Subgroups — #29571171

Order 15616.

414 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21444567	C₂	1	2	yes	maximal
2	#21444102	C₂	1	2	yes	maximal
3	#21443652	C₂	1	2	yes	maximal
4	C₄²×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	#16839815	C₂²	1	4	yes	—
6	#16839515	C₄	1	4	yes	complemented
7	#16839513	C₄	1	4	yes	complemented
8	#16839483	—	2	2	no	—
9	C₈×#92418	—	2	2	no	—
10	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₄³×C₆₁	C₂²	1	1	yes	characteristic
15	C₄⋊#92418	C₂×C₄	1	8	yes	—
16	C₂²⋊#92418	—	2	4	no	—
17	C₄×#92418	—	2	4	no	—
18	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₆₁	—	2	2	no	—
22	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₆₁	—	2	4	no	—
25	C₄×C₈×C₆₁	C₈	1	8	yes	complemented
26	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	Q₈	1	2	yes	—
31	C₂²×C₄×C₆₁	—	2	2	no	—
32	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	#41	1	2	yes	—
34	C₂×C₈×C₆₁	—	2	2	no	—
35	C₂×C₈×C₆₁	C₄²	1	2	yes	—
36	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
43	C₄²×C₆₁	—	2	8	no	—
44	C₄²×C₆₁	—	2	2	no	—
45	C₄²×C₆₁	C₂×C₈	1	4	yes	—
46	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
47	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×#92418	—	2	8	no	—
50	C₂³×C₆₁	#117	1	1	yes	characteristic
51	C₂×C₄×C₆₁	—	2	8	no	—
52	C₂×C₄×C₆₁	—	2	4	no	—
53	C₂×C₄×C₆₁	—	2	4	no	—
54	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
58	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
59	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
60	C₈×C₆₁	—	2	2	no	—
61	C₈×C₆₁	C₄×C₈	1	2	yes	—
62	C₈×C₆₁	C₄×C₈	1	2	yes	—
63	C₈×C₆₁	C₄⋊C₈	1	4	yes	—
64	C₈×C₆₁	C₄⋊C₈	1	4	yes	—
65	#92418	—	4	8	no	—
66	#8266	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	—	2	2	no	—
68	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₆₁	#431	1	2	yes	—
70	C₄×C₆₁	—	2	8	no	—
71	C₄×C₆₁	—	2	2	no	—
72	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
73	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
74	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
75	C₄×C₆₁	#443	1	4	yes	—
76	C₄²×C₈	D₆₁	1	1	yes	characteristic
77	#1564	—	61	2	no	—
78	#1450	—	61	2	no	—
79	#1276	—	61	2	no	—
80	C₂×C₆₁	—	2	2	no	—
81	C₂×C₆₁	C₄×C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₆₁	#1772	1	2	yes	—
83	#431	—	61	4	no	—
84	C₂×C₄×C₈	Dic₆₁	1	2	yes	—
85	C₂×C₄×C₈	Dic₆₁	1	2	yes	—
86	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
88	C₄³	C₂×D₆₁	1	1	yes	characteristic
89	#335	—	61	4	no	—
90	#334	—	61	4	no	—
91	#322	—	122	2	no	—
92	C₈²	—	122	2	no	—
93	C₆₁	#8266	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
95	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
96	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
97	C₂×C₄²	—	2	2	no	—
98	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
99	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
100	C₄⋊C₈	—	61	8	no	—
101	C₂²⋊C₈	—	122	4	no	—
102	C₄×C₈	—	2	4	no	—
103	C₄×C₈	—	122	4	no	—
104	C₄×C₈	#92418	1	8	yes	complemented
105	C₄×C₈	Dic₁₂₂	1	2	yes	—
106	C₄×C₈	C₄×D₆₁	1	2	yes	—
107	C₄×C₈	C₄×D₆₁	1	2	yes	—
108	C₄×C₈	D₂₄₄	1	1	yes	characteristic
109	C₄×C₈	D₂₄₄	1	1	yes	characteristic
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	#11758222	1	2	yes	—
112	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	122	8	no	—
115	C₂×C₈	C₂×#92418	1	4	yes	—
116	C₂×C₈	C₂×#92418	1	4	yes	—
117	C₂×C₈	#11758194	1	4	yes	—
118	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
119	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
120	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
121	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
122	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
123	C₄²	—	2	8	no	—
124	C₄²	—	2	2	no	—
125	C₄²	C₂×#92418	1	4	yes	—
126	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
127	C₄²	C₂×C₄×D₆₁	1	2	yes	—
128	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
129	C₂³	#14519736	1	1	yes	characteristic
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
135	C₂×C₄	C₄×D₂₄₄	1	2	yes	—
136	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
137	C₂×C₄	C₂²×#92418	1	2	yes	—
138	C₂×C₄	C₂×#11758194	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	244	8	no	—
141	C₈	C₄×#92418	1	2	yes	—
142	C₈	C₄×#92418	1	2	yes	—
143	C₈	C₄⋊#92418	1	4	yes	—
144	C₈	C₄⋊#92418	1	4	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
147	C₂²	#16839815	1	2	yes	—
148	C₄	—	2	8	no	—
149	C₄	—	2	2	no	—
150	C₄	C₂×C₄×#92418	1	1	yes	characteristic
151	C₄	C₄×#11758194	1	1	yes	characteristic
152	C₄	C₂×C₄⋊#92418	1	2	yes	—
153	C₄	#16839880	1	4	yes	—
154	C₂	—	2	2	no	—
155	C₂	C₄×C₄⋊#92418	1	1	yes	characteristic
156	C₂	#21445698	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	D₂₄₄
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄²×C₈×C₆₁	C₂
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄²×C₈	D₆₁
hypocenter	C₆₁	#8266
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
6. #16839515	C₄	semidirect
7. #16839513	C₄	semidirect
25. C₄×C₈×C₆₁	C₈	semidirect
93. C₆₁	#8266	semidirect
104. C₄×C₈	#92418	semidirect

Maximal subgroups

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

subgroup	index
#21444567	2
#21444102	2
#21443652	2
C₄²×C₈×C₆₁	2
#8266	61