Subgroups — C₄×#16839588 #29595412

Order 15616.

366 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	C₂×#16839588	C₂	1	2	yes	maximal
2	C₄×#14519711	C₂	1	2	yes	maximal
3	#21443911	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519711	C₂²	1	4	yes	—
6	C₄×C₄⋊Dic₆₁	—	2	2	no	—
7	#16839588	C₄	1	8	yes	direct product factor
8	#16839525	—	2	2	no	—
9	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₆₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊Dic₆₁	—	2	4	no	—
14	C₂×C₄×Dic₆₁	—	4	2	no	—
15	#14519711	—	2	4	no	—
16	#14519711	C₂×C₄	1	8	yes	—
17	#14519684	—	4	2	no	—
18	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₆₁	—	2	2	no	—
22	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
24	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₆₁	Q₈	1	2	yes	—
27	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
30	C₂²×Dic₆₁	—	4	4	no	—
31	C₂×C₈×C₆₁	—	2	2	no	—
32	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
33	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₆₁	#41	1	2	yes	—
37	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
38	C₄²×C₆₁	Q₁₆	1	2	yes	—
39	C₄⋊Dic₆₁	—	2	8	no	—
40	C₄⋊Dic₆₁	—	4	4	no	—
41	C₄×Dic₆₁	—	8	4	no	—
42	C₁₆×C₆₁	—	2	2	no	—
43	C₁₆×C₆₁	C₄²	1	4	yes	complemented
44	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₆₁	—	2	2	no	—
46	C₂×C₄×C₆₁	#107	1	2	yes	—
47	C₂×C₄×C₆₁	#107	1	2	yes	—
48	C₂×C₄×C₆₁	D₁₆	1	2	yes	—
49	C₂×C₄×C₆₁	Q₃₂	1	2	yes	—
50	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
52	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#135	1	2	yes	—
54	C₂×Dic₆₁	—	4	8	no	—
55	C₂×Dic₆₁	—	8	4	no	—
56	C₂×Dic₆₁	—	8	4	no	—
57	C₈×C₆₁	—	2	2	no	—
58	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
59	C₈×C₆₁	#117	1	1	yes	characteristic
60	C₈×C₆₁	C₄×Q₈	1	2	yes	—
61	C₄×#366	—	61	1	no	maximal, Sylow
62	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
63	C₂²×C₆₁	C₄×D₈	1	2	yes	—
64	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
65	C₂²×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
66	C₂²×C₆₁	#508	1	2	yes	—
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	#366	1	4	yes	complemented
69	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
70	C₄×C₆₁	#427	1	1	yes	characteristic
71	C₄×C₆₁	C₄×Q₁₆	1	2	yes	—
72	Dic₆₁	—	8	8	no	—
73	C₂×#366	—	61	2	no	—
74	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic
75	C₄×#107	—	61	2	no	—
76	#1380	—	61	2	no	—
77	C₂×C₆₁	C₄×#107	1	1	yes	characteristic
78	C₂×C₆₁	C₂×#366	1	1	yes	characteristic
79	C₂×C₆₁	#2158	1	1	yes	characteristic
80	C₂×C₆₁	C₄×D₁₆	1	2	yes	—
81	C₂×C₆₁	C₄×Q₃₂	1	2	yes	—
82	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
83	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
84	C₂×#107	—	61	4	no	—
85	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
86	C₄×C₄⋊C₄	—	122	2	no	—
87	#366	—	61	8	no	—
88	C₄×C₁₆	Dic₆₁	1	4	yes	complemented
89	#340	—	122	2	no	—
90	C₆₁	C₄×#366	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	122	4	no	—
94	C₂×C₄²	—	244	2	no	—
95	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	C₄×D₆₁	1	4	yes	—
98	C₂×C₁₆	C₂×Dic₆₁	1	2	yes	—
99	C₂×C₁₆	C₂×Dic₆₁	1	2	yes	—
100	#107	—	61	8	no	—
101	#107	—	122	4	no	—
102	C₄×C₈	Dic₁₂₂	1	2	yes	—
103	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
104	#95	—	244	2	no	—
105	C₂²×C₄	—	244	4	no	—
106	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
107	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
108	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
111	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
112	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
113	C₂×C₈	#11758222	1	2	yes	—
114	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
115	C₄⋊C₄	—	122	8	no	—
116	C₄⋊C₄	—	244	4	no	—
117	C₄²	—	488	4	no	—
118	C₄²	Dic₂₄₄	1	2	yes	—
119	C₄²	C₄⋊Dic₆₁	1	2	yes	—
120	C₁₆	—	2	2	no	—
121	C₁₆	C₄×Dic₆₁	1	4	yes	complemented
122	C₂³	C₂×D₄₈₈	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	488	4	no	—
125	C₂×C₄	—	488	4	no	—
126	C₂×C₄	—	244	8	no	—
127	C₂×C₄	#14519711	1	2	yes	—
128	C₂×C₄	#14519711	1	2	yes	—
129	C₂×C₄	D₉₇₆	1	2	yes	—
130	C₂×C₄	Dic₄₈₈	1	2	yes	—
131	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
132	C₂×C₄	C₄×D₂₄₄	1	2	yes	—
133	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
134	C₂×C₄	#14519799	1	2	yes	—
135	C₈	—	2	2	no	—
136	C₈	C₂×C₄×Dic₆₁	1	1	yes	characteristic
137	C₈	#14519736	1	1	yes	characteristic
138	C₈	C₄×Dic₁₂₂	1	2	yes	—
139	C₂²	C₂×#14519711	1	1	yes	characteristic
140	C₂²	C₄×D₄₈₈	1	2	yes	—
141	C₂²	C₂×D₉₇₆	1	1	yes	characteristic
142	C₂²	C₂×Dic₄₈₈	1	1	yes	characteristic
143	C₂²	#16840194	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	488	8	no	—
146	C₄	#16839588	1	4	yes	direct product factor
147	C₄	C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
148	C₄	#16839800	1	1	yes	characteristic
149	C₄	C₄×Dic₂₄₄	1	2	yes	—
150	C₂	C₄×#14519711	1	1	yes	characteristic
151	C₂	C₂×#16839588	1	1	yes	characteristic
152	C₂	#21447603	1	1	yes	characteristic
153	C₂	C₄×D₉₇₆	1	2	yes	—
154	C₂	C₄×Dic₄₈₈	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄₈₈
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₁
hypocenter	C₆₁	C₄×#366
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
7. #16839588	C₄	direct (N × H)
12. C₄×C₁₆×C₆₁	C₄	semidirect
43. C₁₆×C₆₁	C₄²	semidirect
68. C₄×C₆₁	#366	semidirect
88. C₄×C₁₆	Dic₆₁	semidirect
90. C₆₁	C₄×#366	semidirect
121. C₁₆	C₄×Dic₆₁	semidirect
146. C₄	#16839588	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×#16839588	2
C₄×#14519711	2
#21443911	2
C₂×C₄×C₁₆×C₆₁	2
C₄×#366	61

Subgroups — C4×#16839588 #29595412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16839588 #29595412