Subgroups — C₂²×#16837807 #29458287

Order 15104.

2398 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273727	C₂	1	2	yes	maximal
2	C₂×#16837807	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14273661	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₁₈	—	2	2	no	—
6	C₂×#14273727	—	2	12	no	—
7	C₂×#14273727	C₂²	1	24	yes	—
8	C₂²×C₄⋊Dic₅₉	—	2	2	no	—
9	C₂²×C₄⋊Dic₅₉	C₂²	1	4	yes	—
10	#16837807	C₂²	1	64	yes	direct product factor
11	C₂×#14273661	C₂²	1	48	yes	—
12	C₂³×C₈×C₅₉	C₂²	1	2	yes	—
13	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₉	C₂²	1	12	yes	—
15	C₂²×Dic₁₁₈	—	2	24	no	—
16	C₂²×Dic₁₁₈	—	4	16	no	—
17	C₂³×Dic₅₉	—	2	4	no	—
18	#14273727	—	2	48	no	—
19	#14273727	C₂³	1	32	yes	—
20	C₂×C₄⋊Dic₅₉	—	2	12	no	—
21	C₂×C₄⋊Dic₅₉	—	2	24	no	—
22	C₂×C₄⋊Dic₅₉	C₂³	1	24	yes	—
23	#14273661	C₂³	1	64	yes	—
24	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	Q₈	1	16	yes	complemented
28	C₂²×C₈×C₅₉	C₂³	1	12	yes	—
29	C₂×C₄²×C₅₉	D₄	1	6	yes	—
30	C₂×C₄²×C₅₉	C₂³	1	6	yes	—
31	C₄×C₈×C₅₉	C₂³	1	16	yes	—
32	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	D₈	1	8	yes	—
34	C₂²×C₄×C₅₉	Q₁₆	1	8	yes	—
35	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
36	C₂²×C₄×C₅₉	C₂×D₄	1	6	yes	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	6	yes	—
38	C₂²×C₄×C₅₉	C₂×Q₈	1	8	yes	—
39	C₂²×C₄×C₅₉	C₂⁴	1	3	yes	—
40	C₂²×Dic₅₉	—	2	24	no	—
41	C₂²×Dic₅₉	—	4	16	no	—
42	C₂×Dic₁₁₈	—	2	32	no	—
43	C₂×Dic₁₁₈	—	4	96	no	—
44	C₂×C₈×C₅₉	C₂×Q₈	1	48	yes	complemented
45	C₂×C₈×C₅₉	C₂⁴	1	8	yes	—
46	C₄²×C₅₉	C₂×D₄	1	12	yes	—
47	C₄²×C₅₉	C₂⁴	1	4	yes	—
48	C₄⋊Dic₅₉	—	2	32	no	—
49	C₄⋊Dic₅₉	—	2	24	no	—
50	C₄⋊Dic₅₉	C₂⁴	1	16	yes	—
51	C₂³×C₅₉	C₄⋊Q₈	1	4	yes	—
52	C₂³×C₅₉	C₂×D₈	1	4	yes	—
53	C₂³×C₅₉	C₂×Q₁₆	1	4	yes	—
54	C₂³×C₅₉	C₂²×D₄	1	3	yes	—
55	C₂×C₄×C₅₉	C₄⋊Q₈	1	12	yes	—
56	C₂×C₄×C₅₉	C₂×D₈	1	24	yes	—
57	C₂×C₄×C₅₉	C₂×Q₁₆	1	24	yes	—
58	C₂×C₄×C₅₉	C₂²×D₄	1	3	yes	—
59	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
60	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
61	C₂×C₄×C₅₉	C₂²×Q₈	1	12	yes	—
62	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
63	C₂×Dic₅₉	—	2	16	no	—
64	C₂×Dic₅₉	—	4	48	no	—
65	Dic₁₁₈	—	4	64	no	—
66	C₈×C₅₉	C₂²×Q₈	1	16	yes	complemented
67	C₂²×#500	—	59	1	no	maximal, Sylow
68	C₂²×C₅₉	#500	1	16	yes	complemented
69	C₂²×C₅₉	C₂×C₄⋊Q₈	1	6	yes	—
70	C₂²×C₅₉	C₂²×D₈	1	6	yes	—
71	C₂²×C₅₉	C₂²×Q₁₆	1	6	yes	—
72	C₂²×C₅₉	C₂³×D₄	1	1	yes	characteristic
73	C₄×C₅₉	C₂×C₄⋊Q₈	1	6	yes	—
74	C₄×C₅₉	C₂²×D₈	1	8	yes	—
75	C₄×C₅₉	C₂²×Q₁₆	1	8	yes	—
76	C₄×C₅₉	C₂³×Q₈	1	2	yes	—
77	Dic₅₉	—	4	16	no	—
78	C₂²×C₄⋊Q₈	—	59	2	no	—
79	C₂×#500	—	59	24	no	—
80	C₂²×#107	—	59	4	no	—
81	C₂²×C₄×C₈	D₅₉	1	1	yes	characteristic
82	C₂×C₅₉	C₂×#500	1	12	yes	complemented
83	C₂×C₅₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₅₉	C₂³×D₈	1	1	yes	characteristic
85	C₂×C₅₉	C₂³×Q₁₆	1	1	yes	characteristic
86	C₂³×Q₈	—	118	2	no	—
87	C₂³×C₈	C₂×D₅₉	1	2	yes	—
88	C₂×C₄⋊Q₈	—	59	24	no	—
89	C₂×C₄⋊Q₈	—	118	12	no	—
90	C₂²×C₄⋊C₄	—	59	4	no	—
91	C₂²×C₄⋊C₄	—	118	2	no	—
92	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
93	#500	—	59	64	no	—
94	C₂×#107	—	59	48	no	—
95	C₂×C₄×C₈	C₂×D₅₉	1	12	yes	—
96	C₅₉	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
97	C₂²×Q₈	—	118	24	no	—
98	C₂²×Q₈	—	236	16	no	—
99	C₂³×C₄	—	118	4	no	—
100	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
101	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
102	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
103	C₂²×C₈	Dic₁₁₈	1	16	yes	complemented
104	C₂²×C₈	C₂²×D₅₉	1	12	yes	—
105	C₄⋊Q₈	—	59	32	no	—
106	C₄⋊Q₈	—	118	48	no	—
107	C₂×C₄⋊C₄	—	59	24	no	—
108	C₂×C₄⋊C₄	—	118	12	no	—
109	C₂×C₄⋊C₄	—	118	24	no	—
110	C₂×C₄²	D₂₃₆	1	6	yes	—
111	C₂×C₄²	C₂²×D₅₉	1	6	yes	—
112	#107	—	59	64	no	—
113	C₄×C₈	C₂²×D₅₉	1	16	yes	—
114	C₂⁴	C₂×D₂₃₆	1	1	yes	characteristic
115	C₂×Q₈	—	118	32	no	—
116	C₂×Q₈	—	236	96	no	—
117	C₂²×C₄	—	118	24	no	—
118	C₂²×C₄	—	236	16	no	—
119	C₂²×C₄	D₄₇₂	1	8	yes	—
120	C₂²×C₄	Dic₂₃₆	1	8	yes	—
121	C₂²×C₄	C₂×Dic₁₁₈	1	8	yes	—
122	C₂²×C₄	C₂×D₂₃₆	1	3	yes	—
123	C₂²×C₄	C₂×D₂₃₆	1	6	yes	—
124	C₂²×C₄	C₂×D₂₃₆	1	6	yes	—
125	C₂²×C₄	C₂³×D₅₉	1	3	yes	—
126	C₂×C₈	C₂×Dic₁₁₈	1	48	yes	complemented
127	C₂×C₈	C₂³×D₅₉	1	8	yes	—
128	C₄⋊C₄	—	59	16	no	—
129	C₄⋊C₄	—	118	24	no	—
130	C₄⋊C₄	—	118	32	no	—
131	C₄²	C₂×D₂₃₆	1	12	yes	—
132	C₄²	C₂³×D₅₉	1	4	yes	—
133	C₂³	#14273727	1	4	yes	—
134	C₂³	C₂×D₄₇₂	1	4	yes	—
135	C₂³	C₂×Dic₂₃₆	1	4	yes	—
136	C₂³	C₂²×D₂₃₆	1	3	yes	—
137	Q₈	—	236	64	no	—
138	C₂×C₄	—	118	16	no	—
139	C₂×C₄	—	236	48	no	—
140	C₂×C₄	#14273727	1	12	yes	—
141	C₂×C₄	C₂×D₄₇₂	1	24	yes	—
142	C₂×C₄	C₂×Dic₂₃₆	1	24	yes	—
143	C₂×C₄	C₂²×D₂₃₆	1	4	yes	—
144	C₂×C₄	C₂²×D₂₃₆	1	4	yes	—
145	C₂×C₄	C₂²×D₂₃₆	1	3	yes	—
146	C₂×C₄	C₂²×Dic₁₁₈	1	12	yes	—
147	C₂×C₄	C₂⁴×D₅₉	1	1	yes	characteristic
148	C₈	C₂²×Dic₁₁₈	1	16	yes	complemented
149	C₂²	#16837807	1	16	yes	direct product factor
150	C₂²	C₂×#14273727	1	6	yes	—
151	C₂²	C₂²×D₄₇₂	1	6	yes	—
152	C₂²	C₂²×Dic₂₃₆	1	6	yes	—
153	C₂²	C₂³×D₂₃₆	1	1	yes	characteristic
154	C₄	—	236	16	no	—
155	C₄	C₂×#14273727	1	6	yes	—
156	C₄	C₂²×D₄₇₂	1	8	yes	—
157	C₄	C₂²×Dic₂₃₆	1	8	yes	—
158	C₄	C₂³×Dic₁₁₈	1	2	yes	—
159	C₂	C₂×#16837807	1	12	yes	direct product factor
160	C₂	C₂²×#14273727	1	1	yes	characteristic
161	C₂	C₂³×D₄₇₂	1	1	yes	characteristic
162	C₂	C₂³×Dic₂₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₂₃₆
commutator, cosocle	C₂×C₄×C₅₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₉
Fitting	C₂²×C₄×C₈×C₅₉	C₂
socle, prime radical	C₂⁴×C₅₉	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₅₉
hypocenter	C₅₉	C₂²×#500
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16837807	C₂	direct (N × H)
10. #16837807	C₂²	direct (N × H)
27. C₂²×C₈×C₅₉	Q₈	semidirect
44. C₂×C₈×C₅₉	C₂×Q₈	semidirect
66. C₈×C₅₉	C₂²×Q₈	semidirect
68. C₂²×C₅₉	#500	semidirect
82. C₂×C₅₉	C₂×#500	semidirect
96. C₅₉	C₂²×#500	semidirect
103. C₂²×C₈	Dic₁₁₈	semidirect
126. C₂×C₈	C₂×Dic₁₁₈	semidirect
148. C₈	C₂²×Dic₁₁₈	semidirect
149. C₂²	#16837807	direct (N × H)
159. C₂	C₂×#16837807	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×#14273727	2
C₂×#16837807	2
C₂²×#14273661	2
C₂²×C₄×C₈×C₅₉	2
C₂²×#500	59

Subgroups — C22×#16837807 #29458287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16837807 #29458287