Subgroups — #29596439

Order 15616.

334 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839585	C₂	1	4	yes	maximal
2	#21445991	C₂	1	1	yes	maximal, characteristic
3	#21445976	C₂	1	1	yes	maximal, characteristic
4	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519713	—	2	1	no	—
6	C₂×#14519713	C₂²	1	2	yes	—
7	C₂×#14519709	—	2	1	no	—
8	C₂×#14519709	C₂²	1	2	yes	—
9	#16839786	—	2	1	no	—
10	#16839632	—	2	1	no	—
11	#16839585	—	2	8	no	—
12	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#346	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₆₁	C₄	1	2	yes	complemented
16	C₂×#11758194	—	2	2	no	—
17	C₂×C₄⋊Dic₆₁	—	2	2	no	—
18	C₂×C₄⋊Dic₆₁	—	4	2	no	—
19	#14519713	—	2	4	no	—
20	#14519713	—	4	2	no	—
21	#14519709	—	2	4	no	—
22	#14519709	—	4	2	no	—
23	C₄⋊#92418	—	4	2	no	—
24	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
27	M₃₂×C₆₁	D₄	1	4	yes	—
28	M₃₂×C₆₁	Q₈	1	4	yes	—
29	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
35	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₆₁	—	4	2	no	—
37	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
40	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
41	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₆₁	SD₁₆	1	2	yes	—
48	C₄⋊Dic₆₁	—	4	2	no	—
49	C₄⋊Dic₆₁	—	4	2	no	—
50	C₄⋊Dic₆₁	—	8	4	no	—
51	#11758194	—	4	2	no	—
52	#11758194	—	4	2	no	—
53	C₂×#92418	—	4	2	no	—
54	C₁₆×C₆₁	C₄⋊C₄	1	8	yes	complemented
55	C₂³×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₆₁	—	2	4	no	—
57	C₂×C₄×C₆₁	#106	1	2	yes	—
58	C₂×C₄×C₆₁	#106	1	2	yes	—
59	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×Dic₆₁	—	8	4	no	—
66	C₂×Dic₆₁	—	8	2	no	—
67	C₈×C₆₁	#106	1	2	yes	—
68	C₈×C₆₁	#106	1	2	yes	—
69	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
70	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
71	#92418	—	8	2	no	—
72	#12903	—	61	1	no	maximal, Sylow
73	C₂²×C₆₁	—	2	2	no	—
74	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
75	C₂²×C₆₁	#492	1	1	yes	characteristic
76	C₂²×C₆₁	#498	1	1	yes	characteristic
77	C₄×C₆₁	—	2	2	no	—
78	C₄×C₆₁	#365	1	4	yes	complemented
79	C₄×C₆₁	#384	1	1	yes	characteristic
80	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
81	C₄×C₆₁	#492	1	1	yes	characteristic
82	C₄×C₆₁	#498	1	1	yes	characteristic
83	Dic₆₁	—	16	2	no	—
84	C₂×#365	—	61	4	no	—
85	C₄×M₃₂	D₆₁	1	1	yes	characteristic
86	#1848	—	61	1	no	—
87	#1844	—	61	1	no	—
88	C₂×C₆₁	—	2	1	no	—
89	C₂×C₆₁	—	2	1	no	—
90	C₂×C₆₁	#1844	1	1	yes	characteristic
91	C₂×C₆₁	C₂×#365	1	2	yes	—
92	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
93	C₂×#108	—	61	2	no	—
94	C₂×#108	—	122	1	no	—
95	C₂×#106	—	61	2	no	—
96	C₂×#106	—	122	1	no	—
97	#423	—	122	1	no	—
98	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
99	#384	—	122	1	no	—
100	#365	—	122	8	no	—
101	#346	Dic₆₁	1	2	yes	complemented
102	C₄×C₁₆	Dic₆₁	1	2	yes	complemented
103	C₆₁	#12903	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₁₆	—	122	2	no	—
105	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
106	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	122	2	no	—
108	C₂×C₄⋊C₄	—	244	2	no	—
109	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
110	M₃₂	Dic₁₂₂	1	4	yes	—
111	M₃₂	D₂₄₄	1	4	yes	—
112	C₂×C₁₆	C₂×Dic₆₁	1	4	yes	—
113	#108	—	122	4	no	—
114	#108	—	244	2	no	—
115	#106	—	122	4	no	—
116	#106	—	244	2	no	—
117	C₄⋊C₈	—	244	2	no	—
118	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
119	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
121	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
122	C₂²×C₄	—	244	2	no	—
123	C₂²×C₄	#11758190	1	2	yes	—
124	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
125	M₁₆	—	244	2	no	—
126	M₁₆	—	244	2	no	—
127	C₂×C₈	—	244	2	no	—
128	C₂×C₈	#11758190	1	2	yes	—
129	C₂×C₈	#11758190	1	2	yes	—
130	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
131	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
132	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
133	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
134	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
135	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
136	C₄⋊C₄	—	488	4	no	—
137	C₄⋊C₄	—	244	2	no	—
138	C₄⋊C₄	—	244	2	no	—
139	C₄²	#11758190	1	2	yes	—
140	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
141	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
142	C₁₆	C₄⋊Dic₆₁	1	8	yes	complemented
143	C₂³	C₂×#11758190	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	488	4	no	—
146	C₂×C₄	—	488	2	no	—
147	C₂×C₄	#14519709	1	2	yes	—
148	C₂×C₄	#14519709	1	2	yes	—
149	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
150	C₂×C₄	#14519774	1	1	yes	characteristic
151	C₂×C₄	#14519777	1	1	yes	characteristic
152	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
153	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
154	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
155	C₈	—	488	2	no	—
156	C₈	#14519709	1	2	yes	—
157	C₈	#14519709	1	2	yes	—
158	C₈	C₂×C₄⋊Dic₆₁	1	2	yes	—
159	C₈	#14519777	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	C₂×#14519709	1	1	yes	characteristic
162	C₂²	#16840135	1	1	yes	characteristic
163	C₂²	#16840160	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	976	2	no	—
166	C₄	#16839585	1	4	yes	complemented
167	C₄	#16839632	1	1	yes	characteristic
168	C₄	C₂×#14519709	1	1	yes	characteristic
169	C₄	#16840135	1	1	yes	characteristic
170	C₄	#16840160	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#21445976	1	1	yes	characteristic
174	C₂	C₂×#16839585	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14519709
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₄×M₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂×#106
prime radical	C₂³×C₆₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₄×M₃₂	D₆₁
hypocenter	C₆₁	#12903
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
14. C₆₁×#346	C₄	semidirect
15. C₄×C₁₆×C₆₁	C₄	semidirect
54. C₁₆×C₆₁	C₄⋊C₄	semidirect
78. C₄×C₆₁	#365	semidirect
101. #346	Dic₆₁	semidirect
102. C₄×C₁₆	Dic₆₁	semidirect
103. C₆₁	#12903	semidirect
142. C₁₆	C₄⋊Dic₆₁	semidirect
166. C₄	#16839585	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839585	2
#21445991	2
#21445976	2
C₄×M₃₂×C₆₁	2
#12903	61