Subgroups — #29566849

Order 15616.

246 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519717	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839538	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519717	C₄	1	2	yes	—
5	C₂²×#11758185	C₄	1	2	yes	—
6	#16839538	—	2	1	no	—
7	#16839538	—	2	1	no	—
8	#16839538	—	2	1	no	—
9	C₄×#11758185	—	2	1	no	—
10	C₆₁×#431	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#322	—	2	1	no	—
13	C₈²×C₆₁	—	2	1	no	—
14	#14519717	—	2	2	no	—
15	#14519717	C₈	1	4	yes	complemented
16	C₂×#11758185	—	2	2	no	—
17	C₂×#11758185	—	2	2	no	—
18	C₂×#11758185	—	2	2	no	—
19	C₂×#11758185	C₈	1	4	yes	complemented
20	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₈×C₆₁	—	2	2	no	—
24	C₂²⋊C₈×C₆₁	C₈	1	4	yes	—
25	C₄×C₈×C₆₁	—	2	2	no	—
26	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₆₁	—	2	4	no	—
33	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
40	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
41	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₆₁	M₁₆	1	2	yes	—
44	#11758185	—	2	4	no	—
45	#11758185	—	2	4	no	—
46	C₂³×C₆₁	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₆₁	—	4	2	no	—
48	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
56	C₈×C₆₁	—	4	4	no	—
57	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
58	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
59	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
60	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
61	#7140	—	61	1	no	maximal, Sylow
62	C₂²×C₆₁	—	2	2	no	—
63	C₂²×C₆₁	C₈²	1	1	yes	characteristic
64	C₂²×C₆₁	#322	1	1	yes	characteristic
65	C₂²×C₆₁	#336	1	1	yes	characteristic
66	C₄×C₆₁	—	2	2	no	—
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	#322	1	1	yes	characteristic
69	C₄×C₆₁	#322	1	1	yes	characteristic
70	C₄×C₆₁	#336	1	1	yes	characteristic
71	C₄×C₆₁	#336	1	1	yes	characteristic
72	C₄×M₃₂	—	61	1	no	—
73	C₂×#346	—	61	1	no	—
74	#1450	D₆₁	1	1	yes	characteristic
75	C₂×C₆₁	—	2	1	no	—
76	C₂×C₆₁	—	2	1	no	—
77	C₂×C₆₁	#1273	1	1	yes	characteristic
78	C₂×C₆₁	#1378	1	2	yes	—
79	C₂×M₃₂	—	61	2	no	—
80	C₂²×C₁₆	—	61	2	no	—
81	#431	Dic₆₁	1	2	yes	—
82	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
83	#346	—	122	1	no	—
84	#346	—	122	1	no	—
85	#346	—	122	1	no	—
86	C₄×C₁₆	—	122	1	no	—
87	#322	—	2	1	no	—
88	C₈²	—	2	1	no	—
89	C₆₁	#7140	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
91	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
92	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
93	M₃₂	—	61	4	no	—
94	M₃₂	—	122	2	no	—
95	C₂×C₁₆	—	61	4	no	—
96	C₂×C₁₆	—	122	2	no	—
97	C₂×C₁₆	—	122	2	no	—
98	C₂×C₁₆	—	122	2	no	—
99	C₄⋊C₈	—	2	2	no	—
100	C₂²⋊C₈	#92418	1	4	yes	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
103	C₄×C₈	D₂₄₄	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
106	C₂²×C₄	C₂×#92418	1	2	yes	—
107	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	C₈×D₆₁	1	2	yes	—
110	C₂×C₈	C₈×D₆₁	1	2	yes	—
111	C₂×C₈	#11758189	1	2	yes	—
112	C₂×C₈	#11758189	1	2	yes	—
113	C₂×C₈	#11758196	1	1	yes	characteristic
114	C₂×C₈	#11758196	1	1	yes	characteristic
115	C₂×C₈	#11758198	1	1	yes	characteristic
116	C₂×C₈	#11758198	1	1	yes	characteristic
117	C₄²	#11758194	1	2	yes	—
118	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
119	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
120	C₁₆	—	122	4	no	—
121	C₁₆	—	122	4	no	—
122	C₂³	C₄×#92418	1	1	yes	characteristic
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	#14519684	1	1	yes	characteristic
125	C₂×C₄	C₈×Dic₆₁	1	1	yes	characteristic
126	C₂×C₄	C₈×Dic₆₁	1	1	yes	characteristic
127	C₂×C₄	#14519687	1	1	yes	characteristic
128	C₂×C₄	#14519687	1	1	yes	characteristic
129	C₂×C₄	#14519688	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊#92418	1	2	yes	—
131	C₂×C₄	C₄⋊#92418	1	2	yes	—
132	C₈	—	4	4	no	—
133	C₈	#14519689	1	2	yes	—
134	C₈	#14519689	1	2	yes	—
135	C₈	#14519706	1	2	yes	—
136	C₈	#14519706	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₈×#92418	1	1	yes	characteristic
139	C₂²	#16839483	1	1	yes	characteristic
140	C₂²	#16839517	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	#16839484	1	1	yes	characteristic
144	C₄	#16839484	1	1	yes	characteristic
145	C₄	#16839516	1	1	yes	characteristic
146	C₄	#16839516	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#21443646	1	1	yes	characteristic
150	C₂	#21443905	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#11758198
commutator	C₂²×C₆₁	C₈²
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₆₁×#1450	C₂
socle	C₂²×C₆₁	#336
prime radical	C₂³×C₆₁	C₄×C₈
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	#1450	D₆₁
hypocenter	C₆₁	#7140
LCS	G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
15. #14519717	C₈	semidirect
19. C₂×#11758185	C₈	semidirect
89. C₆₁	#7140	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519717	2
C₂×#16839538	2
C₆₁×#1450	2
#7140	61