Subgroups — #29573694

Order 15616.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445991	C₂	1	1	yes	maximal, characteristic
2	#21445983	C₂	1	1	yes	maximal, characteristic
3	#21443678	C₂	1	4	yes	maximal
4	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519713	—	2	1	no	—
6	C₂×#14519713	—	2	1	no	—
7	C₂×#14519711	—	2	1	no	—
8	C₂×#14519709	—	2	1	no	—
9	#16839786	C₂²	1	2	yes	—
10	#16839628	C₂²	1	2	yes	—
11	C₆₁×#431	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#322	C₄	1	2	yes	complemented
14	C₈²×C₆₁	C₄	1	2	yes	complemented
15	C₂×#11758194	—	2	2	no	—
16	C₂×C₄⋊Dic₆₁	—	2	2	no	—
17	#14519713	—	4	2	no	—
18	#14519713	—	4	2	no	—
19	#14519711	—	4	2	no	—
20	#14519709	—	4	2	no	—
21	C₄⋊#92418	—	2	4	no	—
22	#14519684	—	2	4	no	—
23	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₆₁	—	2	2	no	—
27	C₂²⋊C₈×C₆₁	—	2	2	no	—
28	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	#41	1	2	yes	—
35	C₂²×Dic₆₁	—	4	2	no	—
36	C₂×C₈×C₆₁	—	2	4	no	—
37	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₁	D₈	1	2	yes	—
42	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
43	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
44	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
45	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₆₁	#41	1	2	yes	—
48	C₄⋊Dic₆₁	—	4	2	no	—
49	C₄⋊Dic₆₁	—	4	2	no	—
50	#11758194	—	4	2	no	—
51	#11758194	—	4	2	no	—
52	C₂×#92418	—	4	2	no	—
53	C₂³×C₆₁	#124	1	1	yes	characteristic
54	C₂×C₄×C₆₁	—	4	2	no	—
55	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#117	1	2	yes	—
57	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
58	C₂×C₄×C₆₁	#125	1	2	yes	—
59	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₆₁	—	8	4	no	—
64	C₂×Dic₆₁	—	8	2	no	—
65	C₈×C₆₁	—	4	4	no	—
66	C₈×C₆₁	#106	1	2	yes	—
67	C₈×C₆₁	#106	1	2	yes	—
68	C₈×C₆₁	#107	1	2	yes	—
69	C₈×C₆₁	#107	1	2	yes	—
70	#92418	—	8	2	no	—
71	#8790	—	61	1	no	maximal, Sylow
72	C₂²×C₆₁	—	2	2	no	—
73	C₂²×C₆₁	#382	1	1	yes	characteristic
74	C₂²×C₆₁	#486	1	1	yes	characteristic
75	C₂²×C₆₁	#487	1	1	yes	characteristic
76	C₄×C₆₁	—	2	2	no	—
77	C₄×C₆₁	—	2	2	no	—
78	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
79	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
80	C₄×C₆₁	#486	1	1	yes	characteristic
81	C₄×C₆₁	#487	1	1	yes	characteristic
82	Dic₆₁	—	16	2	no	—
83	#1848	—	61	1	no	—
84	#1846	—	61	1	no	—
85	#1450	D₆₁	1	1	yes	characteristic
86	#1286	—	61	4	no	—
87	C₂×C₆₁	—	2	1	no	—
88	C₂×C₆₁	—	2	1	no	—
89	C₂×C₆₁	#1833	1	2	yes	—
90	C₂×C₆₁	#1834	1	1	yes	characteristic
91	#431	C₂×D₆₁	1	2	yes	—
92	C₂×#108	—	122	1	no	—
93	C₂×#108	—	122	1	no	—
94	C₂×#107	—	122	1	no	—
95	C₂×#106	—	122	1	no	—
96	#423	—	61	2	no	—
97	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
98	#382	—	61	2	no	—
99	#322	Dic₆₁	1	2	yes	complemented
100	C₈²	Dic₆₁	1	2	yes	complemented
101	C₆₁	#8790	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	122	2	no	—
103	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
104	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	122	2	no	—
106	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
107	#108	—	244	2	no	—
108	#108	—	244	2	no	—
109	#107	—	244	2	no	—
110	#106	—	244	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	122	4	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
115	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
117	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
119	#95	—	122	4	no	—
120	C₂²×C₄	—	244	2	no	—
121	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
122	C₂²×C₄	#11758222	1	2	yes	—
123	M₁₆	—	244	2	no	—
124	M₁₆	—	244	2	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	244	2	no	—
127	C₂×C₈	#11758190	1	2	yes	—
128	C₂×C₈	#11758190	1	2	yes	—
129	C₂×C₈	D₄₈₈	1	2	yes	—
130	C₂×C₈	Dic₂₄₄	1	2	yes	—
131	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
132	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
134	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
135	C₄⋊C₄	—	244	2	no	—
136	C₄⋊C₄	—	244	2	no	—
137	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
138	C₄²	#11758222	1	2	yes	—
139	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
140	C₂³	#14519765	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	488	4	no	—
143	C₂×C₄	—	488	2	no	—
144	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
145	C₂×C₄	#14519736	1	2	yes	—
146	C₂×C₄	#14519765	1	1	yes	characteristic
147	C₂×C₄	#14519769	1	2	yes	—
148	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
149	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
150	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
152	C₈	—	4	4	no	—
153	C₈	—	488	2	no	—
154	C₈	#14519709	1	2	yes	—
155	C₈	#14519709	1	2	yes	—
156	C₈	#14519711	1	2	yes	—
157	C₈	#14519711	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#16839628	1	1	yes	characteristic
160	C₂²	#16840109	1	1	yes	characteristic
161	C₂²	#16840113	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	976	2	no	—
165	C₄	C₂×#14519709	1	1	yes	characteristic
166	C₄	C₂×#14519711	1	1	yes	characteristic
167	C₄	#16840109	1	1	yes	characteristic
168	C₄	#16840113	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#21445943	1	2	yes	—
172	C₂	#21445947	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16839628
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1450	C₂
socle	C₂²×C₆₁	#382
prime radical	C₂³×C₆₁	#124
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1450	D₆₁
hypocenter	C₆₁	#8790
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
13. C₆₁×#322	C₄	semidirect
14. C₈²×C₆₁	C₄	semidirect
99. #322	Dic₆₁	semidirect
100. C₈²	Dic₆₁	semidirect
101. C₆₁	#8790	semidirect

Maximal subgroups

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

subgroup	index
#21445991	2
#21445983	2
#21443678	2
C₆₁×#1450	2
#8790	61