Finite Group Databasebeta

← #27301052

Subgroups — #27301052

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283326C211yesmaximal, characteristic, complemented
2#20276443C211yesmaximal, characteristic, complemented
3C2×#16810430C211yesmaximal, characteristic, complemented
4C2×#16810407C211yesmaximal, characteristic, complemented
5#20272828C211yesmaximal, characteristic, complemented
6#20272808C211yesmaximal, characteristic, complemented
7C53×#3067C211yesmaximal, characteristic
8#16811390—21no—
9C2×#13133157C2211yescharacteristic
10C22×#11719908C2211yescharacteristic
11#16810672—22no—
12#16810672—21no—
13C2×#13133153—22no—
14C2×#13133153C2211yescharacteristic
15C2×#13133153C2211yescharacteristic
16#16810430—22no—
17#16810430—22no—
18#16810407—22no—
19C53×#575—21no—
20C2×#135×C53C2211yescharacteristic
21C2×#131×C53C2211yescharacteristic
22C53×#473C412yes—
23C53×#472C412yes—
24C53×#471—22no—
25C2×#108×C53C2211yescharacteristic
26C53×#413—21no—
27#13133224—42no—
28#13133224—42no—
29C2×#11719917—24no—
30C2×#11719917—21no—
31C2×#11719917—21no—
32#13133157—21no—
33#13133157—21no—
34C2×#11719908—21no—
35C2×#11719908—21no—
36C2×#11719908—21no—
37C2×#11719908—21no—
38#13133153—24no—
39#13133153—21no—
40#13133153—21no—
41#13133153D412yes—
42#13133153D412yes—
43C2×#41×C53—21no—
44C2×#41×C53D411yescharacteristic
45#137×C53—22no—
46#137×C53—41no—
47AGL(1,ℤ/8)×C53—22no—
48AGL(1,ℤ/8)×C53—41no—
49#135×C53—21no—
50#135×C53—21no—
51#135×C53—21no—
52#135×C53—21no—
53#135×C53—42no—
54C2×Q16×C53C2×C411yescharacteristic
55C2×SD16×C53—21no—
56C2×D8×C53C2×C411yescharacteristic
57#131×C53—21no—
58#131×C53—21no—
59#131×C53C2×C412yes—
60#131×C53D412yes—
61C2×M16×C53—21no—
62C2×M16×C53—21no—
63C2×M16×C53D411yescharacteristic
64C22×C8×C53—21no—
65C22×C8×C53C2311yescharacteristic
66#108×C53—21no—
67#108×C53C2×C412yes—
68#101×C53—22no—
69#100×C53—22no—
70#41×C53—21no—
71#41×C53—21no—
72#41×C53—41no—
73#41×C53—41no—
74#41×C53—41no—
75#41×C53—41no—
76#41×C53C22⋊C412yes—
77#41×C53D812yes—
78C2×Q8×C53—21no—
79C2×Q8×C53SD1611yescharacteristic
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—41no—
83C2×D4×C53SD1611yescharacteristic
84C22×C4×C53—21no—
85C22×C4×C53—41no—
86C22×C4×C53C2×D411yescharacteristic
87Q16×C53—21no—
88Q16×C53—21no—
89Q16×C53—42no—
90SD16×C53—42no—
91SD16×C53—42no—
92SD16×C53—41no—
93SD16×C53—41no—
94D8×C53—21no—
95D8×C53—21no—
96D8×C53—42no—
97M16×C53—21no—
98M16×C53—22no—
99M16×C53—22no—
100M16×C53—22no—
101M16×C53—41no—
102M16×C53C22⋊C412yes—
103C2×C8×C53—21no—
104C2×C8×C53—22no—
105C2×C8×C53—22no—
106C2×C8×C53—41no—
107C2×C8×C53C22×C411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53#4111yescharacteristic
112C2×C8×C53#4111yescharacteristic
113#11719917—22no—
114#11719917—22no—
115#11719917—44no—
116#11719917—41no—
117#11719917—41no—
118C2×#88322—44no—
119C2×#88322—41no—
120C2×#88322—41no—
121#11719908—41no—
122#11719908—41no—
123C23×C53—21no—
124C23×C53—41no—
125C23×C53#13711yescharacteristic
126Q8×C53—21no—
127Q8×C53—41no—
128Q8×C53—41no—
129Q8×C53#10212yes—
130D4×C53—22no—
131D4×C53—22no—
132D4×C53—21no—
133D4×C53—41no—
134D4×C53—41no—
135D4×C53—81no—
136D4×C53—81no—
137D4×C53#10212yes—
138C2×C4×C53—21no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—41no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53C2×C22⋊C411yescharacteristic
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12311yescharacteristic
147C2×C4×C53C2×D811yescharacteristic
148C2×C4×C53C2×SD1611yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C8×C53—22no—
151C8×C53—42no—
152C8×C53C4×D411yescharacteristic
153C8×C53C4×D411yescharacteristic
154C8×C53#12111yescharacteristic
155C8×C53#12111yescharacteristic
156#88322—42no—
157#88322—42no—
158#12995—531nomaximal, Sylow
159C22×C53—22no—
160C22×C53—41no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53—161no—
164C22×C53#41711yescharacteristic
165C22×C53#44811yescharacteristic
166C22×C53#48011yescharacteristic
167C4×C53—22no—
168C4×C53—81no—
169C4×C53#38611yescharacteristic
170C4×C53C2×#10211yescharacteristic
171C4×C53#44711yescharacteristic
172C4×C53#48111yescharacteristic
173#3067D5311yescharacteristic
174#2916—531no—
175#2143—531no—
176C2×#359—531no—
177C2×#350—531no—
178#1388—531no—
179#1381—531no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—42no—
184C2×C53—161no—
185C2×C53#187411yescharacteristic
186#575—21no—
187C2×#135C2×D5311yescharacteristic
188#568—1061no—
189C2×#131C2×D5311yescharacteristic
190C2×M32—531no—
191C22×C16—531no—
192#473Dic5312yes—
193#472Dic5312yes—
194#471—22no—
195#430—1062no—
196#430—1061no—
197C2×#108—531no—
198C2×#108—531no—
199C2×#108—1062no—
200C2×#108C2×D5311yescharacteristic
201#413—21no—
202#359—1062no—
203#359—1062no—
204#350—1062no—
205C53#1299511yescharacteristic, complemented, Sylow
206C2×#41—21no—
207C2×#41C22⋊D5311yescharacteristic
208#137—22no—
209#137—41no—
210AGL(1,ℤ/8)—22no—
211AGL(1,ℤ/8)—41no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—42no—
217C2×Q16C2×Dic5311yescharacteristic
218C2×SD16—21no—
219C2×D8C2×Dic5311yescharacteristic
220#131—21no—
221#131—21no—
222#131—2122no—
223#131—2122no—
224#131C2×Dic5312yes—
225#131C22⋊D5312yes—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—1064no—
229C2×M16—1061no—
230C2×M16—1061no—
231C2×M16C22⋊D5311yescharacteristic
232C22×C8—21no—
233C22×C8C22×D5311yescharacteristic
234M32—1061no—
235M32—1061no—
236C2×C16—1061no—
237C2×C16—1061no—
238C2×C16—1061no—
239C2×C16—1061no—
240#108—21no—
241#108—532no—
242#108—532no—
243#108—1064no—
244#108—1061no—
245#108—1061no—
246#108C2×Dic5312yes—
247#101—22no—
248#100—22no—
249#41—21no—
250#41—21no—
251#41—41no—
252#41—41no—
253#41—41no—
254#41—41no—
255#41#1171992212yes—
256#41C22⋊Dic5312yes—
257C2×Q8—21no—
258C2×Q8#1171992311yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4#1171992311yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4C2×C22⋊D5311yescharacteristic
266Q16—21no—
267Q16—21no—
268Q16—42no—
269SD16—42no—
270SD16—42no—
271SD16—41no—
272SD16—41no—
273D8—21no—
274D8—21no—
275D8—42no—
276M16—21no—
277M16—22no—
278M16—22no—
279M16—22no—
280M16—41no—
281M16—1062no—
282M16—1062no—
283M16—2124no—
284M16—2121no—
285M16—2121no—
286M16C22⋊Dic5312yes—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—2124no—
292C2×C8—2121no—
293C2×C8—2121no—
294C2×C8D4×D5311yescharacteristic
295C2×C8D4×D5311yescharacteristic
296C2×C8#1171994711yescharacteristic
297C2×C8#1171994711yescharacteristic
298C2×C8C22×Dic5311yescharacteristic
299C2×C8C2×C22⋊D5311yescharacteristic
300C16—2121no—
301C16—2121no—
302C23—21no—
303C23—41no—
304C23#1313324711yescharacteristic
305Q8—21no—
306Q8—41no—
307Q8—41no—
308Q8#1313313812yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—41no—
313D4—41no—
314D4—81no—
315D4—81no—
316D4#1313313812yes—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4C2×C22⋊Dic5311yescharacteristic
324C2×C4#1313318511yescharacteristic
325C2×C4#1313319811yescharacteristic
326C2×C4C2×#1171992211yescharacteristic
327C2×C4C2×#1171992311yescharacteristic
328C2×C4#1313324211yescharacteristic
329C8—22no—
330C8—42no—
331C8—2122no—
332C8—2122no—
333C8D4×Dic5311yescharacteristic
334C8D4×Dic5311yescharacteristic
335C8#1313318911yescharacteristic
336C8#1313318911yescharacteristic
337C22—22no—
338C22—41no—
339C22—81no—
340C22—81no—
341C22—161no—
342C22#1681062411yescharacteristic
343C22#1681076111yescharacteristic
344C22#1681093411yescharacteristic
345C4—22no—
346C4—81no—
347C4#1681050311yescharacteristic
348C4C2×#1313313811yescharacteristic
349C4#1681075611yescharacteristic
350C4#1681093911yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—161no—
356C2#2027503311yescharacteristic
subgroupG/H
centerC4#16810756
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3067C2
socleC2×C53#1874
cosocleC22×C8×C53C23
hypercenter#3067D53
hypocenterC53#12995
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #20283326C2semidirect
2. #20276443C2semidirect
3. C2×#16810430C2semidirect
4. C2×#16810407C2semidirect
5. #20272828C2semidirect
6. #20272808C2semidirect
205. C53#12995semidirect

Maximal subgroups

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

subgroupindex
#202833262
#202764432
C2×#168104302
C2×#168104072
#202728282
#202728082
C53×#30672
#1299553