Subgroups — C₂×#336×C₁₂₇ #30634679

Order 16256.

528 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₁₂₇	C₂	1	2	yes	maximal
2	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#336×C₁₂₇	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₄×C₁₂₇	C₄	1	2	yes	—
5	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₁₂₇	—	2	4	no	—
7	C₂²×C₈×C₁₂₇	—	2	4	no	—
8	C₂²×C₈×C₁₂₇	C₄	1	8	yes	complemented
9	C₂²×C₈×C₁₂₇	C₂²	1	4	yes	—
10	C₂×C₄²×C₁₂₇	—	2	4	no	—
11	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
12	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
13	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₂₇	—	2	4	no	—
15	C₂²×C₄×C₁₂₇	—	2	4	no	—
16	C₂²×C₄×C₁₂₇	C₈	1	8	yes	complemented
17	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
22	C₂²×C₄×C₁₂₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₂₇	—	2	16	no	—
25	C₂×C₈×C₁₂₇	—	2	4	no	—
26	C₂×C₈×C₁₂₇	—	2	4	no	—
27	C₂×C₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
28	C₄²×C₁₂₇	—	2	4	no	—
29	C₄²×C₁₂₇	—	2	4	no	—
30	C₂³×C₁₂₇	C₄²	1	2	yes	—
31	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
32	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
33	C₂³×C₁₂₇	C₂×C₈	1	4	yes	—
34	C₂³×C₁₂₇	M₁₆	1	4	yes	—
35	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	—	2	16	no	—
37	C₂×C₄×C₁₂₇	—	2	4	no	—
38	C₂×C₄×C₁₂₇	—	2	4	no	—
39	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
40	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	8	yes	—
42	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
44	C₂×C₄×C₁₂₇	C₂×C₈	1	8	yes	complemented
45	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
49	C₈×C₁₂₇	—	2	8	no	—
50	C₂²×C₁₂₇	#95	1	4	yes	—
51	C₂²×C₁₂₇	C₄×C₈	1	4	yes	—
52	C₂²×C₁₂₇	#97	1	4	yes	—
53	C₂²×C₁₂₇	C₂²⋊C₈	1	8	yes	—
54	C₂²×C₁₂₇	C₄⋊C₈	1	8	yes	—
55	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
56	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₂₇	C₂²×C₈	1	2	yes	—
59	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
60	C₄×C₁₂₇	—	2	8	no	—
61	C₄×C₁₂₇	#95	1	4	yes	—
62	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
63	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
64	C₂×C₁₂₇	#336	1	8	yes	direct product factor
65	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
66	C₂×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
67	C₂×C₁₂₇	C₂×#97	1	1	yes	characteristic
68	C₂×C₁₂₇	C₂×C₂²⋊C₈	1	2	yes	—
69	C₂×C₁₂₇	C₂×C₄⋊C₈	1	2	yes	—
70	C₂×#336	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₁₂₇	C₂×#336	1	1	yes	characteristic, direct product factor, Sylow
72	C₂³×C₈	C₂×C₁₂₇	1	2	yes	—
73	C₂²×C₄²	C₂×C₁₂₇	1	1	yes	characteristic
74	#336	C₂×C₁₂₇	1	4	yes	direct product factor
75	C₂³×C₄	C₄×C₁₂₇	1	2	yes	—
76	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
77	C₂²×C₈	—	2	4	no	—
78	C₂²×C₈	—	2	4	no	—
79	C₂²×C₈	C₄×C₁₂₇	1	8	yes	complemented
80	C₂²×C₈	C₂²×C₁₂₇	1	4	yes	—
81	C₂×C₄²	—	2	4	no	—
82	C₂×C₄²	C₄×C₁₂₇	1	2	yes	—
83	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
84	C₂⁴	C₂×C₄×C₁₂₇	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	2	4	no	—
87	C₂²×C₄	C₈×C₁₂₇	1	8	yes	complemented
88	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
89	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
90	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
91	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
92	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
93	C₂²×C₄	Q₈×C₁₂₇	1	2	yes	—
94	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
95	C₂×C₈	—	2	16	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	C₂×C₄×C₁₂₇	1	8	yes	complemented
99	C₄²	—	2	4	no	—
100	C₄²	—	2	4	no	—
101	C₂³	C₄²×C₁₂₇	1	2	yes	—
102	C₂³	C₂²⋊C₄×C₁₂₇	1	2	yes	—
103	C₂³	C₄⋊C₄×C₁₂₇	1	2	yes	—
104	C₂³	C₂×C₈×C₁₂₇	1	4	yes	—
105	C₂³	M₁₆×C₁₂₇	1	4	yes	—
106	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
107	C₂×C₄	—	2	16	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	C₄²×C₁₂₇	1	2	yes	—
111	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	2	yes	—
112	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	8	yes	—
113	C₂×C₄	C₄⋊C₄×C₁₂₇	1	2	yes	—
114	C₂×C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
115	C₂×C₄	C₂×C₈×C₁₂₇	1	8	yes	complemented
116	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
117	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
118	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
119	C₂×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
120	C₈	—	2	8	no	—
121	C₂²	#95×C₁₂₇	1	4	yes	—
122	C₂²	C₄×C₈×C₁₂₇	1	4	yes	—
123	C₂²	#97×C₁₂₇	1	4	yes	—
124	C₂²	C₂²⋊C₈×C₁₂₇	1	8	yes	—
125	C₂²	C₄⋊C₈×C₁₂₇	1	8	yes	—
126	C₂²	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
127	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
128	C₂²	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
129	C₂²	C₂²×C₈×C₁₂₇	1	2	yes	—
130	C₂²	C₂×M₁₆×C₁₂₇	1	2	yes	—
131	C₄	—	2	8	no	—
132	C₄	#95×C₁₂₇	1	4	yes	—
133	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
134	C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
135	C₂	#336×C₁₂₇	1	8	yes	direct product factor
136	C₂	C₂×#95×C₁₂₇	1	1	yes	characteristic
137	C₂	C₂×C₄×C₈×C₁₂₇	1	1	yes	characteristic
138	C₂	C₂×#97×C₁₂₇	1	1	yes	characteristic
139	C₂	C₂×C₂²⋊C₈×C₁₂₇	1	2	yes	—
140	C₂	C₂×C₄⋊C₈×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₂₇	C₂²
commutator	C₂	C₂×C₄×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₂₇
socle, prime radical	C₂⁴×C₁₂₇	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #336×C₁₂₇	C₂	direct (N × H)
8. C₂²×C₈×C₁₂₇	C₄	semidirect
16. C₂²×C₄×C₁₂₇	C₈	semidirect
27. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
44. C₂×C₄×C₁₂₇	C₂×C₈	semidirect
64. C₂×C₁₂₇	#336	direct (N × H)
70. C₂×#336	C₁₂₇	direct (N × H)
71. C₁₂₇	C₂×#336	direct (N × H)
74. #336	C₂×C₁₂₇	direct (N × H)
79. C₂²×C₈	C₄×C₁₂₇	semidirect
87. C₂²×C₄	C₈×C₁₂₇	semidirect
98. C₂×C₈	C₂×C₄×C₁₂₇	semidirect
115. C₂×C₄	C₂×C₈×C₁₂₇	semidirect
135. C₂	#336×C₁₂₇	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×C₈×C₁₂₇	2
C₂²×C₄²×C₁₂₇	2
#336×C₁₂₇	2
C₂×#336	127

Subgroups — C2×#336×C127 #30634679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#336×C₁₂₇ #30634679