Subgroups — C₂×#21445570 #29677482

Order 15616.

1134 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519688	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839608	C₂	1	1	yes	maximal, characteristic
3	#21445570	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839516	C₂	1	4	yes	maximal
5	C₂²×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×#92418	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₆₁	C₂²	1	2	yes	—
8	C₂×#14519688	C₄	1	8	yes	complemented
9	C₂×#14519688	C₂²	1	4	yes	—
10	#16839608	C₄	1	4	yes	—
11	#16839608	C₂²	1	4	yes	—
12	C₂×#14519684	C₄	1	4	yes	—
13	#16839516	C₂²	1	16	yes	—
14	C₂³×C₈×C₆₁	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₁	C₄	1	8	yes	complemented
17	C₂×C₄×C₈×C₆₁	C₂²	1	4	yes	—
18	C₂³×Dic₆₁	C₂×C₄	1	4	yes	—
19	C₂²×#92418	—	2	8	no	—
20	C₂²×#92418	C₂×C₄	1	8	yes	—
21	C₂²×#92418	C₂³	1	4	yes	—
22	C₂×C₄×Dic₆₁	—	2	8	no	—
23	C₂×C₄×Dic₆₁	C₂×C₄	1	4	yes	—
24	C₂×C₄×Dic₆₁	C₂³	1	4	yes	—
25	#14519688	C₂×C₄	1	16	yes	complemented
26	#14519684	C₂×C₄	1	16	yes	—
27	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	—	2	8	no	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
32	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
36	C₄×C₈×C₆₁	C₂×C₄	1	16	yes	complemented
37	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	—	2	8	no	—
39	C₂²×C₄×C₆₁	C₄²	1	8	yes	—
40	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₆₁	#41	1	8	yes	—
45	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
46	C₂²×Dic₆₁	—	2	8	no	—
47	C₂²×Dic₆₁	—	2	16	no	—
48	C₂²×Dic₆₁	C₂²×C₄	1	8	yes	—
49	C₂×C₈×C₆₁	—	2	16	no	—
50	C₂×C₈×C₆₁	—	2	8	no	—
51	C₂×C₈×C₆₁	C₂²×C₄	1	8	yes	—
52	C₄²×C₆₁	C₄²	1	8	yes	—
53	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
54	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
55	C₄×Dic₆₁	—	2	8	no	—
56	C₄×Dic₆₁	—	2	8	no	—
57	C₂×#92418	—	2	16	no	—
58	C₂×#92418	—	2	8	no	—
59	C₂×#92418	C₂²×C₄	1	8	yes	—
60	C₂³×C₆₁	C₂×C₄²	1	2	yes	—
61	C₂³×C₆₁	#117	1	4	yes	—
62	C₂³×C₆₁	#131	1	8	yes	—
63	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₁	—	2	8	no	—
65	C₂×C₄×C₆₁	—	2	16	no	—
66	C₂×C₄×C₆₁	C₂×C₄²	1	8	yes	—
67	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
68	C₂×C₄×C₆₁	#117	1	8	yes	—
69	C₂×C₄×C₆₁	#117	1	8	yes	—
70	C₂×C₄×C₆₁	#117	1	4	yes	—
71	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
74	C₂×Dic₆₁	—	2	8	no	—
75	C₂×Dic₆₁	—	2	16	no	—
76	C₂×Dic₆₁	—	4	16	no	—
77	C₈×C₆₁	—	2	8	no	—
78	#92418	—	2	8	no	—
79	C₂×#1746	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	#376	1	4	yes	—
81	C₂²×C₆₁	#405	1	8	yes	—
82	C₂²×C₆₁	#433	1	16	yes	—
83	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
84	C₂²×C₆₁	C₂×#117	1	2	yes	—
85	C₂²×C₆₁	C₂×#131	1	4	yes	—
86	C₄×C₆₁	—	2	8	no	—
87	C₄×C₆₁	#376	1	4	yes	—
88	C₄×C₆₁	C₂×#117	1	2	yes	—
89	C₄×C₆₁	C₂×#117	1	2	yes	—
90	Dic₆₁	—	4	8	no	—
91	C₂²×#97	—	61	1	no	—
92	C₂²×C₄×C₈	D₆₁	1	1	yes	characteristic
93	C₂×#376	—	61	1	no	—
94	#1746	—	61	8	no	—
95	C₂×#336	—	61	4	no	—
96	C₂×C₆₁	#1746	1	8	yes	complemented
97	C₂×C₆₁	C₂×#376	1	1	yes	characteristic
98	C₂×C₆₁	C₂×#405	1	2	yes	—
99	C₂×C₆₁	C₂×#433	1	4	yes	—
100	C₂³×C₈	—	61	2	no	—
101	C₂³×C₈	C₂×D₆₁	1	2	yes	—
102	C₂²×C₄²	—	61	2	no	—
103	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
104	C₂×#97	—	61	8	no	—
105	C₂×#97	—	61	4	no	—
106	C₂×C₄×C₈	Dic₆₁	1	8	yes	complemented
107	C₂×C₄×C₈	C₂×D₆₁	1	4	yes	—
108	#376	—	61	4	no	—
109	#376	—	61	4	no	—
110	C₂×#95	—	61	4	no	—
111	#336	—	61	16	no	—
112	C₆₁	C₂×#1746	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₄	—	61	4	no	—
114	C₂³×C₄	C₄×D₆₁	1	2	yes	—
115	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
116	C₂²×C₈	—	2	8	no	—
117	C₂²×C₈	—	61	8	no	—
118	C₂²×C₈	—	61	4	no	—
119	C₂²×C₈	—	122	8	no	—
120	C₂²×C₈	C₂×Dic₆₁	1	8	yes	—
121	C₂²×C₈	C₂²×D₆₁	1	4	yes	—
122	C₂×C₄²	—	61	4	no	—
123	C₂×C₄²	—	61	4	no	—
124	C₂×C₄²	—	122	8	no	—
125	C₂×C₄²	C₄×D₆₁	1	2	yes	—
126	C₂×C₄²	C₄×D₆₁	1	4	yes	—
127	C₂×C₄²	C₂×Dic₆₁	1	4	yes	—
128	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
129	#97	—	61	16	no	—
130	C₄×C₈	C₂×Dic₆₁	1	16	yes	complemented
131	#95	—	61	16	no	—
132	C₂⁴	C₂×C₄×D₆₁	1	1	yes	characteristic
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	61	8	no	—
135	C₂²×C₄	—	122	8	no	—
136	C₂²×C₄	—	122	16	no	—
137	C₂²×C₄	C₄×Dic₆₁	1	8	yes	—
138	C₂²×C₄	C₂×C₄×D₆₁	1	4	yes	—
139	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
140	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
141	C₂²×C₄	#11758222	1	8	yes	—
142	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
143	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	2	16	no	—
146	C₂×C₈	—	61	8	no	—
147	C₂×C₈	—	122	8	no	—
148	C₂×C₈	—	122	16	no	—
149	C₂×C₈	C₂²×Dic₆₁	1	8	yes	—
150	C₄²	—	122	8	no	—
151	C₄²	—	122	8	no	—
152	C₄²	C₄×Dic₆₁	1	8	yes	—
153	C₄²	C₂×C₄×D₆₁	1	4	yes	—
154	C₄²	C₂²×Dic₆₁	1	4	yes	—
155	C₂³	C₂×C₄×Dic₆₁	1	2	yes	—
156	C₂³	#14519734	1	4	yes	—
157	C₂³	#14519783	1	8	yes	—
158	C₂³	C₂²×C₄×D₆₁	1	1	yes	characteristic
159	C₂×C₄	—	2	16	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	122	8	no	—
162	C₂×C₄	—	122	16	no	—
163	C₂×C₄	—	244	16	no	—
164	C₂×C₄	C₂×C₄×Dic₆₁	1	8	yes	—
165	C₂×C₄	C₂×C₄×Dic₆₁	1	2	yes	—
166	C₂×C₄	#14519734	1	8	yes	—
167	C₂×C₄	#14519734	1	4	yes	—
168	C₂×C₄	#14519736	1	8	yes	—
169	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
170	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
171	C₂×C₄	C₂×#11758222	1	4	yes	—
172	C₈	—	2	8	no	—
173	C₈	—	122	8	no	—
174	C₂²	#16839608	1	4	yes	—
175	C₂²	#16839717	1	8	yes	—
176	C₂²	#16839821	1	16	yes	—
177	C₂²	C₂²×C₄×Dic₆₁	1	1	yes	characteristic
178	C₂²	C₂×#14519734	1	2	yes	—
179	C₂²	C₂×#14519783	1	4	yes	—
180	C₄	—	2	8	no	—
181	C₄	—	244	8	no	—
182	C₄	#16839608	1	4	yes	—
183	C₄	C₂×#14519734	1	2	yes	—
184	C₄	C₂×#14519736	1	2	yes	—
185	C₂	#21445570	1	8	yes	direct product factor
186	C₂	C₂×#16839608	1	1	yes	characteristic
187	C₂	C₂×#16839717	1	2	yes	—
188	C₂	C₂×#16839821	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂²×C₄×C₈×C₆₁	C₂
socle, prime radical	C₂⁴×C₆₁	C₂²×C₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂²×C₄×C₈	D₆₁
hypocenter	C₆₁	C₂×#1746
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #21445570	C₂	direct (N × H)
8. C₂×#14519688	C₄	semidirect
16. C₂×C₄×C₈×C₆₁	C₄	semidirect
25. #14519688	C₂×C₄	semidirect
36. C₄×C₈×C₆₁	C₂×C₄	semidirect
96. C₂×C₆₁	#1746	semidirect
106. C₂×C₄×C₈	Dic₆₁	semidirect
112. C₆₁	C₂×#1746	semidirect
130. C₄×C₈	C₂×Dic₆₁	semidirect
185. C₂	#21445570	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×#14519688	2
C₂×#16839608	2
#21445570	2
C₂×#16839516	2
C₂²×C₄×C₈×C₆₁	2
C₂×#1746	61

Subgroups — C2×#21445570 #29677482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21445570 #29677482