Subgroups — #29571139

Order 15616.

502 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519706	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839483	C₂	1	1	yes	maximal, characteristic
4	#21443646	C₂	1	2	yes	maximal
5	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519706	C₄	1	4	yes	—
7	C₂×C₄×#92418	C₂²	1	2	yes	—
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
10	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
11	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
12	#16839516	C₄	1	4	yes	—
13	#16839483	—	2	2	no	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#322	—	2	2	no	—
17	C₂²×#92418	C₂×C₄	1	4	yes	—
18	C₂²×#92418	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
23	#14519706	—	2	8	no	—
24	C₄×#92418	—	2	2	no	—
25	C₄×#92418	—	2	2	no	—
26	C₄×#92418	—	2	2	no	—
27	C₈×Dic₆₁	—	2	4	no	—
28	C₈×Dic₆₁	C₈	1	8	yes	complemented
29	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₆₁	—	2	2	no	—
33	C₄×C₈×C₆₁	—	2	2	no	—
34	C₄×C₈×C₆₁	D₄	1	2	yes	—
35	C₄×C₈×C₆₁	Q₈	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
38	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
39	C₂²×Dic₆₁	C₄²	1	2	yes	—
40	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₆₁	—	2	4	no	—
42	C₂×C₈×C₆₁	—	2	4	no	—
43	C₂×C₈×C₆₁	—	2	4	no	—
44	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
45	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
47	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₆₁	#41	1	2	yes	—
50	C₄×Dic₆₁	—	2	2	no	—
51	C₄×Dic₆₁	—	2	4	no	—
52	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
53	C₄×Dic₆₁	M₁₆	1	8	yes	—
54	C₂×#92418	—	2	2	no	—
55	C₂×#92418	—	2	2	no	—
56	C₂×#92418	—	2	4	no	—
57	C₂×#92418	—	2	4	no	—
58	C₂×#92418	—	2	4	no	—
59	C₂×#92418	C₂×C₈	1	4	yes	—
60	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
68	C₂×C₄×C₆₁	#131	1	4	yes	—
69	C₂×Dic₆₁	—	2	2	no	—
70	C₂×Dic₆₁	—	2	2	no	—
71	C₂×Dic₆₁	—	2	2	no	—
72	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
73	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
74	C₂×Dic₆₁	#97	1	4	yes	—
75	C₂×Dic₆₁	#97	1	4	yes	—
76	C₈×C₆₁	—	4	4	no	—
77	C₈×C₆₁	C₄⋊C₈	1	8	yes	complemented
78	#92418	—	2	4	no	—
79	#92418	—	4	4	no	—
80	#8260	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₆₁	C₂×#97	1	2	yes	—
84	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
85	C₂²×C₆₁	#405	1	2	yes	—
86	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
87	C₄×C₆₁	#431	1	2	yes	—
88	C₄×C₆₁	#436	1	4	yes	—
89	C₄×C₆₁	#446	1	4	yes	—
90	Dic₆₁	—	2	2	no	—
91	Dic₆₁	—	2	4	no	—
92	Dic₆₁	#322	1	4	yes	complemented
93	C₄×C₄⋊C₈	—	61	2	no	—
94	C₄²×C₈	—	61	1	no	—
95	C₂×#322	—	61	1	no	—
96	C₂×#322	D₆₁	1	1	yes	characteristic
97	#1273	—	61	2	no	—
98	C₂×C₆₁	C₂×#322	1	1	yes	characteristic
99	C₂×C₆₁	#1450	1	1	yes	characteristic
100	C₂×C₆₁	#1451	1	2	yes	—
101	C₂×C₆₁	C₄×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₆₁	#1770	1	2	yes	—
103	C₂×C₄⋊C₈	—	61	4	no	—
104	C₂×C₄×C₈	—	61	2	no	—
105	C₂×C₄×C₈	—	61	2	no	—
106	C₂×C₄×C₈	—	61	2	no	—
107	C₂×C₄×C₈	—	61	1	no	—
108	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
110	C₄³	—	61	1	no	—
111	#336	—	61	4	no	—
112	#322	—	2	2	no	—
113	#322	—	122	2	no	—
114	C₆₁	#8260	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	61	2	no	—
116	C₂²×C₈	—	61	4	no	—
117	C₂²×C₈	C₄×D₆₁	1	4	yes	—
118	C₂²×C₈	C₄×D₆₁	1	2	yes	—
119	C₂×C₄²	—	61	1	no	—
120	C₂×C₄²	—	61	2	no	—
121	C₂×C₄²	—	61	2	no	—
122	C₂×C₄²	—	61	1	no	—
123	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
124	C₄⋊C₈	—	122	8	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	61	8	no	—
128	C₄×C₈	—	122	4	no	—
129	C₄×C₈	—	122	2	no	—
130	C₄×C₈	—	122	2	no	—
131	C₄×C₈	—	122	2	no	—
132	C₄×C₈	Dic₁₂₂	1	2	yes	—
133	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
134	C₂²×C₄	—	61	1	no	—
135	C₂²×C₄	—	61	1	no	—
136	C₂²×C₄	—	61	2	no	—
137	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	61	4	no	—
143	C₂×C₈	—	122	2	no	—
144	C₂×C₈	—	122	2	no	—
145	C₂×C₈	—	122	4	no	—
146	C₂×C₈	—	122	4	no	—
147	C₂×C₈	—	122	4	no	—
148	C₂×C₈	C₈×D₆₁	1	4	yes	—
149	C₂×C₈	#11758189	1	4	yes	—
150	C₂×C₈	#11758196	1	4	yes	—
151	C₄²	—	61	4	no	—
152	C₄²	—	61	8	no	—
153	C₄²	—	122	2	no	—
154	C₄²	—	122	4	no	—
155	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
156	C₄²	#11758222	1	2	yes	—
157	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
158	C₂³	C₄²×D₆₁	1	1	yes	characteristic
159	C₂×C₄	—	61	2	no	—
160	C₂×C₄	—	61	4	no	—
161	C₂×C₄	—	61	2	no	—
162	C₂×C₄	—	61	4	no	—
163	C₂×C₄	—	122	2	no	—
164	C₂×C₄	—	122	2	no	—
165	C₂×C₄	—	122	2	no	—
166	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
167	C₂×C₄	#14519734	1	1	yes	characteristic
168	C₂×C₄	C₄×C₂²⋊D₆₁	1	2	yes	—
169	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
170	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
171	C₂×C₄	M₁₆×D₆₁	1	4	yes	—
172	C₂×C₄	C₂×#11758189	1	2	yes	—
173	C₂×C₄	#14519785	1	4	yes	—
174	C₈	—	4	4	no	—
175	C₈	—	122	4	no	—
176	C₈	—	244	4	no	—
177	C₈	#14519706	1	8	yes	complemented
178	C₂²	C₄×#11758196	1	1	yes	characteristic
179	C₂²	C₄×C₈×D₆₁	1	1	yes	characteristic
180	C₂²	#97×D₆₁	1	2	yes	—
181	C₂²	C₄×#11758189	1	1	yes	characteristic
182	C₂²	#16839719	1	2	yes	—
183	C₄	—	61	4	no	—
184	C₄	—	122	2	no	—
185	C₄	—	122	4	no	—
186	C₄	C₂×#14519706	1	2	yes	—
187	C₄	#16839813	1	2	yes	—
188	C₄	#16839838	1	4	yes	—
189	C₄	#16839890	1	4	yes	—
190	C₂	#322×D₆₁	1	1	yes	characteristic
191	C₂	#21444104	1	1	yes	characteristic
192	C₂	#21444105	1	2	yes	—
193	C₂	C₄×#14519706	1	1	yes	characteristic
194	C₂	#21445687	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₂×C₆₁×#322	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄²
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#322	D₆₁
hypocenter	C₆₁	#8260
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
28. C₈×Dic₆₁	C₈	semidirect
77. C₈×C₆₁	C₄⋊C₈	semidirect
92. Dic₆₁	#322	semidirect
114. C₆₁	#8260	semidirect
177. C₈	#14519706	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519706	2
C₄×C₈×Dic₆₁	2
C₂×#16839483	2
#21443646	2
C₂×C₆₁×#322	2
#8260	61