Subgroups — C₂×#21437199 #29495158

Order 15104.

2074 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437199	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16837819	C₂	1	8	yes	maximal, complemented
3	C₂²×#14273667	C₂	1	6	yes	maximal, complemented
4	C₂²×#131×C₅₉	C₂	1	1	yes	maximal, characteristic
5	#16837819	C₂²	1	64	yes	complemented
6	C₂×#14273667	C₂²	1	4	yes	—
7	C₂×#14273667	C₂²	1	12	yes	complemented
8	C₂×#14273667	C₂²	1	48	yes	complemented
9	C₂²×C₅₉⋊C₁₆	C₂²	1	12	yes	complemented
10	C₂²×#41×C₅₉	C₄	1	1	yes	characteristic
11	C₂×#131×C₅₉	C₄	1	4	yes	—
12	C₂×#131×C₅₉	C₄	1	8	yes	—
13	C₂×#131×C₅₉	C₂²	1	4	yes	—
14	C₂×#131×C₅₉	C₂²	1	8	yes	—
15	C₂²×M₁₆×C₅₉	C₄	1	3	yes	—
16	C₂³×C₈×C₅₉	C₂²	1	3	yes	—
17	#14273667	C₂³	1	16	yes	—
18	#14273667	C₂³	1	48	yes	complemented
19	C₂×C₅₉⋊C₁₆	C₂³	1	8	yes	—
20	C₂×C₅₉⋊C₁₆	C₂³	1	48	yes	complemented
21	C₂×#41×C₅₉	C₈	1	4	yes	—
22	C₂×#41×C₅₉	C₈	1	8	yes	—
23	C₂×#41×C₅₉	C₂×C₄	1	4	yes	—
24	C₂×#41×C₅₉	C₂×C₄	1	8	yes	—
25	C₂²×Q₈×C₅₉	C₈	1	1	yes	characteristic
26	C₂²×D₄×C₅₉	C₈	1	3	yes	—
27	C₂³×C₄×C₅₉	C₂×C₄	1	3	yes	—
28	#131×C₅₉	C₂×C₄	1	16	yes	—
29	#131×C₅₉	C₂×C₄	1	32	yes	—
30	#131×C₅₉	C₂³	1	16	yes	—
31	C₂×M₁₆×C₅₉	C₂×C₄	1	12	yes	—
32	C₂×M₁₆×C₅₉	C₂×C₄	1	24	yes	—
33	C₂²×C₈×C₅₉	C₂×C₄	1	3	yes	—
34	C₂²×C₈×C₅₉	C₂×C₄	1	6	yes	—
35	C₂²×C₈×C₅₉	C₂×C₄	1	12	yes	—
36	C₂²×C₈×C₅₉	C₂³	1	6	yes	—
37	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	C₂³	1	12	yes	—
39	C₂⁴×C₅₉	C₂×C₈	1	3	yes	—
40	#41×C₅₉	C₂×C₈	1	16	yes	—
41	#41×C₅₉	C₂×C₈	1	32	yes	—
42	#41×C₅₉	C₂²×C₄	1	16	yes	—
43	C₂×Q₈×C₅₉	C₂×C₈	1	8	yes	—
44	C₂×Q₈×C₅₉	C₂×C₈	1	4	yes	—
45	C₂×D₄×C₅₉	C₂×C₈	1	12	yes	—
46	C₂×D₄×C₅₉	C₂×C₈	1	24	yes	—
47	C₂²×C₄×C₅₉	C₂×C₈	1	3	yes	—
48	C₂²×C₄×C₅₉	C₂×C₈	1	6	yes	—
49	C₂²×C₄×C₅₉	C₂×C₈	1	12	yes	—
50	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	C₂²×C₄	1	6	yes	—
52	C₂²×C₄×C₅₉	C₂²×C₄	1	12	yes	—
53	M₁₆×C₅₉	C₂²×C₄	1	48	yes	—
54	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
57	C₂×C₈×C₅₉	C₂²×C₄	1	12	yes	—
58	C₂×C₈×C₅₉	C₂²×C₄	1	12	yes	—
59	C₂×C₈×C₅₉	C₂²×C₄	1	24	yes	—
60	C₂×C₈×C₅₉	C₂⁴	1	2	yes	—
61	C₂×C₈×C₅₉	C₂⁴	1	12	yes	—
62	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
63	C₅₉⋊C₁₆	C₂⁴	1	16	yes	—
64	C₂³×C₅₉	—	4	6	no	—
65	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
66	C₂³×C₅₉	C₂²×C₈	1	6	yes	—
67	C₂³×C₅₉	C₂²×C₈	1	12	yes	—
68	Q₈×C₅₉	C₂²×C₈	1	16	yes	—
69	D₄×C₅₉	C₂²×C₈	1	48	yes	—
70	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂²×C₈	1	12	yes	—
73	C₂×C₄×C₅₉	C₂²×C₈	1	6	yes	—
74	C₂×C₄×C₅₉	C₂²×C₈	1	24	yes	—
75	C₂×C₄×C₅₉	C₂²×C₈	1	12	yes	—
76	C₂×C₄×C₅₉	C₂³×C₄	1	2	yes	—
77	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂³×C₄	1	12	yes	—
79	C₈×C₅₉	C₂³×C₄	1	2	yes	—
80	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₅₉	C₂³×C₄	1	12	yes	—
82	C₈×C₅₉	C₂⁵	1	1	yes	characteristic
83	C₂×#3407	—	59	1	no	maximal, Sylow
84	C₂²×C₅₉	—	2	2	no	—
85	C₂²×C₅₉	—	2	12	no	—
86	C₂²×C₅₉	—	4	12	no	—
87	C₂²×C₅₉	C₂³×C₈	1	2	yes	—
88	C₂²×C₅₉	C₂³×C₈	1	12	yes	—
89	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₅₉	C₂³×C₈	1	2	yes	—
91	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
92	C₄×C₅₉	C₂³×C₈	1	12	yes	—
93	C₄×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
94	C₂²×#131	D₅₉	1	1	yes	characteristic
95	#3407	—	59	16	no	—
96	C₂×#504	—	59	8	no	—
97	C₂²×M₃₂	—	59	6	no	—
98	C₂×C₅₉	—	2	2	no	—
99	C₂×C₅₉	—	2	12	no	—
100	C₂×C₅₉	#3407	1	2	yes	complemented
101	C₂×C₅₉	C₂⁴×C₈	1	1	yes	characteristic
102	C₂²×#41	Dic₅₉	1	1	yes	characteristic
103	C₂×#131	Dic₅₉	1	4	yes	—
104	C₂×#131	Dic₅₉	1	8	yes	—
105	C₂×#131	C₂×D₅₉	1	4	yes	—
106	C₂×#131	C₂×D₅₉	1	8	yes	—
107	C₂²×M₁₆	Dic₅₉	1	3	yes	—
108	C₂³×C₈	C₂×D₅₉	1	3	yes	—
109	#504	—	59	64	no	—
110	C₂×M₃₂	—	59	4	no	—
111	C₂×M₃₂	—	59	12	no	—
112	C₂×M₃₂	—	59	48	no	—
113	C₂²×C₁₆	—	59	12	no	—
114	C₅₉	C₂×#3407	1	1	yes	characteristic, complemented, Sylow
115	C₂×#41	C₅₉⋊C₈	1	4	yes	—
116	C₂×#41	C₅₉⋊C₈	1	8	yes	—
117	C₂×#41	C₂×Dic₅₉	1	4	yes	—
118	C₂×#41	C₂×Dic₅₉	1	8	yes	—
119	C₂²×Q₈	C₅₉⋊C₈	1	1	yes	characteristic
120	C₂²×D₄	C₅₉⋊C₈	1	3	yes	—
121	C₂³×C₄	C₂×Dic₅₉	1	3	yes	—
122	#131	C₂×Dic₅₉	1	16	yes	—
123	#131	C₂×Dic₅₉	1	32	yes	—
124	#131	C₂²×D₅₉	1	16	yes	—
125	C₂×M₁₆	C₂×Dic₅₉	1	12	yes	—
126	C₂×M₁₆	C₂×Dic₅₉	1	24	yes	—
127	C₂²×C₈	C₂×Dic₅₉	1	3	yes	—
128	C₂²×C₈	C₂×Dic₅₉	1	6	yes	—
129	C₂²×C₈	C₂×Dic₅₉	1	12	yes	—
130	C₂²×C₈	C₂²×D₅₉	1	12	yes	—
131	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₅₉	1	6	yes	—
133	M₃₂	—	59	16	no	—
134	M₃₂	—	59	48	no	—
135	C₂×C₁₆	—	59	48	no	—
136	C₂×C₁₆	—	59	8	no	—
137	C₂⁴	C₂×C₅₉⋊C₈	1	3	yes	—
138	#41	C₂×C₅₉⋊C₈	1	16	yes	—
139	#41	C₂×C₅₉⋊C₈	1	32	yes	—
140	#41	C₂²×Dic₅₉	1	16	yes	—
141	C₂×Q₈	C₂×C₅₉⋊C₈	1	8	yes	—
142	C₂×Q₈	C₂×C₅₉⋊C₈	1	4	yes	—
143	C₂×D₄	C₂×C₅₉⋊C₈	1	12	yes	—
144	C₂×D₄	C₂×C₅₉⋊C₈	1	24	yes	—
145	C₂²×C₄	C₂×C₅₉⋊C₈	1	3	yes	—
146	C₂²×C₄	C₂×C₅₉⋊C₈	1	6	yes	—
147	C₂²×C₄	C₂×C₅₉⋊C₈	1	12	yes	—
148	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
149	C₂²×C₄	C₂²×Dic₅₉	1	12	yes	—
150	C₂²×C₄	C₂²×Dic₅₉	1	6	yes	—
151	M₁₆	C₂²×Dic₅₉	1	48	yes	—
152	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
153	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
154	C₂×C₈	C₂²×Dic₅₉	1	12	yes	—
155	C₂×C₈	C₂²×Dic₅₉	1	6	yes	—
156	C₂×C₈	C₂²×Dic₅₉	1	24	yes	—
157	C₂×C₈	C₂²×Dic₅₉	1	12	yes	—
158	C₂×C₈	C₂³×D₅₉	1	12	yes	—
159	C₂×C₈	C₂³×D₅₉	1	2	yes	—
160	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
161	C₁₆	—	59	16	no	—
162	C₂³	—	4	6	no	—
163	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
164	C₂³	C₂²×C₅₉⋊C₈	1	6	yes	—
165	C₂³	C₂²×C₅₉⋊C₈	1	12	yes	—
166	Q₈	C₂²×C₅₉⋊C₈	1	16	yes	—
167	D₄	C₂²×C₅₉⋊C₈	1	48	yes	—
168	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
169	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₅₉⋊C₈	1	12	yes	—
171	C₂×C₄	C₂²×C₅₉⋊C₈	1	6	yes	—
172	C₂×C₄	C₂²×C₅₉⋊C₈	1	24	yes	—
173	C₂×C₄	C₂²×C₅₉⋊C₈	1	12	yes	—
174	C₂×C₄	C₂³×Dic₅₉	1	2	yes	—
175	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
176	C₂×C₄	C₂³×Dic₅₉	1	12	yes	—
177	C₈	C₂³×Dic₅₉	1	2	yes	—
178	C₈	C₂³×Dic₅₉	1	1	yes	characteristic
179	C₈	C₂³×Dic₅₉	1	12	yes	—
180	C₈	C₂⁴×D₅₉	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	2	12	no	—
183	C₂²	—	4	12	no	—
184	C₂²	C₂³×C₅₉⋊C₈	1	2	yes	—
185	C₂²	C₂³×C₅₉⋊C₈	1	12	yes	—
186	C₂²	C₂³×C₅₉⋊C₈	1	1	yes	characteristic
187	C₄	C₂³×C₅₉⋊C₈	1	2	yes	—
188	C₄	C₂³×C₅₉⋊C₈	1	1	yes	characteristic
189	C₄	C₂³×C₅₉⋊C₈	1	12	yes	—
190	C₄	C₂⁴×Dic₅₉	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	12	no	—
193	C₂	#21437199	1	2	yes	direct product factor
194	C₂	C₂⁴×C₅₉⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂³×D₅₉
commutator	C₂×C₅₉	C₂⁴×C₈
Frattini	C₈	C₂⁴×D₅₉
Fitting	C₂²×#131×C₅₉	C₂
socle	C₂²×C₅₉	C₂³×C₈
prime radical	C₂²×#41×C₅₉	C₄
cosocle	C₈×C₅₉	C₂⁵
hypercenter	C₂²×#131	D₅₉
hypocenter	C₅₉	C₂×#3407
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. #21437199	C₂	direct (N × H)
2. C₂×#16837819	C₂	semidirect
3. C₂²×#14273667	C₂	semidirect
5. #16837819	C₂²	semidirect
7. C₂×#14273667	C₂²	semidirect
8. C₂×#14273667	C₂²	semidirect
9. C₂²×C₅₉⋊C₁₆	C₂²	semidirect
18. #14273667	C₂³	semidirect
20. C₂×C₅₉⋊C₁₆	C₂³	semidirect
100. C₂×C₅₉	#3407	semidirect
114. C₅₉	C₂×#3407	semidirect
193. C₂	#21437199	direct (N × H)

Maximal subgroups

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

subgroup	index
#21437199	2
C₂×#16837819	2
C₂²×#14273667	2
C₂²×#131×C₅₉	2
C₂×#3407	59

Subgroups — C2×#21437199 #29495158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21437199 #29495158