Finite Group Databasebeta

← #28492409

Subgroups — #28492409

Order 15104.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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#21433901C211yesmaximal, characteristic
2#21422851C211yesmaximal, characteristic
3#21422787C211yesmaximal, characteristic
4#21422550C211yesmaximal, characteristic
5C2×Q8⋊(C59⋊C8)C211yesmaximal, characteristic
6#21421970C211yesmaximal, characteristic
7C59×#1801C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837709—22no—
10#16837697—21no—
11#16837687—21no—
12#16837442—21no—
13#16837426C2211yescharacteristic
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15#16837411—21no—
16C2×C4×C59⋊C8C2211yescharacteristic
17#16837152—22no—
18#16837150—21no—
19#16837147—21no—
20Q8⋊(C59⋊C8)—21no—
21Q8⋊(C59⋊C8)—21no—
22#16837132—21no—
23#16837132—21no—
24C2×C4×Q8×C59C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C59×#423C2211yescharacteristic
27C2×#101×C59—21no—
28C59×#343—21no—
29C2×#11745664—21no—
30C22×C59⋊C8—21no—
31C22×C59⋊C8—21no—
32#14273727D411yescharacteristic
33#14273727D411yescharacteristic
34#14273719—21no—
35#14273706—21no—
36#14273706—21no—
37C4×Dic118—21no—
38C4×Dic118—21no—
39#14273686—21no—
40#14273660—22no—
41#14273658—22no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)D412yes—
46#14273651—22no—
47#14273651—22no—
48C4×C59⋊C8—21no—
49C4×C59⋊C8D411yescharacteristic
50C4×C59⋊C8D411yescharacteristic
51C22×Q8×C59D411yescharacteristic
52C2×M16×C59—21no—
53C2×M16×C59—21no—
54C4×Q8×C59—21no—
55C4×Q8×C59—21no—
56C4×Q8×C59—21no—
57C4×Q8×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59D411yescharacteristic
60C2×C42×C59—21no—
61C2×C42×C59C2311yescharacteristic
62C4⋊C8×C59—21no—
63C4⋊C8×C59—21no—
64C4⋊C8×C59—21no—
65C4⋊C8×C59—21no—
66#101×C59—42no—
67C2×Q8×C59—21no—
68C2×Q8×C59—21no—
69C2×Q8×C59—21no—
70C2×Q8×C59—21no—
71C2×Q8×C59—21no—
72C2×Q8×C59SD1612yes—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59#4111yescharacteristic
77C22×C4×C59#4111yescharacteristic
78C2×Dic118—21no—
79C2×Dic118—21no—
80M16×C59—41no—
81M16×C59—41no—
82M16×C59—41no—
83M16×C59—41no—
84C2×C8×C59—41no—
85C2×C8×C59—41no—
86C4⋊C4×C59—21no—
87C4⋊C4×C59—21no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C42×C59—41no—
91C42×C59—41no—
92C42×C59C2×D411yescharacteristic
93C42×C59C2×D411yescharacteristic
94C42×C59C2×D411yescharacteristic
95C42×C59#4111yescharacteristic
96C22⋊Dic59—41no—
97C4⋊Dic59—21no—
98C4⋊Dic59—21no—
99C4⋊Dic59—21no—
100C4⋊Dic59—21no—
101Dic59⋊C4—41no—
102Dic59⋊C4—41no—
103C4×Dic59—41no—
104#11745664—41no—
105#11745664—41no—
106C2×C59⋊C8—21no—
107C2×C59⋊C8—22no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—42no—
110C2×C59⋊C8—41no—
111C2×C59⋊C8—81no—
112C23×C59#12311yescharacteristic
113Q8×C59—22no—
114Q8×C59—22no—
115Q8×C59—41no—
116Q8×C59—41no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—22no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—81no—
125C2×C4×C59#12011yescharacteristic
126C2×C4×C59#12111yescharacteristic
127C2×C4×C59#12111yescharacteristic
128C2×C4×C59#12111yescharacteristic
129C2×C4×C59#12311yescharacteristic
130C2×C4×C59#12411yescharacteristic
131C2×C4×C59C2×SD1611yescharacteristic
132C2×C4×C59#13511yescharacteristic
133C2×C4×C59#13711yescharacteristic
134C2×C4×C59#13711yescharacteristic
135C2×Dic59—41no—
136C2×Dic59—41no—
137Dic118—41no—
138Dic118—41no—
139Dic118—81no—
140C8×C59—81no—
141C8×C59—81no—
142C59⋊C8—22no—
143C59⋊C8—41no—
144C59⋊C8—42no—
145C59⋊C8—81no—
146#10462—591nomaximal, Sylow
147C22×C59—21no—
148C22×C59—21no—
149C22×C59#39411yescharacteristic
150C22×C59#48311yescharacteristic
151C22×C59#48511yescharacteristic
152C4×C59—21no—
153C4×C59—21no—
154C4×C59—21no—
155C4×C59—42no—
156C4×C59—81no—
157C4×C59#44811yescharacteristic
158C4×C59#45211yescharacteristic
159C4×C59#46511yescharacteristic
160C4×C59#47011yescharacteristic
161C4×C59#47111yescharacteristic
162C4×C59#47311yescharacteristic
163Dic59—81no—
164Dic59—81no—
165#3078—591no—
166#1801D5911yescharacteristic
167#1566—591no—
168#1550—591no—
169#1493—591no—
170C2×Q8⋊C8—591no—
171#1292—591no—
172C2×C59—21no—
173C2×C59—21no—
174C2×C59#203511yescharacteristic
175C2×C59#205011yescharacteristic
176C2×C59#205211yescharacteristic
177#533—591no—
178C2×C4×Q8C2×D5911yescharacteristic
179#479—1182no—
180#477—1181no—
181#475—1181no—
182#428—1181no—
183#423—591no—
184#423C2×D5911yescharacteristic
185#423C2×D5911yescharacteristic
186C2×C4⋊C8—591no—
187#419—1181no—
188C2×#101—21no—
189C2×C4×C8—591no—
190#343—21no—
191#334—1182no—
192#333—1181no—
193#332—1181no—
194Q8⋊C8—1181no—
195Q8⋊C8—1181no—
196Q8⋊C8—1181no—
197Q8⋊C8—1181no—
198C59#1046211yescharacteristic, complemented, Sylow
199C22×Q8C22⋊D5911yescharacteristic
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—1181no—
203C22×C8—1181no—
204C22×C8—1181no—
205C4⋊Q8—591no—
206C4⋊Q8—591no—
207#125—1181no—
208C22⋊Q8—1181no—
209C22⋊Q8—1181no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—1181no—
215C4×Q8—1181no—
216#117—1181no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C22⋊D5911yescharacteristic
219C2×C42—21no—
220C2×C42C22×D5911yescharacteristic
221#107—1182no—
222#106—1182no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—592no—
228C4⋊C8—1181no—
229C4⋊C8—1181no—
230C4⋊C8—1181no—
231#103—1182no—
232#103—1182no—
233#101—42no—
234C4×C8—591no—
235C4×C8—591no—
236C4×C8—1181no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—1181no—
243C2×Q8—1181no—
244C2×Q8#1174567012yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4#1174568511yescharacteristic
248C22×C4#1174568511yescharacteristic
249C22×C4C2×C22⋊D5911yescharacteristic
250M16—41no—
251M16—41no—
252M16—41no—
253M16—41no—
254M16—2361no—
255M16—2361no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—1181no—
259C2×C8—1182no—
260C2×C8—1181no—
261C2×C8—4721no—
262C2×C8—2362no—
263C2×C8—2361no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—1181no—
269C4⋊C4—1181no—
270C4⋊C4—1181no—
271C4⋊C4—1181no—
272C4⋊C4—2361no—
273C4⋊C4—2361no—
274C22⋊C4—2361no—
275C42—41no—
276C42—41no—
277C42—2361no—
278C42D4×D5911yescharacteristic
279C42D4×D5911yescharacteristic
280C42D4×D5911yescharacteristic
281C42#1174568711yescharacteristic
282C23#1427371011yescharacteristic
283Q8—22no—
284Q8—22no—
285Q8—41no—
286Q8—41no—
287Q8—4721no—
288Q8—2361no—
289Q8—2361no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—81no—
298C2×C4—2361no—
299C2×C4—2361no—
300C2×C4#1427369511yescharacteristic
301C2×C4#1427369911yescharacteristic
302C2×C4#1427370011yescharacteristic
303C2×C4#1427370011yescharacteristic
304C2×C4#1427370911yescharacteristic
305C2×C4#1427371311yescharacteristic
306C2×C4C2×#1174567011yescharacteristic
307C2×C4#1427374711yescharacteristic
308C2×C4#1427375711yescharacteristic
309C2×C4#1427375811yescharacteristic
310C8—81no—
311C8—81no—
312C8—1182no—
313C8—4721no—
314C8—2361no—
315C8—2362no—
316C22—21no—
317C22—21no—
318C22#1683731711yescharacteristic
319C22#1683773211yescharacteristic
320C22#1683774311yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—42no—
325C4—81no—
326C4—4721no—
327C4—4721no—
328C4#1683754011yescharacteristic
329C4#1683756211yescharacteristic
330C4#1683762911yescharacteristic
331C4#1683765811yescharacteristic
332C4#1683766811yescharacteristic
333C4#1683767611yescharacteristic
334C2—21no—
335C2—21no—
336C2#2142529211yescharacteristic
337C2#2142538111yescharacteristic
338C2#2142539511yescharacteristic
subgroupG/H
centerC22#16837317
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1801C2
socleC22×C59#394
prime radicalC23×C59#123
hypercenter#1801D59
hypocenterC59#10462
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
198. C59#10462semidirect

Maximal subgroups

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

subgroupindex
#214339012
#214228512
#214227872
#214225502
C2×Q8⋊(C59⋊C8)2
#214219702
C59×#18012
#1046259