Finite Group Databasebeta

← #27382128

Subgroups — #27382128

Order 13568.

1010 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810661C211yesmaximal, characteristic
2#20283133C212yesmaximal
3C2×#16810578C211yesmaximal, characteristic
4#20274602C214yesmaximal, complemented
5#20274597C214yesmaximal, complemented
6#20274426C212yesmaximal
7C22×#108×C53C211yesmaximal, characteristic
8C22×#11719917C2211yescharacteristic
9C2×#13133176C2212yes—
10C2×#13133176C2211yescharacteristic
11#16810661C412yescomplemented
12#16810661C412yescomplemented
13#16810661C2212yes—
14#16810661C2212yes—
15C2×#13133151C412yescomplemented
16C2×#13133149C412yescomplemented
17#16810581C2218yes—
18#16810578—24no—
19#16810578C418yescomplemented
20M16×Dic53—24no—
21C2×#13133127C2212yes—
22C2×C8×Dic53C2212yes—
23#16810384C2218yes—
24C22×M16×C53C2212yes—
25C23×C8×C53C2211yescharacteristic
26C2×#108×C53C418yescomplemented
27C2×#108×C53C2212yes—
28C2×#108×C53C2212yes—
29C2×#11719917C2×C414yes—
30C2×#11719917D412yes—
31C2×#11719917D412yes—
32C2×#11719917Q814yes—
33C22×#88322C2312yes—
34#13133176—22no—
35#13133176—22no—
36#13133176—22no—
37#13133176C2×C414yes—
38#13133176C2×C414yes—
39#13133176C2314yes—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53C2×C414yes—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—22no—
44C2×C4×Dic53—22no—
45C2×C4×Dic53C2312yes—
46#13133151C2×C414yes—
47#13133151C2×C414yes—
48#13133149C2×C414yes—
49#13133149C2×C414yes—
50#13133127—28no—
51#13133127C2×C418yescomplemented
52C8×Dic53—28no—
53C8×Dic53C2×C418yescomplemented
54C23×C4×C53C2311yescharacteristic
55C2×M16×C53—22no—
56C2×M16×C53—24no—
57C2×M16×C53C2×C418yes—
58C2×M16×C53C2314yes—
59C22×C8×C53—22no—
60C22×C8×C53C2×C414yes—
61C22×C8×C53C2×C411yescharacteristic
62C22×C8×C53C2×C411yescharacteristic
63C22×C8×C53C2×C411yescharacteristic
64C22×C8×C53C2×C411yescharacteristic
65C22×C8×C53C2×C414yes—
66C22×C8×C53C2311yescharacteristic
67C22×C8×C53C2311yescharacteristic
68#108×C53C2×C418yescomplemented
69#108×C53C2×C418yescomplemented
70C24×C53#4111yescharacteristic
71C22×C4×C53C22×C412yes—
72C22×C4×C53C22×C412yes—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×Q812yes—
78C22×C4×C53#4111yescharacteristic
79C22×C4×C53#4112yes—
80C22×C4×C53C2411yescharacteristic
81C22×Dic53—22no—
82C22×Dic53—24no—
83M16×C53—28no—
84M16×C53—24no—
85M16×C53C22×C418yes—
86C2×C8×C53—28no—
87C2×C8×C53—24no—
88C2×C8×C53C4214yes—
89C2×C8×C53C4214yes—
90C2×C8×C53C4214yes—
91C2×C8×C53C4214yes—
92C2×C8×C53C22×C412yes—
93C2×C8×C53C22×C412yes—
94C2×C8×C53C22×C411yescharacteristic
95C2×C8×C53C22×C411yescharacteristic
96C2×C8×C53C22×C411yescharacteristic
97C2×C8×C53C22×C411yescharacteristic
98C2×C8×C53C22×C412yes—
99C2×C8×C53C22×C412yes—
100C22⋊Dic53—24no—
101C22⋊Dic53—24no—
102C22⋊Dic53—24no—
103C4⋊Dic53—24no—
104C4⋊Dic53—24no—
105C4⋊Dic53C22×C418yes—
106C4×Dic53—24no—
107C4×Dic53—24no—
108C4×Dic53—22no—
109C4×Dic53C22×C414yes—
110#11719917C4⋊C418yes—
111#11719917C4⋊C418yes—
112C2×#88322C22×C414yes—
113C2×#88322C2×D414yes—
114C2×#88322C2×Q814yes—
115C23×C53—42no—
116C23×C53#11712yes—
117C23×C53C4×D411yescharacteristic
118C23×C53C4×D411yescharacteristic
119C23×C53C4×Q812yes—
120C23×C53C2×#4111yescharacteristic
121C2×C4×C53C2×C4212yes—
122C2×C4×C53C2×C4212yes—
123C2×C4×C53C2×C4⋊C412yes—
124C2×C4×C53C2×C4⋊C412yes—
125C2×C4×C53#11712yes—
126C2×C4×C53C4×D411yescharacteristic
127C2×C4×C53C4×D411yescharacteristic
128C2×C4×C53C4×D412yes—
129C2×C4×C53C4×D412yes—
130C2×C4×C53C4×Q812yes—
131C2×C4×C53C4×Q812yes—
132C2×C4×C53C4×Q812yes—
133C2×C4×C53C23×C411yescharacteristic
134C2×C4×C53C23×C411yescharacteristic
135C2×C4×C53C23×C411yescharacteristic
136C2×C4×C53C22×D411yescharacteristic
137C2×C4×C53C22×Q811yescharacteristic
138C2×C4×C53C2×#4111yescharacteristic
139C2×Dic53—24no—
140C2×Dic53—28no—
141C2×Dic53—44no—
142C2×Dic53—84no—
143C8×C53—28no—
144C8×C53C2×C4212yes—
145C8×C53C2×C4212yes—
146C8×C53C2×C4212yes—
147C8×C53C2×C4212yes—
148#88322C2×C4⋊C418yes—
149#20446—531nomaximal, Sylow
150C22×C53—22no—
151C22×C53—22no—
152C22×C53—22no—
153C22×C53—42no—
154C22×C53—42no—
155C22×C53C4×C4⋊C412yes—
156C22×C53C4×C4⋊C412yes—
157C22×C53C2×#11711yescharacteristic
158C22×C53C2×C4×D411yescharacteristic
159C22×C53C2×C4×Q811yescharacteristic
160C4×C53C4×C4⋊C412yes—
161C4×C53C4×C4⋊C412yes—
162C4×C53C22×C4211yescharacteristic
163C4×C53C22×C4⋊C411yescharacteristic
164C4×C53C2×C4×D411yescharacteristic
165C4×C53C2×C4×Q811yescharacteristic
166Dic53—44no—
167Dic53—48no—
168C22×#108D5311yescharacteristic
169C2×#427—531no—
170#2873—532no—
171C2×#405—531no—
172#1773—534no—
173#1772—534no—
174#1733—532no—
175C2×C53—21no—
176C2×C53—21no—
177C2×C53—22no—
178C2×C53—22no—
179C2×C53C2×C4×C4⋊C411yescharacteristic
180C2×C53#295712yes—
181C22×M16—531no—
182C22×M16C2×D5312yes—
183C23×C8C2×D5311yescharacteristic
184C2×#117—531no—
185C2×#117—532no—
186C2×#108Dic5318yescomplemented
187C2×#108C2×D5312yes—
188C2×#108C2×D5312yes—
189#427—532no—
190#427—532no—
191#427—532no—
192#427—532no—
193C2×#107—532no—
194C2×#106—532no—
195#405—538no—
196#405—538no—
197#405—1064no—
198C4×M16—1064no—
199C2×#97—532no—
200C2×C4×C8—532no—
201#339—538no—
202C53#2044611yescharacteristic, complemented, Sylow
203C23×C4C22×D5311yescharacteristic
204C2×M16—22no—
205C2×M16—24no—
206C2×M16—534no—
207C2×M16—532no—
208C2×M16—532no—
209C2×M16—534no—
210C2×M16C2×Dic5318yes—
211C2×M16C22×D5314yes—
212C22×C8—22no—
213C22×C8—532no—
214C22×C8C4×D5314yes—
215C22×C8C4×D5311yescharacteristic
216C22×C8C4×D5311yescharacteristic
217C22×C8C4×D5311yescharacteristic
218C22×C8C4×D5311yescharacteristic
219C22×C8C2×Dic5314yes—
220C22×C8C22×D5311yescharacteristic
221C22×C8C22×D5311yescharacteristic
222#117—534no—
223#117—534no—
224#117—534no—
225#117—1062no—
226#117—1062no—
227#117—1062no—
228C2×C4⋊C4—534no—
229C2×C4⋊C4—1061no—
230C2×C22⋊C4—1061no—
231C2×C22⋊C4—1062no—
232C2×C42—532no—
233C2×C42—1062no—
234#108C2×Dic5318yescomplemented
235#108C2×Dic5318yescomplemented
236#107—534no—
237#107—534no—
238#106—534no—
239#106—534no—
240#97—538no—
241#97—1068no—
242C4×C8—538no—
243C4×C8—1068no—
244C24#1171994911yescharacteristic
245C22×C4—1064no—
246C22×C4—1062no—
247C22×C4C2×C4×D5312yes—
248C22×C4C2×C4×D5311yescharacteristic
249C22×C4C2×C4×D5311yescharacteristic
250C22×C4D4×D5311yescharacteristic
251C22×C4D4×D5311yescharacteristic
252C22×C4#1171994712yes—
253C22×C4Q8×D5312yes—
254C22×C4#1171994911yescharacteristic
255C22×C4C22×Dic5312yes—
256C22×C4C23×D5311yescharacteristic
257M16—28no—
258M16—24no—
259M16—538no—
260M16—538no—
261M16C22×Dic5318yes—
262C2×C8—28no—
263C2×C8—24no—
264C2×C8—534no—
265C2×C8—534no—
266C2×C8—534no—
267C2×C8C4×Dic5314yes—
268C2×C8C4×Dic5314yes—
269C2×C8C4×Dic5314yes—
270C2×C8C4×Dic5314yes—
271C2×C8C2×C4×D5312yes—
272C2×C8C2×C4×D5312yes—
273C2×C8C2×C4×D5311yescharacteristic
274C2×C8C2×C4×D5311yescharacteristic
275C2×C8C2×C4×D5311yescharacteristic
276C2×C8C2×C4×D5311yescharacteristic
277C2×C8C22×Dic5312yes—
278C2×C8C22×Dic5312yes—
279C4⋊C4—538no—
280C4⋊C4—1064no—
281C4⋊C4—1064no—
282C22⋊C4—1064no—
283C22⋊C4—1064no—
284C22⋊C4—1064no—
285C42—534no—
286C42—1064no—
287C42—1064no—
288C42—1062no—
289C23—42no—
290C23#1313317512yes—
291C23#1313317911yescharacteristic
292C23#1313317911yescharacteristic
293C23Q8×Dic5312yes—
294C23C2×#1171994911yescharacteristic
295C2×C4—1064no—
296C2×C4—1068no—
297C2×C4—4244no—
298C2×C4—2124no—
299C2×C4C2×C4×Dic5312yes—
300C2×C4C2×C4×Dic5312yes—
301C2×C4C4⋊C4×D5312yes—
302C2×C4C4⋊C4×D5312yes—
303C2×C4#1313317512yes—
304C2×C4#1313317911yescharacteristic
305C2×C4#1313317911yescharacteristic
306C2×C4D4×Dic5312yes—
307C2×C4D4×Dic5312yes—
308C2×C4#1313318212yes—
309C2×C4#1313318212yes—
310C2×C4Q8×Dic5312yes—
311C2×C4C22×C4×D5311yescharacteristic
312C2×C4C22×C4×D5311yescharacteristic
313C2×C4C23×Dic5311yescharacteristic
314C2×C4C2×D4×D5311yescharacteristic
315C2×C4C2×Q8×D5311yescharacteristic
316C2×C4C2×#1171994711yescharacteristic
317C8—28no—
318C8—538no—
319C8C2×C4×Dic5312yes—
320C8C2×C4×Dic5312yes—
321C8C2×C4×Dic5312yes—
322C8C2×C4×Dic5312yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22C4⋊C4×Dic5312yes—
329C22C4⋊C4×Dic5312yes—
330C22C2×#1313317511yescharacteristic
331C22C2×#1313317911yescharacteristic
332C22C2×Q8×Dic5311yescharacteristic
333C4—2124no—
334C4—2128no—
335C4C4⋊C4×Dic5312yes—
336C4C4⋊C4×Dic5312yes—
337C4C22×C4×Dic5311yescharacteristic
338C4C2×C4⋊C4×D5311yescharacteristic
339C4C2×D4×Dic5311yescharacteristic
340C4C2×#1313318211yescharacteristic
341C2—21no—
342C2—21no—
343C2—22no—
344C2—22no—
345C2C2×C4⋊C4×Dic5311yescharacteristic
346C2#2028363112yes—
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC4×C53C22×C42
FrattiniC22×C4C23×D53
FittingC22×#108×C53C2
socleC22×C53C2×C4×D4
prime radicalC24×C53#41
cosocleC22×C4×C53C24
hypercenterC22×#108D53
hypocenterC53#20446
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
4. #20274602C2semidirect
5. #20274597C2semidirect
11. #16810661C4semidirect
12. #16810661C4semidirect
15. C2×#13133151C4semidirect
16. C2×#13133149C4semidirect
19. #16810578C4semidirect
26. C2×#108×C53C4semidirect
51. #13133127C2×C4semidirect
53. C8×Dic53C2×C4semidirect
68. #108×C53C2×C4semidirect
69. #108×C53C2×C4semidirect
186. C2×#108Dic53semidirect
202. C53#20446semidirect
234. #108C2×Dic53semidirect
235. #108C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106612
#202831332
C2×#168105782
#202746022
#202745972
#202744262
C22×#108×C532
#2044653