Finite Group Databasebeta

← #27299704

Subgroups — #27299704

Order 13568.

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

normal subgroups only

This group has 670 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#20276410C211yesmaximal, characteristic
2#20276390C211yesmaximal, characteristic
3#20276369C212yesmaximal
4#20274819C212yesmaximal
5C53×#3509C211yesmaximal, characteristic
6#16810607C2212yes—
7#16810597C2212yes—
8#16810422—22no—
9#16810418—22no—
10#16810413—24no—
11#16810383—22no—
12#16810379—22no—
13C53×#557—22no—
14C53×#556—22no—
15C53×#552—24no—
16C53×#541—22no—
17C53×#536C2211yescharacteristic
18C53×#536C2211yescharacteristic
19C53×#533—22no—
20C53×#533C411yescharacteristic, complemented
21C2×C4⋊Q8×C53C411yescharacteristic, complemented
22C53×#519C411yescharacteristic, complemented
23C53×#519C411yescharacteristic, complemented
24C4×#41×C53C2211yescharacteristic
25C2×#11719917—22no—
26#13133176—22no—
27#13133136—22no—
28#13133134—22no—
29#13133132—22no—
30#13133132—22no—
31C2×#41×C53C2311yescharacteristic
32C22×Q8×C53C2×C411yescharacteristic
33C22×Q8×C53C2×C411yescharacteristic
34C4⋊Q8×C53—21no—
35C4⋊Q8×C53—21no—
36C4⋊Q8×C53—21no—
37C4⋊Q8×C53—22no—
38C4⋊Q8×C53—21no—
39C4⋊Q8×C53D412yes—
40#126×C53—24no—
41#126×C53—24no—
42#125×C53—24no—
43#125×C53—24no—
44#125×C53—24no—
45#125×C53—21no—
46#124×C53—21no—
47#124×C53D412yes—
48#123×C53—22no—
49#123×C53D414yes—
50C22⋊Q8×C53—24no—
51C22⋊Q8×C53—22no—
52C22⋊Q8×C53—22no—
53C22⋊Q8×C53—22no—
54C4×Q8×C53—24no—
55C4×Q8×C53—22no—
56C4×Q8×C53—22no—
57C4×Q8×C53—22no—
58C4×Q8×C53—22no—
59C4×Q8×C53—22no—
60C4×Q8×C53—21no—
61C4×D4×C53—22no—
62C4×D4×C53—21no—
63#117×C53—24no—
64#117×C53—22no—
65#117×C53—22no—
66#117×C53C2×C411yescharacteristic
67#117×C53C2×C411yescharacteristic
68#117×C53C2×C411yescharacteristic
69#117×C53D412yes—
70C2×C4⋊C4×C53—22no—
71C2×C42×C53C2×C411yescharacteristic
72C2×C42×C53D412yes—
73#41×C53—42no—
74C2×Q8×C53—22no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—42no—
79C2×Q8×C53—42no—
80C2×Q8×C53C22⋊C412yes—
81C2×Q8×C53C2×D411yescharacteristic
82C2×D4×C53C2×D411yescharacteristic
83C2×D4×C53C2×D412yes—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53C22⋊C412yes—
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C2×D412yes—
90C4⋊C4×C53—24no—
91C4⋊C4×C53—24no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—24no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—24no—
102C4⋊C4×C53—24no—
103C22⋊C4×C53—21no—
104C22⋊C4×C53—21no—
105C22⋊C4×C53—24no—
106C22⋊C4×C53—24no—
107C22⋊C4×C53—22no—
108C22⋊C4×C53C22⋊C412yes—
109C42×C53—24no—
110C42×C53—24no—
111C42×C53—21no—
112C42×C53—21no—
113C42×C53—22no—
114C42×C53C22⋊C411yescharacteristic
115C42×C53C22⋊C411yescharacteristic
116C42×C53C22⋊C412yes—
117C42×C53C22⋊C412yes—
118C22⋊Dic53—42no—
119C4⋊Dic53—42no—
120C4×Dic53—42no—
121#11719917—42no—
122#11719917—42no—
123C2×#88322—42no—
124C23×C53C2×C22⋊C411yescharacteristic
125C23×C53#12012yes—
126Q8×C53—41no—
127Q8×C53—44no—
128Q8×C53—42no—
129Q8×C53—42no—
130Q8×C53—42no—
131Q8×C53—42no—
132D4×C53—42no—
133D4×C53—41no—
134C2×C4×C53—21no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—21no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—42no—
145C2×C4×C53—44no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53C2×C22⋊C411yescharacteristic
149C2×C4×C53C2×C22⋊C411yescharacteristic
150C2×C4×C53#12012yes—
151C2×Dic53—82no—
152#88322—82no—
153#12724—531nomaximal, Sylow
154C22×C53—22no—
155C22×C53—21no—
156C22×C53—41no—
157C22×C53—42no—
158C22×C53#37911yescharacteristic
159C4×C53—22no—
160C4×C53—22no—
161C4×C53—21no—
162C4×C53—42no—
163C4×C53—41no—
164C4×C53—42no—
165C4×C53—42no—
166C4×C53—42no—
167C4×C53—42no—
168C4×C53#41012yes—
169Dic53—162no—
170#3509D5311yescharacteristic
171#2135—531no—
172#2129—531no—
173#2123—532no—
174#1827—532no—
175C2×C53—21no—
176C2×C53—42no—
177C2×C53—41no—
178C2×C53#178211yescharacteristic
179#557—22no—
180#556—22no—
181#552—24no—
182#541—22no—
183#536C2×D5311yescharacteristic
184#536C2×D5311yescharacteristic
185#533—22no—
186#533Dic5311yescharacteristic, complemented
187C2×C4⋊Q8Dic5311yescharacteristic, complemented
188#519Dic5311yescharacteristic, complemented
189#519Dic5311yescharacteristic, complemented
190C4×#41C2×D5311yescharacteristic
191#413—532no—
192#410—532no—
193#356—1062no—
194#354—1062no—
195#352—1064no—
196#339—1062no—
197H(3,ℤ/4)—1062no—
198C53#1272411yescharacteristic, complemented, Sylow
199C2×#41C22×D5311yescharacteristic
200C22×Q8C2×Dic5311yescharacteristic
201C22×Q8C2×Dic5311yescharacteristic
202C2×M16—1062no—
203C4⋊Q8—21no—
204C4⋊Q8—22no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8C22⋊D5312yes—
209#126—24no—
210#126—24no—
211#125—24no—
212#125—24no—
213#125—24no—
214#125—21no—
215#124—21no—
216#124C22⋊D5312yes—
217#123—22no—
218#123C22⋊D5314yes—
219C22⋊Q8—24no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C4×Q8—24no—
224C4×Q8—21no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×D4—22no—
231C4×D4—21no—
232#117—24no—
233#117—22no—
234#117—22no—
235#117—1062no—
236#117C2×Dic5311yescharacteristic
237#117C2×Dic5311yescharacteristic
238#117C2×Dic5311yescharacteristic
239#117C22⋊D5312yes—
240C2×C4⋊C4—22no—
241C2×C42C2×Dic5311yescharacteristic
242C2×C42C22⋊D5312yes—
243#101—1062no—
244#100—1062no—
245#99—1062no—
246#99—1062no—
247#41—42no—
248C2×Q8—22no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8C22⋊Dic5312yes—
255C2×Q8C2×C22⋊D5311yescharacteristic
256C2×D4C2×C22⋊D5311yescharacteristic
257C2×D4C2×C22⋊D5312yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4C22⋊Dic5312yes—
262C22×C4C22×Dic5311yescharacteristic
263C22×C4C2×C22⋊D5312yes—
264M16—2122no—
265M16—2122no—
266C2×C8—2122no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—2122no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—22no—
286C22⋊C4—2122no—
287C22⋊C4C22⋊Dic5312yes—
288C42—24no—
289C42—24no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42—2122no—
294C42C22⋊Dic5312yes—
295C42C22⋊Dic5311yescharacteristic
296C42C22⋊Dic5311yescharacteristic
297C42C22⋊Dic5312yes—
298C23C2×C22⋊Dic5311yescharacteristic
299C23#1313318612yes—
300Q8—41no—
301Q8—42no—
302Q8—44no—
303Q8—42no—
304Q8—42no—
305Q8—42no—
306D4—42no—
307D4—41no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—4242no—
323C2×C4C2×C22⋊Dic5311yescharacteristic
324C2×C4C2×C22⋊Dic5311yescharacteristic
325C2×C4#1313318612yes—
326C8—4242no—
327C22—21no—
328C22—22no—
329C22—41no—
330C22—42no—
331C22#1681047911yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—42no—
336C4—41no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—8482no—
342C4#1681059712yes—
343C2—21no—
344C2—42no—
345C2—41no—
346C2#2027463311yescharacteristic
subgroupG/H
centerC2#20274633
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
FittingC53×#3509C2
socleC2×C53#1782
prime radical, cosocleC2×#41×C53C23
hypercenter#3509D53
hypocenterC53#12724
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×#41 ◁ #3509

Complements

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

NH ≅ G/Ntype
20. C53×#533C4semidirect
21. C2×C4⋊Q8×C53C4semidirect
22. C53×#519C4semidirect
23. C53×#519C4semidirect
186. #533Dic53semidirect
187. C2×C4⋊Q8Dic53semidirect
188. #519Dic53semidirect
189. #519Dic53semidirect
198. C53#12724semidirect

Maximal subgroups

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

subgroupindex
#202764102
#202763902
#202763692
#202748192
C53×#35092
#1272453