Subgroups — C₂²×#16837834 #29495176

Order 15104.

2118 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273749	C₂	1	2	yes	maximal, complemented
2	C₂×#16837834	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#14273672	C₂	1	2	yes	maximal, complemented
5	C₂²×D₉₄₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11745685	—	2	2	no	—
8	C₂×#14273749	—	2	12	no	—
9	C₂×#14273749	C₂²	1	24	yes	complemented
10	C₂²×Dic₂₃₆	C₂²	1	2	yes	—
11	C₂²×#11745660	—	2	2	no	—
12	C₂²×D₄₇₂	C₂²	1	2	yes	—
13	#16837834	C₂²	1	64	yes	direct product factor
14	C₂×Dic₄₇₂	C₂²	1	12	yes	complemented
15	C₂×#14273672	C₂²	1	24	yes	complemented
16	C₂×D₉₄₄	C₂²	1	12	yes	complemented
17	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₅₉	C₂²	1	12	yes	complemented
21	C₂×#11745685	—	2	24	no	—
22	C₂×#11745685	—	4	12	no	—
23	C₂²×Dic₁₁₈	—	2	2	no	—
24	C₂²×D₂₃₆	—	2	2	no	—
25	C₂²×C₂²⋊D₅₉	—	4	2	no	—
26	C₂²×C₄×D₅₉	—	4	2	no	—
27	#14273749	—	2	48	no	—
28	#14273749	C₂³	1	32	yes	complemented
29	C₂×Dic₂₃₆	—	2	6	no	—
30	C₂×Dic₂₃₆	C₂³	1	12	yes	—
31	C₂×#11745660	—	2	24	no	—
32	C₂×D₄₇₂	—	2	6	no	—
33	C₂×D₄₇₂	C₂³	1	12	yes	—
34	Dic₄₇₂	C₂³	1	16	yes	complemented
35	#14273672	C₂³	1	32	yes	complemented
36	D₉₄₄	C₂³	1	16	yes	complemented
37	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₉	D₄	1	6	yes	—
40	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₅₉	C₂³	1	6	yes	—
42	C₂×C₁₆×C₅₉	C₂³	1	6	yes	—
43	C₂×C₁₆×C₅₉	C₂³	1	6	yes	—
44	C₂×C₁₆×C₅₉	C₂³	1	16	yes	complemented
45	C₂⁴×C₅₉	D₈	1	1	yes	characteristic
46	C₂³×D₅₉	—	4	2	no	—
47	C₂²×C₄×C₅₉	D₈	1	6	yes	—
48	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂×D₄	1	6	yes	—
51	C₂×C₂²⋊D₅₉	—	4	24	no	—
52	C₂²×Dic₅₉	—	4	2	no	—
53	#11745685	—	2	32	no	—
54	#11745685	—	4	48	no	—
55	C₂×D₂₃₆	—	2	12	no	—
56	C₂×D₂₃₆	—	4	6	no	—
57	C₂×C₄×D₅₉	—	4	24	no	—
58	C₂×Dic₁₁₈	—	2	12	no	—
59	C₂×Dic₁₁₈	—	4	6	no	—
60	C₂×C₈×C₅₉	C₂×D₄	1	3	yes	—
61	C₂×C₈×C₅₉	C₂×D₄	1	6	yes	—
62	C₂×C₈×C₅₉	C₂×D₄	1	12	yes	—
63	C₂×C₈×C₅₉	C₂⁴	1	3	yes	—
64	C₂×C₈×C₅₉	C₂⁴	1	4	yes	—
65	Dic₂₃₆	—	2	12	no	—
66	Dic₂₃₆	C₂⁴	1	8	yes	—
67	D₄₇₂	—	2	12	no	—
68	D₄₇₂	C₂⁴	1	8	yes	—
69	#11745660	—	2	32	no	—
70	C₁₆×C₅₉	C₂⁴	1	4	yes	—
71	C₁₆×C₅₉	C₂⁴	1	4	yes	—
72	C₂³×C₅₉	—	2	4	no	—
73	C₂³×C₅₉	C₂×D₈	1	6	yes	—
74	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
75	C₂²×D₅₉	—	4	12	no	—
76	C₂²×D₅₉	—	8	8	no	—
77	C₂×C₄×C₅₉	C₂×D₈	1	3	yes	—
78	C₂×C₄×C₅₉	C₂×D₈	1	6	yes	—
79	C₂×C₄×C₅₉	C₂×D₈	1	12	yes	—
80	C₂×C₄×C₅₉	C₂²×D₄	1	3	yes	—
81	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
82	C₂²⋊D₅₉	—	4	32	no	—
83	C₂×Dic₅₉	—	4	12	no	—
84	D₂₃₆	—	2	8	no	—
85	D₂₃₆	—	4	12	no	—
86	C₄×D₅₉	—	4	32	no	—
87	Dic₁₁₈	—	2	8	no	—
88	Dic₁₁₈	—	4	12	no	—
89	C₈×C₅₉	C₂²×D₄	1	4	yes	—
90	C₈×C₅₉	C₂²×D₄	1	3	yes	—
91	C₈×C₅₉	C₂⁵	1	1	yes	characteristic
92	C₂²×#508	—	59	1	no	maximal, Sylow
93	C₂²×C₅₉	—	2	12	no	—
94	C₂²×C₅₉	#508	1	4	yes	complemented
95	C₂²×C₅₉	C₂²×D₈	1	4	yes	—
96	C₂²×C₅₉	C₂²×D₈	1	3	yes	—
97	C₂×D₅₉	—	4	8	no	—
98	C₂×D₅₉	—	8	24	no	—
99	C₄×C₅₉	C₂²×D₈	1	3	yes	—
100	C₄×C₅₉	C₂²×D₈	1	4	yes	—
101	C₄×C₅₉	C₂³×D₄	1	1	yes	characteristic
102	Dic₅₉	—	4	8	no	—
103	C₂²×#135	—	59	2	no	—
104	C₂×#508	—	59	24	no	—
105	C₂²×Q₃₂	—	59	1	no	—
106	C₂²×SD₃₂	—	59	2	no	—
107	C₂²×D₁₆	—	59	1	no	—
108	C₂³×C₁₆	D₅₉	1	1	yes	characteristic, complemented
109	C₂×C₅₉	—	2	4	no	—
110	C₂×C₅₉	C₂×#508	1	6	yes	complemented
111	C₂×C₅₉	C₂³×D₈	1	1	yes	characteristic
112	D₅₉	—	8	8	no	—
113	C₂²×#41	—	118	2	no	—
114	C₂×#135	—	59	24	no	—
115	C₂×#135	—	118	12	no	—
116	C₂²×Q₁₆	—	59	2	no	—
117	C₂²×SD₁₆	—	118	2	no	—
118	C₂²×D₈	—	59	2	no	—
119	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
120	#508	—	59	64	no	—
121	C₂×Q₃₂	—	59	12	no	—
122	C₂×SD₃₂	—	59	24	no	—
123	C₂×D₁₆	—	59	12	no	—
124	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
125	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
126	C₂²×C₁₆	C₂×D₅₉	1	12	yes	complemented
127	C₅₉	C₂²×#508	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	—	118	24	no	—
129	C₂×#41	—	236	12	no	—
130	C₂²×Q₈	—	118	2	no	—
131	C₂²×D₄	—	118	2	no	—
132	C₂²×D₄	—	236	2	no	—
133	C₂³×C₄	—	236	2	no	—
134	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
135	#135	—	59	32	no	—
136	#135	—	118	48	no	—
137	C₂×Q₁₆	—	59	12	no	—
138	C₂×Q₁₆	—	118	6	no	—
139	C₂×SD₁₆	—	118	24	no	—
140	C₂×D₈	—	59	12	no	—
141	C₂×D₈	—	118	6	no	—
142	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
143	C₂²×C₈	D₂₃₆	1	6	yes	—
144	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
145	C₂²×C₈	C₂²×D₅₉	1	6	yes	—
146	Q₃₂	—	59	16	no	—
147	SD₃₂	—	59	32	no	—
148	D₁₆	—	59	16	no	—
149	C₂×C₁₆	C₂²×D₅₉	1	6	yes	—
150	C₂×C₁₆	C₂²×D₅₉	1	6	yes	—
151	C₂×C₁₆	C₂²×D₅₉	1	16	yes	complemented
152	C₂⁴	—	236	2	no	—
153	C₂⁴	D₄₇₂	1	1	yes	characteristic
154	#41	—	118	32	no	—
155	#41	—	236	48	no	—
156	C₂×Q₈	—	118	12	no	—
157	C₂×Q₈	—	236	6	no	—
158	C₂×D₄	—	118	12	no	—
159	C₂×D₄	—	236	24	no	—
160	C₂×D₄	—	236	6	no	—
161	C₂²×C₄	—	236	2	no	—
162	C₂²×C₄	—	236	24	no	—
163	C₂²×C₄	D₄₇₂	1	6	yes	—
164	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₂₃₆	1	6	yes	—
167	Q₁₆	—	59	8	no	—
168	Q₁₆	—	118	12	no	—
169	SD₁₆	—	118	32	no	—
170	D₈	—	59	8	no	—
171	D₈	—	118	12	no	—
172	C₂×C₈	C₂×D₂₃₆	1	3	yes	—
173	C₂×C₈	C₂×D₂₃₆	1	6	yes	—
174	C₂×C₈	C₂×D₂₃₆	1	12	yes	—
175	C₂×C₈	C₂³×D₅₉	1	3	yes	—
176	C₂×C₈	C₂³×D₅₉	1	4	yes	—
177	C₁₆	C₂³×D₅₉	1	4	yes	—
178	C₁₆	C₂³×D₅₉	1	4	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	—	472	8	no	—
181	C₂³	—	236	12	no	—
182	C₂³	C₂×D₄₇₂	1	6	yes	—
183	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
184	Q₈	—	118	8	no	—
185	Q₈	—	236	12	no	—
186	D₄	—	118	8	no	—
187	D₄	—	236	32	no	—
188	D₄	—	236	12	no	—
189	C₂×C₄	—	236	12	no	—
190	C₂×C₄	—	236	32	no	—
191	C₂×C₄	C₂×D₄₇₂	1	6	yes	—
192	C₂×C₄	C₂×D₄₇₂	1	3	yes	—
193	C₂×C₄	C₂×D₄₇₂	1	12	yes	—
194	C₂×C₄	C₂²×D₂₃₆	1	3	yes	—
195	C₂×C₄	C₂²×D₂₃₆	1	4	yes	—
196	C₈	C₂²×D₂₃₆	1	3	yes	—
197	C₈	C₂²×D₂₃₆	1	4	yes	—
198	C₈	C₂⁴×D₅₉	1	1	yes	characteristic
199	C₂²	—	2	12	no	—
200	C₂²	—	472	24	no	—
201	C₂²	—	236	8	no	—
202	C₂²	#16837834	1	4	yes	direct product factor
203	C₂²	C₂²×D₄₇₂	1	4	yes	—
204	C₂²	C₂²×D₄₇₂	1	3	yes	—
205	C₄	—	236	8	no	—
206	C₄	C₂²×D₄₇₂	1	4	yes	—
207	C₄	C₂²×D₄₇₂	1	3	yes	—
208	C₄	C₂³×D₂₃₆	1	1	yes	characteristic
209	C₂	—	2	4	no	—
210	C₂	—	472	8	no	—
211	C₂	C₂×#16837834	1	6	yes	direct product factor
212	C₂	C₂³×D₄₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄₇₂
commutator, cosocle	C₈×C₅₉	C₂⁵
Frattini	C₈	C₂⁴×D₅₉
Fitting	C₂³×C₁₆×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₈
hypercenter	C₂³×C₁₆	D₅₉
hypocenter	C₅₉	C₂²×#508
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#14273749	C₂	semidirect
2. C₂×#16837834	C₂	direct (N × H)
3. C₂²×Dic₄₇₂	C₂	semidirect
4. C₂²×#14273672	C₂	semidirect
5. C₂²×D₉₄₄	C₂	semidirect
6. C₂³×C₁₆×C₅₉	C₂	semidirect
9. C₂×#14273749	C₂²	semidirect
13. #16837834	C₂²	direct (N × H)
14. C₂×Dic₄₇₂	C₂²	semidirect
15. C₂×#14273672	C₂²	semidirect
16. C₂×D₉₄₄	C₂²	semidirect
20. C₂²×C₁₆×C₅₉	C₂²	semidirect
28. #14273749	C₂³	semidirect
34. Dic₄₇₂	C₂³	semidirect
35. #14273672	C₂³	semidirect
36. D₉₄₄	C₂³	semidirect
44. C₂×C₁₆×C₅₉	C₂³	semidirect
94. C₂²×C₅₉	#508	semidirect
108. C₂³×C₁₆	D₅₉	semidirect
110. C₂×C₅₉	C₂×#508	semidirect
126. C₂²×C₁₆	C₂×D₅₉	semidirect
127. C₅₉	C₂²×#508	semidirect
151. C₂×C₁₆	C₂²×D₅₉	semidirect
202. C₂²	#16837834	direct (N × H)
211. C₂	C₂×#16837834	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×#14273749	2
C₂×#16837834	2
C₂²×Dic₄₇₂	2
C₂²×#14273672	2
C₂²×D₉₄₄	2
C₂³×C₁₆×C₅₉	2
C₂²×#508	59

Subgroups — C22×#16837834 #29495176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16837834 #29495176