristic
150C2×Dic53—41no—
151C2×Dic53—81no—
152C8×C53—21no—
153C8×C53—21no—
154#88322—22no—
155#88322—41no—
156#88322#12011yescharacteristic
157#88322#12111yescharacteristic
158#88322#12111yescharacteristic
159#88322#12311yescharacteristic
160#12986—531nomaximal, Sylow
161C22×C53—22no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53#40811yescharacteristic
167C22×C53C8×D411yescharacteristic
168C22×C53#43511yescharacteristic
169C4×C53—22no—
170C4×C53—41no—
171C4×C53#38611yescharacteristic
172C4×C53C2×C22⋊C811yescharacteristic
173C4×C53C8×D411yescharacteristic
174C4×C53#43511yescharacteristic
175Dic53—81no—
176#3407—531no—
177C2×#444—531no—
178#2117—531no—
179#2117—531no—
180#2115—531no—
181#1377—531no—
182#1375—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—42no—
187C2×C53—81no—
188C2×C53#186911yescharacteristic
189C2×#135#615711yescharacteristic
190C2×#131—531no—
191C2×#131—531no—
192#504—1062no—
193#504—1062no—
194C2×M32—531no—
195C2×M32—531no—
196C2×M32—531no—
197C2×M32—531no—
198C2×M32—1062no—
199#444—1062no—
200#444—1062no—
201C2×#108—531no—
202C2×#104—531no—
203C2×#104—531no—
204C2×#97—531no—
205#350—1062no—
206#350—1062no—
207#349—1062no—
208C53#1298611yescharacteristic, complemented, Sylow
209C2×#41C2×#615711yescharacteristic
210C2×#41C2×#615711yescharacteristic
211#135—22no—
212#135—22no—
213C2×Q16#8832411yescharacteristic
214C2×SD16#8832411yescharacteristic
215C2×SD16#8832411yescharacteristic
216C2×D8#8832411yescharacteristic
217#131—532no—
218#131—532no—
219#131—1061no—
220#131—1061no—
221#131—1061no—
222#131—1061no—
223#131—1061no—
224#131—1061no—
225C2×M16—531no—
226C2×M16—531no—
227C2×M16—1061no—
228C2×M16—1061no—
229C22×C8—531no—
230C22×C8—1061no—
231C22×C8—1061no—
232C22×C8C2×#615711yescharacteristic
233C2×C42—531no—
234M32—532no—
235M32—1061no—
236M32—1061no—
237M32—1061no—
238M32—1064no—
239M32—1061no—
240M32—1061no—
241M32—1061no—
242M32—1061no—
243C2×C16—532no—
244C2×C16—1061no—
245C2×C16—1064no—
246C2×C16—1061no—
247C2×C16—1061no—
248#108—1062no—
249#104—1062no—
250#104—1062no—
251#97—1061no—
252#97—1061no—
253#41—21no—
254#41—21no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41#1171994012yes—
260#41C22⋊#615712yes—
261C2×Q8C2×#8832411yescharacteristic
262C2×Q8C2×#8832411yescharacteristic
263C2×D4—21no—
264C2×D4—21no—
265C2×D4C2×#8832411yescharacteristic
266C2×D4C2×#8832411yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—1061no—
270C22×C4C22×#615711yescharacteristic
271Q16—22no—
272SD16—22no—
273SD16—22no—
274D8—22no—
275M16—532no—
276M16—1062no—
277M16—1062no—
278M16—1061no—
279M16—1061no—
280M16—1061no—
281M16—2121no—
282M16—2121no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—531no—
286C2×C8—531no—
287C2×C8—531no—
288C2×C8—531no—
289C2×C8—531no—
290C2×C8—531no—
291C2×C8—1062no—
292C2×C8—1062no—
293C2×C8—1061no—
294C2×C8—1061no—
295C2×C8—2121no—
296C2×C8—2121no—
297C2×C8C2×#8832411yescharacteristic
298C2×C8C2×#8832411yescharacteristic
299C42—1061no—
300C42—1061no—
301C16—1062no—
302C16—1062no—
303C16—2121no—
304C16—2121no—
305C23—21no—
306C23—21no—
307C23#1313330411yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8C22⋊#8832412yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—41no—
318D4—41no—
319D4C22⋊#8832412yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—4241no—
327C2×C4—2121no—
328C2×C4C2×C22⋊#615711yescharacteristic
329C2×C4D4×#615711yescharacteristic
330C2×C4D4×#615711yescharacteristic
331C2×C4C22×#8832411yescharacteristic
332C2×C4C2×#1171994011yescharacteristic
333C2×C4#1313330411yescharacteristic
334C8—21no—
335C8—21no—
336C8—531no—
337C8—531no—
338C8—531no—
339C8—531no—
340C8—1062no—
341C8—2121no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22#1681159611yescharacteristic
348C22D4×#8832411yescharacteristic
349C22#1681165411yescharacteristic
350C4—22no—
351C4—41no—
352C4—4241no—
353C4#1681157111yescharacteristic
354C4C2×C22⋊#8832411yescharacteristic
355C4D4×#8832411yescharacteristic
356C4#1681165411yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2#2029068811yescharacteristic
subgroupG/H
centerC4#16811571
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#135×C53C4
socleC2×C53#1869
cosocleC22×#88322C23
hypercenterC2×#135#6157
hypocenterC53#12986
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #20291682C2semidirect
2. #20290957C2semidirect
3. #20290957C2semidirect
4. #20290951C2semidirect
5. #20289763C2semidirect
6. #20289757C2semidirect
208. C53#12986semidirect

Maximal subgroups

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

subgroupindex
#202916822
#202909572
#202909512
#202897632
#202897572
C2×#168107452
#1298653