Subgroups — #29596469

Order 15616.

266 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839592	C₂	1	1	yes	maximal, characteristic
2	#21447603	C₂	1	1	yes	maximal, characteristic
3	#21447592	C₂	1	2	yes	maximal
4	#21445943	C₂	1	2	yes	maximal
5	C₆₁×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519713	C₂²	1	2	yes	—
7	#16839800	C₂²	1	2	yes	—
8	#16839592	—	2	1	no	—
9	#16839592	—	2	1	no	—
10	#16839590	—	2	1	no	—
11	#16839588	—	2	1	no	—
12	#16839585	—	2	4	no	—
13	#16839533	—	2	2	no	—
14	#16839518	—	2	2	no	—
15	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#405	C₂²	1	1	yes	characteristic
18	C₆₁×#346	C₄	1	2	yes	complemented
19	C₄×C₁₆×C₆₁	C₄	1	2	yes	complemented
20	C₂×#11758194	—	2	2	no	—
21	#14519736	—	2	2	no	—
22	#14519713	—	2	2	no	—
23	#14519713	—	2	2	no	—
24	#14519711	—	2	2	no	—
25	#14519709	—	2	2	no	—
26	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
28	#117×C₆₁	D₄	1	1	yes	characteristic
29	M₃₂×C₆₁	D₄	1	2	yes	—
30	M₃₂×C₆₁	Q₈	1	2	yes	—
31	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₆₁	Q₈	1	1	yes	characteristic
37	C₂×C₁₆×C₆₁	Q₈	1	1	yes	characteristic
38	#97×C₆₁	Q₈	1	2	yes	—
39	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₆₁	—	2	1	no	—
42	M₁₆×C₆₁	—	2	1	no	—
43	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
50	C₄⋊C₄×C₆₁	—	2	1	no	—
51	C₂²⋊C₄×C₆₁	—	2	1	no	—
52	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
53	C₂²⋊Dic₆₁	—	4	2	no	—
54	C₄⋊Dic₆₁	—	4	2	no	—
55	C₄×Dic₆₁	—	4	2	no	—
56	#11758194	—	4	2	no	—
57	#11758194	—	4	2	no	—
58	C₂×#92418	—	4	2	no	—
59	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
61	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
62	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
63	C₁₆×C₆₁	C₄⋊C₄	1	4	yes	—
64	C₂³×C₆₁	#135	1	1	yes	characteristic
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
70	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
72	C₂×Dic₆₁	—	8	2	no	—
73	C₈×C₆₁	—	2	2	no	—
74	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
77	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
78	#92418	—	8	2	no	—
79	#12911	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	—	4	1	no	—
81	C₂²×C₆₁	#427	1	1	yes	characteristic
82	C₂²×C₆₁	#495	1	1	yes	characteristic
83	C₂²×C₆₁	#499	1	1	yes	characteristic
84	C₄×C₆₁	—	4	2	no	—
85	C₄×C₆₁	#384	1	1	yes	characteristic
86	C₄×C₆₁	#427	1	1	yes	characteristic
87	C₄×C₆₁	#495	1	1	yes	characteristic
88	C₄×C₆₁	#499	1	1	yes	characteristic
89	Dic₆₁	—	16	2	no	—
90	C₂×#368	—	61	1	no	—
91	#2158	—	61	1	no	—
92	#2155	—	61	2	no	—
93	#2108	D₆₁	1	1	yes	characteristic
94	#1833	—	61	2	no	—
95	C₂×C₆₁	—	2	1	no	—
96	C₂×C₆₁	—	2	1	no	—
97	C₂×C₆₁	—	2	1	no	—
98	C₂×C₆₁	#1846	1	1	yes	characteristic
99	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
100	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
101	C₂×#108	—	61	2	no	—
102	#427	—	61	2	no	—
103	#405	C₂×D₆₁	1	1	yes	characteristic
104	#368	—	122	1	no	—
105	#368	—	122	1	no	—
106	#367	—	122	1	no	—
107	#366	—	122	1	no	—
108	#365	—	122	4	no	—
109	#346	Dic₆₁	1	2	yes	complemented
110	C₄×C₁₆	Dic₆₁	1	2	yes	complemented
111	#344	—	122	2	no	—
112	H(3,ℤ/4)	—	122	2	no	—
113	C₆₁	#12911	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	122	2	no	—
115	C₂×M₁₆	D₂₄₄	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
117	#117	—	122	2	no	—
118	#117	D₂₄₄	1	1	yes	characteristic
119	M₃₂	Dic₁₂₂	1	2	yes	—
120	M₃₂	D₂₄₄	1	2	yes	—
121	C₂×C₁₆	Dic₁₂₂	1	1	yes	characteristic
122	C₂×C₁₆	Dic₁₂₂	1	1	yes	characteristic
123	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
124	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
125	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
126	C₂×C₁₆	C₂×Dic₆₁	1	2	yes	—
127	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
128	#108	—	122	2	no	—
129	#108	—	122	2	no	—
130	#107	—	122	2	no	—
131	#106	—	122	2	no	—
132	#97	Dic₁₂₂	1	2	yes	—
133	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
134	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
135	M₁₆	—	2	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	244	2	no	—
138	M₁₆	—	244	2	no	—
139	C₂×C₈	—	244	2	no	—
140	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
141	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
144	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
145	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
146	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	244	2	no	—
149	C₂²⋊C₄	—	2	1	no	—
150	C₂²⋊C₄	—	244	2	no	—
151	C₄²	—	244	2	no	—
152	C₄²	C₄⋊Dic₆₁	1	2	yes	—
153	C₁₆	C₄⋊Dic₆₁	1	1	yes	characteristic
154	C₁₆	C₄⋊Dic₆₁	1	1	yes	characteristic
155	C₁₆	C₄⋊Dic₆₁	1	1	yes	characteristic
156	C₁₆	C₄⋊Dic₆₁	1	1	yes	characteristic
157	C₁₆	C₄⋊Dic₆₁	1	4	yes	—
158	C₂³	#14519799	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	488	2	no	—
161	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
162	C₂×C₄	#14519774	1	1	yes	characteristic
163	C₂×C₄	#14519777	1	1	yes	characteristic
164	C₂×C₄	#14519799	1	1	yes	characteristic
165	C₂×C₄	#14519799	1	1	yes	characteristic
166	C₂×C₄	#14519799	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	488	2	no	—
169	C₈	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
170	C₈	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
171	C₈	#14519777	1	1	yes	characteristic
172	C₈	#14519777	1	1	yes	characteristic
173	C₂²	—	4	1	no	—
174	C₂²	#16839800	1	1	yes	characteristic
175	C₂²	#16840146	1	1	yes	characteristic
176	C₂²	#16840164	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	976	2	no	—
179	C₄	#16839632	1	1	yes	characteristic
180	C₄	#16839800	1	1	yes	characteristic
181	C₄	#16840146	1	1	yes	characteristic
182	C₄	#16840164	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#21445983	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16839800
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2108	C₂
socle	C₂×C₆₁	#1846
prime radical	C₂³×C₆₁	#135
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#2108	D₆₁
hypocenter	C₆₁	#12911
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
18. C₆₁×#346	C₄	semidirect
19. C₄×C₁₆×C₆₁	C₄	semidirect
109. #346	Dic₆₁	semidirect
110. C₄×C₁₆	Dic₆₁	semidirect
113. C₆₁	#12911	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839592	2
#21447603	2
#21447592	2
#21445943	2
C₆₁×#2108	2
#12911	61