Subgroups — #29832654

Order 15616.

826 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	C₂²×#14519717	C₂	1	2	yes	maximal
2	#21447416	C₂	1	8	yes	maximal, complemented
3	C₄×#14519717	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519717	C₄	1	16	yes	complemented
6	C₂×#14519717	C₂²	1	4	yes	—
7	C₂×#14519717	C₂²	1	4	yes	—
8	C₂²×#11758185	C₂²	1	4	yes	—
9	#16839538	C₂²	1	8	yes	complemented
10	C₄×#11758185	C₂²	1	8	yes	complemented
11	C₂²×M₁₆×C₆₁	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
14	C₆₁×#405	C₄	1	4	yes	—
15	C₆₁×#405	C₂²	1	4	yes	—
16	C₂×#97×C₆₁	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
18	#14519717	C₂×C₄	1	16	yes	—
19	#14519717	C₂×C₄	1	16	yes	complemented
20	C₂×#11758185	C₂×C₄	1	16	yes	complemented
21	C₂×#11758185	C₂³	1	8	yes	—
22	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₆₁	C₈	1	8	yes	—
24	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₁	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
31	#117×C₆₁	C₈	1	4	yes	—
32	#117×C₆₁	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
34	C₂×C₂²⋊C₄×C₆₁	C₈	1	2	yes	—
35	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
36	#97×C₆₁	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₆₁	C₂³	1	4	yes	—
39	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₄²	1	4	yes	—
41	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
47	M₁₆×C₆₁	C₂×C₈	1	16	yes	—
48	C₂×C₈×C₆₁	C₄²	1	4	yes	—
49	C₂×C₈×C₆₁	C₄²	1	8	yes	—
50	C₂×C₈×C₆₁	C₂×C₈	1	8	yes	—
51	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
56	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
59	C₂²⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
60	C₄²×C₆₁	C₂×C₈	1	4	yes	—
61	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
62	#11758185	C₂²×C₄	1	16	yes	—
63	C₂³×C₆₁	—	4	2	no	—
64	C₂³×C₆₁	C₄×C₈	1	4	yes	—
65	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
66	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	4	4	no	—
68	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
69	C₂×C₄×C₆₁	C₄×C₈	1	8	yes	—
70	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
71	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
72	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	—
73	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
75	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
78	C₈×C₆₁	C₂×C₄²	1	2	yes	—
79	C₈×C₆₁	C₂×C₄²	1	4	yes	—
80	C₈×C₆₁	C₂²×C₈	1	8	yes	—
81	C₈×C₆₁	C₂³×C₄	1	2	yes	—
82	#33324	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	—	2	2	no	—
84	C₂²×C₆₁	—	2	4	no	—
85	C₂²×C₆₁	—	4	4	no	—
86	C₂²×C₆₁	C₂×C₄×C₈	1	2	yes	—
87	C₂²×C₆₁	C₂×C₄×C₈	1	4	yes	—
88	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
89	C₄×C₆₁	—	2	8	no	—
90	C₄×C₆₁	C₂×C₄×C₈	1	2	yes	—
91	C₄×C₆₁	C₂×C₄×C₈	1	4	yes	—
92	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
93	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
94	C₂²×M₃₂	—	61	2	no	—
95	C₂×#405	D₆₁	1	1	yes	characteristic
96	#2108	—	61	8	no	—
97	C₄×M₃₂	—	61	4	no	—
98	C₂×C₆₁	—	2	2	no	—
99	C₂×C₆₁	—	2	4	no	—
100	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
101	C₂×C₆₁	#3407	1	2	yes	—
102	C₂²×M₁₆	Dic₆₁	1	1	yes	characteristic
103	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
104	C₂×#117	Dic₆₁	1	1	yes	characteristic
105	C₂×M₃₂	—	61	16	no	—
106	C₂×M₃₂	—	61	4	no	—
107	C₂×M₃₂	—	61	4	no	—
108	C₂²×C₁₆	—	61	4	no	—
109	#405	Dic₆₁	1	4	yes	—
110	#405	C₂×D₆₁	1	4	yes	—
111	C₂×#97	Dic₆₁	1	2	yes	—
112	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
113	#346	—	61	8	no	—
114	C₄×C₁₆	—	61	8	no	—
115	C₆₁	#33324	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₄	C₂×Dic₆₁	1	1	yes	characteristic
117	C₂×M₁₆	#92418	1	8	yes	—
118	C₂×M₁₆	C₂×Dic₆₁	1	4	yes	—
119	C₂²×C₈	C₄×D₆₁	1	8	yes	—
120	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
121	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
122	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
123	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
125	#117	#92418	1	4	yes	—
126	#117	C₂×Dic₆₁	1	4	yes	—
127	C₂×C₄⋊C₄	#92418	1	2	yes	—
128	C₂×C₂²⋊C₄	#92418	1	2	yes	—
129	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
130	M₃₂	—	61	16	no	—
131	M₃₂	—	61	16	no	—
132	C₂×C₁₆	—	61	16	no	—
133	C₂×C₁₆	—	61	8	no	—
134	#97	C₂×Dic₆₁	1	8	yes	—
135	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
136	C₄×C₈	C₂²×D₆₁	1	4	yes	—
137	C₂⁴	C₂×#92418	1	1	yes	characteristic
138	C₂²×C₄	C₂×#92418	1	4	yes	—
139	C₂²×C₄	C₂×#92418	1	1	yes	characteristic
140	C₂²×C₄	C₂×#92418	1	2	yes	—
141	C₂²×C₄	C₂×#92418	1	4	yes	—
142	C₂²×C₄	C₄×Dic₆₁	1	4	yes	—
143	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
144	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
145	M₁₆	C₂×#92418	1	16	yes	—
146	C₂×C₈	C₂×#92418	1	8	yes	—
147	C₂×C₈	C₄×Dic₆₁	1	4	yes	—
148	C₂×C₈	C₄×Dic₆₁	1	8	yes	—
149	C₂×C₈	C₂×C₄×D₆₁	1	4	yes	—
150	C₂×C₈	C₂×C₄×D₆₁	1	4	yes	—
151	C₂×C₈	C₂×C₄×D₆₁	1	4	yes	—
152	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
153	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
154	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
155	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
156	C₄⋊C₄	C₂×#92418	1	8	yes	—
157	C₂²⋊C₄	C₂×#92418	1	8	yes	—
158	C₄²	C₂×#92418	1	4	yes	—
159	C₄²	C₂²×Dic₆₁	1	4	yes	—
160	C₁₆	—	61	16	no	—
161	C₂³	—	4	2	no	—
162	C₂³	C₄×#92418	1	4	yes	—
163	C₂³	C₂²×#92418	1	2	yes	—
164	C₂³	C₂²×#92418	1	1	yes	characteristic
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	C₄×#92418	1	4	yes	—
167	C₂×C₄	C₄×#92418	1	8	yes	—
168	C₂×C₄	C₂×C₄×Dic₆₁	1	4	yes	—
169	C₂×C₄	C₂×C₄×Dic₆₁	1	2	yes	—
170	C₂×C₄	C₂²×#92418	1	8	yes	—
171	C₂×C₄	C₂²×#92418	1	1	yes	characteristic
172	C₂×C₄	C₂²×#92418	1	2	yes	—
173	C₂×C₄	C₂²×#92418	1	2	yes	—
174	C₂×C₄	C₂²×#92418	1	4	yes	—
175	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
176	C₈	C₂×C₄×Dic₆₁	1	4	yes	—
177	C₈	C₂×C₄×Dic₆₁	1	2	yes	—
178	C₈	C₂²×#92418	1	8	yes	—
179	C₈	C₂²×C₄×D₆₁	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×C₄×#92418	1	2	yes	—
184	C₂²	C₂×C₄×#92418	1	4	yes	—
185	C₂²	C₂³×#92418	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	C₂×C₄×#92418	1	2	yes	—
188	C₄	C₂×C₄×#92418	1	4	yes	—
189	C₄	C₂²×C₄×Dic₆₁	1	1	yes	characteristic
190	C₄	C₂³×#92418	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	4	no	—
193	C₂	C₂²×C₄×#92418	1	1	yes	characteristic
194	C₂	#21458922	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator	C₂×C₆₁	C₂²×C₄×C₈
Fitting	C₂×C₆₁×#405	C₂
socle	C₂²×C₆₁	C₂³×C₈
prime radical	C₂⁴×C₆₁	C₂×C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×#405	D₆₁
hypocenter	C₆₁	#33324
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #21447416	C₂	semidirect
3. C₄×#14519717	C₂	semidirect
5. C₂×#14519717	C₄	semidirect
9. #16839538	C₂²	semidirect
10. C₄×#11758185	C₂²	semidirect
19. #14519717	C₂×C₄	semidirect
20. C₂×#11758185	C₂×C₄	semidirect
115. C₆₁	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14519717	2
#21447416	2
C₄×#14519717	2
C₂×C₆₁×#405	2
#33324	61