Subgroups — C₂×#433×C₁₂₇ #30635871

Order 16256.

412 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	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#433×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊C₈×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₁₂₇	C₂	1	2	yes	maximal
5	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
9	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
10	#117×C₁₂₇	C₄	1	4	yes	—
11	#117×C₁₂₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
13	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₂₇	C₂²	1	8	yes	complemented
17	C₂²⋊C₈×C₁₂₇	C₂²	1	8	yes	—
18	#97×C₁₂₇	C₂²	1	4	yes	complemented
19	C₄×C₈×C₁₂₇	C₂²	1	4	yes	complemented
20	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₂₇	—	2	4	no	—
22	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₂₇	—	2	4	no	—
29	C₂×C₈×C₁₂₇	—	2	4	no	—
30	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
31	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
32	C₄⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
33	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
34	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
35	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
36	C₄²×C₁₂₇	C₂³	1	2	yes	—
37	C₄²×C₁₂₇	C₂³	1	2	yes	—
38	C₂³×C₁₂₇	—	2	2	no	—
39	C₂³×C₁₂₇	—	2	4	no	—
40	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
41	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	—	2	2	no	—
43	C₂×C₄×C₁₂₇	—	2	4	no	—
44	C₂×C₄×C₁₂₇	—	2	4	no	—
45	C₂×C₄×C₁₂₇	—	2	4	no	—
46	C₂×C₄×C₁₂₇	—	2	4	no	—
47	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	#41	1	4	yes	—
52	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
53	C₈×C₁₂₇	—	2	4	no	—
54	C₈×C₁₂₇	—	2	4	no	—
55	C₂²×C₁₂₇	—	2	2	no	—
56	C₂²×C₁₂₇	—	2	4	no	—
57	C₂²×C₁₂₇	—	4	4	no	—
58	C₂²×C₁₂₇	#117	1	2	yes	—
59	C₂²×C₁₂₇	#131	1	4	yes	—
60	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₁₂₇	—	2	2	no	—
62	C₄×C₁₂₇	—	2	4	no	—
63	C₄×C₁₂₇	—	2	4	no	—
64	C₄×C₁₂₇	#117	1	2	yes	—
65	C₄×C₁₂₇	C₂×#41	1	2	yes	—
66	C₂×C₁₂₇	—	4	2	no	—
67	C₂×C₁₂₇	#433	1	4	yes	direct product factor
68	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
69	C₂×C₁₂₇	C₂×#131	1	2	yes	—
70	C₂×#433	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₁₂₇	C₂×#433	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×#117	C₂×C₁₂₇	1	1	yes	characteristic
73	#433	C₂×C₁₂₇	1	8	yes	direct product factor
74	C₂×C₄⋊C₈	C₂×C₁₂₇	1	2	yes	complemented
75	C₂×C₂²⋊C₈	C₂×C₁₂₇	1	2	yes	—
76	C₂×#97	C₂×C₁₂₇	1	1	yes	characteristic, complemented
77	C₂×C₄×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
78	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
80	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
81	#117	C₄×C₁₂₇	1	4	yes	—
82	#117	C₂²×C₁₂₇	1	4	yes	—
83	C₂×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
84	C₂×C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
85	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
86	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
87	C₄⋊C₈	C₂²×C₁₂₇	1	8	yes	complemented
88	C₂²⋊C₈	C₂²×C₁₂₇	1	8	yes	—
89	#97	C₂²×C₁₂₇	1	4	yes	complemented
90	C₄×C₈	C₂²×C₁₂₇	1	4	yes	complemented
91	C₂⁴	C₂×C₄×C₁₂₇	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
94	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
95	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
96	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
97	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
98	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	C₂³×C₁₂₇	1	4	yes	—
102	C₂×C₈	C₂³×C₁₂₇	1	4	yes	—
103	C₄⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
104	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
105	C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
106	C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
107	C₄²	C₂³×C₁₂₇	1	2	yes	—
108	C₄²	C₂³×C₁₂₇	1	2	yes	—
109	C₂³	—	2	2	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂²×C₄×C₁₂₇	1	2	yes	—
112	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
119	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
120	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
121	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
122	C₂×C₄	#41×C₁₂₇	1	4	yes	—
123	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
124	C₈	—	2	4	no	—
125	C₈	—	2	4	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	2	4	no	—
128	C₂²	—	4	4	no	—
129	C₂²	#117×C₁₂₇	1	2	yes	—
130	C₂²	#131×C₁₂₇	1	4	yes	—
131	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	4	no	—
134	C₄	—	2	4	no	—
135	C₄	#117×C₁₂₇	1	2	yes	—
136	C₄	C₂×#41×C₁₂₇	1	2	yes	—
137	C₂	—	4	2	no	—
138	C₂	#433×C₁₂₇	1	4	yes	direct product factor
139	C₂	C₂×#117×C₁₂₇	1	1	yes	characteristic
140	C₂	C₂×#131×C₁₂₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #433×C₁₂₇	C₂	direct (N × H)
3. C₂×C₄⋊C₈×C₁₂₇	C₂	semidirect
5. C₂×#97×C₁₂₇	C₂	semidirect
6. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
16. C₄⋊C₈×C₁₂₇	C₂²	semidirect
18. #97×C₁₂₇	C₂²	semidirect
19. C₄×C₈×C₁₂₇	C₂²	semidirect
67. C₂×C₁₂₇	#433	direct (N × H)
70. C₂×#433	C₁₂₇	direct (N × H)
71. C₁₂₇	C₂×#433	direct (N × H)
73. #433	C₂×C₁₂₇	direct (N × H)
74. C₂×C₄⋊C₈	C₂×C₁₂₇	semidirect
76. C₂×#97	C₂×C₁₂₇	semidirect
77. C₂×C₄×C₈	C₂×C₁₂₇	semidirect
87. C₄⋊C₈	C₂²×C₁₂₇	semidirect
89. #97	C₂²×C₁₂₇	semidirect
90. C₄×C₈	C₂²×C₁₂₇	semidirect
138. C₂	#433×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
C₂×#117×C₁₂₇	2
#433×C₁₂₇	2
C₂×C₄⋊C₈×C₁₂₇	2
C₂×C₂²⋊C₈×C₁₂₇	2
C₂×#97×C₁₂₇	2
C₂×C₄×C₈×C₁₂₇	2
C₂×#433	127

Subgroups — C2×#433×C127 #30635871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#433×C₁₂₇ #30635871