Subgroups — C₂²×#16839580 #29832987

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11758185	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839580	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×#92418	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₆₁	C₄	1	1	yes	characteristic
7	C₂²×#11758185	—	2	4	no	—
8	C₂²×#11758185	C₂²	1	6	yes	—
9	C₂×C₈×Dic₆₁	C₄	1	6	yes	—
10	C₂×C₈×Dic₆₁	C₂²	1	6	yes	—
11	#16839580	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₆₁	—	2	4	no	—
14	C₂²×C₁₆×C₆₁	C₂²	1	6	yes	—
15	C₂³×Dic₆₁	C₈	1	1	yes	characteristic
16	C₂³×Dic₆₁	C₈	1	1	yes	characteristic
17	C₂²×#92418	—	2	4	no	—
18	C₂²×#92418	C₂×C₄	1	6	yes	—
19	C₂×C₄×Dic₆₁	C₈	1	6	yes	—
20	C₂×C₄×Dic₆₁	C₂×C₄	1	6	yes	—
21	C₂×#11758185	—	2	12	no	—
22	C₂×#11758185	C₂³	1	4	yes	—
23	C₈×Dic₆₁	C₂×C₄	1	12	yes	—
24	C₈×Dic₆₁	C₂³	1	4	yes	—
25	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
27	C₂²×C₈×C₆₁	D₄	1	4	yes	—
28	C₂²×C₈×C₆₁	Q₈	1	4	yes	—
29	C₂²×C₈×C₆₁	C₂³	1	3	yes	—
30	C₂×C₁₆×C₆₁	—	2	12	no	—
31	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
32	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
34	C₂²×C₄×C₆₁	C₂×C₈	1	3	yes	—
35	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
36	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
37	C₂²×Dic₆₁	—	2	4	no	—
38	C₂²×Dic₆₁	C₁₆	1	8	yes	complemented
39	C₂²×Dic₆₁	C₂×C₈	1	6	yes	—
40	C₂²×Dic₆₁	C₂×C₈	1	6	yes	—
41	C₂×C₈×C₆₁	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
43	C₂×C₈×C₆₁	C₂×D₄	1	6	yes	—
44	C₂×C₈×C₆₁	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
46	C₄×Dic₆₁	C₂×C₈	1	12	yes	—
47	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
48	C₂×#92418	—	2	12	no	—
49	C₂×#92418	C₂²×C₄	1	4	yes	—
50	C₁₆×C₆₁	—	2	4	no	—
51	#11758185	—	2	4	no	—
52	C₂³×C₆₁	C₄⋊C₈	1	4	yes	—
53	C₂³×C₆₁	C₂×C₁₆	1	4	yes	—
54	C₂³×C₆₁	M₃₂	1	4	yes	—
55	C₂³×C₆₁	C₂²×C₈	1	3	yes	—
56	C₂×C₄×C₆₁	C₄⋊C₈	1	12	yes	—
57	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
58	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
59	C₂×C₄×C₆₁	C₂×M₁₆	1	6	yes	—
60	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
61	C₂×Dic₆₁	—	2	12	no	—
62	C₂×Dic₆₁	C₂×C₁₆	1	24	yes	complemented
63	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
64	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
65	C₈×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
66	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
67	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
68	#92418	—	2	4	no	—
69	C₂²×C₄⋊C₁₆	—	61	1	no	maximal, Sylow
70	C₂²×C₆₁	C₄⋊C₁₆	1	16	yes	complemented
71	C₂²×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
72	C₂²×C₆₁	C₂²×C₁₆	1	6	yes	—
73	C₂²×C₆₁	C₂×M₃₂	1	6	yes	—
74	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
75	C₄×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
76	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
78	Dic₆₁	—	2	4	no	—
79	Dic₆₁	C₂²×C₁₆	1	8	yes	complemented
80	C₂³×C₁₆	—	61	1	no	—
81	C₂³×C₁₆	D₆₁	1	1	yes	characteristic
82	C₂²×C₄×C₈	—	61	1	no	—
83	C₂×C₄⋊C₁₆	—	61	12	no	—
84	C₂×C₆₁	C₂×C₄⋊C₁₆	1	12	yes	complemented
85	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
86	C₂×C₆₁	C₂³×C₁₆	1	1	yes	characteristic
87	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
88	C₂³×C₈	—	61	1	no	—
89	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
90	C₂²×C₄²	—	61	1	no	—
91	C₂²×C₁₆	—	2	4	no	—
92	C₂²×C₁₆	—	61	6	no	—
93	C₂²×C₁₆	—	122	4	no	—
94	C₂²×C₁₆	C₂×D₆₁	1	6	yes	—
95	C₂×C₄×C₈	—	61	6	no	—
96	C₂×C₄×C₈	—	61	6	no	—
97	C₄⋊C₁₆	—	61	16	no	—
98	C₆₁	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	—	61	1	no	—
100	C₂³×C₄	—	61	1	no	—
101	C₂³×C₄	C₄×D₆₁	1	1	yes	characteristic
102	C₂²×C₈	—	61	6	no	—
103	C₂²×C₈	—	122	4	no	—
104	C₂²×C₈	Dic₁₂₂	1	4	yes	—
105	C₂²×C₈	C₄×D₆₁	1	3	yes	—
106	C₂²×C₈	C₂²⋊D₆₁	1	4	yes	—
107	C₂²×C₈	C₂²×D₆₁	1	3	yes	—
108	C₂×C₄²	—	61	6	no	—
109	C₂×C₄²	—	61	6	no	—
110	C₂×C₁₆	—	2	12	no	—
111	C₂×C₁₆	—	61	4	no	—
112	C₂×C₁₆	—	122	12	no	—
113	C₂×C₁₆	C₂²×D₆₁	1	4	yes	—
114	C₄×C₈	—	61	12	no	—
115	C₄×C₈	—	61	4	no	—
116	C₂⁴	C₈×D₆₁	1	1	yes	characteristic
117	C₂²×C₄	—	61	8	no	—
118	C₂²×C₄	—	61	6	no	—
119	C₂²×C₄	—	61	6	no	—
120	C₂²×C₄	—	122	4	no	—
121	C₂²×C₄	C₈×D₆₁	1	3	yes	—
122	C₂²×C₄	#11758189	1	4	yes	—
123	C₂²×C₄	#11758196	1	4	yes	—
124	C₂²×C₄	C₂×C₄×D₆₁	1	3	yes	—
125	C₂×C₈	—	61	4	no	—
126	C₂×C₈	—	122	12	no	—
127	C₂×C₈	#11758196	1	12	yes	—
128	C₂×C₈	C₂×Dic₁₂₂	1	6	yes	—
129	C₂×C₈	C₂×C₄×D₆₁	1	3	yes	—
130	C₂×C₈	C₂×C₂²⋊D₆₁	1	6	yes	—
131	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
132	C₄²	—	61	4	no	—
133	C₄²	—	61	12	no	—
134	C₁₆	—	2	4	no	—
135	C₁₆	—	122	4	no	—
136	C₂³	#14519706	1	4	yes	—
137	C₂³	C₁₆×D₆₁	1	4	yes	—
138	C₂³	#14519716	1	4	yes	—
139	C₂³	C₂×C₈×D₆₁	1	3	yes	—
140	C₂×C₄	—	61	4	no	—
141	C₂×C₄	—	61	24	no	—
142	C₂×C₄	—	61	4	no	—
143	C₂×C₄	—	122	12	no	—
144	C₂×C₄	#14519706	1	12	yes	—
145	C₂×C₄	C₂×#11758196	1	6	yes	—
146	C₂×C₄	C₂×C₈×D₆₁	1	3	yes	—
147	C₂×C₄	C₂×#11758189	1	6	yes	—
148	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
149	C₈	—	122	4	no	—
150	C₈	C₂×#11758196	1	6	yes	—
151	C₈	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
152	C₈	C₂²×Dic₁₂₂	1	1	yes	characteristic
153	C₂²	#16839580	1	16	yes	direct product factor
154	C₂²	C₂×#14519706	1	6	yes	—
155	C₂²	C₂×C₁₆×D₆₁	1	6	yes	—
156	C₂²	C₂×#14519716	1	6	yes	—
157	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
158	C₄	—	61	8	no	—
159	C₄	—	122	4	no	—
160	C₄	C₂×#14519706	1	6	yes	—
161	C₄	C₂²×#11758196	1	1	yes	characteristic
162	C₄	C₂²×#11758189	1	1	yes	characteristic
163	C₂	C₂×#16839580	1	12	yes	direct product factor
164	C₂	C₂²×#14519706	1	1	yes	characteristic
165	C₂	C₂²×C₁₆×D₆₁	1	1	yes	characteristic
166	C₂	C₂²×#14519716	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₆₁
commutator	C₂×C₆₁	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂³×C₁₆×C₆₁	C₂
socle, prime radical	C₂⁴×C₆₁	C₂×C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂³×C₁₆	D₆₁
hypocenter	C₆₁	C₂²×C₄⋊C₁₆
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#16839580	C₂	direct (N × H)
11. #16839580	C₂²	direct (N × H)
38. C₂²×Dic₆₁	C₁₆	semidirect
62. C₂×Dic₆₁	C₂×C₁₆	semidirect
70. C₂²×C₆₁	C₄⋊C₁₆	semidirect
79. Dic₆₁	C₂²×C₁₆	semidirect
84. C₂×C₆₁	C₂×C₄⋊C₁₆	semidirect
98. C₆₁	C₂²×C₄⋊C₁₆	semidirect
153. C₂²	#16839580	direct (N × H)
163. C₂	C₂×#16839580	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11758185	2
C₂²×C₈×Dic₆₁	2
C₂×#16839580	2
C₂³×C₁₆×C₆₁	2
C₂²×C₄⋊C₁₆	61

Subgroups — C22×#16839580 #29832987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16839580 #29832987