Subgroups — C₂×#21448114 #29838123

Order 15616.

470 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840178	C₂	1	1	yes	maximal, characteristic
2	#21448114	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16839595	C₂	1	2	yes	maximal, complemented
4	C₂²×#14519683	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₂×D₆₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₆₁×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14519783	C₄	1	1	yes	characteristic
8	#16840178	C₄	1	4	yes	—
9	#16840178	C₂²	1	4	yes	—
10	C₂×#14519717	C₄	1	1	yes	characteristic
11	C₂×#14519716	C₄	1	2	yes	—
12	C₂×C₁₆×D₆₁	C₂²	1	2	yes	—
13	#16839595	C₂²	1	8	yes	complemented
14	C₂×#14519683	C₂²	1	1	yes	characteristic
15	C₂×#14519683	C₂²	1	1	yes	characteristic
16	C₂×#14519683	C₂²	1	4	yes	complemented
17	C₃₂×D₆₁	C₂²	1	8	yes	complemented
18	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×M₆₄	C₂²	1	4	yes	complemented
20	C₂×C₃₂×C₆₁	C₂²	1	2	yes	—
21	C₂×#11758222	C₈	1	1	yes	characteristic
22	#14519783	C₈	1	4	yes	—
23	#14519783	C₂×C₄	1	4	yes	—
24	C₂×#11758194	C₈	1	1	yes	characteristic
25	C₂×#11758189	C₈	1	2	yes	—
26	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
27	#14519717	C₂×C₄	1	4	yes	—
28	#14519716	C₂×C₄	1	8	yes	—
29	C₂×#11758185	C₂×C₄	1	2	yes	—
30	C₁₆×D₆₁	C₂×C₄	1	4	yes	—
31	C₁₆×D₆₁	C₂³	1	4	yes	—
32	#14519683	C₂³	1	2	yes	—
33	#14519683	C₂³	1	2	yes	—
34	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
37	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
38	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
39	C₃₂×C₆₁	C₂³	1	4	yes	—
40	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
41	C₂×C₂²⋊D₆₁	C₁₆	1	2	yes	—
42	#11758222	C₁₆	1	4	yes	—
43	#11758222	C₂×C₈	1	4	yes	—
44	C₂×D₂₄₄	C₁₆	1	1	yes	characteristic
45	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
46	C₂×Dic₁₂₂	C₁₆	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
51	#11758194	C₂×C₈	1	4	yes	—
52	C₂×#92418	C₂×C₈	1	2	yes	—
53	#11758189	C₂×C₈	1	8	yes	—
54	C₈×D₆₁	C₂×C₈	1	4	yes	—
55	C₈×D₆₁	C₂²×C₄	1	4	yes	—
56	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
58	C₁₆×C₆₁	C₂⁴	1	1	yes	characteristic
59	#11758185	C₂²×C₄	1	4	yes	—
60	C₂³×C₆₁	C₂×C₁₆	1	1	yes	characteristic
61	C₂²×D₆₁	C₂×C₁₆	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
64	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
66	C₂²⋊D₆₁	C₂×C₁₆	1	8	yes	—
67	C₂×Dic₆₁	C₂×C₁₆	1	2	yes	—
68	D₂₄₄	C₂×C₁₆	1	4	yes	—
69	C₄×D₆₁	C₂×C₁₆	1	4	yes	—
70	C₄×D₆₁	C₂²×C₈	1	4	yes	—
71	Dic₁₂₂	C₂×C₁₆	1	4	yes	—
72	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
73	C₈×C₆₁	C₂²×C₈	1	2	yes	—
74	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
75	#92418	C₂²×C₈	1	4	yes	—
76	C₂×#2258	—	61	1	no	maximal, Sylow
77	C₂²×C₆₁	—	2	2	no	—
78	C₂²×C₆₁	C₂²×C₁₆	1	2	yes	—
79	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
80	C₂×D₆₁	—	2	4	no	—
81	C₂×D₆₁	C₂²×C₁₆	1	4	yes	—
82	C₄×C₆₁	C₂²×C₁₆	1	2	yes	—
83	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
84	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
85	Dic₆₁	C₂²×C₁₆	1	4	yes	—
86	C₂×#504	—	61	1	no	—
87	#2258	—	61	8	no	—
88	C₂×M₆₄	—	61	2	no	—
89	C₂×M₆₄	D₆₁	1	1	yes	characteristic, complemented
90	C₂²×C₃₂	—	61	1	no	—
91	C₂²×C₃₂	—	61	2	no	—
92	C₂×C₆₁	—	2	2	no	—
93	C₂×C₆₁	#2258	1	2	yes	complemented
94	C₂×C₆₁	C₂³×C₁₆	1	1	yes	characteristic
95	D₆₁	—	2	4	no	—
96	C₂×#131	—	61	1	no	—
97	#504	—	61	4	no	—
98	#504	—	61	4	no	—
99	C₂×M₃₂	—	61	1	no	—
100	C₂×M₃₂	—	61	2	no	—
101	C₂²×C₁₆	—	61	2	no	—
102	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
103	M₆₄	—	61	8	no	—
104	M₆₄	C₂×D₆₁	1	4	yes	complemented
105	C₂×C₃₂	—	61	4	no	—
106	C₂×C₃₂	—	61	1	no	—
107	C₂×C₃₂	—	61	1	no	—
108	C₂×C₃₂	—	61	8	no	—
109	C₂×C₃₂	C₂×D₆₁	1	2	yes	—
110	C₆₁	C₂×#2258	1	1	yes	characteristic, complemented, Sylow
111	C₂×#41	—	61	1	no	—
112	#131	—	61	4	no	—
113	#131	—	61	4	no	—
114	C₂×M₁₆	—	61	1	no	—
115	C₂×M₁₆	—	61	2	no	—
116	C₂²×C₈	—	61	2	no	—
117	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
118	M₃₂	—	61	4	no	—
119	M₃₂	—	61	8	no	—
120	C₂×C₁₆	—	61	2	no	—
121	C₂×C₁₆	—	61	4	no	—
122	C₂×C₁₆	—	61	4	no	—
123	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
124	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
125	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
126	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
127	C₃₂	—	61	2	no	—
128	C₃₂	—	61	2	no	—
129	C₃₂	C₂²×D₆₁	1	4	yes	—
130	#41	—	61	4	no	—
131	#41	—	61	4	no	—
132	C₂×Q₈	—	61	1	no	—
133	C₂×D₄	—	61	2	no	—
134	C₂×D₄	—	61	1	no	—
135	C₂²×C₄	—	61	2	no	—
136	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
137	M₁₆	—	61	4	no	—
138	M₁₆	—	61	8	no	—
139	C₂×C₈	—	61	2	no	—
140	C₂×C₈	—	61	4	no	—
141	C₂×C₈	—	61	4	no	—
142	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
143	C₂×C₈	C₈×D₆₁	1	2	yes	—
144	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
145	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
146	C₁₆	—	61	4	no	—
147	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
148	C₁₆	C₂×C₄×D₆₁	1	2	yes	—
149	C₁₆	C₂³×D₆₁	1	1	yes	characteristic
150	C₂³	—	61	2	no	—
151	C₂³	C₁₆×D₆₁	1	1	yes	characteristic
152	Q₈	—	61	4	no	—
153	D₄	—	61	8	no	—
154	D₄	—	61	4	no	—
155	C₂×C₄	—	61	2	no	—
156	C₂×C₄	—	61	4	no	—
157	C₂×C₄	—	61	4	no	—
158	C₂×C₄	C₁₆×D₆₁	1	1	yes	characteristic
159	C₂×C₄	C₁₆×D₆₁	1	2	yes	—
160	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
161	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
162	C₈	—	61	4	no	—
163	C₈	C₂×C₈×D₆₁	1	1	yes	characteristic
164	C₈	C₂×C₈×D₆₁	1	2	yes	—
165	C₈	C₂²×C₄×D₆₁	1	1	yes	characteristic
166	C₂²	—	2	2	no	—
167	C₂²	—	61	4	no	—
168	C₂²	—	122	4	no	—
169	C₂²	C₂×C₁₆×D₆₁	1	2	yes	—
170	C₂²	C₂×C₁₆×D₆₁	1	1	yes	characteristic
171	C₄	—	61	4	no	—
172	C₄	C₂×C₁₆×D₆₁	1	1	yes	characteristic
173	C₄	C₂×C₁₆×D₆₁	1	2	yes	—
174	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	122	4	no	—
177	C₂	#21448114	1	2	yes	direct product factor
178	C₂	C₂²×C₁₆×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₆₁
commutator	C₂×C₆₁	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₆₁
Fitting	C₂×C₆₁×M₆₄	C₂
socle	C₂²×C₆₁	C₂²×C₁₆
prime radical	C₂×#11758222	C₈
cosocle	C₁₆×C₆₁	C₂⁴
hypercenter	C₂×M₆₄	D₆₁
hypocenter	C₆₁	C₂×#2258
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. #21448114	C₂	direct (N × H)
3. C₂×#16839595	C₂	semidirect
4. C₂²×#14519683	C₂	semidirect
5. C₂×C₃₂×D₆₁	C₂	semidirect
6. C₂×C₆₁×M₆₄	C₂	semidirect
13. #16839595	C₂²	semidirect
16. C₂×#14519683	C₂²	semidirect
17. C₃₂×D₆₁	C₂²	semidirect
19. C₆₁×M₆₄	C₂²	semidirect
89. C₂×M₆₄	D₆₁	semidirect
93. C₂×C₆₁	#2258	semidirect
104. M₆₄	C₂×D₆₁	semidirect
110. C₆₁	C₂×#2258	semidirect
177. C₂	#21448114	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×#16840178	2
#21448114	2
C₂×#16839595	2
C₂²×#14519683	2
C₂×C₃₂×D₆₁	2
C₂×C₆₁×M₆₄	2
C₂×#2258	61

Subgroups — C2×#21448114 #29838123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21448114 #29838123