Subgroups — C₂×#22549831 #30647142

Order 16256.

592 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	C₂²×#14523232	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861168	C₂	1	1	yes	maximal, characteristic
3	#22549831	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16861119	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14523232	C₄	1	8	yes	complemented
8	C₂×#14523232	C₂²	1	4	yes	—
9	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
10	#16861168	C₄	1	4	yes	—
11	#16861168	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₁₂₇	C₄	1	2	yes	—
13	C₂×C₂²⋊Dic₁₂₇	C₄	1	2	yes	—
14	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
15	#16861119	C₂²	1	8	yes	complemented
16	C₈×Dic₁₂₇	C₂²	1	8	yes	complemented
17	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₂₇	C₄	1	8	yes	complemented
19	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₁₂₇	C₂²	1	4	yes	—
22	C₂²×Dic₁₂₇	C₂×C₄	1	4	yes	—
23	#14523232	C₂×C₄	1	16	yes	complemented
24	C₂×C₁₂₇⋊C₈	C₂×C₄	1	8	yes	—
25	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
26	C₄⋊Dic₁₂₇	C₂×C₄	1	8	yes	—
27	C₂²⋊Dic₁₂₇	C₂×C₄	1	8	yes	—
28	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	—
29	C₄×Dic₁₂₇	C₂³	1	4	yes	—
30	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
37	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
38	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
39	C₂×C₈×C₁₂₇	C₂×C₄	1	16	yes	complemented
40	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
41	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
42	C₂×Dic₁₂₇	—	2	8	no	—
43	C₂×Dic₁₂₇	C₂²×C₄	1	8	yes	—
44	C₁₂₇⋊C₈	C₂²×C₄	1	8	yes	—
45	C₂³×C₁₂₇	—	2	4	no	—
46	C₂³×C₁₂₇	C₄²	1	4	yes	—
47	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
48	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₄²	1	8	yes	—
50	C₂×C₄×C₁₂₇	C₄²	1	4	yes	—
51	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
55	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
58	C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
59	C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₁₂₇	—	2	4	no	—
61	C₂²×C₁₂₇	—	2	8	no	—
62	C₂²×C₁₂₇	C₂×C₄²	1	4	yes	—
63	C₂²×C₁₂₇	C₂×C₄²	1	2	yes	—
64	C₂²×C₁₂₇	#131	1	4	yes	—
65	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
67	C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
68	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
70	Dic₁₂₇	—	2	8	no	—
71	C₂×C₁₂₇	—	2	4	no	—
72	C₂×C₁₂₇	#405	1	4	yes	complemented
73	C₂×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
74	C₂×C₁₂₇	C₂×#131	1	2	yes	—
75	C₂×#405	—	127	1	no	maximal, Sylow
76	C₁₂₇	C₂×#405	1	1	yes	characteristic, complemented, Sylow
77	C₂²×M₁₆	—	127	1	no	—
78	C₂³×C₈	D₁₂₇	1	1	yes	characteristic
79	C₂×#117	—	127	1	no	—
80	#405	—	127	8	no	—
81	C₂×#97	—	127	2	no	—
82	C₂×C₄×C₈	—	127	2	no	—
83	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
84	C₂×M₁₆	—	127	8	no	—
85	C₂×M₁₆	—	127	4	no	—
86	C₂²×C₈	—	127	2	no	—
87	C₂²×C₈	Dic₁₂₇	1	8	yes	complemented
88	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₁₂₇	1	4	yes	—
90	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
91	#117	—	127	4	no	—
92	#117	—	127	4	no	—
93	C₂×C₄⋊C₄	—	127	2	no	—
94	C₂×C₂²⋊C₄	—	127	2	no	—
95	C₂×C₄²	—	127	2	no	—
96	#97	—	127	8	no	—
97	C₄×C₈	—	127	8	no	—
98	C₂⁴	C₄×D₁₂₇	1	1	yes	characteristic
99	C₂²×C₄	—	127	4	no	—
100	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
101	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
102	C₂²×C₄	C₄×D₁₂₇	1	4	yes	—
103	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
104	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
106	M₁₆	—	127	16	no	—
107	C₂×C₈	—	127	8	no	—
108	C₂×C₈	—	127	4	no	—
109	C₂×C₈	C₂×Dic₁₂₇	1	4	yes	—
110	C₂×C₈	C₂×Dic₁₂₇	1	4	yes	—
111	C₂×C₈	C₂×Dic₁₂₇	1	16	yes	complemented
112	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
113	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
114	C₄⋊C₄	—	127	8	no	—
115	C₂²⋊C₄	—	127	8	no	—
116	C₄²	—	127	4	no	—
117	C₄²	—	127	4	no	—
118	C₂³	—	2	4	no	—
119	C₂³	C₄×Dic₁₂₇	1	4	yes	—
120	C₂³	C₂×C₄×D₁₂₇	1	2	yes	—
121	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
122	C₂×C₄	—	127	8	no	—
123	C₂×C₄	—	254	8	no	—
124	C₂×C₄	C₄×Dic₁₂₇	1	8	yes	—
125	C₂×C₄	C₄×Dic₁₂₇	1	4	yes	—
126	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
127	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
128	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
130	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
131	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
132	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
133	C₈	—	127	8	no	—
134	C₈	C₂²×Dic₁₂₇	1	4	yes	—
135	C₈	C₂²×Dic₁₂₇	1	4	yes	—
136	C₂²	—	2	8	no	—
137	C₂²	—	2	4	no	—
138	C₂²	C₂×C₄×Dic₁₂₇	1	4	yes	—
139	C₂²	C₂×C₄×Dic₁₂₇	1	2	yes	—
140	C₂²	#16861215	1	4	yes	—
141	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
142	C₄	—	254	8	no	—
143	C₄	C₂×C₄×Dic₁₂₇	1	2	yes	—
144	C₄	C₂×C₄×Dic₁₂₇	1	4	yes	—
145	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
146	C₄	C₂³×Dic₁₂₇	1	1	yes	characteristic
147	C₂	—	2	4	no	—
148	C₂	#22549831	1	4	yes	direct product factor
149	C₂	C₂²×C₄×Dic₁₂₇	1	1	yes	characteristic
150	C₂	C₂×#16861215	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂³×C₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂³×C₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#405
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. #22549831	C₂	direct (N × H)
4. C₂×#16861119	C₂	semidirect
5. C₂×C₈×Dic₁₂₇	C₂	semidirect
7. C₂×#14523232	C₄	semidirect
15. #16861119	C₂²	semidirect
16. C₈×Dic₁₂₇	C₂²	semidirect
18. C₂²×C₈×C₁₂₇	C₄	semidirect
23. #14523232	C₂×C₄	semidirect
39. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
72. C₂×C₁₂₇	#405	semidirect
76. C₁₂₇	C₂×#405	semidirect
87. C₂²×C₈	Dic₁₂₇	semidirect
111. C₂×C₈	C₂×Dic₁₂₇	semidirect
148. C₂	#22549831	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×#14523232	2
C₂×#16861168	2
#22549831	2
C₂×#16861119	2
C₂×C₈×Dic₁₂₇	2
C₂³×C₈×C₁₂₇	2
C₂×#405	127

Subgroups — C2×#22549831 #30647142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549831 #30647142