Subgroups — #29573816

Order 15616.

354 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445990	C₂	1	2	yes	maximal
2	#21444578	C₂	1	1	yes	maximal, characteristic
3	#21444577	C₂	1	2	yes	maximal
4	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1567	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519712	—	2	4	no	—
7	#16839786	C₂²	1	2	yes	—
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	#16839515	—	2	2	no	—
11	#16839514	—	2	4	no	—
12	C₈×#92418	C₄	1	1	yes	characteristic
13	C₈×#92418	C₄	1	1	yes	characteristic
14	C₈×#92418	C₄	1	1	yes	characteristic
15	C₈×#92418	C₄	1	1	yes	characteristic
16	C₆₁×#423	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#335	—	2	2	no	—
19	C₂×#11758194	—	2	2	no	—
20	C₂²×#92418	D₄	1	2	yes	—
21	C₂²×#92418	Q₈	1	2	yes	—
22	#14519712	—	4	8	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄×#92418	C₂×C₄	1	1	yes	characteristic
26	C₄×#92418	C₂×C₄	1	1	yes	characteristic
27	C₄×#92418	C₂×C₄	1	1	yes	characteristic
28	C₄×#92418	C₂×C₄	1	1	yes	characteristic
29	C₄×#92418	D₄	1	2	yes	—
30	C₄×#92418	Q₈	1	2	yes	—
31	C₂×M₁₆×C₆₁	—	2	2	no	—
32	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₆₁	—	2	2	no	—
35	C₄⋊C₈×C₆₁	—	2	2	no	—
36	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₆₁	D₄	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₆₁	—	4	4	no	—
43	C₂×C₈×C₆₁	—	2	4	no	—
44	C₂×C₈×C₆₁	—	4	2	no	—
45	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
47	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
51	#11758194	—	4	4	no	—
52	C₂×#92418	—	2	4	no	—
53	C₂×#92418	—	4	2	no	—
54	C₂×#92418	C₄⋊C₄	1	2	yes	—
55	C₂×#92418	C₄⋊C₄	1	2	yes	—
56	C₂×#92418	C₄⋊C₄	1	2	yes	—
57	C₂×#92418	C₄⋊C₄	1	2	yes	—
58	C₂×#92418	D₈	1	4	yes	—
59	C₂×#92418	Q₁₆	1	4	yes	—
60	C₂³×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
71	C₈×C₆₁	—	2	2	no	—
72	C₈×C₆₁	—	8	2	no	—
73	C₈×C₆₁	#108	1	4	yes	complemented
74	#92418	—	2	2	no	—
75	#92418	—	8	2	no	—
76	#92418	#107	1	4	yes	—
77	#92418	#107	1	4	yes	—
78	#92418	#108	1	4	yes	complemented
79	#8823	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	—	2	1	no	—
81	C₂²×C₆₁	—	2	1	no	—
82	C₂²×C₆₁	#384	1	1	yes	characteristic
83	C₂²×C₆₁	#500	1	2	yes	—
84	C₄×C₆₁	—	2	1	no	—
85	C₄×C₆₁	—	2	1	no	—
86	C₄×C₆₁	C₂×#107	1	2	yes	—
87	C₄×C₆₁	C₂×#108	1	2	yes	—
88	C₄×C₆₁	C₂×#108	1	2	yes	—
89	C₄×C₆₁	#493	1	1	yes	characteristic
90	C₄×C₆₁	#494	1	1	yes	characteristic
91	#1848	—	61	2	no	—
92	#1567	—	61	2	no	—
93	#1567	—	61	1	no	—
94	#1567	D₆₁	1	1	yes	characteristic
95	C₂×C₈²	—	61	1	no	—
96	C₂×C₆₁	—	2	2	no	—
97	C₂×C₆₁	#1845	1	1	yes	characteristic
98	C₂×C₆₁	#1848	1	1	yes	characteristic
99	C₂×C₆₁	#1848	1	1	yes	characteristic
100	C₂×#108	—	122	4	no	—
101	#423	—	61	2	no	—
102	#423	C₂×D₆₁	1	2	yes	—
103	C₂×C₄×C₈	—	61	1	no	—
104	C₂×C₄×C₈	—	61	1	no	—
105	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
106	#335	—	2	2	no	—
107	#335	—	122	4	no	—
108	#335	—	122	2	no	—
109	C₈²	—	61	1	no	—
110	C₈²	—	61	1	no	—
111	C₈²	—	61	1	no	—
112	C₈²	—	61	1	no	—
113	C₆₁	#8823	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	122	2	no	—
116	C₂²×C₈	—	61	2	no	—
117	C₂²×C₈	—	61	2	no	—
118	C₂²×C₈	Dic₁₂₂	1	2	yes	—
119	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
120	#108	—	244	8	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	122	2	no	—
124	C₄⋊C₈	—	122	2	no	—
125	C₄×C₈	—	61	1	no	—
126	C₄×C₈	—	61	1	no	—
127	C₄×C₈	—	61	1	no	—
128	C₄×C₈	—	61	1	no	—
129	C₄×C₈	—	61	2	no	—
130	C₄×C₈	—	61	2	no	—
131	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
132	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
134	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
135	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
136	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
137	M₁₆	—	4	4	no	—
138	M₁₆	—	244	4	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	61	2	no	—
142	C₂×C₈	—	61	2	no	—
143	C₂×C₈	—	61	2	no	—
144	C₂×C₈	—	61	4	no	—
145	C₂×C₈	—	61	4	no	—
146	C₂×C₈	—	61	2	no	—
147	C₂×C₈	—	122	4	no	—
148	C₂×C₈	—	244	2	no	—
149	C₂×C₈	#11758196	1	2	yes	—
150	C₂×C₈	#11758196	1	2	yes	—
151	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
152	C₄²	D₄×D₆₁	1	1	yes	characteristic
153	C₄²	Q₈×D₆₁	1	1	yes	characteristic
154	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
155	C₂³	#14519776	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
160	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
161	C₂×C₄	#14519773	1	1	yes	characteristic
162	C₂×C₄	#14519775	1	1	yes	characteristic
163	C₂×C₄	#14519776	1	1	yes	characteristic
164	C₂×C₄	D₈×D₆₁	1	2	yes	—
165	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
166	C₈	—	2	2	no	—
167	C₈	—	8	2	no	—
168	C₈	—	61	4	no	—
169	C₈	—	61	4	no	—
170	C₈	—	61	4	no	—
171	C₈	—	122	2	no	—
172	C₈	—	488	2	no	—
173	C₈	#14519712	1	4	yes	complemented
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#16839631	1	1	yes	characteristic
177	C₂²	#16840166	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	#107×D₆₁	1	2	yes	—
181	C₄	#108×D₆₁	1	2	yes	—
182	C₄	C₂×#14519712	1	2	yes	—
183	C₄	#16840137	1	1	yes	characteristic
184	C₄	#16840140	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#21445977	1	1	yes	characteristic
187	C₂	#21445989	1	1	yes	characteristic
188	C₂	#21445990	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519773
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1567	C₂
socle	C₂²×C₆₁	#384
prime radical	C₂³×C₆₁	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1567	D₆₁
hypocenter	C₆₁	#8823
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
73. C₈×C₆₁	#108	semidirect
78. #92418	#108	semidirect
113. C₆₁	#8823	semidirect
173. C₈	#14519712	semidirect

Maximal subgroups

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

subgroup	index
#21445990	2
#21444578	2
#21444577	2
C₂×C₈×#92418	2
C₆₁×#1567	2
#8823	61