Finite Group Databasebeta

← #28502384

Subgroups — #28502384

Order 15104.

312 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 69 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#21436964C211yesmaximal, characteristic, complemented
2#21426319C211yesmaximal, characteristic, complemented
3#21426128C211yesmaximal, characteristic
4#21422155C211yesmaximal, characteristic, complemented
5Q8⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
6#21422056C211yesmaximal, characteristic, complemented
7C59×#1668C211yesmaximal, characteristic, complemented
8#16838112—21no—
9C2×#14273668—21no—
10#16837807C2211yescharacteristic
11#16837804—21no—
12#16837703—22no—
13#16837693—21no—
14C4×D472—21no—
15C4×D472C2211yescharacteristic
16C4⋊(C59⋊C16)C2211yescharacteristic
17#16837203—22no—
18#16837203—21no—
19C4×C59⋊C16C2211yescharacteristic
20C59×#493C2211yescharacteristic
21C59×#460—21no—
22C8×Q8×C59C2211yescharacteristic
23C59×#335C2211yescharacteristic
24C59×#331—21no—
25C2×D472D411yescharacteristic
26#14273727—21no—
27#14273719—22no—
28#14273705—42no—
29#14273703—41no—
30C4×D236—22no—
31C4×D236—21no—
32C4⋊C4×D59—41no—
33#14273668—42no—
34C2×C59⋊C16—21no—
35C2×C59⋊C16—21no—
36#14273661—21no—
37#14273661D412yes—
38#14273661D411yescharacteristic
39#14273659—22no—
40#14273649—22no—
41#14273649—21no—
42C2×SD16×C59—41no—
43C2×D8×C59—21no—
44C2×D8×C59—21no—
45#127×C59—21no—
46C4×Q8×C59D411yescharacteristic
47C4⋊C8×C59—21no—
48C4⋊C8×C59—21no—
49C4⋊C8×C59D411yescharacteristic
50#102×C59—41no—
51C4×C8×C59—21no—
52C4×C8×C59C2311yescharacteristic
53C2×Q8×C59D811yescharacteristic
54C2×D4×C59—41no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D236—21no—
58C2×C4×D59—41no—
59C2×C4×D59—42no—
60C2×Dic118—41no—
61SD16×C59—82no—
62D8×C59—42no—
63D8×C59—41no—
64D8×C59—41no—
65C2×C8×C59—21no—
66C2×C8×C59—41no—
67C2×C8×C59C2×D411yescharacteristic
68C2×C8×C59C2×D411yescharacteristic
69C4⋊C4×C59—21no—
70C4⋊C4×C59D811yescharacteristic
71C42×C59—21no—
72C42×C59C2×D411yescharacteristic
73(C2×D59)⋊C4—42no—
74(C2×D59)⋊C4—41no—
75C4⋊Dic59—21no—
76C4⋊Dic59—22no—
77C4⋊Dic59—41no—
78Dic59⋊C4—42no—
79Dic59⋊C4—42no—
80D472—21no—
81D472SD1612yescomplemented
82C59⋊C16—41no—
83C59⋊C16—41no—
84C23×C59—81no—
85C22×D59—41no—
86Q8×C59—21no—
87Q8×C59—21no—
88D4×C59—81no—
89D4×C59—81no—
90D4×C59—81no—
91D4×C59—81no—
92D4×C59—161no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59#12011yescharacteristic
96C2×C4×C59C2×D811yescharacteristic
97C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
98C2×Dic59—41no—
99C2×Dic59—42no—
100D236—22no—
101D236—41no—
102C4×D59—81no—
103C4×D59—81no—
104C4×D59—162no—
105Dic118—161no—
106C8×C59—21no—
107C8×C59—41no—
108C8×C59—81no—
109C8×C59C2×SD1611yescharacteristic
110C8×C59#13711yescharacteristic
111#12355—591nomaximal, Sylow
112C22×C59—161no—
113C22×C59—161no—
114C22×C59—161no—
115C22×C59#44711yescharacteristic
116C2×D59—42no—
117C2×D59—81no—
118C2×D59—81no—
119C4×C59—21no—
120C4×C59—41no—
121C4×C59—41no—
122C4×C59#44811yescharacteristic
123C4×C59#45411yescharacteristic
124C4×C59#50811yescharacteristic
125C4×C59#50911yescharacteristic
126Dic59—81no—
127Dic59—82no—
128#3380—591no—
129#2240—591no—
130#2206—591no—
131#1668D5911yescharacteristic, complemented
132#1365—591no—
133Q8⋊C16—591no—
134#1331—591no—
135C2×C59—321no—
136C2×C59#162711yescharacteristic
137C2×C59#218611yescharacteristic
138C2×C59#219211yescharacteristic
139D59—82no—
140#554—1181no—
141C2×D16—1181no—
142#500—591no—
143#499—1181no—
144#493C2×D5911yescharacteristic
145#478—1182no—
146#476—1181no—
147#460—21no—
148C8×Q8C2×D5911yescharacteristic
149C4×D8—591no—
150C4×D8—1181no—
151C4⋊C16—591no—
152#357—1181no—
153#357—1182no—
154C4×C16—591no—
155#335C2×D5911yescharacteristic
156#331—21no—
157C59#1235511yescharacteristic, complemented, Sylow
158C2×SD16—41no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—591no—
162C4⋊Q8—1181no—
163#127—21no—
164#125—1182no—
165C22⋊Q8—2362no—
166C22⋊Q8—2361no—
167C4×Q8C22⋊D5911yescharacteristic
168C4×D4—1182no—
169C4×D4—1181no—
170C2×C4⋊C4—2361no—
171D16—2362no—
172C2×C16—1181no—
173C2×C16—1181no—
174#107—591no—
175#107—592no—
176#107—1181no—
177#106—1182no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8C22⋊D5911yescharacteristic
181#102—41no—
182#102—1182no—
183#102—1181no—
184C4×C8—21no—
185C4×C8C22×D5911yescharacteristic
186C2×Q8—2361no—
187C2×Q8#1174566911yescharacteristic
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—1181no—
192C22×C4—2362no—
193C22×C4—2361no—
194SD16—82no—
195D8—42no—
196D8—41no—
197D8—41no—
198D8—592no—
199D8—1181no—
200C2×C8—21no—
201C2×C8—41no—
202C2×C8D4×D5911yescharacteristic
203C2×C8D4×D5911yescharacteristic
204C4⋊C4—21no—
205C4⋊C4—1181no—
206C4⋊C4—1182no—
207C4⋊C4—2362no—
208C4⋊C4—2362no—
209C4⋊C4—2361no—
210C4⋊C4#1174566911yescharacteristic
211C22⋊C4—2362no—
212C22⋊C4—2361no—
213C42—21no—
214C42C2×C22⋊D5911yescharacteristic
215C16—2361no—
216C16—2361no—
217C23—81no—
218C23—2361no—
219Q8—21no—
220Q8—21no—
221Q8—9441no—
222D4—81no—
223D4—81no—
224D4—81no—
225D4—81no—
226D4—161no—
227D4—1182no—
228D4—2361no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—9442no—
232C2×C4—4721no—
233C2×C4—4721no—
234C2×C4—2361no—
235C2×C4—2362no—
236C2×C4#1427369511yescharacteristic
237C2×C4C2×#1174566911yescharacteristic
238C2×C4#1427375211yescharacteristic
239C8—21no—
240C8—41no—
241C8—81no—
242C8SD16×D5911yescharacteristic
243C8#1427375611yescharacteristic
244C22—161no—
245C22—161no—
246C22—161no—
247C22—4721no—
248C22—4721no—
249C22—2362no—
250C22#1683753511yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—4721no—
255C4—4722no—
256C4#1683753911yescharacteristic
257C4#1683757311yescharacteristic
258C4#1683783211yescharacteristic
259C4#1683783811yescharacteristic
260C2—321no—
261C2—4722no—
262C2#2142319711yescharacteristic
263C2#2142601711yescharacteristic
264C2#2142605111yescharacteristic
subgroupG/H
centerC22#16837535
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1668C2
socleC22×C59#447
prime radical#21426128C2
hypercenter#1668D59
hypocenterC59#12355
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1668

Complements

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

NH ≅ G/Ntype
1. #21436964C2semidirect
2. #21426319C2semidirect
4. #21422155C2semidirect
5. Q8⋊(C59⋊C16)C2semidirect
6. #21422056C2semidirect
7. C59×#1668C2semidirect
81. D472SD16semidirect
131. #1668D59semidirect
157. C59#12355semidirect

Maximal subgroups

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

subgroupindex
#214369642
#214263192
#214261282
#214221552
Q8⋊(C59⋊C16)2
#214220562
C59×#16682
#1235559