Finite Group Databasebeta

← #27299723

Subgroups — #27299723

Order 13568.

714 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276384C212yesmaximal
2C2×#16810418C212yesmaximal, complemented
3#20274823C212yesmaximal, complemented
4C53×#3510C211yesmaximal, characteristic
5C2×#13133132C2214yes—
6#16810418—22no—
7#16810418—22no—
8#16810418—22no—
9#16810416—22no—
10#16810389—24no—
11C53×#559—22no—
12C53×#556—22no—
13C53×#551—24no—
14C53×#542—22no—
15C53×#536C411yescharacteristic, complemented
16C53×#535C2211yescharacteristic
17C53×#533C411yescharacteristic, complemented
18C53×#529—22no—
19C2×#124×C53C2211yescharacteristic
20C53×#519C411yescharacteristic, complemented
21C53×#518C2211yescharacteristic
22C4×#41×C53C411yescharacteristic, complemented
23C2×C22⋊Dic53—24no—
24#13133132—24no—
25#13133132—24no—
26C2×#41×C53C2×C411yescharacteristic
27C22×Q8×C53C2×C411yescharacteristic
28C22×D4×C53C2311yescharacteristic
29C4⋊Q8×C53—21no—
30C4⋊Q8×C53—21no—
31#127×C53D412yes—
32#126×C53—24no—
33#126×C53—24no—
34#125×C53—21no—
35#124×C53—24no—
36#124×C53—21no—
37#124×C53—22no—
38#124×C53D412yes—
39#124×C53D412yes—
40#124×C53D412yes—
41#123×C53—24no—
42#123×C53—24no—
43#123×C53—22no—
44C22⋊Q8×C53—24no—
45C22⋊Q8×C53—22no—
46C22⋊Q8×C53—22no—
47#121×C53—22no—
48#120×C53—22no—
49C4×Q8×C53—22no—
50C4×Q8×C53—22no—
51C4×Q8×C53—21no—
52C4×Q8×C53—22no—
53C4×D4×C53—24no—
54C4×D4×C53—22no—
55C4×D4×C53—21no—
56C4×D4×C53—22no—
57C4×D4×C53—22no—
58#117×C53—24no—
59#117×C53—22no—
60#117×C53—21no—
61#117×C53C2×C411yescharacteristic
62#117×C53C2×C411yescharacteristic
63#117×C53C2×C411yescharacteristic
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53D414yes—
66C2×C42×C53—21no—
67C2×C42×C53C2×C411yescharacteristic
68C24×C53C2×D412yes—
69#41×C53—42no—
70C2×Q8×C53—22no—
71C2×Q8×C53—21no—
72C2×Q8×C53—42no—
73C2×Q8×C53C22⋊C412yes—
74C2×Q8×C53C22⋊C411yescharacteristic
75C2×D4×C53—21no—
76C2×D4×C53—24no—
77C2×D4×C53—42no—
78C2×D4×C53C22⋊C411yescharacteristic
79C2×D4×C53C2×D412yes—
80C2×D4×C53C2×D412yes—
81C22×C4×C53—21no—
82C22×C4×C53—22no—
83C22×C4×C53—21no—
84C22×C4×C53—24no—
85C22×C4×C53—21no—
86C22×C4×C53C22×C411yescharacteristic
87C22×Dic53—44no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53—24no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53—24no—
99C22⋊C4×C53—21no—
100C22⋊C4×C53—24no—
101C22⋊C4×C53—24no—
102C22⋊C4×C53—24no—
103C22⋊C4×C53—24no—
104C22⋊C4×C53C22⋊C412yes—
105C22⋊C4×C53C22⋊C412yes—
106C42×C53—22no—
107C42×C53—24no—
108C42×C53—21no—
109C42×C53—21no—
110C42×C53—21no—
111C42×C53C22⋊C411yescharacteristic
112C42×C53C22⋊C411yescharacteristic
113C42×C53C22⋊C412yes—
114C22⋊Dic53—44no—
115C22⋊Dic53—44no—
116C23×C53—24no—
117C23×C53—21no—
118C23×C53—82no—
119C23×C53C2×C22⋊C411yescharacteristic
120C23×C53#12014yes—
121Q8×C53—42no—
122Q8×C53—42no—
123Q8×C53—41no—
124Q8×C53—82no—
125D4×C53—42no—
126D4×C53—42no—
127D4×C53—41no—
128D4×C53—42no—
129D4×C53—84no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—21no—
133C2×C4×C53—22no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—82no—
141C2×C4×C53—84no—
142C2×C4×C53—81no—
143C2×C4×C53C2×C22⋊C411yescharacteristic
144C2×C4×C53C2×C22⋊C411yescharacteristic
145C2×Dic53—84no—
146C2×Dic53—164no—
147#12728—531nomaximal, Sylow
148C22×C53—24no—
149C22×C53—41no—
150C22×C53—44no—
151C22×C53—42no—
152C22×C53—41no—
153C22×C53—84no—
154C22×C53#37911yescharacteristic
155C22×C53#41012yes—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—42no—
159C4×C53—41no—
160C4×C53—42no—
161C4×C53—41no—
162C4×C53—42no—
163C4×C53—41no—
164C4×C53—42no—
165Dic53—164no—
166#3510D5311yescharacteristic
167#2127—532no—
168C2×#354—532no—
169#1828—532no—
170C2×C53—21no—
171C2×C53—22no—
172C2×C53—44no—
173C2×C53—81no—
174C2×C53#178211yescharacteristic
175#559—22no—
176#556—22no—
177#551—24no—
178#542—22no—
179#536Dic5311yescharacteristic, complemented
180#535C2×D5311yescharacteristic
181#533Dic5311yescharacteristic, complemented
182#529—22no—
183C2×#124C2×D5311yescharacteristic
184#519Dic5311yescharacteristic, complemented
185#518C2×D5311yescharacteristic
186C4×#41Dic5311yescharacteristic, complemented
187C2×#99—534no—
188#354—1062no—
189#354—1062no—
190#354—1062no—
191#353—1062no—
192#342—1064no—
193C53#1272811yescharacteristic, complemented, Sylow
194C2×#41C2×Dic5311yescharacteristic
195C22×Q8C2×Dic5311yescharacteristic
196C22×D4C22×D5311yescharacteristic
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199#127C22⋊D5312yes—
200#126—24no—
201#126—24no—
202#125—21no—
203#124—24no—
204#124—22no—
205#124—21no—
206#124C22⋊D5312yes—
207#124C22⋊D5312yes—
208#124C22⋊D5312yes—
209#123—24no—
210#123—24no—
211#123—22no—
212C22⋊Q8—24no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215#121—22no—
216#120—22no—
217C4×Q8—21no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×D4—24no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—21no—
225C4×D4—22no—
226#117—24no—
227#117—22no—
228#117—21no—
229#117C2×Dic5311yescharacteristic
230#117C2×Dic5311yescharacteristic
231#117C2×Dic5311yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—1064no—
234C2×C22⋊C4C22⋊D5314yes—
235C2×C42—21no—
236C2×C42C2×Dic5311yescharacteristic
237#99—1064no—
238#99—1064no—
239C24C2×C22⋊D5312yes—
240#41—42no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8C22⋊Dic5312yes—
245C2×Q8C22⋊Dic5311yescharacteristic
246C2×D4—21no—
247C2×D4—24no—
248C2×D4—42no—
249C2×D4C22⋊Dic5311yescharacteristic
250C2×D4C2×C22⋊D5312yes—
251C2×D4C2×C22⋊D5312yes—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—2124no—
258C22×C4C22×Dic5311yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C22⋊C4—22no—
269C22⋊C4—24no—
270C22⋊C4—21no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—2124no—
276C22⋊C4—2124no—
277C22⋊C4C22⋊Dic5312yes—
278C22⋊C4C22⋊Dic5312yes—
279C42—22no—
280C42—24no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42C22⋊Dic5311yescharacteristic
285C42C22⋊Dic5311yescharacteristic
286C42C22⋊Dic5312yes—
287C23—24no—
288C23—21no—
289C23—82no—
290C23C2×C22⋊Dic5311yescharacteristic
291C23#1313318614yes—
292Q8—42no—
293Q8—41no—
294Q8—42no—
295Q8—82no—
296D4—42no—
297D4—42no—
298D4—41no—
299D4—42no—
300D4—84no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—81no—
314C2×C4—8484no—
315C2×C4—4244no—
316C2×C4C2×C22⋊Dic5311yescharacteristic
317C2×C4C2×C22⋊Dic5311yescharacteristic
318C22—24no—
319C22—41no—
320C22—44no—
321C22—42no—
322C22—41no—
323C22—84no—
324C22#1681047911yescharacteristic
325C22#1681059712yes—
326C4—21no—
327C4—21no—
328C4—42no—
329C4—41no—
330C4—42no—
331C4—41no—
332C4—42no—
333C4—41no—
334C4—42no—
335C4—8484no—
336C2—21no—
337C2—22no—
338C2—44no—
339C2—81no—
340C2#2027463311yescharacteristic
subgroupG/H
centerC2#20274633
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#3510C2
socleC2×C53#1782
prime radicalC53×#535C22
cosocleC22×D4×C53C23
hypercenter#3510D53
hypocenterC53#12728
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ #518 ◁ #3510

Complements

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

NH ≅ G/Ntype
2. C2×#16810418C2semidirect
3. #20274823C2semidirect
15. C53×#536C4semidirect
17. C53×#533C4semidirect
20. C53×#519C4semidirect
22. C4×#41×C53C4semidirect
179. #536Dic53semidirect
181. #533Dic53semidirect
184. #519Dic53semidirect
186. C4×#41Dic53semidirect
193. C53#12728semidirect

Maximal subgroups

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

subgroupindex
#202763842
C2×#168104182
#202748232
C53×#35102
#1272853