Subgroups — C₂×#22550359 #30651927

Order 16256.

1640 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550359	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×Dic₁₂₇	C₂	1	6	yes	maximal, complemented
4	C₂×#16861168	C₂	1	3	yes	maximal, complemented
5	C₂²×C₄⋊Dic₁₂₇	C₂	1	3	yes	maximal, complemented
6	C₂²×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₂₇	C₂²	1	6	yes	—
8	Q₈×Dic₁₂₇	C₂²	1	16	yes	complemented
9	D₄×Dic₁₂₇	C₂²	1	48	yes	complemented
10	#16861168	C₂²	1	24	yes	complemented
11	C₂×C₄⋊Dic₁₂₇	C₂²	1	3	yes	—
12	C₂×C₄⋊Dic₁₂₇	C₂²	1	6	yes	—
13	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₂₇	C₂²	1	24	yes	complemented
15	C₂×C₂²⋊Dic₁₂₇	C₂²	1	6	yes	—
16	C₂×C₄×Dic₁₂₇	C₂²	1	6	yes	—
17	C₂×#41×C₁₂₇	C₄	1	16	yes	complemented
18	C₂×#41×C₁₂₇	C₂²	1	8	yes	—
19	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂²×D₄×C₁₂₇	C₂²	1	3	yes	—
21	C₂³×C₄×C₁₂₇	C₂²	1	3	yes	—
22	C₂²×Dic₁₂₇	—	4	12	no	—
23	C₂²×Dic₁₂₇	C₂³	1	6	yes	—
24	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
25	C₂²×Dic₁₂₇	C₂³	1	24	yes	—
26	C₄⋊Dic₁₂₇	C₂³	1	12	yes	—
27	C₄⋊Dic₁₂₇	C₂³	1	24	yes	—
28	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
29	C₂²⋊Dic₁₂₇	C₂³	1	24	yes	—
30	C₄×Dic₁₂₇	C₂³	1	24	yes	—
31	C₂⁴×C₁₂₇	C₂³	1	3	yes	—
32	#41×C₁₂₇	C₂×C₄	1	64	yes	complemented
33	C₂×Q₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
34	C₂×Q₈×C₁₂₇	C₂³	1	4	yes	—
35	C₂×D₄×C₁₂₇	C₂×C₄	1	24	yes	complemented
36	C₂×D₄×C₁₂₇	C₂³	1	12	yes	—
37	C₂²×C₄×C₁₂₇	C₂×C₄	1	24	yes	complemented
38	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
40	C₂²×C₄×C₁₂₇	C₂³	1	12	yes	—
41	C₂×Dic₁₂₇	—	2	12	no	—
42	C₂×Dic₁₂₇	—	2	4	no	—
43	C₂×Dic₁₂₇	—	4	24	no	—
44	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
45	C₂×Dic₁₂₇	C₂⁴	1	12	yes	—
46	C₂³×C₁₂₇	—	2	6	no	—
47	C₂³×C₁₂₇	—	2	6	no	—
48	C₂³×C₁₂₇	C₂²×C₄	1	12	yes	—
49	C₂³×C₁₂₇	C₂⁴	1	6	yes	—
50	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
51	Q₈×C₁₂₇	C₂²×C₄	1	16	yes	complemented
52	D₄×C₁₂₇	C₂²×C₄	1	48	yes	complemented
53	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
55	C₂×C₄×C₁₂₇	C₂²×C₄	1	48	yes	complemented
56	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₂⁴	1	6	yes	—
58	C₂²×C₁₂₇	—	2	24	no	—
59	C₂²×C₁₂₇	—	2	6	no	—
60	C₂²×C₁₂₇	—	2	6	no	—
61	C₂²×C₁₂₇	C₂³×C₄	1	12	yes	—
62	C₂²×C₁₂₇	C₂³×C₄	1	2	yes	—
63	C₂²×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
64	C₂²×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
65	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
66	C₄×C₁₂₇	C₂³×C₄	1	4	yes	—
67	C₄×C₁₂₇	C₂³×C₄	1	12	yes	—
68	Dic₁₂₇	—	2	12	no	—
69	Dic₁₂₇	—	2	4	no	—
70	C₂×C₁₂₇	—	2	12	no	—
71	C₂×C₁₂₇	#520	1	4	yes	complemented
72	C₂×C₁₂₇	C₂⁴×C₄	1	1	yes	characteristic
73	C₂×C₁₂₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
74	C₂×C₁₂₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
75	C₂×#520	—	127	1	no	maximal, Sylow
76	C₁₂₇	C₂×#520	1	1	yes	characteristic, complemented, Sylow
77	C₂²×#41	D₁₂₇	1	1	yes	characteristic
78	#520	—	127	16	no	—
79	C₂×C₄×Q₈	—	127	2	no	—
80	C₂×C₄×D₄	—	127	6	no	—
81	C₂×#117	—	127	3	no	—
82	C₂²×C₄⋊C₄	—	127	3	no	—
83	C₂×#41	Dic₁₂₇	1	16	yes	complemented
84	C₂×#41	C₂×D₁₂₇	1	8	yes	—
85	C₂²×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
86	C₂²×D₄	C₂×D₁₂₇	1	3	yes	—
87	C₂³×C₄	—	127	6	no	—
88	C₂³×C₄	C₂×D₁₂₇	1	3	yes	—
89	C₄×Q₈	—	127	16	no	—
90	C₄×D₄	—	127	48	no	—
91	#117	—	127	24	no	—
92	C₂×C₄⋊C₄	—	127	3	no	—
93	C₂×C₄⋊C₄	—	127	6	no	—
94	C₂×C₄⋊C₄	—	127	24	no	—
95	C₂×C₄⋊C₄	—	127	1	no	—
96	C₂×C₂²⋊C₄	—	127	6	no	—
97	C₂×C₄²	—	127	6	no	—
98	C₂⁴	C₂²×D₁₂₇	1	3	yes	—
99	#41	C₂×Dic₁₂₇	1	64	yes	complemented
100	C₂×Q₈	C₂×Dic₁₂₇	1	8	yes	complemented
101	C₂×Q₈	C₂²×D₁₂₇	1	4	yes	—
102	C₂×D₄	C₂×Dic₁₂₇	1	24	yes	complemented
103	C₂×D₄	C₂²×D₁₂₇	1	12	yes	—
104	C₂²×C₄	—	127	24	no	—
105	C₂²×C₄	—	127	2	no	—
106	C₂²×C₄	—	127	6	no	—
107	C₂²×C₄	—	508	12	no	—
108	C₂²×C₄	C₂×Dic₁₂₇	1	24	yes	complemented
109	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
110	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
111	C₂²×C₄	C₂²×D₁₂₇	1	12	yes	—
112	C₄⋊C₄	—	127	12	no	—
113	C₄⋊C₄	—	127	24	no	—
114	C₄⋊C₄	—	127	4	no	—
115	C₂²⋊C₄	—	127	24	no	—
116	C₄²	—	127	24	no	—
117	C₂³	—	2	6	no	—
118	C₂³	—	2	6	no	—
119	C₂³	C₂²×Dic₁₂₇	1	12	yes	—
120	C₂³	C₂³×D₁₂₇	1	6	yes	—
121	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
122	Q₈	C₂²×Dic₁₂₇	1	16	yes	complemented
123	D₄	C₂²×Dic₁₂₇	1	48	yes	complemented
124	C₂×C₄	—	127	4	no	—
125	C₂×C₄	—	127	12	no	—
126	C₂×C₄	—	508	24	no	—
127	C₂×C₄	—	254	12	no	—
128	C₂×C₄	—	254	4	no	—
129	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
130	C₂×C₄	C₂²×Dic₁₂₇	1	12	yes	—
131	C₂×C₄	C₂²×Dic₁₂₇	1	48	yes	complemented
132	C₂×C₄	C₂³×D₁₂₇	1	6	yes	—
133	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
134	C₂²	—	2	24	no	—
135	C₂²	—	2	6	no	—
136	C₂²	—	2	6	no	—
137	C₂²	C₂³×Dic₁₂₇	1	12	yes	—
138	C₂²	C₂³×Dic₁₂₇	1	2	yes	—
139	C₂²	#16861255	1	2	yes	—
140	C₂²	#16861256	1	2	yes	—
141	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
142	C₄	—	254	12	no	—
143	C₄	—	254	4	no	—
144	C₄	C₂³×Dic₁₂₇	1	4	yes	—
145	C₄	C₂³×Dic₁₂₇	1	12	yes	—
146	C₂	—	2	12	no	—
147	C₂	#22550359	1	4	yes	direct product factor
148	C₂	C₂⁴×Dic₁₂₇	1	1	yes	characteristic
149	C₂	C₂×#16861255	1	1	yes	characteristic
150	C₂	C₂×#16861256	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₂₇
commutator	C₂×C₁₂₇	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting, prime radical	C₂²×#41×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂⁴
cosocle	C₂²×C₁₂₇	C₂⁵
hypercenter	C₂²×#41	D₁₂₇
hypocenter	C₁₂₇	C₂×#520
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #22550359	C₂	direct (N × H)
2. C₂×Q₈×Dic₁₂₇	C₂	semidirect
3. C₂×D₄×Dic₁₂₇	C₂	semidirect
4. C₂×#16861168	C₂	semidirect
5. C₂²×C₄⋊Dic₁₂₇	C₂	semidirect
8. Q₈×Dic₁₂₇	C₂²	semidirect
9. D₄×Dic₁₂₇	C₂²	semidirect
10. #16861168	C₂²	semidirect
14. C₂×C₄⋊Dic₁₂₇	C₂²	semidirect
17. C₂×#41×C₁₂₇	C₄	semidirect
32. #41×C₁₂₇	C₂×C₄	semidirect
33. C₂×Q₈×C₁₂₇	C₂×C₄	semidirect
35. C₂×D₄×C₁₂₇	C₂×C₄	semidirect
37. C₂²×C₄×C₁₂₇	C₂×C₄	semidirect
51. Q₈×C₁₂₇	C₂²×C₄	semidirect
52. D₄×C₁₂₇	C₂²×C₄	semidirect
55. C₂×C₄×C₁₂₇	C₂²×C₄	semidirect
71. C₂×C₁₂₇	#520	semidirect
76. C₁₂₇	C₂×#520	semidirect
83. C₂×#41	Dic₁₂₇	semidirect
99. #41	C₂×Dic₁₂₇	semidirect
100. C₂×Q₈	C₂×Dic₁₂₇	semidirect
102. C₂×D₄	C₂×Dic₁₂₇	semidirect
108. C₂²×C₄	C₂×Dic₁₂₇	semidirect
122. Q₈	C₂²×Dic₁₂₇	semidirect
123. D₄	C₂²×Dic₁₂₇	semidirect
131. C₂×C₄	C₂²×Dic₁₂₇	semidirect
147. C₂	#22550359	direct (N × H)

Maximal subgroups

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

subgroup	index
#22550359	2
C₂×Q₈×Dic₁₂₇	2
C₂×D₄×Dic₁₂₇	2
C₂×#16861168	2
C₂²×C₄⋊Dic₁₂₇	2
C₂²×#41×C₁₂₇	2
C₂×#520	127

Subgroups — C2×#22550359 #30651927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550359 #30651927