Subgroups — C₂×#21422190 #28503576

Order 15104.

574 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273663	C₂	1	2	yes	maximal
2	#21422190	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11745664	—	2	2	no	—
5	C₂×#14273663	—	2	2	no	—
6	C₂×#14273663	—	2	4	no	—
7	C₂×#14273663	C₄	1	8	yes	—
8	C₂×#14273663	C₂²	1	4	yes	—
9	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₉	—	2	4	no	—
11	C₂²×C₁₆×C₅₉	C₄	1	2	yes	—
12	C₂²×C₁₆×C₅₉	C₄	1	2	yes	—
13	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
14	C₂×#11745664	—	2	8	no	—
15	C₂×#11745664	—	2	4	no	—
16	C₂×#11745664	—	4	2	no	—
17	C₂×#11745664	—	4	4	no	—
18	C₂²×C₅₉⋊C₈	—	4	2	no	—
19	#14273663	—	2	8	no	—
20	#14273663	—	2	4	no	—
21	#14273663	C₂×C₄	1	8	yes	—
22	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	D₄	1	4	yes	—
28	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₅₉	—	2	4	no	—
32	C₂×C₁₆×C₅₉	—	2	4	no	—
33	C₂×C₁₆×C₅₉	—	2	4	no	—
34	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
35	C₂⁴×C₅₉	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
37	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	SD₁₆	1	4	yes	—
42	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₄²	1	2	yes	—
45	C₂×C₈×C₅₉	C₄²	1	2	yes	—
46	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
47	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
52	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
57	#11745664	—	2	8	no	—
58	#11745664	—	4	8	no	—
59	#11745664	—	4	4	no	—
60	C₂×C₅₉⋊C₈	—	4	8	no	—
61	C₂×C₅₉⋊C₈	—	4	4	no	—
62	C₁₆×C₅₉	—	2	4	no	—
63	C₂³×C₅₉	—	2	4	no	—
64	C₂³×C₅₉	#103	1	4	yes	—
65	C₂³×C₅₉	#107	1	1	yes	characteristic
66	C₂³×C₅₉	#107	1	1	yes	characteristic
67	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₉	#95	1	2	yes	—
69	C₂×C₄×C₅₉	#95	1	2	yes	—
70	C₂×C₄×C₅₉	#102	1	4	yes	—
71	C₂×C₄×C₅₉	#102	1	4	yes	—
72	C₂×C₄×C₅₉	#103	1	4	yes	—
73	C₂×C₄×C₅₉	#106	1	2	yes	—
74	C₂×C₄×C₅₉	#106	1	2	yes	—
75	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
81	C₈×C₅₉	#95	1	2	yes	—
82	C₈×C₅₉	#95	1	2	yes	—
83	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
84	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
86	C₅₉⋊C₈	—	4	8	no	—
87	C₂×#1381	—	59	1	no	maximal, Sylow
88	C₂²×C₅₉	—	2	8	no	—
89	C₂²×C₅₉	—	2	2	no	—
90	C₂²×C₅₉	—	2	2	no	—
91	C₂²×C₅₉	#340	1	2	yes	—
92	C₂²×C₅₉	#340	1	2	yes	—
93	C₂²×C₅₉	#368	1	4	yes	—
94	C₂²×C₅₉	C₂×#103	1	2	yes	—
95	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
96	C₄×C₅₉	#340	1	2	yes	—
97	C₄×C₅₉	#340	1	2	yes	—
98	C₄×C₅₉	C₂×#95	1	1	yes	characteristic
99	C₄×C₅₉	C₂×#102	1	2	yes	—
100	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
101	C₂³×C₁₆	D₅₉	1	1	yes	characteristic
102	C₂²×#108	—	59	2	no	—
103	#1381	—	59	4	no	—
104	C₂×C₅₉	—	2	2	no	—
105	C₂×C₅₉	—	2	2	no	—
106	C₂×C₅₉	#1381	1	4	yes	complemented
107	C₂×C₅₉	C₂×#340	1	1	yes	characteristic
108	C₂×C₅₉	C₂×#368	1	2	yes	—
109	C₂²×M₁₆	—	118	2	no	—
110	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
111	C₂²×C₁₆	—	2	4	no	—
112	C₂²×C₁₆	Dic₅₉	1	2	yes	—
113	C₂²×C₁₆	Dic₅₉	1	2	yes	—
114	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
115	C₂×#108	—	59	8	no	—
116	C₂×#108	—	59	4	no	—
117	C₂×#108	—	118	2	no	—
118	C₂×#108	—	118	4	no	—
119	C₅₉	C₂×#1381	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
121	C₂×M₁₆	—	118	8	no	—
122	C₂×M₁₆	—	118	4	no	—
123	C₂×M₁₆	—	236	4	no	—
124	C₂×M₁₆	—	236	2	no	—
125	C₂²×C₈	—	236	2	no	—
126	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
127	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
128	C₂²×C₈	C₄×D₅₉	1	4	yes	—
129	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
131	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
132	C₂²×C₈	C₂²⋊D₅₉	1	4	yes	—
133	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
134	C₂×C₁₆	—	2	4	no	—
135	C₂×C₁₆	—	2	4	no	—
136	C₂×C₁₆	—	2	4	no	—
137	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
138	#108	—	59	8	no	—
139	#108	—	118	8	no	—
140	#108	—	118	4	no	—
141	C₂⁴	Dic₂₃₆	1	1	yes	characteristic
142	C₂²×C₄	#11745660	1	4	yes	—
143	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
144	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
145	C₂²×C₄	Dic₂₃₆	1	1	yes	characteristic
146	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
147	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
148	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	4	yes	—
149	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
150	M₁₆	—	118	8	no	—
151	M₁₆	—	236	8	no	—
152	M₁₆	—	236	4	no	—
153	C₂×C₈	—	236	8	no	—
154	C₂×C₈	—	236	4	no	—
155	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
156	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
157	C₂×C₈	Dic₅₉⋊C₄	1	4	yes	—
158	C₂×C₈	Dic₅₉⋊C₄	1	4	yes	—
159	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
160	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
161	C₂×C₈	(C₂×D₅₉)⋊C₄	1	4	yes	—
162	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
163	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
164	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
165	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
166	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₂³	—	2	4	no	—
170	C₂³	#14273652	1	4	yes	—
171	C₂³	#14273661	1	1	yes	characteristic
172	C₂³	#14273661	1	1	yes	characteristic
173	C₂³	C₂×Dic₂₃₆	1	1	yes	characteristic
174	C₂×C₄	C₅₉⋊#95	1	2	yes	—
175	C₂×C₄	C₅₉⋊#95	1	2	yes	—
176	C₂×C₄	#14273649	1	4	yes	—
177	C₂×C₄	#14273649	1	4	yes	—
178	C₂×C₄	#14273652	1	4	yes	—
179	C₂×C₄	#14273659	1	2	yes	—
180	C₂×C₄	#14273659	1	2	yes	—
181	C₂×C₄	#14273661	1	1	yes	characteristic
182	C₂×C₄	#14273661	1	1	yes	characteristic
183	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	2	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
185	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
186	C₂×C₄	C₂×#11745660	1	2	yes	—
187	C₈	—	236	8	no	—
188	C₈	C₅₉⋊#95	1	2	yes	—
189	C₈	C₅₉⋊#95	1	2	yes	—
190	C₈	C₂×C₄×Dic₅₉	1	1	yes	characteristic
191	C₈	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
192	C₈	C₂×Dic₅₉⋊C₄	1	2	yes	—
193	C₂²	—	2	8	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#16837165	1	2	yes	—
197	C₂²	#16837165	1	2	yes	—
198	C₂²	#16837232	1	4	yes	—
199	C₂²	C₂×#14273652	1	2	yes	—
200	C₂²	C₂×#14273661	1	1	yes	characteristic
201	C₄	#16837165	1	2	yes	—
202	C₄	#16837165	1	2	yes	—
203	C₄	C₂×C₅₉⋊#95	1	1	yes	characteristic
204	C₄	C₂×#14273649	1	2	yes	—
205	C₄	C₂×#14273659	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	#21422190	1	4	yes	direct product factor
209	C₂	C₂×#16837165	1	1	yes	characteristic
210	C₂	C₂×#16837232	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄₇₂
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂³×C₁₆×C₅₉	C₂
socle	C₂³×C₅₉	#107
prime radical	C₂⁴×C₅₉	Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂³×C₁₆	D₅₉
hypocenter	C₅₉	C₂×#1381
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
2. #21422190	C₂	direct (N × H)
106. C₂×C₅₉	#1381	semidirect
119. C₅₉	C₂×#1381	semidirect
208. C₂	#21422190	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×#14273663	2
#21422190	2
C₂³×C₁₆×C₅₉	2
C₂×#1381	59

Subgroups — C2×#21422190 #28503576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422190 #28503576