Subgroups — C₁₂₇×#2879 #30635831

Order 16256.

764 conjugacy classes of subgroups.

The subgroup lattice has 168 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	C₂²×M₁₆×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	#408×C₁₂₇	C₂	1	8	yes	maximal, complemented
4	#407×C₁₂₇	C₂	1	4	yes	maximal, complemented
5	C₂×#41×C₁₂₇	—	2	8	no	—
6	C₂×#41×C₁₂₇	C₄	1	4	yes	—
7	C₂×#41×C₁₂₇	C₂²	1	4	yes	—
8	C₂²×Q₈×C₁₂₇	C₄	1	1	yes	characteristic
9	C₂²×D₄×C₁₂₇	C₄	1	2	yes	—
10	C₂²×D₄×C₁₂₇	C₄	1	1	yes	characteristic
11	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
12	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₂₇	—	2	8	no	—
14	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	—
15	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	complemented
16	C₂²×C₈×C₁₂₇	C₂²	1	4	yes	complemented
17	C₂²⋊C₈×C₁₂₇	C₂²	1	16	yes	complemented
18	C₂⁴×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	#41×C₁₂₇	—	2	16	no	—
21	#41×C₁₂₇	—	2	8	no	—
22	#41×C₁₂₇	—	2	8	no	—
23	C₂×Q₈×C₁₂₇	—	2	4	no	—
24	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₂×D₄×C₁₂₇	—	2	8	no	—
26	C₂×D₄×C₁₂₇	—	2	4	no	—
27	C₂×D₄×C₁₂₇	C₂×C₄	1	8	yes	—
28	C₂×D₄×C₁₂₇	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₁₂₇	—	2	8	no	—
30	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₂₇	D₄	1	8	yes	—
35	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
36	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	M₁₆×C₁₂₇	—	2	8	no	—
39	M₁₆×C₁₂₇	—	2	8	no	—
40	C₂×C₈×C₁₂₇	—	2	8	no	—
41	C₂×C₈×C₁₂₇	C₂³	1	8	yes	—
42	C₂³×C₁₂₇	—	2	4	no	—
43	C₂³×C₁₂₇	—	4	2	no	—
44	C₂³×C₁₂₇	—	4	4	no	—
45	C₂³×C₁₂₇	C₂²⋊C₄	1	4	yes	—
46	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
47	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
49	Q₈×C₁₂₇	—	2	8	no	—
50	D₄×C₁₂₇	—	2	16	no	—
51	D₄×C₁₂₇	—	2	8	no	—
52	C₂×C₄×C₁₂₇	—	2	4	no	—
53	C₂×C₄×C₁₂₇	—	2	8	no	—
54	C₂×C₄×C₁₂₇	—	2	8	no	—
55	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	8	yes	—
57	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
63	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
64	C₈×C₁₂₇	—	2	8	no	—
65	C₂²×C₁₂₇	—	2	2	no	—
66	C₂²×C₁₂₇	—	2	4	no	—
67	C₂²×C₁₂₇	—	2	4	no	—
68	C₂²×C₁₂₇	—	2	8	no	—
69	C₂²×C₁₂₇	—	4	4	no	—
70	C₂²×C₁₂₇	—	4	8	no	—
71	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
72	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
73	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₁₂₇	—	2	4	no	—
75	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
76	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
77	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
78	C₂×C₁₂₇	—	2	2	no	—
79	C₂×C₁₂₇	—	2	4	no	—
80	C₂×C₁₂₇	—	4	4	no	—
81	C₂×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₁₂₇	#568	1	2	yes	—
83	#2879	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₁₂₇	#2879	1	1	yes	characteristic, direct product factor, Sylow
85	C₂²×#41	C₂×C₁₂₇	1	1	yes	characteristic
86	C₂²×M₁₆	C₂×C₁₂₇	1	2	yes	complemented
87	#408	C₂×C₁₂₇	1	8	yes	complemented
88	#407	C₂×C₁₂₇	1	4	yes	complemented
89	C₂×#41	—	2	8	no	—
90	C₂×#41	C₄×C₁₂₇	1	4	yes	—
91	C₂×#41	C₂²×C₁₂₇	1	4	yes	—
92	C₂²×Q₈	C₄×C₁₂₇	1	1	yes	characteristic
93	C₂²×D₄	C₄×C₁₂₇	1	2	yes	—
94	C₂²×D₄	C₄×C₁₂₇	1	1	yes	characteristic
95	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
96	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
97	C₂×M₁₆	—	2	8	no	—
98	C₂×M₁₆	C₂²×C₁₂₇	1	4	yes	—
99	C₂×M₁₆	C₂²×C₁₂₇	1	4	yes	complemented
100	C₂²×C₈	C₂²×C₁₂₇	1	4	yes	complemented
101	C₂²⋊C₈	C₂²×C₁₂₇	1	16	yes	complemented
102	C₂⁴	C₂×C₄×C₁₂₇	1	2	yes	—
103	C₂⁴	C₂×C₄×C₁₂₇	1	1	yes	characteristic
104	#41	—	2	16	no	—
105	#41	—	2	8	no	—
106	#41	—	2	8	no	—
107	C₂×Q₈	—	2	4	no	—
108	C₂×Q₈	C₂×C₄×C₁₂₇	1	4	yes	—
109	C₂×D₄	—	2	8	no	—
110	C₂×D₄	—	2	4	no	—
111	C₂×D₄	C₂×C₄×C₁₂₇	1	8	yes	—
112	C₂×D₄	C₂×C₄×C₁₂₇	1	4	yes	—
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
115	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
116	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
117	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
118	C₂²×C₄	D₄×C₁₂₇	1	8	yes	—
119	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
120	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
121	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
122	M₁₆	—	2	8	no	—
123	M₁₆	—	2	8	no	—
124	C₂×C₈	—	2	8	no	—
125	C₂×C₈	C₂³×C₁₂₇	1	8	yes	—
126	C₂³	—	2	4	no	—
127	C₂³	—	4	2	no	—
128	C₂³	—	4	4	no	—
129	C₂³	C₂²⋊C₄×C₁₂₇	1	4	yes	—
130	C₂³	C₂²×C₄×C₁₂₇	1	4	yes	—
131	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
132	C₂³	C₂²×C₄×C₁₂₇	1	2	yes	—
133	Q₈	—	2	8	no	—
134	D₄	—	2	16	no	—
135	D₄	—	2	8	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
140	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	8	yes	—
141	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
142	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
143	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
144	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
145	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
146	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
147	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
148	C₈	—	2	8	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	4	no	—
152	C₂²	—	2	8	no	—
153	C₂²	—	4	4	no	—
154	C₂²	—	4	8	no	—
155	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
156	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
157	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
158	C₄	—	2	4	no	—
159	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
160	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
161	C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
162	C₂	—	2	2	no	—
163	C₂	—	2	4	no	—
164	C₂	—	4	4	no	—
165	C₂	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
166	C₂	#568×C₁₂₇	1	2	yes	—

	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. C₂²×M₁₆×C₁₂₇	C₂	semidirect
3. #408×C₁₂₇	C₂	semidirect
4. #407×C₁₂₇	C₂	semidirect
15. C₂×M₁₆×C₁₂₇	C₂²	semidirect
16. C₂²×C₈×C₁₂₇	C₂²	semidirect
17. C₂²⋊C₈×C₁₂₇	C₂²	semidirect
83. #2879	C₁₂₇	direct (N × H)
84. C₁₂₇	#2879	direct (N × H)
86. C₂²×M₁₆	C₂×C₁₂₇	semidirect
87. #408	C₂×C₁₂₇	semidirect
88. #407	C₂×C₁₂₇	semidirect
99. C₂×M₁₆	C₂²×C₁₂₇	semidirect
100. C₂²×C₈	C₂²×C₁₂₇	semidirect
101. C₂²⋊C₈	C₂²×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
C₂²×M₁₆×C₁₂₇	2
#408×C₁₂₇	2
#407×C₁₂₇	2
#2879	127

Subgroups — C127×#2879 #30635831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2879 #30635831