Finite Group Databasebeta

← #28477697

Subgroups — #28477697

Order 15104.

436 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 85 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#21432058C211yesmaximal, characteristic
2#21425234C211yesmaximal, characteristic
3#21425196C211yesmaximal, characteristic
4#21421921C211yesmaximal, characteristic
5#21421921C211yesmaximal, characteristic
6#21421919C211yesmaximal, characteristic
7C59×#2057C211yesmaximal, characteristic
8C22×#11745672—21no—
9C2×#14273651—21no—
10C2×#14273650—21no—
11#16837335—21no—
12#16837334—21no—
13#16837322—21no—
14#16837308C2211yescharacteristic
15#16837308C2211yescharacteristic
16#16837306—21no—
17#16837301—21no—
18#16837288—21no—
19#16837164—21no—
20#16837157C2211yescharacteristic
21#16837157C2211yescharacteristic
22C2×C4⋊Q8×C59C2211yescharacteristic
23C2×#107×C59—21no—
24C59×#389C2211yescharacteristic
25C59×#340—21no—
26C59×#336C2211yescharacteristic
27C22×Dic118D411yescharacteristic
28C22×Dic118D411yescharacteristic
29C2×#11745672—42no—
30C2×#11745672—42no—
31C2×#11745672—42no—
32C22×C59⋊C8—21no—
33C22×C59⋊C8—21no—
34C2×C4⋊Dic59—21no—
35C2×C4⋊Dic59—21no—
36C2×C4⋊Dic59D411yescharacteristic
37C2×C4⋊Dic59D411yescharacteristic
38C2×Dic59⋊C4—21no—
39C2×Dic59⋊C4—21no—
40C2×C4×Dic59—21no—
41#14273651—42no—
42#14273650—42no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C22×Q8×C59—21no—
51C22×C8×C59—21no—
52C4⋊Q8×C59—42no—
53C2×C4⋊C4×C59—21no—
54C2×C4⋊C4×C59—21no—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59D411yescharacteristic
57C2×C4⋊C4×C59D411yescharacteristic
58C2×C42×C59C2311yescharacteristic
59#107×C59—42no—
60#95×C59—21no—
61C2×Q8×C59—22no—
62C2×Q8×C59—41no—
63C2×Q8×C59—41no—
64C2×Q8×C59—41no—
65C2×Q8×C59—41no—
66C2×Q8×C59—41no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—41no—
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C2×Dic118—21no—
78C2×Dic118—21no—
79C2×Dic118—21no—
80C2×Dic118—41no—
81C2×Dic118—41no—
82C2×Dic118—41no—
83C2×Dic118—41no—
84C2×Dic118Q1612yescomplemented
85C2×C8×C59—22no—
86C2×C8×C59—41no—
87C2×C8×C59—41no—
88C4⋊C4×C59—21no—
89C4⋊C4×C59—22no—
90C4⋊C4×C59—22no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59—41no—
95C4⋊C4×C59—41no—
96C4⋊C4×C59Q1612yescomplemented
97C42×C59—41no—
98#11745672—48no—
99C4⋊Dic59—41no—
100C4⋊Dic59—41no—
101C4⋊Dic59—81no—
102C4⋊Dic59—81no—
103Dic59⋊C4—81no—
104Dic59⋊C4—81no—
105C4×Dic59—81no—
106C2×C59⋊C8—22no—
107C2×C59⋊C8—41no—
108C2×C59⋊C8—41no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8—41no—
111C2×C59⋊C8—41no—
112C23×C59#12011yescharacteristic
113Q8×C59—42no—
114Q8×C59—42no—
115Q8×C59—81no—
116Q8×C59—81no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—22no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—42no—
130C2×C4×C59—81no—
131C2×C4×C59—81no—
132C2×C4×C59C2×Q1611yescharacteristic
133C2×C4×C59C2×Q1611yescharacteristic
134C2×C4×C59#13511yescharacteristic
135C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C59#13711yescharacteristic
137C2×C4×C59#13711yescharacteristic
138C2×Dic59—22no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150Dic118—42no—
151Dic118—42no—
152Dic118—81no—
153Dic118—81no—
154C8×C59—42no—
155C59⋊C8—42no—
156C59⋊C8—81no—
157#7784—591nomaximal, Sylow
158C22×C59#44911yescharacteristic
159C22×C59#45111yescharacteristic
160C22×C59#45111yescharacteristic
161C22×C59#45311yescharacteristic
162C22×C59#45511yescharacteristic
163C22×C59#45611yescharacteristic
164C22×C59B(4,𝔽2)11yescharacteristic
165C4×C59—21no—
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—21no—
170C4×C59—21no—
171C4×C59—42no—
172C4×C59—81no—
173C4×C59—82no—
174Dic59—42no—
175Dic59—81no—
176Dic59—81no—
177Dic59—81no—
178#2806—591no—
179#2057D5911yescharacteristic
180#2024—591no—
181#2016—591no—
182#1272—591no—
183#1272—591no—
184#1271—591no—
185C2×C59#160011yescharacteristic
186C2×C59#160211yescharacteristic
187C2×C59#160511yescharacteristic
188C2×C59#163311yescharacteristic
189C2×C59#164411yescharacteristic
190C2×C59#164811yescharacteristic
191C2×C59#166111yescharacteristic
192C22×Q16—1181no—
193C2×C4⋊Q8C2×D5911yescharacteristic
194C2×#107—21no—
195C2×#103—1181no—
196C2×#103—1181no—
197#398—1181no—
198#398—1181no—
199#395—1181no—
200#391—591no—
201#391—591no—
202#391—1181no—
203#389—1181no—
204#389C2×D5911yescharacteristic
205#387—1181no—
206#340—21no—
207#340—1181no—
208#336—591no—
209#336—591no—
210#336C2×D5911yescharacteristic
211C59#778411yescharacteristic, complemented, Sylow
212C22×Q8—21no—
213C22×Q8—591no—
214C22×Q8—591no—
215C2×Q16—2362no—
216C2×Q16—2362no—
217C2×Q16—2362no—
218C22×C8—21no—
219C22×C8—1181no—
220C22×C8—1181no—
221C4⋊Q8—42no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—591no—
226C2×C4⋊C4—591no—
227C2×C4⋊C4—1181no—
228C2×C4⋊C4—1181no—
229C2×C4⋊C4—1181no—
230C2×C4⋊C4—1181no—
231C2×C4⋊C4C22⋊D5911yescharacteristic
232C2×C4⋊C4C22⋊D5911yescharacteristic
233C2×C42—1181no—
234C2×C42C22×D5911yescharacteristic
235#107—42no—
236#103—2362no—
237#103—2362no—
238#95—21no—
239#95—1181no—
240#95—1181no—
241#95—1181no—
242#95—1181no—
243#95—1181no—
244#95—1181no—
245#95—1181no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—592no—
253C2×Q8—1181no—
254C2×Q8—1181no—
255C2×Q8—1181no—
256C2×Q8—2361no—
257C2×Q8—2361no—
258C2×Q8—2361no—
259C2×Q8—2361no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—1181no—
264C22×C4—1181no—
265C22×C4—1181no—
266C22×C4—1181no—
267C22×C4D4×D5911yescharacteristic
268C22×C4D4×D5911yescharacteristic
269C22×C4C2×C22⋊D5911yescharacteristic
270Q16—2368no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—1182no—
275C2×C8—2361no—
276C2×C8—2361no—
277C2×C8—2361no—
278C2×C8—2361no—
279C2×C8—2361no—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—4721no—
289C4⋊C4—4721no—
290C4⋊C4—4721no—
291C4⋊C4—4721no—
292C4⋊C4—2361no—
293C4⋊C4—2361no—
294C4⋊C4#1174567212yescomplemented
295C42—41no—
296C42—4721no—
297C23#1427369511yescharacteristic
298Q8—42no—
299Q8—42no—
300Q8—81no—
301Q8—81no—
302Q8—4721no—
303Q8—4721no—
304Q8—2362no—
305Q8—2362no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—1182no—
322C2×C4—2361no—
323C2×C4—2361no—
324C2×C4—2361no—
325C2×C4—2361no—
326C2×C4—2361no—
327C2×C4—2361no—
328C2×C4—2361no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4—2361no—
332C2×C4—2361no—
333C2×C4Q16×D5911yescharacteristic
334C2×C4C2×#1174567211yescharacteristic
335C2×C4#1427374611yescharacteristic
336C2×C4#1427375111yescharacteristic
337C2×C4#1427375711yescharacteristic
338C2×C4#1427375911yescharacteristic
339C8—42no—
340C8—4721no—
341C8—2362no—
342C22#1683754411yescharacteristic
343C22#1683755611yescharacteristic
344C22#1683755711yescharacteristic
345C22#1683756811yescharacteristic
346C22#1683757811yescharacteristic
347C22#1683758411yescharacteristic
348C22#1683758811yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—42no—
356C4—81no—
357C4—82no—
358C4—4721no—
359C4—4721no—
360C4—4721no—
361C4—2362no—
362C2#2142303911yescharacteristic
363C2#2142305111yescharacteristic
364C2#2142306911yescharacteristic
365C2#2142323511yescharacteristic
366C2#2142330911yescharacteristic
367C2#2142333511yescharacteristic
368C2#2142341111yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2057C2
socle, prime radicalC23×C59#120
hypercenter#2057D59
hypocenterC59#7784
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
84. C2×Dic118Q16semidirect
96. C4⋊C4×C59Q16semidirect
211. C59#7784semidirect
294. C4⋊C4#11745672semidirect

Maximal subgroups

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

subgroupindex
#214320582
#214252342
#214251962
#214219212
#214219192
C59×#20572
#778459