Subgroups — #29571151

Order 15616.

470 conjugacy classes of subgroups.

The subgroup lattice has 194 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×#14519687	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839483	C₂	1	2	yes	maximal
4	#21443646	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519706	C₄	1	4	yes	—
7	C₂×#14519687	C₄	1	2	yes	—
8	C₂×#14519687	C₄	1	2	yes	—
9	C₂×C₄×#92418	C₂²	1	2	yes	—
10	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
11	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
12	#16839516	C₄	1	4	yes	—
13	#16839483	—	2	4	no	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×#92418	C₂×C₄	1	4	yes	—
17	C₂²×#92418	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
22	#14519706	—	2	8	no	—
23	#14519687	—	2	4	no	—
24	#14519687	—	2	4	no	—
25	C₄×#92418	—	2	2	no	—
26	C₄×#92418	—	2	2	no	—
27	C₄×#92418	D₄	1	2	yes	—
28	C₄×#92418	Q₈	1	2	yes	—
29	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	4	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	—	2	2	no	—
35	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
37	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
38	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
39	C₂²×Dic₆₁	C₄²	1	2	yes	—
40	C₂×C₈×C₆₁	—	2	2	no	—
41	C₂×C₈×C₆₁	—	2	4	no	—
42	C₂×C₈×C₆₁	—	2	2	no	—
43	C₂×C₈×C₆₁	—	2	4	no	—
44	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
45	C₂×C₈×C₆₁	M₁₆	1	8	yes	—
46	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₆₁	#41	1	2	yes	—
49	C₄×Dic₆₁	—	2	2	no	—
50	C₄×Dic₆₁	—	2	4	no	—
51	C₄×Dic₆₁	—	2	4	no	—
52	C₄×Dic₆₁	M₁₆	1	4	yes	—
53	C₂×#92418	—	2	2	no	—
54	C₂×#92418	—	2	2	no	—
55	C₂×#92418	—	2	4	no	—
56	C₂×#92418	—	2	4	no	—
57	C₂×#92418	—	2	4	no	—
58	C₂×#92418	C₄⋊C₄	1	4	yes	—
59	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
67	C₂×C₄×C₆₁	#131	1	4	yes	—
68	C₂×Dic₆₁	—	2	2	no	—
69	C₂×Dic₆₁	—	2	2	no	—
70	C₂×Dic₆₁	—	2	1	no	—
71	C₂×Dic₆₁	—	2	2	no	—
72	C₂×Dic₆₁	—	2	1	no	—
73	C₂×Dic₆₁	—	2	2	no	—
74	C₂×Dic₆₁	#97	1	4	yes	—
75	C₈×C₆₁	—	2	4	no	—
76	C₈×C₆₁	—	2	8	no	—
77	#92418	—	2	4	no	—
78	#92418	—	4	4	no	—
79	#8263	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
82	C₂²×C₆₁	C₄×M₁₆	1	2	yes	—
83	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
84	C₂²×C₆₁	#405	1	2	yes	—
85	C₄×C₆₁	#423	1	2	yes	—
86	C₄×C₆₁	#432	1	2	yes	—
87	C₄×C₆₁	#435	1	4	yes	—
88	C₄×C₆₁	#446	1	4	yes	—
89	Dic₆₁	—	2	4	no	—
90	Dic₆₁	—	4	1	no	—
91	Dic₆₁	—	4	1	no	—
92	C₄×C₄⋊C₈	—	61	2	no	—
93	C₄×#97	—	61	1	no	—
94	C₂×#322	—	61	2	no	—
95	#1273	—	61	1	no	—
96	#1273	D₆₁	1	1	yes	characteristic
97	C₂×C₆₁	#1451	1	2	yes	—
98	C₂×C₆₁	#1453	1	1	yes	characteristic
99	C₂×C₆₁	#1455	1	1	yes	characteristic
100	C₂×C₆₁	#1767	1	1	yes	characteristic
101	C₂×C₆₁	#1771	1	2	yes	—
102	C₂×C₄⋊C₈	—	61	4	no	—
103	C₂×#97	—	61	2	no	—
104	C₂×#97	—	61	2	no	—
105	C₂×C₄×C₈	—	61	1	no	—
106	C₂×C₄×C₈	—	61	2	no	—
107	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
109	C₄³	—	61	1	no	—
110	#336	—	61	4	no	—
111	#322	—	122	4	no	—
112	C₆₁	#8263	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	61	2	no	—
114	C₂²×C₈	—	61	4	no	—
115	C₂²×C₈	C₄×D₆₁	1	4	yes	—
116	C₂²×C₈	C₄×D₆₁	1	2	yes	—
117	C₂×C₄²	—	61	1	no	—
118	C₂×C₄²	—	61	2	no	—
119	C₂×C₄²	—	61	1	no	—
120	C₂×C₄²	—	61	2	no	—
121	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
122	C₄⋊C₈	—	122	8	no	—
123	#97	—	122	4	no	—
124	#97	—	122	4	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	61	2	no	—
129	C₄×C₈	—	61	2	no	—
130	C₄×C₈	—	122	2	no	—
131	C₄×C₈	—	122	2	no	—
132	C₂²×C₄	—	61	2	no	—
133	C₂²×C₄	—	61	1	no	—
134	C₂²×C₄	—	61	1	no	—
135	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	61	4	no	—
142	C₂×C₈	—	122	2	no	—
143	C₂×C₈	—	122	2	no	—
144	C₂×C₈	—	122	4	no	—
145	C₂×C₈	—	122	4	no	—
146	C₂×C₈	—	122	4	no	—
147	C₂×C₈	#11758189	1	4	yes	—
148	C₂×C₈	#11758189	1	8	yes	—
149	C₄²	—	61	4	no	—
150	C₄²	—	122	2	no	—
151	C₄²	—	122	4	no	—
152	C₄²	—	122	4	no	—
153	C₄²	D₄×D₆₁	1	1	yes	characteristic
154	C₄²	#11758224	1	2	yes	—
155	C₄²	Q₈×D₆₁	1	1	yes	characteristic
156	C₂³	C₄²×D₆₁	1	1	yes	characteristic
157	C₂×C₄	—	61	4	no	—
158	C₂×C₄	—	122	2	no	—
159	C₂×C₄	—	122	2	no	—
160	C₂×C₄	—	122	1	no	—
161	C₂×C₄	—	122	2	no	—
162	C₂×C₄	—	122	1	no	—
163	C₂×C₄	—	122	2	no	—
164	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
165	C₂×C₄	#14519733	1	1	yes	characteristic
166	C₂×C₄	#14519737	1	2	yes	—
167	C₂×C₄	#14519742	1	2	yes	—
168	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
169	C₂×C₄	C₂×#11758189	1	2	yes	—
170	C₂×C₄	C₂×#11758189	1	4	yes	—
171	C₂×C₄	#14519785	1	4	yes	—
172	C₈	—	2	4	no	—
173	C₈	—	2	8	no	—
174	C₈	—	122	4	no	—
175	C₈	—	244	4	no	—
176	C₂²	#16839613	1	1	yes	characteristic
177	C₂²	#97×D₆₁	1	1	yes	characteristic
178	C₂²	C₄×#11758189	1	2	yes	—
179	C₂²	C₄×#11758189	1	1	yes	characteristic
180	C₂²	#16839719	1	2	yes	—
181	C₄	—	122	4	no	—
182	C₄	—	244	1	no	—
183	C₄	—	244	1	no	—
184	C₄	#16839784	1	2	yes	—
185	C₄	#16839816	1	2	yes	—
186	C₄	#16839830	1	4	yes	—
187	C₄	#16839889	1	4	yes	—
188	C₂	#21444105	1	2	yes	—
189	C₂	#21444117	1	1	yes	characteristic
190	C₂	#21444121	1	1	yes	characteristic
191	C₂	#21445671	1	1	yes	characteristic
192	C₂	#21445690	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂×C₄²
Fitting	C₆₁×#1273	C₂
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1273	D₆₁
hypocenter	C₆₁	#8263
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
112. C₆₁	#8263	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×#14519687	2
C₂×#16839483	2
#21443646	2
C₆₁×#1273	2
#8263	61