Subgroups — C₂×#21425998 #28742169

Order 15104.

582 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	#21425998	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₅₉⋊C₁₆	C₂	1	2	yes	maximal
4	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₆×Dic₅₉	C₂	1	2	yes	maximal
6	C₂×C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₁₁₈	C₄	1	1	yes	characteristic
8	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₈×Dic₁₁₈	C₄	1	4	yes	—
11	C₈×Dic₁₁₈	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₅₉⋊C₈)	C₄	1	1	yes	characteristic
13	C₂×Dic₅₉⋊C₈	C₄	1	2	yes	—
14	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
15	Dic₅₉⋊C₁₆	C₂²	1	8	yes	—
16	C₄×C₅₉⋊C₁₆	C₂²	1	4	yes	—
17	C₁₆×Dic₅₉	C₂²	1	8	yes	—
18	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₅₉×C₄⋊C₁₆	C₂²	1	4	yes	—
21	C₂²×Dic₁₁₈	C₈	1	1	yes	characteristic
22	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×Dic₁₁₈	C₈	1	4	yes	—
24	C₄×Dic₁₁₈	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₅₉	C₈	1	1	yes	characteristic
26	C₂×Dic₅₉⋊C₄	C₈	1	2	yes	—
27	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
28	C₂×C₅₉⋊C₁₆	—	2	2	no	—
29	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	complemented
30	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
31	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
32	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
33	Dic₅₉⋊C₈	C₂×C₄	1	8	yes	—
34	C₈×Dic₅₉	C₂×C₄	1	4	yes	—
35	C₈×Dic₅₉	C₂³	1	4	yes	—
36	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₅₉	—	2	4	no	—
40	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
41	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₅₉	C₂×C₈	1	2	yes	—
47	C₂²×Dic₅₉	C₂×C₈	1	2	yes	—
48	C₂×Dic₁₁₈	C₁₆	1	8	yes	complemented
49	C₂×Dic₁₁₈	C₂×C₈	1	4	yes	—
50	C₂×C₈×C₅₉	—	2	2	no	—
51	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₅₉	#41	1	2	yes	—
55	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
56	C₄²×C₅₉	C₂×C₈	1	2	yes	—
57	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
58	C₄⋊Dic₅₉	C₂×C₈	1	4	yes	—
59	Dic₅₉⋊C₄	C₂×C₈	1	8	yes	—
60	C₄×Dic₅₉	C₂×C₈	1	4	yes	—
61	C₄×Dic₅₉	C₂²×C₄	1	4	yes	—
62	C₂×C₅₉⋊C₈	—	2	4	no	—
63	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
64	C₁₆×C₅₉	—	2	4	no	—
65	C₅₉⋊C₁₆	—	2	2	no	—
66	C₅₉⋊C₁₆	C₂×Q₈	1	4	yes	complemented
67	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	—	2	2	no	—
69	C₂×C₄×C₅₉	C₂×C₁₆	1	4	yes	—
70	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
71	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₅₉	#131	1	2	yes	—
75	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
76	C₂×Dic₅₉	—	2	4	no	—
77	C₂×Dic₅₉	C₂×C₁₆	1	8	yes	—
78	C₂×Dic₅₉	C₂²×C₈	1	4	yes	—
79	C₂×Dic₅₉	C₂²×C₈	1	4	yes	—
80	Dic₁₁₈	C₂×C₁₆	1	16	yes	complemented
81	C₈×C₅₉	—	2	2	no	—
82	C₈×C₅₉	C₄×Q₈	1	2	yes	—
83	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
84	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
85	C₅₉⋊C₈	—	2	4	no	—
86	C₂×Q₈×C₁₆	—	59	1	no	maximal, Sylow
87	C₂²×C₅₉	C₈×Q₈	1	2	yes	—
88	C₂²×C₅₉	C₂²×C₁₆	1	2	yes	—
89	C₂²×C₅₉	#504	1	2	yes	—
90	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
91	C₄×C₅₉	—	2	2	no	—
92	C₄×C₅₉	C₈×Q₈	1	2	yes	—
93	C₄×C₅₉	C₂²×C₁₆	1	4	yes	—
94	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
95	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
96	Dic₅₉	—	2	4	no	—
97	Dic₅₉	C₂²×C₁₆	1	8	yes	—
98	C₂×C₈×Q₈	—	59	1	no	—
99	Q₈×C₁₆	—	59	8	no	—
100	C₂×C₄⋊C₁₆	—	59	2	no	—
101	C₂×C₄⋊C₁₆	D₅₉	1	1	yes	characteristic
102	C₂×C₄×C₁₆	—	59	2	no	—
103	C₂×C₄×C₁₆	—	59	1	no	—
104	C₂×C₅₉	Q₈×C₁₆	1	4	yes	complemented
105	C₂×C₅₉	C₂×C₈×Q₈	1	1	yes	characteristic
106	C₂×C₅₉	C₂³×C₁₆	1	1	yes	characteristic
107	C₂×C₅₉	C₂×#504	1	1	yes	characteristic
108	C₂×C₄×Q₈	—	59	1	no	—
109	C₂²×C₁₆	—	59	1	no	—
110	C₂²×C₁₆	—	59	1	no	—
111	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
112	C₈×Q₈	—	59	4	no	—
113	C₈×Q₈	—	59	4	no	—
114	C₂×C₄⋊C₈	—	59	2	no	—
115	C₂×C₄⋊C₈	—	59	1	no	—
116	C₂×C₄×C₈	—	59	2	no	—
117	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
118	C₄⋊C₁₆	—	59	8	no	—
119	C₄⋊C₁₆	C₂×D₅₉	1	4	yes	—
120	C₄×C₁₆	—	59	8	no	—
121	C₄×C₁₆	—	59	4	no	—
122	C₅₉	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
123	C₂²×Q₈	—	59	1	no	—
124	C₂²×C₈	—	59	2	no	—
125	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
127	C₄×Q₈	—	59	4	no	—
128	C₄×Q₈	—	59	4	no	—
129	C₂×C₄⋊C₄	—	59	2	no	—
130	C₂×C₄⋊C₄	—	59	1	no	—
131	C₂×C₄²	—	59	2	no	—
132	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
133	C₂×C₁₆	—	2	4	no	—
134	C₂×C₁₆	—	59	4	no	—
135	C₂×C₁₆	—	59	2	no	—
136	C₂×C₁₆	—	59	2	no	—
137	C₂×C₁₆	—	118	2	no	—
138	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
139	C₄⋊C₈	—	59	8	no	—
140	C₄⋊C₈	—	59	4	no	—
141	C₄×C₈	—	59	4	no	—
142	C₄×C₈	—	59	4	no	—
143	C₄×C₈	C₄×D₅₉	1	2	yes	—
144	C₄×C₈	C₂²×D₅₉	1	2	yes	—
145	C₂×Q₈	—	59	8	no	—
146	C₂×Q₈	—	59	4	no	—
147	C₂²×C₄	—	59	2	no	—
148	C₂²×C₄	—	59	2	no	—
149	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
150	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	59	4	no	—
154	C₂×C₈	—	118	4	no	—
155	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
156	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
157	C₂×C₈	#11745687	1	2	yes	—
158	C₂×C₈	Q₈×D₅₉	1	2	yes	—
159	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
160	C₄⋊C₄	—	59	8	no	—
161	C₄⋊C₄	—	59	4	no	—
162	C₄²	—	59	4	no	—
163	C₄²	—	59	4	no	—
164	C₄²	C₈×D₅₉	1	2	yes	—
165	C₄²	C₂×C₄×D₅₉	1	2	yes	—
166	C₁₆	—	2	4	no	—
167	C₁₆	—	59	4	no	—
168	C₁₆	—	118	2	no	—
169	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
170	Q₈	—	59	16	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	59	4	no	—
173	C₂×C₄	—	59	8	no	—
174	C₂×C₄	—	59	4	no	—
175	C₂×C₄	—	118	4	no	—
176	C₂×C₄	C₁₆×D₅₉	1	4	yes	—
177	C₂×C₄	#14273692	1	2	yes	—
178	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
179	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
180	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
181	C₂×C₄	#14273735	1	2	yes	—
182	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	118	4	no	—
185	C₈	#14273692	1	2	yes	—
186	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
187	C₈	C₂×#11745687	1	1	yes	characteristic
188	C₂²	#16837526	1	2	yes	—
189	C₂²	C₂×C₁₆×D₅₉	1	2	yes	—
190	C₂²	#16837820	1	2	yes	—
191	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	59	8	no	—
194	C₄	—	118	4	no	—
195	C₄	#16837526	1	2	yes	—
196	C₄	C₂×C₁₆×D₅₉	1	4	yes	—
197	C₄	C₂×#14273692	1	1	yes	characteristic
198	C₄	C₂×#14273735	1	1	yes	characteristic
199	C₂	#21425998	1	4	yes	direct product factor
200	C₂	C₂×#16837526	1	1	yes	characteristic
201	C₂	C₂²×C₁₆×D₅₉	1	1	yes	characteristic
202	C₂	C₂×#16837820	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₉
commutator	C₂×C₅₉	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₉	C₂²×C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #21425998	C₂	direct (N × H)
29. C₂×C₅₉⋊C₁₆	Q₈	semidirect
48. C₂×Dic₁₁₈	C₁₆	semidirect
66. C₅₉⋊C₁₆	C₂×Q₈	semidirect
80. Dic₁₁₈	C₂×C₁₆	semidirect
104. C₂×C₅₉	Q₈×C₁₆	semidirect
122. C₅₉	C₂×Q₈×C₁₆	semidirect
199. C₂	#21425998	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₁₈	2
#21425998	2
C₂×Dic₅₉⋊C₁₆	2
C₂×C₄×C₅₉⋊C₁₆	2
C₂×C₁₆×Dic₅₉	2
C₂×C₅₉×C₄⋊C₁₆	2
C₂×Q₈×C₁₆	59

Subgroups — C2×#21425998 #28742169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21425998 #28742169