Finite Group Databasebeta

← #28502030

Subgroups — #28502030

Order 15104.

342 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 55 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#21434961C211yesmaximal, characteristic
2#21426310C212yesmaximal, complemented
3#21422080C212yesmaximal, complemented
4#21422034C211yesmaximal, characteristic, complemented
5C59×#2098C211yesmaximal, characteristic, complemented
6#16837966—21no—
7#16837727—22no—
8#16837655C2212yes—
9#16837638—21no—
10#16837538—22no—
11#16837226—22no—
12#16837214—22no—
13#16837184C2212yescomplemented
14C59×#430C2211yescharacteristic, complemented
15C59×#424—21no—
16C59×#421—21no—
17C59×#408C2211yescharacteristic
18C59×H(3,ℤ/4)C2211yescharacteristic, complemented
19C22×D236—21no—
20C2×D472—21no—
21C2×D472D412yes—
22#14273723—41no—
23#14273716—41no—
24#14273701—22no—
25#14273701—22no—
26#14273695—41no—
27#14273695—41no—
28#14273686—21no—
29#14273667—22no—
30#14273661—21no—
31#14273659D412yes—
32#14273649—22no—
33#14273649—22no—
34C2×#41×C59D411yescharacteristic
35C2×M16×C59—21no—
36C2×M16×C59C2311yescharacteristic
37C22×C8×C59D411yescharacteristic
38#117×C59—21no—
39#108×C59—22no—
40C4⋊C8×C59—42no—
41#104×C59—42no—
42C22⋊C8×C59—22no—
43C23×D59—41no—
44#41×C59—41no—
45#41×C59—41no—
46C2×Q8×C59SD1611yescharacteristic
47C2×D4×C59—21no—
48C2×D4×C59SD1611yescharacteristic
49C22×C4×C59—21no—
50C22×C4×C59C2×D411yescharacteristic
51C2×C22⋊D59—42no—
52C2×C22⋊D59—42no—
53C2×D236—22no—
54C2×D236—41no—
55C2×D236—81no—
56C2×D236—81no—
57M16×C59—41no—
58M16×C59—41no—
59M16×C59—41no—
60C2×C8×C59—21no—
61C2×C8×C59—41no—
62C2×C8×C59—41no—
63C2×C8×C59C2×D412yes—
64C4⋊C4×C59—41no—
65C22⋊C4×C59—41no—
66C42×C59—22no—
67C22⋊Dic59—41no—
68(C2×D59)⋊C4—42no—
69(C2×D59)⋊C4—42no—
70C4⋊Dic59—22no—
71C4×Dic59—41no—
72D472—42no—
73D472—81no—
74C59⋊C16—42no—
75C23×C59—21no—
76C23×C59AGL(1,ℤ/8)11yescharacteristic
77C22×D59—42no—
78C22×D59—82no—
79C22×D59—162no—
80Q8×C59—41no—
81D4×C59—41no—
82D4×C59—81no—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59—42no—
89C2×C4×C59#12011yescharacteristic
90C2×C4×C59C2×SD1611yescharacteristic
91C22⋊D59—162no—
92C22⋊D59—162no—
93C2×Dic59—42no—
94D236—42no—
95D236—42no—
96D236—82no—
97C8×C59—22no—
98C8×C59—41no—
99C8×C59—81no—
100#12301—591nomaximal, Sylow
101C22×C59—21no—
102C22×C59—41no—
103C22×C59—42no—
104C22×C59—41no—
105C22×C59#45011yescharacteristic
106C2×D59—82no—
107C2×D59—82no—
108C2×D59—82no—
109C2×D59—164no—
110C4×C59—21no—
111C4×C59—41no—
112C4×C59—82no—
113C4×C59#45311yescharacteristic
114C4×C59#45811yescharacteristic
115Dic59—82no—
116#3199—591no—
117#2238—592no—
118#2098D5911yescharacteristic, complemented
119#1339—592no—
120#1321—591no—
121C2×C59—21no—
122C2×C59—41no—
123C2×C59—81no—
124C2×C59#161311yescharacteristic
125D59—162no—
126#535—1181no—
127#482—1182no—
128#469—592no—
129#466—1181no—
130#447—1182no—
131#430C2×D5911yescharacteristic, complemented
132#424—21no—
133#421—21no—
134#408C2×D5911yescharacteristic
135#365—1182no—
136#361—1182no—
137#349—592no—
138H(3,ℤ/4)C2×D5911yescharacteristic, complemented
139C59#1230111yescharacteristic, complemented, Sylow
140C2×#41C22⋊D5911yescharacteristic
141C22×D4—1181no—
142C2×D8—592no—
143C2×D8—1181no—
144C2×M16—21no—
145C2×M16C22×D5911yescharacteristic
146C22×C8C22⋊D5911yescharacteristic
147#127—2361no—
148#124—2361no—
149#121—1182no—
150#121—1182no—
151#120—2361no—
152#120—2361no—
153#117—21no—
154#117—1181no—
155M32—1182no—
156#108—22no—
157#107—1181no—
158#106—592no—
159C4⋊C8—42no—
160#104—42no—
161#102—1182no—
162#102—1182no—
163C22⋊C8—22no—
164C24—2361no—
165#41—41no—
166#41—41no—
167C2×Q8#1174567111yescharacteristic
168C2×D4—21no—
169C2×D4—1182no—
170C2×D4—4721no—
171C2×D4—4721no—
172C2×D4—2361no—
173C2×D4—2362no—
174C2×D4—2362no—
175C2×D4#1174567111yescharacteristic
176C22×C4—21no—
177C22×C4C2×C22⋊D5911yescharacteristic
178D8—4721no—
179D8—2362no—
180M16—41no—
181M16—41no—
182M16—41no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8D4×D5912yes—
187C4⋊C4—41no—
188C4⋊C4—1182no—
189C22⋊C4—41no—
190C22⋊C4—2362no—
191C22⋊C4—2361no—
192C22⋊C4—2362no—
193C42—22no—
194C42—2361no—
195C16—2362no—
196C23—21no—
197C23—9442no—
198C23—4722no—
199C23—2362no—
200C23#1427375211yescharacteristic
201Q8—41no—
202D4—41no—
203D4—81no—
204D4—9442no—
205D4—9442no—
206D4—4722no—
207D4—2362no—
208D4—2362no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—2362no—
216C2×C4#1427369511yescharacteristic
217C2×C4C2×#1174567111yescharacteristic
218C8—22no—
219C8—41no—
220C8—81no—
221C22—21no—
222C22—41no—
223C22—42no—
224C22—41no—
225C22—9444no—
226C22—4722no—
227C22—4722no—
228C22—4722no—
229C22#1683755211yescharacteristic
230C4—21no—
231C4—41no—
232C4—82no—
233C4—4722no—
234C4#1683756911yescharacteristic
235C4#1683759111yescharacteristic
236C2—21no—
237C2—41no—
238C2—81no—
239C2—9442no—
240C2#2142311511yescharacteristic
subgroupG/H
centerC2#21423115
commutator, cosocleC2×M16×C59C23
FrattiniC2×M16C22×D59
FittingC59×#2098C2
socleC2×C59#1613
prime radical#21434961C2
hypercenter#2098D59
hypocenterC59#12301
LCSG ▷ C2×M16×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2098

Complements

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

NH ≅ G/Ntype
2. #21426310C2semidirect
3. #21422080C2semidirect
4. #21422034C2semidirect
5. C59×#2098C2semidirect
13. #16837184C22semidirect
14. C59×#430C22semidirect
18. C59×H(3,ℤ/4)C22semidirect
118. #2098D59semidirect
131. #430C2×D59semidirect
138. H(3,ℤ/4)C2×D59semidirect
139. C59#12301semidirect

Maximal subgroups

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

subgroupindex
#214349612
#214263102
#214220802
#214220342
C59×#20982
#1230159