Subgroups — C₂²×#16837124 #28554258

Order 15104.

1182 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837124	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₅₉⋊C₈	C₄	1	2	yes	—
6	C₂³×C₅₉⋊C₈	C₄	1	2	yes	—
7	C₂×C₄×C₅₉⋊C₈	C₄	1	6	yes	—
8	C₂×C₄×C₅₉⋊C₈	C₄	1	6	yes	—
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	6	yes	—
10	C₂×C₄×C₅₉⋊C₈	C₂²	1	6	yes	—
11	#16837124	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₅₉	C₄	1	2	yes	—
13	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₉	C₄	1	6	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	6	yes	—
16	C₂²×C₅₉⋊C₈	—	2	8	no	—
17	C₂²×C₅₉⋊C₈	C₈	1	16	yes	complemented
18	C₂²×C₅₉⋊C₈	C₂×C₄	1	12	yes	—
19	C₂²×C₅₉⋊C₈	C₂×C₄	1	12	yes	—
20	C₄×C₅₉⋊C₈	C₂×C₄	1	12	yes	—
21	C₄×C₅₉⋊C₈	C₂×C₄	1	12	yes	—
22	C₄×C₅₉⋊C₈	C₂³	1	4	yes	—
23	C₄×C₅₉⋊C₈	C₂³	1	4	yes	—
24	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	—	2	8	no	—
28	C₂²×C₈×C₅₉	C₂×C₄	1	12	yes	—
29	C₂×C₄²×C₅₉	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₅₉	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₅₉	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₅₉	C₂³	1	3	yes	—
33	C₄×C₈×C₅₉	C₂×C₄	1	12	yes	—
34	C₄×C₈×C₅₉	C₂³	1	4	yes	—
35	C₂⁴×C₅₉	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₄²	1	3	yes	—
37	C₂²×C₄×C₅₉	C₄²	1	3	yes	—
38	C₂²×C₄×C₅₉	C₄²	1	3	yes	—
39	C₂²×C₄×C₅₉	C₂×C₈	1	8	yes	—
40	C₂²×C₄×C₅₉	M₁₆	1	8	yes	—
41	C₂²×C₄×C₅₉	M₁₆	1	8	yes	—
42	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
43	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
44	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
45	C₂×C₈×C₅₉	—	2	24	no	—
46	C₂×C₈×C₅₉	C₂²×C₄	1	8	yes	—
47	C₄²×C₅₉	C₄²	1	6	yes	—
48	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
49	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
50	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
51	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
52	C₂×C₅₉⋊C₈	—	2	24	no	—
53	C₂×C₅₉⋊C₈	C₂×C₈	1	48	yes	complemented
54	C₂×C₅₉⋊C₈	C₂²×C₄	1	8	yes	—
55	C₂×C₅₉⋊C₈	C₂²×C₄	1	8	yes	—
56	C₂³×C₅₉	C₄×C₈	1	4	yes	—
57	C₂³×C₅₉	#97	1	4	yes	—
58	C₂³×C₅₉	#97	1	4	yes	—
59	C₂³×C₅₉	C₂×C₄²	1	3	yes	—
60	C₂×C₄×C₅₉	C₄×C₈	1	12	yes	—
61	C₂×C₄×C₅₉	#97	1	12	yes	—
62	C₂×C₄×C₅₉	#97	1	12	yes	—
63	C₂×C₄×C₅₉	C₂×C₄²	1	3	yes	—
64	C₂×C₄×C₅₉	C₂×C₄²	1	3	yes	—
65	C₂×C₄×C₅₉	C₂×C₄²	1	3	yes	—
66	C₂×C₄×C₅₉	C₂²×C₈	1	12	yes	—
67	C₂×C₄×C₅₉	C₂×M₁₆	1	12	yes	—
68	C₂×C₄×C₅₉	C₂×M₁₆	1	12	yes	—
69	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
72	C₈×C₅₉	—	2	8	no	—
73	C₅₉⋊C₈	—	2	8	no	—
74	C₅₉⋊C₈	C₂²×C₈	1	16	yes	complemented
75	C₂²×#322	—	59	1	no	maximal, Sylow
76	C₂²×C₅₉	#322	1	16	yes	complemented
77	C₂²×C₅₉	C₂×C₄×C₈	1	6	yes	—
78	C₂²×C₅₉	C₂×#97	1	6	yes	—
79	C₂²×C₅₉	C₂×#97	1	6	yes	—
80	C₂²×C₅₉	C₂²×C₄²	1	1	yes	characteristic
81	C₄×C₅₉	C₂×C₄×C₈	1	6	yes	—
82	C₄×C₅₉	C₂×#97	1	6	yes	—
83	C₄×C₅₉	C₂×#97	1	6	yes	—
84	C₄×C₅₉	C₂³×C₈	1	2	yes	—
85	C₄×C₅₉	C₂²×M₁₆	1	2	yes	—
86	C₄×C₅₉	C₂²×M₁₆	1	2	yes	—
87	C₂²×C₄×C₈	—	59	1	no	—
88	C₂²×C₄×C₈	—	59	1	no	—
89	C₂²×C₄×C₈	D₅₉	1	1	yes	characteristic
90	C₂×#322	—	59	12	no	—
91	C₂×C₅₉	C₂×#322	1	12	yes	complemented
92	C₂×C₅₉	C₂²×C₄×C₈	1	1	yes	characteristic
93	C₂×C₅₉	C₂²×#97	1	1	yes	characteristic
94	C₂×C₅₉	C₂²×#97	1	1	yes	characteristic
95	C₂³×C₈	—	59	2	no	—
96	C₂³×C₈	—	59	2	no	—
97	C₂³×C₈	Dic₅₉	1	2	yes	—
98	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
99	C₂×C₄×C₈	—	59	6	no	—
100	C₂×C₄×C₈	—	59	6	no	—
101	C₂×C₄×C₈	—	59	6	no	—
102	C₂×C₄×C₈	—	59	6	no	—
103	C₂×C₄×C₈	Dic₅₉	1	6	yes	—
104	C₂×C₄×C₈	C₂×D₅₉	1	6	yes	—
105	#322	—	59	16	no	—
106	C₅₉	C₂²×#322	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
108	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
109	C₂³×C₄	C₂×Dic₅₉	1	1	yes	characteristic
110	C₂²×C₈	—	2	8	no	—
111	C₂²×C₈	—	59	16	no	—
112	C₂²×C₈	—	59	12	no	—
113	C₂²×C₈	—	59	12	no	—
114	C₂²×C₈	—	118	8	no	—
115	C₂²×C₈	C₂×Dic₅₉	1	12	yes	—
116	C₂×C₄²	C₄×D₅₉	1	3	yes	—
117	C₂×C₄²	C₄×D₅₉	1	3	yes	—
118	C₂×C₄²	C₂×Dic₅₉	1	3	yes	—
119	C₂×C₄²	C₂²×D₅₉	1	3	yes	—
120	C₄×C₈	—	59	12	no	—
121	C₄×C₈	—	59	12	no	—
122	C₄×C₈	—	59	4	no	—
123	C₄×C₈	—	59	4	no	—
124	C₄×C₈	C₂×Dic₅₉	1	12	yes	—
125	C₄×C₈	C₂²×D₅₉	1	4	yes	—
126	C₂⁴	C₄×Dic₅₉	1	1	yes	characteristic
127	C₂²×C₄	C₈×D₅₉	1	8	yes	—
128	C₂²×C₄	#11745659	1	8	yes	—
129	C₂²×C₄	#11745664	1	8	yes	—
130	C₂²×C₄	C₄×Dic₅₉	1	3	yes	—
131	C₂²×C₄	C₄×Dic₅₉	1	3	yes	—
132	C₂²×C₄	C₄×Dic₅₉	1	3	yes	—
133	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
134	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
135	C₂²×C₄	C₂²×Dic₅₉	1	3	yes	—
136	C₂×C₈	—	2	24	no	—
137	C₂×C₈	—	59	48	no	—
138	C₂×C₈	—	59	8	no	—
139	C₂×C₈	—	59	8	no	—
140	C₂×C₈	—	118	24	no	—
141	C₂×C₈	C₂²×Dic₅₉	1	8	yes	—
142	C₄²	C₄×Dic₅₉	1	6	yes	—
143	C₄²	C₂×C₄×D₅₉	1	3	yes	—
144	C₄²	C₂×C₄×D₅₉	1	3	yes	—
145	C₄²	C₂²×Dic₅₉	1	3	yes	—
146	C₄²	C₂³×D₅₉	1	1	yes	characteristic
147	C₂³	C₈×Dic₅₉	1	4	yes	—
148	C₂³	#14273637	1	4	yes	—
149	C₂³	#14273638	1	4	yes	—
150	C₂³	C₂×C₄×Dic₅₉	1	3	yes	—
151	C₂×C₄	C₈×Dic₅₉	1	12	yes	—
152	C₂×C₄	#14273637	1	12	yes	—
153	C₂×C₄	#14273638	1	12	yes	—
154	C₂×C₄	C₂×C₄×Dic₅₉	1	3	yes	—
155	C₂×C₄	C₂×C₄×Dic₅₉	1	3	yes	—
156	C₂×C₄	C₂×C₄×Dic₅₉	1	3	yes	—
157	C₂×C₄	C₂×C₈×D₅₉	1	12	yes	—
158	C₂×C₄	C₂×#11745659	1	12	yes	—
159	C₂×C₄	C₂×#11745664	1	12	yes	—
160	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
162	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
163	C₈	—	2	8	no	—
164	C₈	—	59	16	no	—
165	C₈	—	118	8	no	—
166	C₂²	#16837124	1	16	yes	direct product factor
167	C₂²	C₂×C₈×Dic₅₉	1	6	yes	—
168	C₂²	C₂×#14273637	1	6	yes	—
169	C₂²	C₂×#14273638	1	6	yes	—
170	C₂²	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
171	C₄	C₂×C₈×Dic₅₉	1	6	yes	—
172	C₄	C₂×#14273637	1	6	yes	—
173	C₄	C₂×#14273638	1	6	yes	—
174	C₄	C₂²×C₈×D₅₉	1	2	yes	—
175	C₄	C₂²×#11745659	1	2	yes	—
176	C₄	C₂²×#11745664	1	2	yes	—
177	C₂	C₂×#16837124	1	12	yes	direct product factor
178	C₂	C₂²×C₈×Dic₅₉	1	1	yes	characteristic
179	C₂	C₂²×#14273637	1	1	yes	characteristic
180	C₂	C₂²×#14273638	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₉
commutator	C₂×C₅₉	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂²×C₄×C₈×C₅₉	C₂
socle, prime radical	C₂⁴×C₅₉	C₄²
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₉
hypocenter	C₅₉	C₂²×#322
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#16837124	C₂	direct (N × H)
11. #16837124	C₂²	direct (N × H)
17. C₂²×C₅₉⋊C₈	C₈	semidirect
53. C₂×C₅₉⋊C₈	C₂×C₈	semidirect
74. C₅₉⋊C₈	C₂²×C₈	semidirect
76. C₂²×C₅₉	#322	semidirect
91. C₂×C₅₉	C₂×#322	semidirect
106. C₅₉	C₂²×#322	semidirect
166. C₂²	#16837124	direct (N × H)
177. C₂	C₂×#16837124	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×C₄×C₅₉⋊C₈	2
C₂×#16837124	2
C₂²×C₄×C₈×C₅₉	2
C₂²×#322	59

Subgroups — C22×#16837124 #28554258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16837124 #28554258