Subgroups — #29833015

Order 15616.

684 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458922	C₂	1	1	yes	maximal, characteristic
2	#131×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#21447584	C₂	1	6	yes	maximal, complemented
4	C₂×#16839582	C₂	1	6	yes	maximal, complemented
5	C₆₁×#3407	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519784	C₄	1	1	yes	characteristic
7	#41×Dic₆₁	C₄	1	1	yes	characteristic
8	#16840179	—	2	4	no	—
9	C₂×#14519717	C₂²	1	6	yes	—
10	#16839720	C₄	1	3	yes	—
11	#16839717	C₂²	1	3	yes	—
12	M₁₆×Dic₆₁	C₄	1	3	yes	—
13	C₂×C₈×Dic₆₁	C₂²	1	3	yes	—
14	#16839582	C₂²	1	4	yes	—
15	#16839582	C₂²	1	12	yes	—
16	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#504	—	2	4	no	—
18	C₂×M₃₂×C₆₁	C₂²	1	6	yes	—
19	#14519784	C₈	1	8	yes	—
20	C₂×#11758194	C₂×C₄	1	3	yes	—
21	C₂²×#92418	C₂×C₄	1	3	yes	—
22	Q₈×Dic₆₁	—	2	1	no	—
23	D₄×Dic₆₁	—	2	3	no	—
24	#14519736	C₂×C₄	1	3	yes	—
25	C₂×C₄×Dic₆₁	C₂×C₄	1	3	yes	—
26	#14519717	—	2	6	no	—
27	#14519717	C₂³	1	4	yes	—
28	C₂×#11758185	—	2	6	no	—
29	#14519687	C₂×C₄	1	6	yes	—
30	C₈×Dic₆₁	C₂×C₄	1	3	yes	—
31	C₈×Dic₆₁	C₂×C₄	1	3	yes	—
32	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
33	C₈×Dic₆₁	C₂³	1	3	yes	—
34	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
35	#131×C₆₁	D₄	1	4	yes	—
36	#131×C₆₁	Q₈	1	4	yes	—
37	C₂×M₁₆×C₆₁	C₂×C₄	1	3	yes	—
38	C₂²×C₈×C₆₁	C₂³	1	3	yes	—
39	M₃₂×C₆₁	—	2	6	no	—
40	M₃₂×C₆₁	C₂³	1	4	yes	—
41	C₂×C₁₆×C₆₁	—	2	6	no	—
42	#41×C₆₁	C₄⋊C₄	1	4	yes	—
43	#41×C₆₁	C₂×C₈	1	4	yes	—
44	C₂×Q₈×C₆₁	M₁₆	1	1	yes	characteristic
45	C₂×D₄×C₆₁	M₁₆	1	3	yes	—
46	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
47	C₂²×Dic₆₁	—	2	3	no	—
48	M₁₆×C₆₁	C₄⋊C₄	1	12	yes	—
49	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₆₁	C₂×D₄	1	6	yes	—
51	C₂×C₈×C₆₁	C₂×Q₈	1	6	yes	—
52	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
53	C₂²⋊Dic₆₁	—	2	3	no	—
54	C₄⋊Dic₆₁	—	2	3	no	—
55	C₄×Dic₆₁	—	2	3	no	—
56	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₆₁	C₂²×C₄	1	3	yes	—
58	#11758194	C₂×C₈	1	12	yes	—
59	C₂×#92418	C₂×C₈	1	12	yes	—
60	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
61	C₂×#92418	C₂²×C₄	1	3	yes	—
62	C₁₆×C₆₁	—	2	4	no	—
63	#11758185	—	2	4	no	—
64	C₂³×C₆₁	C₂×M₁₆	1	3	yes	—
65	Q₈×C₆₁	C₄⋊C₈	1	4	yes	—
66	D₄×C₆₁	C₄⋊C₈	1	12	yes	—
67	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
68	C₂×C₄×C₆₁	C₂²×C₈	1	6	yes	—
69	C₂×C₄×C₆₁	C₂×M₁₆	1	3	yes	—
70	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
71	C₂×Dic₆₁	—	2	1	no	—
72	C₂×Dic₆₁	—	2	3	no	—
73	C₂×Dic₆₁	—	4	6	no	—
74	C₈×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
75	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
76	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
77	#92418	C₂²×C₈	1	2	yes	—
78	#92418	C₂²×C₈	1	6	yes	—
79	#33401	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	—	4	3	no	—
81	C₂²×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
82	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
83	C₄×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
84	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
86	Dic₆₁	—	2	2	no	—
87	Dic₆₁	—	4	3	no	—
88	#3407	—	61	1	no	—
89	#3407	D₆₁	1	1	yes	characteristic
90	C₄×#131	—	61	1	no	—
91	#2153	—	61	6	no	—
92	C₂×#364	—	61	6	no	—
93	C₂×C₆₁	—	2	1	no	—
94	C₂×C₆₁	—	2	6	no	—
95	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
96	C₂×#131	—	61	1	no	—
97	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
98	C₄×#41	—	61	1	no	—
99	#504	—	2	4	no	—
100	#504	—	122	4	no	—
101	C₂×M₃₂	—	61	6	no	—
102	C₂×M₃₂	C₂×D₆₁	1	6	yes	—
103	#405	—	61	3	no	—
104	#405	—	61	3	no	—
105	C₄×M₁₆	—	61	3	no	—
106	C₂×C₄×C₈	—	61	3	no	—
107	#364	—	61	4	no	—
108	#364	—	61	12	no	—
109	C₆₁	#33401	1	1	yes	characteristic, complemented, Sylow
110	C₂×#41	C₄×D₆₁	1	1	yes	characteristic
111	#131	—	61	8	no	—
112	#131	Dic₁₂₂	1	4	yes	—
113	#131	C₂²⋊D₆₁	1	4	yes	—
114	C₂×M₁₆	—	61	3	no	—
115	C₂×M₁₆	C₄×D₆₁	1	3	yes	—
116	C₂²×C₈	—	61	3	no	—
117	C₂²×C₈	C₂²×D₆₁	1	3	yes	—
118	C₄×Q₈	—	122	1	no	—
119	C₄×D₄	—	122	3	no	—
120	#117	—	61	3	no	—
121	C₂×C₄²	—	61	3	no	—
122	M₃₂	—	2	6	no	—
123	M₃₂	—	61	4	no	—
124	M₃₂	—	122	6	no	—
125	M₃₂	C₂²×D₆₁	1	4	yes	—
126	C₂×C₁₆	—	2	6	no	—
127	C₂×C₁₆	—	122	6	no	—
128	#97	—	61	6	no	—
129	C₄×C₈	—	61	3	no	—
130	C₄×C₈	—	61	3	no	—
131	C₄×C₈	—	61	1	no	—
132	C₄×C₈	—	61	3	no	—
133	#41	C₈×D₆₁	1	4	yes	—
134	#41	#11758196	1	4	yes	—
135	C₂×Q₈	#11758189	1	1	yes	characteristic
136	C₂×D₄	#11758189	1	3	yes	—
137	C₂²×C₄	—	122	3	no	—
138	C₂²×C₄	C₂×C₄×D₆₁	1	3	yes	—
139	M₁₆	—	61	12	no	—
140	M₁₆	#11758196	1	12	yes	—
141	C₂×C₈	—	61	1	no	—
142	C₂×C₈	—	61	3	no	—
143	C₂×C₈	—	61	12	no	—
144	C₂×C₈	C₂×Dic₁₂₂	1	6	yes	—
145	C₂×C₈	C₂×C₄×D₆₁	1	3	yes	—
146	C₂×C₈	C₂×C₂²⋊D₆₁	1	6	yes	—
147	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
148	C₄⋊C₄	—	122	3	no	—
149	C₂²⋊C₄	—	122	3	no	—
150	C₄²	—	61	1	no	—
151	C₄²	—	61	3	no	—
152	C₄²	—	122	3	no	—
153	C₁₆	—	2	4	no	—
154	C₁₆	—	122	4	no	—
155	C₂³	C₂×#11758189	1	3	yes	—
156	Q₈	#14519706	1	4	yes	—
157	D₄	#14519706	1	12	yes	—
158	C₂×C₄	—	122	1	no	—
159	C₂×C₄	—	122	3	no	—
160	C₂×C₄	—	244	6	no	—
161	C₂×C₄	C₂×#11758196	1	6	yes	—
162	C₂×C₄	C₂×C₈×D₆₁	1	6	yes	—
163	C₂×C₄	C₂×#11758189	1	3	yes	—
164	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
165	C₈	—	61	2	no	—
166	C₈	—	61	6	no	—
167	C₈	C₂×#11758196	1	6	yes	—
168	C₈	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
169	C₈	C₂²×Dic₁₂₂	1	1	yes	characteristic
170	C₂²	—	4	3	no	—
171	C₂²	C₂×#14519706	1	6	yes	—
172	C₂²	C₂²×#11758189	1	1	yes	characteristic
173	C₄	—	122	2	no	—
174	C₄	—	244	3	no	—
175	C₄	C₂×#14519706	1	6	yes	—
176	C₄	C₂²×#11758196	1	1	yes	characteristic
177	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	6	no	—
180	C₂	C₂²×#14519706	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#3407	C₂
socle	C₂×C₆₁	C₂²×C₄⋊C₈
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#3407	D₆₁
hypocenter	C₆₁	#33401
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ #3407

Complements

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

N	H ≅ G/N	type
3. #21447584	C₂	semidirect
4. C₂×#16839582	C₂	semidirect
109. C₆₁	#33401	semidirect

Maximal subgroups

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

subgroup	index
#21458922	2
#131×Dic₆₁	2
#21447584	2
C₂×#16839582	2
C₆₁×#3407	2
#33401	61