Subgroups — #30597374

Order 15616.

574 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×#92420	C₂	1	2	yes	maximal
2	#21460641	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#92420	C₄	1	4	yes	—
5	C₂×C₄×#92420	C₄	1	4	yes	—
6	C₂×C₄×#92420	C₂²	1	4	yes	—
7	C₂×C₄×#92418	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
9	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
10	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
12	C₂²×#92420	C₂×C₄	1	8	yes	—
13	C₄×#92420	—	2	8	no	—
14	C₄×#92420	—	2	8	no	—
15	C₄×#92420	C₈	1	16	yes	complemented
16	C₂²×#92418	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
21	C₄×#92418	C₈	1	4	yes	complemented
22	C₈×Dic₆₁	—	2	4	no	—
23	C₈×Dic₆₁	—	2	4	no	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₁	C₈	1	4	yes	complemented
27	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
30	C₂²×Dic₆₁	C₄²	1	2	yes	—
31	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
32	C₂×#92420	—	2	8	no	—
33	C₂×#92420	—	2	8	no	—
34	C₂×#92420	C₂×C₈	1	16	yes	—
35	C₂×C₈×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	—	2	2	no	—
37	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
38	C₄²×C₆₁	C₂×C₈	1	2	yes	—
39	C₄²×C₆₁	C₂×C₈	1	2	yes	—
40	C₄×Dic₆₁	C₂×C₈	1	8	yes	—
41	C₄×Dic₆₁	M₁₆	1	4	yes	—
42	C₄×Dic₆₁	M₁₆	1	8	yes	—
43	C₄×Dic₆₁	#41	1	4	yes	—
44	C₂×#92418	—	2	2	no	—
45	C₂×#92418	—	2	2	no	—
46	C₂×#92418	C₂×C₈	1	4	yes	—
47	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
49	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
51	C₂×C₄×C₆₁	#97	1	4	yes	—
52	C₂×C₄×C₆₁	#117	1	2	yes	—
53	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
56	C₂×Dic₆₁	C₄×C₈	1	4	yes	—
57	C₂×Dic₆₁	#97	1	4	yes	—
58	C₂×Dic₆₁	#97	1	4	yes	—
59	C₂×Dic₆₁	#117	1	4	yes	—
60	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
61	C₂×Dic₆₁	C₂×M₁₆	1	4	yes	—
62	#92420	—	2	16	no	—
63	C₈×C₆₁	—	2	4	no	—
64	#92418	—	2	4	no	—
65	#8131	—	61	1	no	maximal, Sylow
66	C₂²×C₆₁	#376	1	1	yes	characteristic
67	C₂²×C₆₁	C₂×C₄×C₈	1	2	yes	—
68	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
69	C₂²×C₆₁	C₂×#97	1	2	yes	—
70	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
71	C₄×C₆₁	C₈²	1	4	yes	—
72	C₄×C₆₁	#322	1	4	yes	—
73	C₄×C₆₁	#431	1	2	yes	—
74	C₄×C₆₁	#431	1	2	yes	—
75	Dic₆₁	#322	1	8	yes	—
76	Dic₆₁	#431	1	8	yes	—
77	C₄²×C₈	—	61	2	no	—
78	C₄²×C₈	—	61	1	no	—
79	#1273	—	61	4	no	—
80	C₂×C₆₁	C₂×C₈²	1	1	yes	characteristic
81	C₂×C₆₁	C₂×#322	1	1	yes	characteristic
82	C₂×C₆₁	C₂×#322	1	2	yes	—
83	C₂×C₆₁	#1744	1	2	yes	—
84	C₂×C₆₁	#1744	1	1	yes	characteristic
85	C₂×C₄×C₈	—	61	4	no	—
86	C₂×C₄×C₈	—	61	1	no	—
87	C₂×C₄×C₈	—	61	4	no	—
88	C₂×C₄×C₈	—	61	4	no	—
89	C₂×C₄×C₈	—	61	2	no	—
90	C₂×C₄×C₈	—	61	2	no	—
91	C₂×C₄×C₈	#6923	1	1	yes	characteristic
92	C₄³	—	61	1	no	—
93	C₆₁	#8131	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	61	8	no	—
95	C₂²×C₈	—	61	2	no	—
96	C₂²×C₈	C₂×#6923	1	2	yes	—
97	C₂×C₄²	—	61	2	no	—
98	C₂×C₄²	—	61	2	no	—
99	C₂×C₄²	—	61	1	no	—
100	C₂×C₄²	—	61	1	no	—
101	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
102	C₄×C₈	—	61	16	no	—
103	C₄×C₈	—	61	4	no	—
104	C₄×C₈	—	122	4	no	—
105	C₄×C₈	—	122	4	no	—
106	C₄×C₈	—	122	8	no	—
107	C₄×C₈	—	122	8	no	—
108	C₄×C₈	#92420	1	4	yes	complemented
109	C₂²×C₄	—	61	2	no	—
110	C₂²×C₄	—	61	2	no	—
111	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
112	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
113	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	61	16	no	—
117	C₂×C₈	—	61	4	no	—
118	C₂×C₈	—	122	8	no	—
119	C₂×C₈	—	122	8	no	—
120	C₂×C₈	—	122	2	no	—
121	C₂×C₈	—	122	2	no	—
122	C₂×C₈	C₂×#92420	1	4	yes	—
123	C₄²	—	61	8	no	—
124	C₄²	—	61	8	no	—
125	C₄²	—	61	4	no	—
126	C₄²	—	61	4	no	—
127	C₄²	D₆₁⋊C₈	1	2	yes	—
128	C₄²	C₂×#92420	1	2	yes	—
129	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
130	C₂×C₄	—	61	4	no	—
131	C₂×C₄	—	61	4	no	—
132	C₂×C₄	—	61	4	no	—
133	C₂×C₄	—	61	4	no	—
134	C₂×C₄	—	61	4	no	—
135	C₂×C₄	—	61	4	no	—
136	C₂×C₄	C₈×#6923	1	4	yes	—
137	C₂×C₄	C₄×#92420	1	2	yes	—
138	C₂×C₄	C₄×#92420	1	2	yes	—
139	C₂×C₄	#14519831	1	4	yes	—
140	C₂×C₄	#14519856	1	2	yes	—
141	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
142	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
143	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	122	16	no	—
146	C₈	—	122	4	no	—
147	C₂²	#16840698	1	1	yes	characteristic
148	C₂²	C₂×C₈×#6923	1	2	yes	—
149	C₂²	C₂×C₄×#92420	1	1	yes	characteristic
150	C₂²	C₂×#14519831	1	2	yes	—
151	C₂²	#16840721	1	1	yes	characteristic
152	C₄	—	61	8	no	—
153	C₄	—	61	8	no	—
154	C₄	C₈×#92420	1	4	yes	—
155	C₄	#16840603	1	4	yes	—
156	C₄	#16840778	1	2	yes	—
157	C₄	#16840780	1	2	yes	—
158	C₂	C₂×C₈×#92420	1	1	yes	characteristic
159	C₂	C₂×#16840603	1	1	yes	characteristic
160	C₂	#21461015	1	2	yes	—
161	C₂	#21461532	1	2	yes	—
162	C₂	#21461533	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×#6923
commutator	C₂×C₆₁	C₂×C₈²
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄²
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#8131
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
15. C₄×#92420	C₈	semidirect
21. C₄×#92418	C₈	semidirect
26. C₄×C₈×C₆₁	C₈	semidirect
93. C₆₁	#8131	semidirect
108. C₄×C₈	#92420	semidirect

Maximal subgroups

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

subgroup	index
C₄²×#92420	2
#21460641	2
C₄×C₈×Dic₆₁	2
#8131	61