Finite Group Databasebeta

← #28502822

Subgroups — #28502822

Order 15104.

256 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 63 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#21426240C211yesmaximal, characteristic, complemented
2#21426132C211yesmaximal, characteristic
3#21423689C211yesmaximal, characteristic, complemented
4#21422890C211yesmaximal, characteristic, complemented
5#21422159C211yesmaximal, characteristic, complemented
6#21422141C211yesmaximal, characteristic, complemented
7C59×#1365C211yesmaximal, characteristic, complemented
8C2×D944—21no—
9#16837779C2211yescharacteristic
10#16837776—21no—
11#16837750—21no—
12C4×D472C2211yescharacteristic
13#16837467—21no—
14#16837229—21no—
15C4⋊(C59⋊C16)C2211yescharacteristic
16#16837204—21no—
17#16837202—21no—
18#16837129—21no—
19C8×C59⋊C8C2211yescharacteristic
20C59×#500C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×#11745671—42no—
24C2×D472—21no—
25C2×D472—21no—
26C2×D472D411yescharacteristic
27C2×C8×D59—41no—
28#14273724—21no—
29C4×D236—21no—
30D944—42no—
31C2×C59⋊C16—21no—
32#14273661D411yescharacteristic
33#14273649—21no—
34#14273647—41no—
35#14273647—41no—
36(C2×D59)⋊C8—41no—
37C4×C59⋊C8D411yescharacteristic
38C4×C59⋊C8D411yescharacteristic
39C4⋊Q8×C59—21no—
40C2×C16×C59—21no—
41#107×C59—21no—
42#107×C59—21no—
43C4⋊C8×C59—21no—
44C4×C8×C59C2311yescharacteristic
45C2×Q8×C59—41no—
46C2×D236—21no—
47C2×D236—41no—
48C2×D236—41no—
49C2×D236—41no—
50C2×C4×D59—41no—
51C2×C8×C59—41no—
52C2×C8×C59C2×D411yescharacteristic
53C2×C8×C59#4111yescharacteristic
54C4⋊C4×C59—41no—
55C4⋊C4×C59—41no—
56C42×C59C2×D411yescharacteristic
57#11745671—84no—
58(C2×D59)⋊C4—41no—
59C4⋊Dic59—21no—
60C2×C59⋊C8—21no—
61C2×C59⋊C8D811yescharacteristic
62C2×C59⋊C8D811yescharacteristic
63D472—21no—
64D472—41no—
65D472—41no—
66D472—81no—
67D472SD1612yescomplemented
68C8×D59—82no—
69C16×C59—41no—
70C59⋊C16—41no—
71C22×D59—41no—
72C22×D59—81no—
73Q8×C59—82no—
74C2×C4×C59—81no—
75C2×C4×C59#12111yescharacteristic
76C2×C4×C59C2×D811yescharacteristic
77C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
78C2×Dic59—41no—
79D236—22no—
80D236—41no—
81D236—82no—
82D236—81no—
83D236—81no—
84D236—81no—
85D236—81no—
86C4×D59—81no—
87C4×D59—81no—
88C8×C59—21no—
89C8×C59—81no—
90C8×C59C2×SD1611yescharacteristic
91C8×C59#13511yescharacteristic
92C59⋊C8—22no—
93C59⋊C8—21no—
94C59⋊C8—21no—
95#12418—591nomaximal, Sylow
96C22×C59#45911yescharacteristic
97C2×D59—42no—
98C2×D59—81no—
99C2×D59—81no—
100C2×D59—161no—
101C2×D59—161no—
102C2×D59—161no—
103C4×C59—21no—
104C4×C59—161no—
105C4×C59#46511yescharacteristic
106C4×C59#47111yescharacteristic
107C4×C59#50811yescharacteristic
108C4×C59#50911yescharacteristic
109Dic59—81no—
110#2226—591no—
111#2206—591no—
112#1706—591no—
113C8×D8—591no—
114#1367—591no—
115#1365D5911yescharacteristic, complemented
116#1361—591no—
117C2×C59#166511yescharacteristic
118C2×C59#221011yescharacteristic
119C2×C59#221311yescharacteristic
120D59—82no—
121D59—321no—
122C2×D16—1181no—
123#500C2×D5911yescharacteristic
124#493—591no—
125#492—1181no—
126#486—1181no—
127C4×D8—591no—
128C8×D4—1181no—
129#367—1181no—
130C4⋊C16—591no—
131C4⋊C16C2×D5911yescharacteristic
132#357—1181no—
133#357—1181no—
134#335C2×D5911yescharacteristic
135D4⋊C8—1181no—
136C82—591no—
137C59#1241811yescharacteristic, complemented, Sylow
138C2×SD16—2362no—
139C2×D8—591no—
140C2×D8—1181no—
141C2×D8—1181no—
142C22×C8—2361no—
143C4⋊Q8—21no—
144#127—1181no—
145C4×D4—1181no—
146D16—2362no—
147C2×C16—21no—
148C2×C16—1181no—
149#107—21no—
150#107—21no—
151#107—591no—
152C4⋊C8—21no—
153#102—1181no—
154#102—2361no—
155#102—2361no—
156C22⋊C8—2361no—
157C4×C8—591no—
158C4×C8—591no—
159C4×C8C22×D5911yescharacteristic
160C2×Q8—41no—
161C2×D4—1181no—
162C2×D4—2361no—
163C2×D4—2361no—
164C2×D4—2361no—
165C22×C4—2361no—
166SD16—4724no—
167D8—592no—
168D8—1181no—
169D8—4721no—
170D8—2361no—
171D8—2361no—
172C2×C8—41no—
173C2×C8—591no—
174C2×C8—591no—
175C2×C8—1181no—
176C2×C8—4722no—
177C2×C8#1174568511yescharacteristic
178C2×C8D4×D5911yescharacteristic
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—1181no—
182C22⋊C4—2361no—
183C42D4×D5911yescharacteristic
184C16—41no—
185C16—2361no—
186C23—4721no—
187C23—2361no—
188Q8—82no—
189D4—1182no—
190D4—4722no—
191D4—4721no—
192D4—4721no—
193D4—4721no—
194D4—4721no—
195D4—2361no—
196C2×C4—81no—
197C2×C4—4721no—
198C2×C4—4721no—
199C2×C4—2361no—
200C2×C4#1427370011yescharacteristic
201C2×C4D8×D5911yescharacteristic
202C2×C4#1427375411yescharacteristic
203C8—21no—
204C8—81no—
205C8—1182no—
206C8—1181no—
207C8—1181no—
208C8SD16×D5911yescharacteristic
209C8#1427375011yescharacteristic
210C22—9441no—
211C22—9441no—
212C22—9441no—
213C22—4721no—
214C22—4721no—
215C22—2362no—
216C22#1683759611yescharacteristic
217C4—21no—
218C4—161no—
219C4—4721no—
220C4#1683763411yescharacteristic
221C4#1683766811yescharacteristic
222C4#1683783511yescharacteristic
223C4#1683783911yescharacteristic
224C2—18881no—
225C2—4722no—
226C2#2142344011yescharacteristic
227C2#2142615411yescharacteristic
228C2#2142617111yescharacteristic
subgroupG/H
centerC22#16837596
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1365C2
socleC22×C59#459
prime radical#21426132C2
hypercenter#1365D59
hypocenterC59#12418
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1365

Complements

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

NH ≅ G/Ntype
1. #21426240C2semidirect
3. #21423689C2semidirect
4. #21422890C2semidirect
5. #21422159C2semidirect
6. #21422141C2semidirect
7. C59×#1365C2semidirect
67. D472SD16semidirect
115. #1365D59semidirect
137. C59#12418semidirect

Maximal subgroups

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

subgroupindex
#214262402
#214261322
#214236892
#214228902
#214221592
#214221412
C59×#13652
#1241859