Subgroups — #29835505

Order 15616.

490 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456333	C₂	1	2	yes	maximal
2	#21448101	C₂	1	2	yes	maximal, complemented
3	#21448098	C₂	1	4	yes	maximal, complemented
4	#21448094	C₂	1	2	yes	maximal, complemented
5	#21447603	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
7	#16840319	—	2	2	no	—
8	#16840167	C₂²	1	4	yes	—
9	#16840164	—	2	2	no	—
10	#16840164	C₂²	1	4	yes	—
11	#16840160	—	2	2	no	—
12	#16839800	—	2	2	no	—
13	#16839800	C₂²	1	4	yes	—
14	#16839590	C₂²	1	8	yes	—
15	#16839588	C₂²	1	8	yes	—
16	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
20	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
21	#14519777	—	2	4	no	—
22	#14519769	—	2	4	no	—
23	#14519756	—	4	4	no	—
24	C₄×Dic₁₂₂	—	4	4	no	—
25	#14519736	—	2	4	no	—
26	#14519711	—	2	4	no	—
27	#14519711	C₂³	1	8	yes	—
28	#14519709	—	2	4	no	—
29	#14519709	—	2	4	no	—
30	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₆₁	Q₈	1	8	yes	complemented
34	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
35	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
36	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
37	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	D₈	1	2	yes	—
41	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₁₂₂	—	4	4	no	—
43	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
44	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
48	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
49	C₄²×C₆₁	D₈	1	2	yes	—
50	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂²⋊Dic₆₁	—	4	4	no	—
53	C₄⋊Dic₆₁	—	2	8	no	—
54	C₄⋊Dic₆₁	—	4	4	no	—
55	#11758196	—	4	8	no	—
56	C₄×Dic₆₁	—	4	4	no	—
57	C₁₆×C₆₁	C₂×Q₈	1	4	yes	—
58	C₁₆×C₆₁	C₂×Q₈	1	4	yes	—
59	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₆₁	—	2	4	no	—
61	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
66	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
67	C₂×Dic₆₁	—	4	8	no	—
68	Dic₁₂₂	—	8	8	no	—
69	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
70	C₈×C₆₁	C₂×Q₁₆	1	2	yes	—
71	C₈×C₆₁	C₂×Q₁₆	1	2	yes	—
72	C₈×C₆₁	C₂²×Q₈	1	2	yes	—
73	#33726	—	61	1	no	maximal, Sylow
74	C₂²×C₆₁	—	2	2	no	—
75	C₂²×C₆₁	#500	1	2	yes	—
76	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
77	C₄×C₆₁	—	2	2	no	—
78	C₄×C₆₁	#500	1	2	yes	—
79	C₄×C₆₁	#508	1	4	yes	—
80	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
82	Dic₆₁	—	8	8	no	—
83	#3160	—	61	2	no	—
84	#2254	—	61	2	no	—
85	#2253	—	61	4	no	—
86	#2252	—	61	2	no	—
87	#2158	—	61	4	no	—
88	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic
89	C₂×C₆₁	—	2	2	no	—
90	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
91	C₂×C₆₁	C₂×#508	1	2	yes	—
92	#533	—	122	2	no	—
93	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
94	#500	—	61	4	no	—
95	#499	—	61	4	no	—
96	#499	—	122	2	no	—
97	#498	—	122	2	no	—
98	#427	—	61	4	no	—
99	#427	—	122	2	no	—
100	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
101	#367	—	61	8	no	—
102	#366	—	61	8	no	—
103	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
104	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
105	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
106	C₆₁	#33726	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
109	C₄⋊Q₈	—	122	4	no	—
110	#125	—	122	4	no	—
111	C₂²⋊Q₈	—	244	4	no	—
112	C₄×Q₈	—	244	4	no	—
113	#117	—	122	4	no	—
114	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
115	C₂×C₁₆	Dic₁₂₂	1	8	yes	complemented
116	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
117	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
118	#107	—	61	8	no	—
119	#107	—	122	4	no	—
120	#106	—	122	4	no	—
121	#106	—	122	4	no	—
122	C₄×C₈	D₂₄₄	1	1	yes	characteristic
123	C₄×C₈	D₂₄₄	1	1	yes	characteristic
124	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
125	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
126	C₂×Q₈	—	244	4	no	—
127	C₂²×C₄	D₄₈₈	1	2	yes	—
128	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
129	C₂×C₈	Dic₂₄₄	1	4	yes	—
130	C₂×C₈	C₂×Dic₁₂₂	1	4	yes	—
131	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
132	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
133	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
134	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
135	C₄⋊C₄	—	122	8	no	—
136	C₄⋊C₄	—	244	8	no	—
137	C₄⋊C₄	—	244	4	no	—
138	C₂²⋊C₄	—	244	4	no	—
139	C₄²	—	244	4	no	—
140	C₄²	D₄₈₈	1	2	yes	—
141	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
142	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
143	C₁₆	C₂×Dic₁₂₂	1	4	yes	—
144	C₁₆	C₂×Dic₁₂₂	1	4	yes	—
145	C₂³	C₂×D₄₈₈	1	1	yes	characteristic
146	Q₈	—	488	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	244	8	no	—
149	C₂×C₄	#14519777	1	2	yes	—
150	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
151	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
152	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
153	C₂×C₄	C₂×Dic₂₄₄	1	2	yes	—
154	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
155	C₈	#14519777	1	2	yes	—
156	C₈	C₂×Dic₂₄₄	1	2	yes	—
157	C₈	C₂×Dic₂₄₄	1	2	yes	—
158	C₈	C₂²×Dic₁₂₂	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#16840167	1	2	yes	—
161	C₂²	C₂²×D₄₈₈	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	488	8	no	—
164	C₄	#16840167	1	2	yes	—
165	C₄	#16840194	1	4	yes	—
166	C₄	C₂×#14519777	1	1	yes	characteristic
167	C₄	C₂²×Dic₂₄₄	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×#16840167	1	1	yes	characteristic
170	C₂	C₂×#16840194	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₈₈
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₆₁
hypocenter	C₆₁	#33726
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×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
2. #21448101	C₂	semidirect
3. #21448098	C₂	semidirect
4. #21448094	C₂	semidirect
33. C₂×C₁₆×C₆₁	Q₈	semidirect
106. C₆₁	#33726	semidirect
115. C₂×C₁₆	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21456333	2
#21448101	2
#21448098	2
#21448094	2
#21447603	2
C₂×C₄×C₁₆×C₆₁	2
#33726	61