Finite Group Databasebeta

← #27301155

Subgroups — #27301155

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133162C211yesmaximal, characteristic
2C2×C4×Dic212C211yesmaximal, characteristic, complemented
3C2×#16810428C211yesmaximal, characteristic, complemented
4C2×#16810425C211yesmaximal, characteristic
5#20275581C211yesmaximal, characteristic
6#20272807C211yesmaximal, characteristic, complemented
7C53×#1374C211yesmaximal, characteristic, complemented
8C22×Dic212C2211yescharacteristic
9C22×D424C2211yescharacteristic
10C2×C4×Dic106—21no—
11C2×#13133162—22no—
12C2×#13133162—22no—
13C2×#13133162C414yescomplemented
14C4×Dic212—22no—
15C4×Dic212—22no—
16C2×#13133151C2211yescharacteristic
17C2×#13133151C2211yescharacteristic
18C2×#13133142—21no—
19C2×#13133139—21no—
20#16810525—21no—
21#16810428—22no—
22#16810425—22no—
23C22×C16×C53C2211yescharacteristic
24C22×C16×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C22×Dic106—21no—
27C22×D212—21no—
28C2×Dic212—21no—
29C2×Dic212—21no—
30C2×Dic212—21no—
31C2×Dic212C2×C412yes—
32C2×Dic212D412yes—
33C2×Dic212D412yes—
34C2×D424—21no—
35C2×D424—21no—
36C2×D424—21no—
37C2×D424—21no—
38C2×D424—21no—
39C2×D424C2×C412yes—
40C4×Dic106—24no—
41C4×Dic106—42no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×#11719919—41no—
45C2×#11719921—41no—
46C2×#11719921—41no—
47C2×C4×Dic53—41no—
48#13133162—28no—
49#13133151—21no—
50#13133151—21no—
51#13133151—21no—
52#13133151D412yes—
53#13133142—24no—
54#13133139—42no—
55C22×C8×C53D411yescharacteristic
56C22×C8×C53C2311yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C16×C53—21no—
59C2×C16×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C2×C16×C53—21no—
63C2×C16×C53C2×C412yes—
64C4×C8×C53—21no—
65C4×C8×C53—21no—
66C23×D53—41no—
67C22×C4×C53D811yescharacteristic
68C22×C4×C53D811yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—41no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C2×D212—22no—
74C2×D212—41no—
75C2×D212—41no—
76C2×D212—41no—
77C2×D212—41no—
78C2×D212—41no—
79C2×Dic106—22no—
80C2×Dic106—22no—
81C2×Dic106—22no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84C2×Dic106—41no—
85C2×C8×C53—21no—
86C2×C8×C53—21no—
87C2×C8×C53—21no—
88C2×C8×C53C22×C411yescharacteristic
89C2×C8×C53C2×D411yescharacteristic
90C2×C8×C53C2×D411yescharacteristic
91C2×C8×C53C2×D411yescharacteristic
92C2×C8×C53#4111yescharacteristic
93C2×C8×C53#4111yescharacteristic
94C42×C53—21no—
95C42×C53—21no—
96#11719921—82no—
97#11719921—82no—
98C4⋊Dic53—22no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101C4⋊Dic53—41no—
102#11719919—44no—
103C4×Dic53—44no—
104Dic212—22no—
105Dic212—22no—
106Dic212—22no—
107Dic212C22⋊C414yescomplemented
108D424—22no—
109D424—22no—
110D424—41no—
111D424—41no—
112C16×C53—22no—
113C16×C53—41no—
114C23×C53C2×D811yescharacteristic
115C22×D53—42no—
116C22×D53—81no—
117C22×D53—81no—
118C22×D53—81no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53—81no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53SD3212yescomplemented
128C2×C4×C53C4×D411yescharacteristic
129C2×C4×C53#12011yescharacteristic
130C2×C4×C53#12111yescharacteristic
131C2×C4×C53#13511yescharacteristic
132C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—81no—
139C2×Dic53—81no—
140C2×Dic53—81no—
141C2×Dic53—81no—
142C2×Dic53—81no—
143D212—42no—
144D212—42no—
145D212—81no—
146D212—81no—
147Dic106—24no—
148Dic106—42no—
149Dic106—42no—
150Dic106—42no—
151C8×C53—41no—
152C8×C53C2×C22⋊C411yescharacteristic
153C8×C53C4×D411yescharacteristic
154C8×C53#12111yescharacteristic
155C8×C53#12311yescharacteristic
156#13013—531nomaximal, Sylow
157C22×C53C4×D811yescharacteristic
158C22×C53#44711yescharacteristic
159C22×C53#45911yescharacteristic
160C22×C53C2×SD3211yescharacteristic
161C22×C53#50811yescharacteristic
162C22×C53#50911yescharacteristic
163C22×C53#51011yescharacteristic
164C2×D53—82no—
165C2×D53—82no—
166C2×D53—82no—
167C2×D53—161no—
168C2×D53—161no—
169C2×D53—161no—
170C2×D53—161no—
171C4×C53—22no—
172C4×C53—41no—
173C4×C53#38611yescharacteristic
174C4×C53#44211yescharacteristic
175C4×C53#44911yescharacteristic
176C4×C53#46311yescharacteristic
177Dic53—44no—
178Dic53—82no—
179Dic53—161no—
180C22×SD32—531no—
181C2×C4×Q16—531no—
182C2×#358—531no—
183C2×#357—531no—
184#1970—531no—
185#1380—531no—
186#1374D5311yescharacteristic, complemented
187C2×C53#187911yescharacteristic
188C2×C53C4×SD3211yescharacteristic
189C2×C53#217511yescharacteristic
190C2×C53#218511yescharacteristic
191C2×C53#218611yescharacteristic
192C2×C53#220711yescharacteristic
193C2×C53#221111yescharacteristic
194D53—162no—
195C22×Q16—531no—
196C22×D8—531no—
197C2×C4×Q8—1061no—
198C2×SD32—534no—
199C2×SD32—1062no—
200C2×SD32—1062no—
201C22×C16C2×D5311yescharacteristic
202C22×C16C2×D5311yescharacteristic
203C4×Q16—1062no—
204C4×Q16—1062no—
205C2×#107—531no—
206C2×#107—531no—
207C2×#103—1061no—
208C2×#102—1061no—
209C2×C4×C8C2×D5311yescharacteristic
210#390—1061no—
211#358—1062no—
212#357—1062no—
213C53#1301311yescharacteristic, complemented, Sylow
214C22×Q8—1061no—
215C22×D4—1061no—
216C2×Q16—532no—
217C2×Q16—532no—
218C2×Q16—532no—
219C2×Q16—1061no—
220C2×Q16—1061no—
221C2×Q16—1061no—
222C2×D8—532no—
223C2×D8—1061no—
224C2×D8—1061no—
225C2×D8—1061no—
226C2×D8—1061no—
227C2×D8—1061no—
228C22×C8D21211yescharacteristic
229C22×C8C22×D5311yescharacteristic
230C4×Q8—1064no—
231C4×Q8—2122no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—2121no—
235C2×C22⋊C4—2121no—
236C2×C22⋊C4—2121no—
237C2×C42—2121no—
238C2×C42D21211yescharacteristic
239SD32—1068no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16C4×D5312yes—
246#107—532no—
247#107—1061no—
248#107—1061no—
249#107—1061no—
250#103—1064no—
251#102—2122no—
252C4×C8—21no—
253C4×C8—21no—
254C24—2121no—
255C2×Q8—1062no—
256C2×Q8—1062no—
257C2×Q8—1062no—
258C2×Q8—2121no—
259C2×Q8—2121no—
260C2×Q8—2121no—
261C2×D4—1062no—
262C2×D4—2121no—
263C2×D4—2121no—
264C2×D4—2121no—
265C2×D4—2121no—
266C2×D4—2121no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4D42411yescharacteristic
271C22×C4D42411yescharacteristic
272C22×C4C2×D21211yescharacteristic
273Q16—534no—
274Q16—1062no—
275Q16—1062no—
276Q16—1062no—
277D8—1062no—
278D8—1062no—
279D8—2121no—
280D8—2121no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8C2×C4×D5311yescharacteristic
285C2×C8#1171994511yescharacteristic
286C2×C8D4×D5311yescharacteristic
287C2×C8D4×D5311yescharacteristic
288C2×C8D4×D5311yescharacteristic
289C2×C8#1171994911yescharacteristic
290C4⋊C4—1062no—
291C4⋊C4—2124no—
292C4⋊C4—2121no—
293C4⋊C4—2121no—
294C4⋊C4—2121no—
295C22⋊C4—4242no—
296C22⋊C4—4242no—
297C42—21no—
298C42—21no—
299C42—2124no—
300C16—22no—
301C16—41no—
302C23—4241no—
303C23—4241no—
304C23—4241no—
305C23—4241no—
306C23—4241no—
307C23—4241no—
308C23—2122no—
309C23C2×D42411yescharacteristic
310Q8—1064no—
311Q8—2122no—
312Q8—2122no—
313Q8—2122no—
314D4—4241no—
315D4—4241no—
316D4—2122no—
317D4—2122no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—4241no—
324C2×C4—4241no—
325C2×C4—4241no—
326C2×C4—4241no—
327C2×C4—4241no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4—2122no—
332C2×C4#1313316212yescomplemented
333C2×C4C4×D21211yescharacteristic
334C2×C4#1313318711yescharacteristic
335C2×C4#1313319211yescharacteristic
336C2×C4#1313323911yescharacteristic
337C2×C4#1313324511yescharacteristic
338C2×C4#1313324511yescharacteristic
339C8—41no—
340C8C22⋊C4×D5311yescharacteristic
341C8#1313317911yescharacteristic
342C8#1313319011yescharacteristic
343C8#1313320211yescharacteristic
344C22—8481no—
345C22—8481no—
346C22—8481no—
347C22—8481no—
348C22—4242no—
349C22—4242no—
350C22—4242no—
351C22C4×D42411yescharacteristic
352C22#1681075911yescharacteristic
353C22#1681081911yescharacteristic
354C22C2×#1313316211yescharacteristic
355C22#1681105511yescharacteristic
356C22#1681106111yescharacteristic
357C22#1681106611yescharacteristic
358C4—22no—
359C4—41no—
360C4—8481no—
361C4—4242no—
362C4—2124no—
363C4#1681050111yescharacteristic
364C4#1681073611yescharacteristic
365C4#1681077011yescharacteristic
366C4#1681083911yescharacteristic
367C2—8482no—
368C2#2027506211yescharacteristic
369C2C4×#1313316211yescharacteristic
370C2#2027658111yescharacteristic
371C2#2027663411yescharacteristic
372C2#2027664011yescharacteristic
373C2#2027675611yescharacteristic
374C2#2027677811yescharacteristic
subgroupG/H
centerC23C2×D424
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1374C2
socleC23×C53C2×D8
prime radicalC22×D424C22
cosocleC22×C8×C53C23
hypercenter#1374D53
hypocenterC53#13013
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1374

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic212C2semidirect
3. C2×#16810428C2semidirect
6. #20272807C2semidirect
7. C53×#1374C2semidirect
13. C2×#13133162C4semidirect
107. Dic212C22⋊C4semidirect
127. C2×C4×C53SD32semidirect
186. #1374D53semidirect
213. C53#13013semidirect
332. C2×C4#13133162semidirect

Maximal subgroups

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

subgroupindex
C22×#131331622
C2×C4×Dic2122
C2×#168104282
C2×#168104252
#202755812
#202728072
C53×#13742
#1301353