Subgroups — C₂×#22550643 #30653440

Order 16256.

1460 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550643	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16861216	C₂	1	8	yes	maximal, complemented
3	C₂²×#14523232	C₂	1	6	yes	maximal, complemented
4	C₂²×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#16861216	C₂²	1	64	yes	complemented
6	C₂×#14523232	C₂²	1	4	yes	—
7	C₂×#14523232	C₂²	1	12	yes	complemented
8	C₂×#14523232	C₂²	1	48	yes	complemented
9	C₂²×C₁₂₇⋊C₈	C₂²	1	12	yes	complemented
10	C₂×#41×C₁₂₇	C₄	1	4	yes	—
11	C₂×#41×C₁₂₇	C₄	1	8	yes	—
12	C₂×#41×C₁₂₇	C₂²	1	4	yes	—
13	C₂×#41×C₁₂₇	C₂²	1	8	yes	—
14	C₂²×Q₈×C₁₂₇	C₄	1	1	yes	characteristic
15	C₂²×D₄×C₁₂₇	C₄	1	3	yes	—
16	C₂³×C₄×C₁₂₇	C₂²	1	3	yes	—
17	#14523232	C₂³	1	16	yes	—
18	#14523232	C₂³	1	48	yes	complemented
19	C₂×C₁₂₇⋊C₈	C₂³	1	8	yes	—
20	C₂×C₁₂₇⋊C₈	C₂³	1	48	yes	complemented
21	C₂⁴×C₁₂₇	C₂×C₄	1	3	yes	—
22	#41×C₁₂₇	C₂×C₄	1	16	yes	—
23	#41×C₁₂₇	C₂×C₄	1	32	yes	—
24	#41×C₁₂₇	C₂³	1	16	yes	—
25	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
26	C₂×Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
27	C₂×D₄×C₁₂₇	C₂×C₄	1	12	yes	—
28	C₂×D₄×C₁₂₇	C₂×C₄	1	24	yes	—
29	C₂²×C₄×C₁₂₇	C₂×C₄	1	3	yes	—
30	C₂²×C₄×C₁₂₇	C₂×C₄	1	6	yes	—
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	12	yes	—
32	C₂²×C₄×C₁₂₇	C₂³	1	6	yes	—
33	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₂₇	C₂³	1	12	yes	—
35	C₁₂₇⋊C₈	C₂⁴	1	16	yes	—
36	C₂³×C₁₂₇	—	4	6	no	—
37	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
38	C₂³×C₁₂₇	C₂²×C₄	1	6	yes	—
39	C₂³×C₁₂₇	C₂²×C₄	1	12	yes	—
40	Q₈×C₁₂₇	C₂²×C₄	1	16	yes	—
41	D₄×C₁₂₇	C₂²×C₄	1	48	yes	—
42	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
45	C₂×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
46	C₂×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
47	C₂×C₄×C₁₂₇	C₂²×C₄	1	24	yes	—
48	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂⁴	1	12	yes	—
50	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
51	C₂²×C₁₂₇	—	2	2	no	—
52	C₂²×C₁₂₇	—	2	12	no	—
53	C₂²×C₁₂₇	—	4	12	no	—
54	C₂²×C₁₂₇	C₂³×C₄	1	2	yes	—
55	C₂²×C₁₂₇	C₂³×C₄	1	12	yes	—
56	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
57	C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
58	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₁₂₇	C₂³×C₄	1	12	yes	—
60	C₄×C₁₂₇	C₂⁵	1	1	yes	characteristic
61	C₂×C₁₂₇	—	2	2	no	—
62	C₂×C₁₂₇	—	2	12	no	—
63	C₂×C₁₂₇	#568	1	2	yes	complemented
64	C₂×C₁₂₇	C₂⁴×C₄	1	1	yes	characteristic
65	C₂×#568	—	127	1	no	maximal, Sylow
66	C₁₂₇	C₂×#568	1	1	yes	characteristic, complemented, Sylow
67	C₂²×#41	D₁₂₇	1	1	yes	characteristic
68	#568	—	127	16	no	—
69	C₂×#131	—	127	8	no	—
70	C₂²×M₁₆	—	127	6	no	—
71	C₂×#41	Dic₁₂₇	1	4	yes	—
72	C₂×#41	Dic₁₂₇	1	8	yes	—
73	C₂×#41	C₂×D₁₂₇	1	4	yes	—
74	C₂×#41	C₂×D₁₂₇	1	8	yes	—
75	C₂²×Q₈	Dic₁₂₇	1	1	yes	characteristic
76	C₂²×D₄	Dic₁₂₇	1	3	yes	—
77	C₂³×C₄	C₂×D₁₂₇	1	3	yes	—
78	#131	—	127	64	no	—
79	C₂×M₁₆	—	127	4	no	—
80	C₂×M₁₆	—	127	12	no	—
81	C₂×M₁₆	—	127	48	no	—
82	C₂²×C₈	—	127	12	no	—
83	C₂⁴	C₂×Dic₁₂₇	1	3	yes	—
84	#41	C₂×Dic₁₂₇	1	16	yes	—
85	#41	C₂×Dic₁₂₇	1	32	yes	—
86	#41	C₂²×D₁₂₇	1	16	yes	—
87	C₂×Q₈	C₂×Dic₁₂₇	1	4	yes	—
88	C₂×Q₈	C₂×Dic₁₂₇	1	8	yes	—
89	C₂×D₄	C₂×Dic₁₂₇	1	12	yes	—
90	C₂×D₄	C₂×Dic₁₂₇	1	24	yes	—
91	C₂²×C₄	C₂×Dic₁₂₇	1	3	yes	—
92	C₂²×C₄	C₂×Dic₁₂₇	1	6	yes	—
93	C₂²×C₄	C₂×Dic₁₂₇	1	12	yes	—
94	C₂²×C₄	C₂²×D₁₂₇	1	12	yes	—
95	C₂²×C₄	C₂²×D₁₂₇	1	6	yes	—
96	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
97	M₁₆	—	127	16	no	—
98	M₁₆	—	127	48	no	—
99	C₂×C₈	—	127	48	no	—
100	C₂×C₈	—	127	8	no	—
101	C₂³	—	4	6	no	—
102	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
103	C₂³	C₂²×Dic₁₂₇	1	6	yes	—
104	C₂³	C₂²×Dic₁₂₇	1	12	yes	—
105	Q₈	C₂²×Dic₁₂₇	1	16	yes	—
106	D₄	C₂²×Dic₁₂₇	1	48	yes	—
107	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
108	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
109	C₂×C₄	C₂²×Dic₁₂₇	1	12	yes	—
110	C₂×C₄	C₂²×Dic₁₂₇	1	6	yes	—
111	C₂×C₄	C₂²×Dic₁₂₇	1	24	yes	—
112	C₂×C₄	C₂²×Dic₁₂₇	1	12	yes	—
113	C₂×C₄	C₂³×D₁₂₇	1	12	yes	—
114	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
115	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
116	C₈	—	127	16	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	12	no	—
119	C₂²	—	4	12	no	—
120	C₂²	C₂³×Dic₁₂₇	1	2	yes	—
121	C₂²	C₂³×Dic₁₂₇	1	12	yes	—
122	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
123	C₄	C₂³×Dic₁₂₇	1	2	yes	—
124	C₄	C₂³×Dic₁₂₇	1	1	yes	characteristic
125	C₄	C₂³×Dic₁₂₇	1	12	yes	—
126	C₄	C₂⁴×D₁₂₇	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	12	no	—
129	C₂	#22550643	1	2	yes	direct product factor
130	C₂	C₂⁴×Dic₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₂₇
commutator	C₂×C₁₂₇	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₁₂₇
Fitting, prime radical	C₂²×#41×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂³×C₄
cosocle	C₄×C₁₂₇	C₂⁵
hypercenter	C₂²×#41	D₁₂₇
hypocenter	C₁₂₇	C₂×#568
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #22550643	C₂	direct (N × H)
2. C₂×#16861216	C₂	semidirect
3. C₂²×#14523232	C₂	semidirect
5. #16861216	C₂²	semidirect
7. C₂×#14523232	C₂²	semidirect
8. C₂×#14523232	C₂²	semidirect
9. C₂²×C₁₂₇⋊C₈	C₂²	semidirect
18. #14523232	C₂³	semidirect
20. C₂×C₁₂₇⋊C₈	C₂³	semidirect
63. C₂×C₁₂₇	#568	semidirect
66. C₁₂₇	C₂×#568	semidirect
129. C₂	#22550643	direct (N × H)

Maximal subgroups

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

subgroup	index
#22550643	2
C₂×#16861216	2
C₂²×#14523232	2
C₂²×#41×C₁₂₇	2
C₂×#568	127

Subgroups — C2×#22550643 #30653440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550643 #30653440