Subgroups — C₂²×#16839589 #29833036

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11758185	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14519711	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839589	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#107×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₆₁	—	2	1	no	—
6	C₂²×#11758185	C₄	1	8	yes	complemented
7	C₂²×#11758185	C₂²	1	6	yes	—
8	C₂×#14519711	—	2	3	no	—
9	C₂×#14519711	C₂²	1	6	yes	—
10	#16839589	C₂²	1	16	yes	direct product factor
11	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
13	C₂×#107×C₆₁	—	2	3	no	—
14	C₂×#107×C₆₁	C₂²	1	6	yes	—
15	C₂³×Dic₆₁	—	4	1	no	—
16	C₂×C₄⋊Dic₆₁	—	2	6	no	—
17	C₂×C₄⋊Dic₆₁	—	4	3	no	—
18	C₂×#11758185	C₂×C₄	1	24	yes	complemented
19	C₂×#11758185	C₂³	1	4	yes	—
20	#14519711	—	2	6	no	—
21	#14519711	C₂³	1	4	yes	—
22	C₂³×C₄×C₆₁	—	4	1	no	—
23	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₆₁	D₄	1	3	yes	—
26	C₂²×C₈×C₆₁	Q₈	1	4	yes	—
27	C₂²×C₈×C₆₁	C₂³	1	3	yes	—
28	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
29	C₂×C₄⋊C₄×C₆₁	—	4	3	no	—
30	#107×C₆₁	—	2	6	no	—
31	#107×C₆₁	C₂³	1	4	yes	—
32	C₂⁴×C₆₁	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	—	4	6	no	—
34	C₂²×C₄×C₆₁	—	8	4	no	—
35	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
36	C₂²×C₄×C₆₁	D₈	1	3	yes	—
37	C₂²×C₄×C₆₁	Q₁₆	1	4	yes	—
38	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
39	C₂²×Dic₆₁	—	4	6	no	—
40	C₂²×Dic₆₁	—	8	4	no	—
41	C₂×C₈×C₆₁	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₆₁	C₂²×C₄	1	6	yes	—
43	C₂×C₈×C₆₁	C₂×D₄	1	3	yes	—
44	C₂×C₈×C₆₁	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
46	C₄⋊C₄×C₆₁	—	2	4	no	—
47	C₄⋊C₄×C₆₁	—	4	6	no	—
48	C₄⋊Dic₆₁	—	2	4	no	—
49	C₄⋊Dic₆₁	—	4	6	no	—
50	#11758185	C₂²×C₄	1	8	yes	complemented
51	C₂³×C₆₁	#107	1	4	yes	—
52	C₂³×C₆₁	SD₃₂	1	4	yes	—
53	C₂³×C₆₁	SD₃₂	1	4	yes	—
54	C₂³×C₆₁	C₂×D₈	1	3	yes	—
55	C₂×C₄×C₆₁	—	4	4	no	—
56	C₂×C₄×C₆₁	—	8	12	no	—
57	C₂×C₄×C₆₁	#107	1	12	yes	—
58	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
59	C₂×C₄×C₆₁	C₂×D₈	1	3	yes	—
60	C₂×C₄×C₆₁	C₂×Q₁₆	1	6	yes	—
61	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
62	C₂×Dic₆₁	—	4	4	no	—
63	C₂×Dic₆₁	—	8	12	no	—
64	C₈×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
65	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
66	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
67	C₂²×#367	—	61	1	no	maximal, Sylow
68	C₂²×C₆₁	#367	1	16	yes	complemented
69	C₂²×C₆₁	C₂×#107	1	6	yes	—
70	C₂²×C₆₁	C₂×SD₃₂	1	6	yes	—
71	C₂²×C₆₁	C₂×SD₃₂	1	6	yes	—
72	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
73	C₄×C₆₁	—	8	4	no	—
74	C₄×C₆₁	C₂×#107	1	6	yes	—
75	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
77	Dic₆₁	—	8	4	no	—
78	C₂³×C₁₆	—	61	1	no	—
79	C₂²×#107	—	61	1	no	—
80	C₂²×#107	D₆₁	1	1	yes	characteristic
81	C₂×#367	—	61	12	no	—
82	C₂×C₆₁	C₂×#367	1	12	yes	complemented
83	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
84	C₂×C₆₁	C₂²×SD₃₂	1	1	yes	characteristic
85	C₂×C₆₁	C₂²×SD₃₂	1	1	yes	characteristic
86	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
87	C₂²×C₄⋊C₄	—	2	1	no	—
88	C₂²×C₄⋊C₄	—	122	1	no	—
89	C₂²×C₁₆	—	61	8	no	—
90	C₂²×C₁₆	—	61	6	no	—
91	C₂×#107	—	2	3	no	—
92	C₂×#107	—	61	6	no	—
93	C₂×#107	—	122	3	no	—
94	C₂×#107	C₂×D₆₁	1	6	yes	—
95	#367	—	61	16	no	—
96	C₆₁	C₂²×#367	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	—	4	1	no	—
98	C₂³×C₄	—	244	1	no	—
99	C₂³×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
100	C₂²×C₈	Dic₁₂₂	1	4	yes	—
101	C₂²×C₈	C₄×D₆₁	1	4	yes	—
102	C₂²×C₈	C₂²⋊D₆₁	1	3	yes	—
103	C₂²×C₈	C₂²×D₆₁	1	3	yes	—
104	C₂×C₄⋊C₄	—	2	6	no	—
105	C₂×C₄⋊C₄	—	4	3	no	—
106	C₂×C₄⋊C₄	—	122	6	no	—
107	C₂×C₄⋊C₄	—	244	3	no	—
108	C₂×C₁₆	—	61	24	no	—
109	C₂×C₁₆	—	61	4	no	—
110	#107	—	2	6	no	—
111	#107	—	61	4	no	—
112	#107	—	122	6	no	—
113	#107	C₂²×D₆₁	1	4	yes	—
114	C₂⁴	#11758199	1	1	yes	characteristic
115	C₂²×C₄	—	4	6	no	—
116	C₂²×C₄	—	8	4	no	—
117	C₂²×C₄	—	488	4	no	—
118	C₂²×C₄	—	244	6	no	—
119	C₂²×C₄	#11758196	1	4	yes	—
120	C₂²×C₄	#11758199	1	3	yes	—
121	C₂²×C₄	#11758202	1	4	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
123	C₂×C₈	#11758196	1	12	yes	—
124	C₂×C₈	C₂×Dic₁₂₂	1	6	yes	—
125	C₂×C₈	C₂×C₄×D₆₁	1	6	yes	—
126	C₂×C₈	C₂×C₂²⋊D₆₁	1	3	yes	—
127	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	4	6	no	—
130	C₄⋊C₄	—	122	4	no	—
131	C₄⋊C₄	—	244	6	no	—
132	C₁₆	—	61	8	no	—
133	C₂³	#14519710	1	4	yes	—
134	C₂³	#14519720	1	4	yes	—
135	C₂³	#14519721	1	4	yes	—
136	C₂³	C₂×#11758199	1	3	yes	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	8	12	no	—
139	C₂×C₄	—	488	12	no	—
140	C₂×C₄	—	244	4	no	—
141	C₂×C₄	#14519710	1	12	yes	—
142	C₂×C₄	C₂×#11758196	1	6	yes	—
143	C₂×C₄	C₂×#11758199	1	3	yes	—
144	C₂×C₄	C₂×#11758202	1	6	yes	—
145	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
146	C₈	C₂×#11758196	1	6	yes	—
147	C₈	C₂²×C₄×D₆₁	1	1	yes	characteristic
148	C₈	C₂²×Dic₁₂₂	1	1	yes	characteristic
149	C₂²	#16839589	1	16	yes	direct product factor
150	C₂²	C₂×#14519710	1	6	yes	—
151	C₂²	C₂×#14519720	1	6	yes	—
152	C₂²	C₂×#14519721	1	6	yes	—
153	C₂²	C₂²×#11758199	1	1	yes	characteristic
154	C₄	—	8	4	no	—
155	C₄	—	488	4	no	—
156	C₄	C₂×#14519710	1	6	yes	—
157	C₄	C₂²×#11758196	1	1	yes	characteristic
158	C₄	C₂²×#11758202	1	1	yes	characteristic
159	C₂	C₂×#16839589	1	12	yes	direct product factor
160	C₂	C₂²×#14519710	1	1	yes	characteristic
161	C₂	C₂²×#14519720	1	1	yes	characteristic
162	C₂	C₂²×#14519721	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11758199
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂²×#107×C₆₁	C₂
socle, prime radical	C₂⁴×C₆₁	D₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂²×#107	D₆₁
hypocenter	C₆₁	C₂²×#367
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. C₂×#16839589	C₂	direct (N × H)
6. C₂²×#11758185	C₄	semidirect
10. #16839589	C₂²	direct (N × H)
18. C₂×#11758185	C₂×C₄	semidirect
50. #11758185	C₂²×C₄	semidirect
68. C₂²×C₆₁	#367	semidirect
82. C₂×C₆₁	C₂×#367	semidirect
96. C₆₁	C₂²×#367	semidirect
149. C₂²	#16839589	direct (N × H)
159. C₂	C₂×#16839589	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11758185	2
C₂²×#14519711	2
C₂×#16839589	2
C₂²×#107×C₆₁	2
C₂²×#367	61

Subgroups — C22×#16839589 #29833036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16839589 #29833036