Subgroups — #28504038

Order 15104.

374 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837178	C₂	1	2	yes	maximal
3	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#21422174	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
8	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₄⋊(C₅₉⋊C₁₆)	C₄	1	4	yes	complemented
10	#16837178	—	2	4	no	—
11	C₄×C₅₉⋊C₁₆	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₈×C₅₉	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₉	C₄	1	1	yes	characteristic
16	C₅₉×#336	C₄	1	2	yes	—
17	C₂×C₅₉⋊C₁₆	—	2	2	no	—
18	C₂×C₅₉⋊C₁₆	—	2	1	no	—
19	C₂×C₅₉⋊C₁₆	—	2	1	no	—
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
23	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
25	C₂×C₅₉⋊C₁₆	Q₈	1	2	yes	—
26	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₉	C₈	1	2	yes	—
30	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₅₉	C₈	1	4	yes	—
33	C₄×C₈×C₅₉	—	2	2	no	—
34	C₄×C₈×C₅₉	—	2	2	no	—
35	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
37	#95×C₅₉	C₈	1	2	yes	—
38	C₂²×C₄×C₅₉	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	—	2	2	no	—
43	C₂×C₈×C₅₉	—	2	1	no	—
44	C₂×C₈×C₅₉	—	2	1	no	—
45	C₂×C₈×C₅₉	—	2	2	no	—
46	C₂×C₈×C₅₉	C₄²	1	2	yes	—
47	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
48	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₅₉	—	2	4	no	—
55	C₄²×C₅₉	—	2	2	no	—
56	C₄²×C₅₉	—	2	2	no	—
57	C₄²×C₅₉	C₄²	1	2	yes	—
58	C₄²×C₅₉	C₂×C₈	1	2	yes	—
59	C₅₉⋊C₁₆	—	2	2	no	—
60	C₅₉⋊C₁₆	—	2	4	no	—
61	C₅₉⋊C₁₆	C₄⋊C₄	1	4	yes	complemented
62	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	—	2	4	no	—
65	C₂×C₄×C₅₉	—	2	4	no	—
66	C₂×C₄×C₅₉	—	2	4	no	—
67	C₂×C₄×C₅₉	C₄×C₈	1	2	yes	—
68	C₂×C₄×C₅₉	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₅₉	M₃₂	1	4	yes	—
70	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
75	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
76	C₈×C₅₉	—	2	2	no	—
77	C₈×C₅₉	—	2	4	no	—
78	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₅₉	#117	1	1	yes	characteristic
80	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
81	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
82	#12662	—	59	1	no	maximal, Sylow
83	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
84	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
85	C₂²×C₅₉	C₈×Q₈	1	1	yes	characteristic
86	C₂²×C₅₉	C₂×M₃₂	1	2	yes	—
87	C₂²×C₅₉	#504	1	2	yes	—
88	C₄×C₅₉	—	2	2	no	—
89	C₄×C₅₉	—	4	4	no	—
90	C₄×C₅₉	#346	1	4	yes	complemented
91	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₅₉	#405	1	1	yes	characteristic
93	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
94	C₄×C₅₉	C₈×Q₈	1	1	yes	characteristic
95	C₂×C₄⋊C₁₆	—	59	1	no	—
96	C₂×#346	—	59	2	no	—
97	C₂×C₄×C₁₆	—	59	1	no	—
98	C₈×C₄⋊C₄	D₅₉	1	1	yes	characteristic
99	#1374	—	59	2	no	—
100	C₂×C₅₉	C₈×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₅₉	C₂×#346	1	1	yes	characteristic
102	C₂×C₅₉	#2108	1	1	yes	characteristic
103	C₂×C₅₉	#2169	1	2	yes	—
104	C₂×C₅₉	#2183	1	2	yes	—
105	C₂²×C₁₆	—	59	1	no	—
106	C₂²×C₁₆	—	59	1	no	—
107	C₂²×C₁₆	—	59	2	no	—
108	C₂×C₄⋊C₈	Dic₅₉	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
110	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
111	C₄×C₄⋊C₄	Dic₅₉	1	1	yes	characteristic
112	C₄⋊C₁₆	—	59	4	no	—
113	#346	—	118	4	no	—
114	C₄×C₁₆	—	59	4	no	—
115	#336	Dic₅₉	1	2	yes	—
116	C₅₉	#12662	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
118	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₅₉⋊C₈	1	2	yes	—
121	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
122	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
123	C₂×C₁₆	—	59	2	no	—
124	C₂×C₁₆	—	59	2	no	—
125	C₂×C₁₆	—	59	2	no	—
126	C₂×C₁₆	—	59	2	no	—
127	C₂×C₁₆	—	59	4	no	—
128	C₂×C₁₆	—	118	2	no	—
129	C₂×C₁₆	—	118	1	no	—
130	C₂×C₁₆	—	118	1	no	—
131	C₂×C₁₆	—	118	2	no	—
132	C₄⋊C₈	C₅₉⋊C₈	1	4	yes	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	C₄×D₅₉	1	2	yes	—
136	C₄×C₈	C₄×D₅₉	1	2	yes	—
137	#95	C₅₉⋊C₈	1	2	yes	—
138	C₂²×C₄	C₂×C₅₉⋊C₈	1	4	yes	—
139	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
141	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	C₂×C₅₉⋊C₈	1	4	yes	—
147	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
148	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
150	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
151	C₂×C₈	#11745687	1	1	yes	characteristic
152	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
153	C₂×C₈	#11745689	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
158	C₄²	C₄×Dic₅₉	1	2	yes	—
159	C₁₆	—	59	4	no	—
160	C₁₆	—	118	2	no	—
161	C₁₆	—	118	4	no	—
162	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	C₄×C₅₉⋊C₈	1	2	yes	—
168	C₂×C₄	C₄×C₅₉⋊C₈	1	2	yes	—
169	C₂×C₄	#14273667	1	4	yes	—
170	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
171	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
172	C₂×C₄	Q₈×Dic₅₉	1	1	yes	characteristic
173	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
174	C₂×C₄	#14273734	1	1	yes	characteristic
175	C₂×C₄	#14273734	1	1	yes	characteristic
176	C₈	—	2	4	no	—
177	C₈	—	2	2	no	—
178	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
179	C₈	#14273685	1	1	yes	characteristic
180	C₈	#14273689	1	1	yes	characteristic
181	C₈	#14273692	1	1	yes	characteristic
182	C₂²	C₂×C₄×C₅₉⋊C₈	1	1	yes	characteristic
183	C₂²	D₄×C₅₉⋊C₈	1	1	yes	characteristic
184	C₂²	Q₈×C₅₉⋊C₈	1	1	yes	characteristic
185	C₂²	C₂×#14273667	1	2	yes	—
186	C₂²	#16837819	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	4	4	no	—
189	C₄	#16837178	1	4	yes	complemented
190	C₄	C₄⋊C₄×Dic₅₉	1	1	yes	characteristic
191	C₄	#16837358	1	1	yes	characteristic
192	C₄	D₄×C₅₉⋊C₈	1	1	yes	characteristic
193	C₄	Q₈×C₅₉⋊C₈	1	1	yes	characteristic
194	C₂	C₄⋊C₄×C₅₉⋊C₈	1	1	yes	characteristic
195	C₂	C₂×#16837178	1	1	yes	characteristic
196	C₂	#21425693	1	1	yes	characteristic
197	C₂	#21425928	1	2	yes	—
198	C₂	#21426005	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂×C₄×C₈
Fitting	C₈×C₄⋊C₄×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂²×C₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₈×C₄⋊C₄	D₅₉
hypocenter	C₅₉	#12662
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₅₉⋊C₁₆)	C₄	semidirect
11. C₄×C₅₉⋊C₁₆	C₄	semidirect
61. C₅₉⋊C₁₆	C₄⋊C₄	semidirect
90. C₄×C₅₉	#346	semidirect
116. C₅₉	#12662	semidirect
189. C₄	#16837178	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₂×#16837178	2
C₂×C₄×C₅₉⋊C₁₆	2
#21422174	2
C₈×C₄⋊C₄×C₅₉	2
#12662	59