Subgroups — C₂²×#16840683 #30624084

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11758187	C₂	1	2	yes	maximal
2	C₂×#16840683	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758187	—	2	8	no	—
5	C₂²×#11758187	C₂²	1	12	yes	—
6	#16840683	C₂²	1	16	yes	direct product factor
7	C₂³×#92418	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₆₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₆₁	C₄	1	6	yes	—
10	C₂×C₈×Dic₆₁	C₂²	1	6	yes	—
11	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
12	C₂×#11758187	—	2	24	no	—
13	C₂×#11758187	C₂³	1	8	yes	—
14	C₂³×Dic₆₁	C₈	1	1	yes	characteristic
15	C₂³×Dic₆₁	C₈	1	1	yes	characteristic
16	C₂²×#92418	C₂×C₄	1	3	yes	—
17	C₂²×#92418	D₄	1	4	yes	—
18	C₂²×#92418	Q₈	1	4	yes	—
19	C₂²×#92418	C₂³	1	3	yes	—
20	C₂×C₄×Dic₆₁	C₈	1	6	yes	—
21	C₂×C₄×Dic₆₁	C₂×C₄	1	6	yes	—
22	C₈×Dic₆₁	C₂×C₄	1	12	yes	—
23	C₈×Dic₆₁	C₂³	1	4	yes	—
24	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	—	2	4	no	—
26	C₂²×C₈×C₆₁	C₂×C₄	1	6	yes	—
27	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
29	C₂²×C₄×C₆₁	C₂×C₈	1	3	yes	—
30	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
31	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
32	C₂²×Dic₆₁	—	2	4	no	—
33	C₂²×Dic₆₁	C₁₆	1	8	yes	complemented
34	C₂²×Dic₆₁	C₂×C₈	1	6	yes	—
35	C₂²×Dic₆₁	C₂×C₈	1	6	yes	—
36	C₂×C₈×C₆₁	—	2	12	no	—
37	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
38	C₄×Dic₆₁	C₂×C₈	1	12	yes	—
39	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
40	C₂×#92418	C₄⋊C₄	1	12	yes	—
41	C₂×#92418	C₂²×C₄	1	3	yes	—
42	C₂×#92418	C₂×D₄	1	6	yes	—
43	C₂×#92418	C₂×Q₈	1	6	yes	—
44	C₂×#92418	C₂⁴	1	1	yes	characteristic
45	#11758187	—	2	8	no	—
46	C₂³×C₆₁	C₄⋊C₈	1	4	yes	—
47	C₂³×C₆₁	C₂×C₁₆	1	4	yes	—
48	C₂³×C₆₁	M₃₂	1	4	yes	—
49	C₂³×C₆₁	C₂²×C₈	1	3	yes	—
50	C₂×C₄×C₆₁	C₄⋊C₈	1	12	yes	—
51	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
52	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
53	C₂×C₄×C₆₁	C₂×M₁₆	1	6	yes	—
54	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
55	C₂×Dic₆₁	—	2	12	no	—
56	C₂×Dic₆₁	C₂×C₁₆	1	24	yes	complemented
57	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
58	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
59	C₈×C₆₁	—	2	4	no	—
60	#92418	C₂×C₄⋊C₄	1	6	yes	—
61	#92418	C₂²×D₄	1	1	yes	characteristic
62	#92418	C₂²×Q₈	1	1	yes	characteristic
63	C₂²×C₄⋊C₁₆	—	61	1	no	maximal, Sylow
64	C₂²×C₆₁	C₄⋊C₁₆	1	16	yes	complemented
65	C₂²×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
66	C₂²×C₆₁	C₂²×C₁₆	1	6	yes	—
67	C₂²×C₆₁	C₂×M₃₂	1	6	yes	—
68	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
69	C₄×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
70	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
72	Dic₆₁	—	2	4	no	—
73	Dic₆₁	C₂²×C₁₆	1	8	yes	complemented
74	C₂³×C₁₆	—	61	2	no	—
75	C₂²×C₄×C₈	—	61	1	no	—
76	C₂×C₄⋊C₁₆	—	61	12	no	—
77	C₂×C₆₁	C₂×C₄⋊C₁₆	1	12	yes	complemented
78	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₆₁	C₂³×C₁₆	1	1	yes	characteristic
80	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
81	C₂³×C₈	—	61	1	no	—
82	C₂³×C₈	#6923	1	1	yes	characteristic
83	C₂²×C₄²	—	61	1	no	—
84	C₂²×C₁₆	—	61	12	no	—
85	C₂²×C₁₆	—	122	8	no	—
86	C₂×C₄×C₈	—	61	6	no	—
87	C₂×C₄×C₈	—	61	6	no	—
88	C₄⋊C₁₆	—	61	16	no	—
89	C₆₁	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	—	61	1	no	—
91	C₂³×C₄	—	61	1	no	—
92	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
93	C₂²×C₈	—	2	4	no	—
94	C₂²×C₈	—	61	4	no	—
95	C₂²×C₈	—	61	4	no	—
96	C₂²×C₈	—	61	3	no	—
97	C₂²×C₈	—	61	3	no	—
98	C₂²×C₈	C₂×#6923	1	6	yes	—
99	C₂×C₄²	—	61	6	no	—
100	C₂×C₄²	—	61	6	no	—
101	C₂×C₁₆	—	61	8	no	—
102	C₂×C₁₆	—	122	24	no	—
103	C₄×C₈	—	61	4	no	—
104	C₄×C₈	—	61	12	no	—
105	C₂⁴	D₆₁⋊C₈	1	1	yes	characteristic
106	C₂²×C₄	—	61	8	no	—
107	C₂²×C₄	—	61	6	no	—
108	C₂²×C₄	—	61	6	no	—
109	C₂²×C₄	—	122	4	no	—
110	C₂²×C₄	D₆₁⋊C₈	1	3	yes	—
111	C₂²×C₄	C₄⋊#6923	1	4	yes	—
112	C₂²×C₄	#11758217	1	4	yes	—
113	C₂²×C₄	C₂²×#6923	1	3	yes	—
114	C₂×C₈	—	2	12	no	—
115	C₂×C₈	—	61	6	no	—
116	C₂×C₈	—	61	12	no	—
117	C₂×C₈	—	61	6	no	—
118	C₂×C₈	—	61	1	no	—
119	C₂×C₈	—	61	3	no	—
120	C₂×C₈	C₂²×#6923	1	4	yes	—
121	C₄²	—	61	12	no	—
122	C₄²	—	61	4	no	—
123	C₁₆	—	122	8	no	—
124	C₂³	#14519844	1	4	yes	—
125	C₂³	D₆₁⋊C₁₆	1	4	yes	—
126	C₂³	#14519852	1	4	yes	—
127	C₂³	C₂×D₆₁⋊C₈	1	3	yes	—
128	C₂×C₄	—	61	24	no	—
129	C₂×C₄	—	61	4	no	—
130	C₂×C₄	—	61	4	no	—
131	C₂×C₄	—	122	12	no	—
132	C₂×C₄	#14519844	1	12	yes	—
133	C₂×C₄	C₂×C₄⋊#6923	1	6	yes	—
134	C₂×C₄	C₂×D₆₁⋊C₈	1	3	yes	—
135	C₂×C₄	C₂×#11758217	1	6	yes	—
136	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
137	C₈	—	2	4	no	—
138	C₈	—	61	6	no	—
139	C₈	—	61	1	no	—
140	C₈	—	61	1	no	—
141	C₂²	#16840683	1	16	yes	direct product factor
142	C₂²	C₂×#14519844	1	6	yes	—
143	C₂²	C₂×D₆₁⋊C₁₆	1	6	yes	—
144	C₂²	C₂×#14519852	1	6	yes	—
145	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
146	C₄	—	61	8	no	—
147	C₄	—	122	4	no	—
148	C₄	C₂×#14519844	1	6	yes	—
149	C₄	C₂²×C₄⋊#6923	1	1	yes	characteristic
150	C₄	C₂²×#11758217	1	1	yes	characteristic
151	C₂	C₂×#16840683	1	12	yes	direct product factor
152	C₂	C₂²×#14519844	1	1	yes	characteristic
153	C₂	C₂²×D₆₁⋊C₁₆	1	1	yes	characteristic
154	C₂	C₂²×#14519852	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#6923
commutator	C₂×C₆₁	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle, prime radical	C₂⁴×C₆₁	C₂×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂³×C₈	#6923
hypocenter	C₆₁	C₂²×C₄⋊C₁₆
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16840683	C₂	direct (N × H)
6. #16840683	C₂²	direct (N × H)
33. C₂²×Dic₆₁	C₁₆	semidirect
56. C₂×Dic₆₁	C₂×C₁₆	semidirect
64. C₂²×C₆₁	C₄⋊C₁₆	semidirect
73. Dic₆₁	C₂²×C₁₆	semidirect
77. C₂×C₆₁	C₂×C₄⋊C₁₆	semidirect
89. C₆₁	C₂²×C₄⋊C₁₆	semidirect
141. C₂²	#16840683	direct (N × H)
151. C₂	C₂×#16840683	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×#11758187	2
C₂×#16840683	2
C₂²×C₈×Dic₆₁	2
C₂²×C₄⋊C₁₆	61

Subgroups — C22×#16840683 #30624084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16840683 #30624084