Subgroups — C₁₂₇×#2880 #30635832

Order 16256.

774 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#131×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	#408×C₁₂₇	C₂	1	6	yes	maximal, complemented
4	#407×C₁₂₇	C₂	1	3	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₁₂₇	C₂	1	3	yes	maximal, complemented
6	C₂×#41×C₁₂₇	—	2	8	no	—
7	C₂×#41×C₁₂₇	C₄	1	1	yes	characteristic
8	C₂×#41×C₁₂₇	C₄	1	3	yes	—
9	C₂×#41×C₁₂₇	C₂²	1	3	yes	—
10	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₁₂₇	C₄	1	1	yes	characteristic
12	C₂²×D₄×C₁₂₇	C₄	1	3	yes	—
13	C₂³×C₄×C₁₂₇	C₂²	1	3	yes	—
14	#131×C₁₂₇	—	2	8	no	—
15	C₂×M₁₆×C₁₂₇	C₂²	1	6	yes	complemented
16	C₂²×C₈×C₁₂₇	C₂²	1	6	yes	complemented
17	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₈×C₁₂₇	C₂²	1	3	yes	—
19	C₂²⋊C₈×C₁₂₇	C₂²	1	6	yes	complemented
20	C₂²⋊C₈×C₁₂₇	C₂²	1	6	yes	complemented
21	C₂⁴×C₁₂₇	C₂×C₄	1	3	yes	—
22	#41×C₁₂₇	—	2	4	no	—
23	#41×C₁₂₇	—	2	12	no	—
24	#41×C₁₂₇	—	2	12	no	—
25	#41×C₁₂₇	D₄	1	8	yes	—
26	C₂×Q₈×C₁₂₇	—	2	4	no	—
27	C₂×Q₈×C₁₂₇	C₂×C₄	1	3	yes	—
28	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₁₂₇	—	2	12	no	—
30	C₂×D₄×C₁₂₇	C₂×C₄	1	6	yes	—
31	C₂×D₄×C₁₂₇	C₂×C₄	1	3	yes	—
32	C₂×D₄×C₁₂₇	C₂×C₄	1	3	yes	—
33	C₂²×C₄×C₁₂₇	—	2	12	no	—
34	C₂²×C₄×C₁₂₇	C₂×C₄	1	3	yes	—
35	C₂²×C₄×C₁₂₇	C₂×C₄	1	3	yes	—
36	C₂²×C₄×C₁₂₇	C₂×C₄	1	3	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
40	M₁₆×C₁₂₇	—	2	12	no	—
41	C₂×C₈×C₁₂₇	—	2	12	no	—
42	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
43	C₂×C₈×C₁₂₇	C₂³	1	6	yes	—
44	C₂³×C₁₂₇	—	2	6	no	—
45	C₂³×C₁₂₇	—	2	6	no	—
46	C₂³×C₁₂₇	—	4	3	no	—
47	C₂³×C₁₂₇	C₂²×C₄	1	3	yes	—
48	C₂³×C₁₂₇	C₂²×C₄	1	3	yes	—
49	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
50	Q₈×C₁₂₇	—	2	6	no	—
51	Q₈×C₁₂₇	C₂²⋊C₄	1	4	yes	—
52	D₄×C₁₂₇	—	2	12	no	—
53	D₄×C₁₂₇	—	2	6	no	—
54	D₄×C₁₂₇	C₂²⋊C₄	1	12	yes	—
55	C₂×C₄×C₁₂₇	—	2	6	no	—
56	C₂×C₄×C₁₂₇	—	2	6	no	—
57	C₂×C₄×C₁₂₇	—	2	6	no	—
58	C₂×C₄×C₁₂₇	—	2	2	no	—
59	C₂×C₄×C₁₂₇	—	2	6	no	—
60	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
62	C₂×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
63	C₂×C₄×C₁₂₇	C₂×D₄	1	12	yes	—
64	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
65	C₈×C₁₂₇	—	2	2	no	—
66	C₈×C₁₂₇	—	2	6	no	—
67	C₂²×C₁₂₇	—	2	1	no	—
68	C₂²×C₁₂₇	—	2	3	no	—
69	C₂²×C₁₂₇	—	2	1	no	—
70	C₂²×C₁₂₇	—	2	3	no	—
71	C₂²×C₁₂₇	—	2	3	no	—
72	C₂²×C₁₂₇	—	2	3	no	—
73	C₂²×C₁₂₇	—	2	3	no	—
74	C₂²×C₁₂₇	—	4	12	no	—
75	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	6	yes	—
76	C₂²×C₁₂₇	#131	1	2	yes	complemented
77	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₁₂₇	—	2	1	no	—
79	C₄×C₁₂₇	—	2	3	no	—
80	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	6	yes	—
81	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
82	C₂×C₁₂₇	—	2	2	no	—
83	C₂×C₁₂₇	—	2	6	no	—
84	C₂×C₁₂₇	—	4	3	no	—
85	C₂×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
87	C₂×C₁₂₇	#568	1	1	yes	characteristic
88	#2880	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₁₂₇	#2880	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×#41	C₂×C₁₂₇	1	1	yes	characteristic
91	C₂×#131	C₂×C₁₂₇	1	2	yes	complemented
92	#408	C₂×C₁₂₇	1	6	yes	complemented
93	#407	C₂×C₁₂₇	1	3	yes	complemented
94	C₂×C₂²⋊C₈	C₂×C₁₂₇	1	3	yes	complemented
95	C₂×#41	—	2	8	no	—
96	C₂×#41	C₄×C₁₂₇	1	1	yes	characteristic
97	C₂×#41	C₄×C₁₂₇	1	3	yes	—
98	C₂×#41	C₂²×C₁₂₇	1	3	yes	—
99	C₂×#41	C₂²×C₁₂₇	1	1	yes	characteristic
100	C₂²×Q₈	C₄×C₁₂₇	1	1	yes	characteristic
101	C₂²×D₄	C₄×C₁₂₇	1	3	yes	—
102	C₂³×C₄	C₂²×C₁₂₇	1	3	yes	—
103	#131	—	2	8	no	—
104	C₂×M₁₆	C₂²×C₁₂₇	1	6	yes	complemented
105	C₂²×C₈	C₂²×C₁₂₇	1	6	yes	complemented
106	C₂²⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
107	C₂²⋊C₈	C₂²×C₁₂₇	1	3	yes	—
108	C₂²⋊C₈	C₂²×C₁₂₇	1	6	yes	complemented
109	C₂²⋊C₈	C₂²×C₁₂₇	1	6	yes	complemented
110	C₂⁴	C₂×C₄×C₁₂₇	1	3	yes	—
111	#41	—	2	4	no	—
112	#41	—	2	12	no	—
113	#41	—	2	12	no	—
114	#41	D₄×C₁₂₇	1	8	yes	—
115	C₂×Q₈	—	2	4	no	—
116	C₂×Q₈	C₂×C₄×C₁₂₇	1	3	yes	—
117	C₂×Q₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
118	C₂×D₄	—	2	12	no	—
119	C₂×D₄	C₂×C₄×C₁₂₇	1	6	yes	—
120	C₂×D₄	C₂×C₄×C₁₂₇	1	3	yes	—
121	C₂×D₄	C₂×C₄×C₁₂₇	1	3	yes	—
122	C₂²×C₄	—	2	12	no	—
123	C₂²×C₄	C₂×C₄×C₁₂₇	1	3	yes	—
124	C₂²×C₄	C₂×C₄×C₁₂₇	1	3	yes	—
125	C₂²×C₄	C₂×C₄×C₁₂₇	1	3	yes	—
126	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₁₂₇	1	3	yes	—
128	C₂²×C₄	C₂³×C₁₂₇	1	3	yes	—
129	M₁₆	—	2	12	no	—
130	C₂×C₈	—	2	12	no	—
131	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
132	C₂×C₈	C₂³×C₁₂₇	1	6	yes	—
133	C₂³	—	2	6	no	—
134	C₂³	—	2	6	no	—
135	C₂³	—	4	3	no	—
136	C₂³	C₂²×C₄×C₁₂₇	1	3	yes	—
137	C₂³	C₂²×C₄×C₁₂₇	1	3	yes	—
138	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
139	Q₈	—	2	6	no	—
140	Q₈	C₂²⋊C₄×C₁₂₇	1	4	yes	—
141	D₄	—	2	12	no	—
142	D₄	—	2	6	no	—
143	D₄	C₂²⋊C₄×C₁₂₇	1	12	yes	—
144	C₂×C₄	—	2	6	no	—
145	C₂×C₄	—	2	6	no	—
146	C₂×C₄	—	2	6	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	6	no	—
149	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
150	C₂×C₄	C₂²×C₄×C₁₂₇	1	3	yes	—
151	C₂×C₄	C₂²×C₄×C₁₂₇	1	3	yes	—
152	C₂×C₄	C₂×D₄×C₁₂₇	1	12	yes	—
153	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	6	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	3	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	3	no	—
160	C₂²	—	2	3	no	—
161	C₂²	—	2	3	no	—
162	C₂²	—	2	3	no	—
163	C₂²	—	4	12	no	—
164	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	6	yes	—
165	C₂²	#131×C₁₂₇	1	2	yes	complemented
166	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	3	no	—
169	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	6	yes	—
170	C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
171	C₂	—	2	2	no	—
172	C₂	—	2	6	no	—
173	C₂	—	4	3	no	—
174	C₂	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
175	C₂	C₂×#131×C₁₂₇	1	1	yes	characteristic
176	C₂	#568×C₁₂₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#131×C₁₂₇	C₂	semidirect
3. #408×C₁₂₇	C₂	semidirect
4. #407×C₁₂₇	C₂	semidirect
5. C₂×C₂²⋊C₈×C₁₂₇	C₂	semidirect
15. C₂×M₁₆×C₁₂₇	C₂²	semidirect
16. C₂²×C₈×C₁₂₇	C₂²	semidirect
19. C₂²⋊C₈×C₁₂₇	C₂²	semidirect
20. C₂²⋊C₈×C₁₂₇	C₂²	semidirect
76. C₂²×C₁₂₇	#131	semidirect
88. #2880	C₁₂₇	direct (N × H)
89. C₁₂₇	#2880	direct (N × H)
91. C₂×#131	C₂×C₁₂₇	semidirect
92. #408	C₂×C₁₂₇	semidirect
93. #407	C₂×C₁₂₇	semidirect
94. C₂×C₂²⋊C₈	C₂×C₁₂₇	semidirect
104. C₂×M₁₆	C₂²×C₁₂₇	semidirect
105. C₂²×C₈	C₂²×C₁₂₇	semidirect
108. C₂²⋊C₈	C₂²×C₁₂₇	semidirect
109. C₂²⋊C₈	C₂²×C₁₂₇	semidirect
165. C₂²	#131×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₁₂₇	2
C₂×#131×C₁₂₇	2
#408×C₁₂₇	2
#407×C₁₂₇	2
C₂×C₂²⋊C₈×C₁₂₇	2
#2880	127

Subgroups — C127×#2880 #30635832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2880 #30635832