Subgroups — C₂²×#16837225 #28741696

Order 15104.

702 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273663	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14273659	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837225	C₂	1	12	yes	maximal, direct product factor
4	C₂²×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11745664	—	2	1	no	—
6	C₂²×C₄⋊Dic₅₉	—	2	1	no	—
7	C₂×#14273663	—	2	3	no	—
8	C₂×#14273663	C₂²	1	6	yes	—
9	C₂×#14273659	—	2	3	no	—
10	C₂×#14273659	C₂²	1	6	yes	—
11	#16837225	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₅₉	C₄	1	6	yes	complemented
14	C₂×M₃₂×C₅₉	C₂²	1	6	yes	—
15	C₂²×C₁₆×C₅₉	C₄	1	2	yes	complemented
16	C₂³×Dic₅₉	—	4	1	no	—
17	C₂×#11745664	—	2	6	no	—
18	C₂×#11745664	—	4	3	no	—
19	C₂²×C₅₉⋊C₈	—	4	1	no	—
20	C₂×C₄⋊Dic₅₉	—	2	6	no	—
21	C₂×C₄⋊Dic₅₉	—	4	3	no	—
22	#14273663	—	2	6	no	—
23	#14273663	C₂³	1	4	yes	—
24	#14273659	—	2	6	no	—
25	#14273659	C₂³	1	4	yes	—
26	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	C₂×C₄	1	3	yes	—
29	C₂²×C₈×C₅₉	D₄	1	3	yes	—
30	C₂²×C₈×C₅₉	Q₈	1	3	yes	—
31	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
33	M₃₂×C₅₉	C₂×C₄	1	12	yes	complemented
34	M₃₂×C₅₉	C₂³	1	4	yes	—
35	C₂×C₁₆×C₅₉	C₂×C₄	1	12	yes	complemented
36	C₂⁴×C₅₉	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₄⋊C₄	1	3	yes	—
38	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	SD₁₆	1	3	yes	—
40	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	SD₁₆	1	3	yes	—
42	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
43	C₂²×Dic₅₉	—	4	6	no	—
44	C₂²×Dic₅₉	—	8	4	no	—
45	C₂×C₈×C₅₉	C₄⋊C₄	1	3	yes	—
46	C₂×C₈×C₅₉	C₄⋊C₄	1	6	yes	—
47	C₂×C₈×C₅₉	C₄⋊C₄	1	3	yes	—
48	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₅₉	C₂×D₄	1	3	yes	—
51	C₂×C₈×C₅₉	C₂×Q₈	1	3	yes	—
52	C₂×C₈×C₅₉	C₂×Q₈	1	3	yes	—
53	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
54	C₄⋊Dic₅₉	—	2	4	no	—
55	C₄⋊Dic₅₉	—	4	6	no	—
56	#11745664	—	2	4	no	—
57	#11745664	—	4	6	no	—
58	C₂×C₅₉⋊C₈	—	4	6	no	—
59	C₁₆×C₅₉	C₂²×C₄	1	8	yes	complemented
60	C₂³×C₅₉	—	2	4	no	—
61	C₂³×C₅₉	#106	1	3	yes	—
62	C₂³×C₅₉	#106	1	1	yes	characteristic
63	C₂³×C₅₉	C₂×SD₁₆	1	3	yes	—
64	C₂×C₄×C₅₉	#106	1	6	yes	—
65	C₂×C₄×C₅₉	#106	1	3	yes	—
66	C₂×C₄×C₅₉	#106	1	3	yes	—
67	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	3	yes	—
68	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₅₉	C₂×SD₁₆	1	3	yes	—
70	C₂×C₄×C₅₉	C₂×SD₁₆	1	3	yes	—
71	C₂×C₄×C₅₉	C₂×SD₁₆	1	3	yes	—
72	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
73	C₂×Dic₅₉	—	4	4	no	—
74	C₂×Dic₅₉	—	8	12	no	—
75	C₈×C₅₉	C₂×C₄⋊C₄	1	3	yes	—
76	C₈×C₅₉	C₂×C₄⋊C₄	1	3	yes	—
77	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
78	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
79	C₅₉⋊C₈	—	4	4	no	—
80	C₂²×#365	—	59	1	no	maximal, Sylow
81	C₂²×C₅₉	—	2	6	no	—
82	C₂²×C₅₉	—	2	6	no	—
83	C₂²×C₅₉	#365	1	4	yes	complemented
84	C₂²×C₅₉	C₂×#106	1	3	yes	—
85	C₂²×C₅₉	C₂×#106	1	3	yes	—
86	C₂²×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
87	C₄×C₅₉	C₂×#106	1	3	yes	—
88	C₄×C₅₉	C₂×#106	1	3	yes	—
89	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
91	Dic₅₉	—	8	4	no	—
92	C₂²×M₃₂	D₅₉	1	1	yes	characteristic
93	C₂²×#108	—	59	1	no	—
94	C₂²×#106	—	59	1	no	—
95	C₂×#365	—	59	12	no	—
96	C₂×C₅₉	—	2	1	no	—
97	C₂×C₅₉	—	2	3	no	—
98	C₂×C₅₉	C₂×#365	1	6	yes	complemented
99	C₂×C₅₉	C₂²×#106	1	1	yes	characteristic
100	C₂²×M₁₆	—	118	1	no	—
101	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
102	C₂²×C₄⋊C₄	—	118	1	no	—
103	C₂×M₃₂	Dic₅₉	1	6	yes	complemented
104	C₂×M₃₂	C₂×D₅₉	1	6	yes	—
105	C₂²×C₁₆	Dic₅₉	1	2	yes	complemented
106	C₂×#108	—	59	6	no	—
107	C₂×#108	—	118	3	no	—
108	C₂×#106	—	59	6	no	—
109	C₂×#106	—	118	3	no	—
110	#365	—	59	16	no	—
111	C₅₉	C₂²×#365	1	1	yes	characteristic, complemented, Sylow
112	C₂³×C₄	—	236	1	no	—
113	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
114	C₂×M₁₆	—	118	6	no	—
115	C₂×M₁₆	—	236	3	no	—
116	C₂²×C₈	—	236	1	no	—
117	C₂²×C₈	Dic₁₁₈	1	3	yes	—
118	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
119	C₂²×C₈	D₂₃₆	1	3	yes	—
120	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
121	C₂²×C₈	C₂×Dic₅₉	1	3	yes	—
122	C₂²×C₈	C₂²×D₅₉	1	3	yes	—
123	C₂×C₄⋊C₄	—	118	6	no	—
124	C₂×C₄⋊C₄	—	236	3	no	—
125	M₃₂	C₂×Dic₅₉	1	12	yes	complemented
126	M₃₂	C₂²×D₅₉	1	4	yes	—
127	C₂×C₁₆	C₂×Dic₅₉	1	12	yes	complemented
128	#108	—	59	4	no	—
129	#108	—	118	6	no	—
130	#106	—	59	4	no	—
131	#106	—	118	6	no	—
132	C₂⁴	#11745660	1	1	yes	characteristic
133	C₂²×C₄	—	472	4	no	—
134	C₂²×C₄	—	236	6	no	—
135	C₂²×C₄	#11745660	1	3	yes	—
136	C₂²×C₄	#11745660	1	3	yes	—
137	C₂²×C₄	#11745660	1	1	yes	characteristic
138	C₂²×C₄	C₄⋊Dic₅₉	1	3	yes	—
139	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₂₃₆	1	3	yes	—
141	M₁₆	—	118	4	no	—
142	M₁₆	—	236	6	no	—
143	C₂×C₈	—	236	6	no	—
144	C₂×C₈	C₄⋊Dic₅₉	1	3	yes	—
145	C₂×C₈	C₄⋊Dic₅₉	1	6	yes	—
146	C₂×C₈	C₄⋊Dic₅₉	1	3	yes	—
147	C₂×C₈	C₂×Dic₁₁₈	1	3	yes	—
148	C₂×C₈	C₂×Dic₁₁₈	1	3	yes	—
149	C₂×C₈	C₂×D₂₃₆	1	3	yes	—
150	C₂×C₈	C₂²×Dic₅₉	1	3	yes	—
151	C₂×C₈	C₂²×Dic₅₉	1	3	yes	—
152	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
153	C₄⋊C₄	—	118	4	no	—
154	C₄⋊C₄	—	236	6	no	—
155	C₁₆	C₂²×Dic₅₉	1	8	yes	complemented
156	C₂³	—	2	4	no	—
157	C₂³	#14273659	1	3	yes	—
158	C₂³	#14273659	1	1	yes	characteristic
159	C₂³	C₂×#11745660	1	3	yes	—
160	C₂×C₄	—	472	12	no	—
161	C₂×C₄	—	236	4	no	—
162	C₂×C₄	#14273659	1	6	yes	—
163	C₂×C₄	#14273659	1	3	yes	—
164	C₂×C₄	#14273659	1	3	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₅₉	1	3	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₅₉	1	3	yes	—
167	C₂×C₄	C₂×#11745660	1	3	yes	—
168	C₂×C₄	C₂×#11745660	1	3	yes	—
169	C₂×C₄	C₂×#11745660	1	3	yes	—
170	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
171	C₈	—	236	4	no	—
172	C₈	C₂×C₄⋊Dic₅₉	1	3	yes	—
173	C₈	C₂×C₄⋊Dic₅₉	1	3	yes	—
174	C₈	C₂³×Dic₅₉	1	1	yes	characteristic
175	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
176	C₂²	—	2	6	no	—
177	C₂²	—	2	6	no	—
178	C₂²	#16837225	1	4	yes	direct product factor
179	C₂²	C₂×#14273659	1	3	yes	—
180	C₂²	C₂×#14273659	1	3	yes	—
181	C₂²	C₂²×#11745660	1	1	yes	characteristic
182	C₄	—	472	4	no	—
183	C₄	C₂×#14273659	1	3	yes	—
184	C₄	C₂×#14273659	1	3	yes	—
185	C₄	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
186	C₄	C₂²×#11745660	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	3	no	—
189	C₂	C₂×#16837225	1	6	yes	direct product factor
190	C₂	C₂²×#14273659	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273659
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂²×M₃₂×C₅₉	C₂
socle	C₂³×C₅₉	#106
prime radical	C₂⁴×C₅₉	SD₁₆
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂²×M₃₂	D₅₉
hypocenter	C₅₉	C₂²×#365
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₂×#16837225	C₂	direct (N × H)
11. #16837225	C₂²	direct (N × H)
13. C₂×M₃₂×C₅₉	C₄	semidirect
15. C₂²×C₁₆×C₅₉	C₄	semidirect
33. M₃₂×C₅₉	C₂×C₄	semidirect
35. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
59. C₁₆×C₅₉	C₂²×C₄	semidirect
83. C₂²×C₅₉	#365	semidirect
98. C₂×C₅₉	C₂×#365	semidirect
103. C₂×M₃₂	Dic₅₉	semidirect
105. C₂²×C₁₆	Dic₅₉	semidirect
111. C₅₉	C₂²×#365	semidirect
125. M₃₂	C₂×Dic₅₉	semidirect
127. C₂×C₁₆	C₂×Dic₅₉	semidirect
155. C₁₆	C₂²×Dic₅₉	semidirect
178. C₂²	#16837225	direct (N × H)
189. C₂	C₂×#16837225	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14273663	2
C₂²×#14273659	2
C₂×#16837225	2
C₂²×M₃₂×C₅₉	2
C₂²×#365	59

Subgroups — C22×#16837225 #28741696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16837225 #28741696