Finite Group Databasebeta

← #28501969

Subgroups — #28501969

Order 15104.

354 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 71 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#21426279C212yesmaximal, complemented
2#21425589C211yesmaximal, characteristic, complemented
3#21422105C212yesmaximal, complemented
4#21422036C211yesmaximal, characteristic, complemented
5C59×#3245C211yesmaximal, characteristic
6C2×#14273661C2211yescharacteristic
7#16837300—21no—
8#16837229—22no—
9#16837203—22no—
10C22⋊(C59⋊C16)C2212yes—
11#16837165—21no—
12#16837165C2211yescharacteristic
13C59×#532—21no—
14C59×#474—22no—
15C59×#466—21no—
16C59×#466C2212yes—
17C59×#459—22no—
18C2×C4⋊C8×C59C2211yescharacteristic
19C2×C4⋊Dic59—21no—
20C2×C4⋊Dic59—21no—
21C2×Dic59⋊C4—41no—
22C2×C59⋊C16—22no—
23#14273661—22no—
24C59⋊#95—41no—
25C2×#41×C59—41no—
26C2×D8×C59—21no—
27C2×D8×C59D412yes—
28C22×C8×C59D411yescharacteristic
29C22×C8×C59C2311yescharacteristic
30C4⋊Q8×C59—21no—
31#127×C59—21no—
32#124×C59—41no—
33#121×C59—22no—
34#121×C59—41no—
35C4×D4×C59—22no—
36C4×D4×C59—41no—
37C2×C42×C59D411yescharacteristic
38#107×C59—21no—
39#107×C59D412yes—
40C4⋊C8×C59—22no—
41#102×C59—22no—
42#102×C59—22no—
43#41×C59—82no—
44#41×C59—82no—
45C2×Q8×C59—41no—
46C2×D4×C59—22no—
47C2×D4×C59—41no—
48C2×D4×C59—41no—
49C2×D4×C59—42no—
50C22×C4×C59—42no—
51C22×C4×C59D811yescharacteristic
52C22×C4×C59D811yescharacteristic
53C22×C4×C59C2×D411yescharacteristic
54C22×Dic59—41no—
55C22×Dic59—41no—
56D8×C59—42no—
57D8×C59—81no—
58C2×C8×C59—21no—
59C2×C8×C59—41no—
60C2×C8×C59—41no—
61C2×C8×C59C2×D412yes—
62C4⋊C4×C59—22no—
63C4⋊C4×C59—41no—
64C22⋊C4×C59—42no—
65C22⋊C4×C59—42no—
66C42×C59—21no—
67C42×C59—21no—
68C4⋊Dic59—22no—
69C4⋊Dic59—41no—
70C4⋊Dic59—41no—
71C4⋊Dic59—41no—
72Dic59⋊C4—82no—
73C59⋊C16—42no—
74C23×C59—42no—
75C23×C59C2×D811yescharacteristic
76Q8×C59—82no—
77D4×C59—42no—
78D4×C59—42no—
79D4×C59—82no—
80D4×C59—82no—
81D4×C59—82no—
82D4×C59—82no—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—21no—
88C2×C4×C59—41no—
89C2×C4×C59—42no—
90C2×C4×C59—84no—
91C2×C4×C59SD3212yes—
92C2×C4×C59#12011yescharacteristic
93C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
94C2×Dic59—42no—
95C2×Dic59—81no—
96C2×Dic59—81no—
97C2×Dic59—81no—
98C2×Dic59—81no—
99C2×Dic59—81no—
100C8×C59—22no—
101C8×C59—41no—
102#12291—591nomaximal, Sylow
103C22×C59—21no—
104C22×C59—21no—
105C22×C59—82no—
106C22×C59—82no—
107C22×C59—82no—
108C22×C59#44711yescharacteristic
109C22×C59C2×SD3211yescharacteristic
110C22×C59#50911yescharacteristic
111C4×C59—21no—
112C4×C59—22no—
113C4×C59—41no—
114C4×C59—82no—
115C4×C59#45611yescharacteristic
116C4×C59#45811yescharacteristic
117Dic59—82no—
118Dic59—161no—
119#3245D5911yescharacteristic
120#2233—592no—
121#2087—591no—
122#1347—592no—
123#1322—591no—
124C2×C59—21no—
125C2×C59—21no—
126C2×C59—162no—
127C2×C59#161011yescharacteristic
128C2×C59#218711yescharacteristic
129C2×C59#219011yescharacteristic
130#532—21no—
131#474—22no—
132#466—21no—
133#466C2×D5912yes—
134#459—22no—
135C2×#107—591no—
136C2×C4⋊C8C2×D5911yescharacteristic
137#389—1181no—
138#367—1182no—
139#357—1182no—
140C22⋊C16—592no—
141#340—591no—
142#340—1181no—
143C59#1229111yescharacteristic, complemented, Sylow
144C2×#41—41no—
145C2×D8—21no—
146C2×D8C22⋊D5912yes—
147C22×C8C22⋊D5911yescharacteristic
148C22×C8C22×D5911yescharacteristic
149C4⋊Q8—21no—
150#127—21no—
151#124—41no—
152#121—22no—
153#121—41no—
154C4×D4—22no—
155C4×D4—41no—
156C2×C4⋊C4—1181no—
157C2×C4⋊C4—1181no—
158C2×C4⋊C4—2361no—
159C2×C42C22⋊D5911yescharacteristic
160C2×C16—1182no—
161#107—21no—
162#107—1182no—
163#107C22⋊D5912yes—
164C4⋊C8—22no—
165#102—22no—
166#102—22no—
167#95—2361no—
168#41—82no—
169#41—82no—
170C2×Q8—41no—
171C2×D4—22no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—42no—
175C22×C4—42no—
176C22×C4—2361no—
177C22×C4—2361no—
178C22×C4#1174566911yescharacteristic
179C22×C4#1174566911yescharacteristic
180C22×C4C2×C22⋊D5911yescharacteristic
181D8—42no—
182D8—81no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8C2×C22⋊D5912yes—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—1182no—
190C4⋊C4—4722no—
191C4⋊C4—2361no—
192C4⋊C4—2361no—
193C4⋊C4—2361no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C42—21no—
197C42—21no—
198C16—2362no—
199C23—42no—
200C23C2×#1174566911yescharacteristic
201Q8—82no—
202D4—42no—
203D4—42no—
204D4—82no—
205D4—82no—
206D4—82no—
207D4—82no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—84no—
216C2×C4—4721no—
217C2×C4—4721no—
218C2×C4—4721no—
219C2×C4—4721no—
220C2×C4—4721no—
221C2×C4—2362no—
222C2×C4#1427367112yes—
223C2×C4#1427369611yescharacteristic
224C2×C4#1427375311yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22#1683753611yescharacteristic
233C22C2×#1427367111yescharacteristic
234C22#1683783811yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—82no—
239C4—9441no—
240C4—4722no—
241C4#1683758511yescharacteristic
242C4#1683759211yescharacteristic
243C2—21no—
244C2—21no—
245C2—162no—
246C2#2142309911yescharacteristic
247C2#2142602511yescharacteristic
248C2#2142604211yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
Fitting, prime radicalC59×#3245C2
socleC22×C59#447
hypercenter#3245D59
hypocenterC59#12291
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #21426279C2semidirect
2. #21425589C2semidirect
3. #21422105C2semidirect
4. #21422036C2semidirect
143. C59#12291semidirect

Maximal subgroups

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

subgroupindex
#214262792
#214255892
#214221052
#214220362
C59×#32452
#1229159