Subgroups — C₂²×#16862252 #30680665

Order 16768.

1504 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523513	C₂	1	1	yes	maximal, characteristic
2	C₂×#16862252	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14523497	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#14523496	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×D₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14523513	C₄	1	12	yes	—
8	C₂×#14523513	C₂²	1	12	yes	—
9	C₂²×Dic₂₆₂	C₄	1	1	yes	characteristic
10	C₂²×D₅₂₄	C₄	1	1	yes	characteristic
11	C₂²×C₂²⋊D₁₃₁	C₄	1	2	yes	—
12	C₂²×C₄×D₁₃₁	C₂²	1	2	yes	—
13	#16862252	C₂²	1	64	yes	direct product factor
14	C₂×#14523497	C₂²	1	12	yes	complemented
15	C₂×#14523496	C₂²	1	24	yes	complemented
16	C₂²×C₁₃₁⋊C₈	C₂²	1	2	yes	—
17	C₂×C₈×D₁₃₁	C₂²	1	24	yes	complemented
18	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₁₃₁	C₂²	1	12	yes	complemented
22	C₂³×D₁₃₁	C₂×C₄	1	2	yes	—
23	#14523513	C₂×C₄	1	48	yes	—
24	#14523513	C₂³	1	16	yes	—
25	C₂×Dic₂₆₂	C₂×C₄	1	12	yes	—
26	C₂×D₅₂₄	C₂×C₄	1	12	yes	—
27	C₂×C₂²⋊D₁₃₁	C₂×C₄	1	24	yes	—
28	C₂²×Dic₁₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₁₃₁	C₂×C₄	1	12	yes	—
30	C₂×C₄×D₁₃₁	C₂³	1	12	yes	—
31	#14523497	C₂³	1	16	yes	complemented
32	#14523496	C₂³	1	32	yes	complemented
33	C₂×C₁₃₁⋊C₈	C₂³	1	12	yes	—
34	C₈×D₁₃₁	C₂³	1	32	yes	complemented
35	C₂⁴×C₁₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₁	C₂×C₄	1	6	yes	—
38	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₃₁	C₂³	1	6	yes	—
40	C₂×C₈×C₁₃₁	C₂³	1	6	yes	—
41	C₂×C₈×C₁₃₁	C₂³	1	6	yes	—
42	C₂×C₈×C₁₃₁	C₂³	1	16	yes	complemented
43	C₂²×D₁₃₁	—	2	8	no	—
44	C₂²×D₁₃₁	C₂²×C₄	1	12	yes	—
45	Dic₂₆₂	C₂²×C₄	1	16	yes	—
46	D₅₂₄	C₂²×C₄	1	16	yes	—
47	C₂²⋊D₁₃₁	C₂²×C₄	1	32	yes	—
48	C₂×Dic₁₃₁	C₂²×C₄	1	12	yes	—
49	C₄×D₁₃₁	C₂²×C₄	1	24	yes	—
50	C₄×D₁₃₁	C₂⁴	1	8	yes	—
51	C₁₃₁⋊C₈	C₂⁴	1	8	yes	—
52	C₂³×C₁₃₁	—	2	4	no	—
53	C₂³×C₁₃₁	C₂²×C₄	1	6	yes	—
54	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
56	C₂×C₄×C₁₃₁	C₂²×C₄	1	6	yes	—
57	C₂×C₄×C₁₃₁	C₂²×C₄	1	12	yes	—
58	C₂×C₄×C₁₃₁	C₂⁴	1	3	yes	—
59	C₂×C₄×C₁₃₁	C₂⁴	1	4	yes	—
60	C₈×C₁₃₁	C₂⁴	1	4	yes	—
61	C₈×C₁₃₁	C₂⁴	1	4	yes	—
62	C₂²×C₁₃₁	—	2	12	no	—
63	C₂²×C₁₃₁	#131	1	4	yes	complemented
64	C₂²×C₁₃₁	C₂³×C₄	1	4	yes	—
65	C₂²×C₁₃₁	C₂³×C₄	1	3	yes	—
66	C₂×D₁₃₁	—	2	24	no	—
67	C₂×D₁₃₁	C₂³×C₄	1	8	yes	—
68	C₄×C₁₃₁	C₂³×C₄	1	3	yes	—
69	C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
70	C₄×C₁₃₁	C₂⁵	1	1	yes	characteristic
71	Dic₁₃₁	C₂³×C₄	1	8	yes	—
72	C₂×C₁₃₁	—	2	4	no	—
73	C₂×C₁₃₁	C₂×#131	1	6	yes	complemented
74	C₂×C₁₃₁	C₂⁴×C₄	1	1	yes	characteristic
75	D₁₃₁	—	2	8	no	—
76	C₁₃₁	C₂²×#131	1	1	yes	characteristic, complemented, Sylow
77	C₂²×#131	—	131	1	no	maximal, Sylow
78	C₂²×#41	—	131	1	no	—
79	C₂×#131	—	131	24	no	—
80	C₂²×M₁₆	—	131	1	no	—
81	C₂²×M₁₆	—	131	2	no	—
82	C₂³×C₈	—	131	2	no	—
83	C₂³×C₈	D₁₃₁	1	1	yes	characteristic, complemented
84	C₂×#41	—	131	12	no	—
85	C₂×#41	—	131	12	no	—
86	C₂²×Q₈	—	131	1	no	—
87	C₂²×D₄	—	131	2	no	—
88	C₂²×D₄	—	131	1	no	—
89	C₂³×C₄	—	131	2	no	—
90	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
91	#131	—	131	64	no	—
92	C₂×M₁₆	—	131	12	no	—
93	C₂×M₁₆	—	131	24	no	—
94	C₂²×C₈	—	131	2	no	—
95	C₂²×C₈	—	131	24	no	—
96	C₂²×C₈	C₂×D₁₃₁	1	1	yes	characteristic
97	C₂²×C₈	C₂×D₁₃₁	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₁₃₁	1	12	yes	complemented
99	C₂⁴	—	131	2	no	—
100	C₂⁴	C₄×D₁₃₁	1	1	yes	characteristic
101	#41	—	131	48	no	—
102	#41	—	131	16	no	—
103	C₂×Q₈	—	131	12	no	—
104	C₂×D₄	—	131	24	no	—
105	C₂×D₄	—	131	12	no	—
106	C₂²×C₄	—	131	2	no	—
107	C₂²×C₄	—	131	12	no	—
108	C₂²×C₄	—	131	12	no	—
109	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
110	C₂²×C₄	C₄×D₁₃₁	1	6	yes	—
111	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
112	C₂²×C₄	C₂²×D₁₃₁	1	6	yes	—
113	M₁₆	—	131	16	no	—
114	M₁₆	—	131	32	no	—
115	C₂×C₈	—	131	12	no	—
116	C₂×C₈	—	131	32	no	—
117	C₂×C₈	C₂²×D₁₃₁	1	6	yes	—
118	C₂×C₈	C₂²×D₁₃₁	1	6	yes	—
119	C₂×C₈	C₂²×D₁₃₁	1	16	yes	complemented
120	C₂³	—	2	4	no	—
121	C₂³	—	262	8	no	—
122	C₂³	—	131	12	no	—
123	C₂³	C₂×C₄×D₁₃₁	1	6	yes	—
124	C₂³	C₂×C₄×D₁₃₁	1	1	yes	characteristic
125	Q₈	—	131	16	no	—
126	D₄	—	131	32	no	—
127	D₄	—	131	16	no	—
128	C₂×C₄	—	131	12	no	—
129	C₂×C₄	—	131	24	no	—
130	C₂×C₄	—	131	8	no	—
131	C₂×C₄	C₂×C₄×D₁₃₁	1	3	yes	—
132	C₂×C₄	C₂×C₄×D₁₃₁	1	6	yes	—
133	C₂×C₄	C₂×C₄×D₁₃₁	1	12	yes	—
134	C₂×C₄	C₂³×D₁₃₁	1	3	yes	—
135	C₂×C₄	C₂³×D₁₃₁	1	4	yes	—
136	C₈	—	131	8	no	—
137	C₈	C₂³×D₁₃₁	1	4	yes	—
138	C₈	C₂³×D₁₃₁	1	4	yes	—
139	C₂²	—	2	12	no	—
140	C₂²	—	262	24	no	—
141	C₂²	—	131	8	no	—
142	C₂²	#16862252	1	4	yes	direct product factor
143	C₂²	C₂²×C₄×D₁₃₁	1	4	yes	—
144	C₂²	C₂²×C₄×D₁₃₁	1	3	yes	—
145	C₄	—	131	8	no	—
146	C₄	C₂²×C₄×D₁₃₁	1	3	yes	—
147	C₄	C₂²×C₄×D₁₃₁	1	4	yes	—
148	C₄	C₂⁴×D₁₃₁	1	1	yes	characteristic
149	C₂	—	2	4	no	—
150	C₂	—	262	8	no	—
151	C₂	C₂×#16862252	1	6	yes	direct product factor
152	C₂	C₂³×C₄×D₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₃₁
commutator	C₂×C₁₃₁	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₁₃₁
Fitting	C₂³×C₈×C₁₃₁	C₂
socle	C₂³×C₁₃₁	C₂²×C₄
prime radical	C₂²×#14523513	C₂
cosocle	C₄×C₁₃₁	C₂⁵
hypercenter	C₂³×C₈	D₁₃₁
hypocenter	C₁₃₁	C₂²×#131
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16862252	C₂	direct (N × H)
3. C₂²×#14523497	C₂	semidirect
4. C₂²×#14523496	C₂	semidirect
5. C₂²×C₈×D₁₃₁	C₂	semidirect
6. C₂³×C₈×C₁₃₁	C₂	semidirect
13. #16862252	C₂²	direct (N × H)
14. C₂×#14523497	C₂²	semidirect
15. C₂×#14523496	C₂²	semidirect
17. C₂×C₈×D₁₃₁	C₂²	semidirect
21. C₂²×C₈×C₁₃₁	C₂²	semidirect
31. #14523497	C₂³	semidirect
32. #14523496	C₂³	semidirect
34. C₈×D₁₃₁	C₂³	semidirect
42. C₂×C₈×C₁₃₁	C₂³	semidirect
63. C₂²×C₁₃₁	#131	semidirect
73. C₂×C₁₃₁	C₂×#131	semidirect
76. C₁₃₁	C₂²×#131	semidirect
83. C₂³×C₈	D₁₃₁	semidirect
98. C₂²×C₈	C₂×D₁₃₁	semidirect
119. C₂×C₈	C₂²×D₁₃₁	semidirect
142. C₂²	#16862252	direct (N × H)
151. C₂	C₂×#16862252	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523513	2
C₂×#16862252	2
C₂²×#14523497	2
C₂²×#14523496	2
C₂²×C₈×D₁₃₁	2
C₂³×C₈×C₁₃₁	2
C₂²×#131	131

Subgroups — C22×#16862252 #30680665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16862252 #30680665