Subgroups — C₁₂₇×#1753 #30634705

Order 16256.

348 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#336×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#336×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₁₂₇	—	2	2	no	—
10	C₂×M₁₆×C₁₂₇	—	2	2	no	—
11	C₂×M₁₆×C₁₂₇	C₄	1	4	yes	complemented
12	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
17	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂²⋊C₈×C₁₂₇	C₄	1	4	yes	complemented
20	#97×C₁₂₇	—	2	2	no	—
21	C₄×C₈×C₁₂₇	—	2	2	no	—
22	#95×C₁₂₇	C₄	1	2	yes	—
23	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₂₇	—	2	1	no	—
25	C₂²×C₄×C₁₂₇	—	2	1	no	—
26	C₂²×C₄×C₁₂₇	—	2	1	no	—
27	C₂²×C₄×C₁₂₇	—	2	1	no	—
28	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	M₁₆×C₁₂₇	—	2	8	no	—
35	C₂×C₈×C₁₂₇	—	2	1	no	—
36	C₂×C₈×C₁₂₇	—	2	1	no	—
37	C₂×C₈×C₁₂₇	—	2	1	no	—
38	C₂×C₈×C₁₂₇	—	2	1	no	—
39	C₂×C₈×C₁₂₇	—	2	1	no	—
40	C₂×C₈×C₁₂₇	—	2	1	no	—
41	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
44	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
45	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
46	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
47	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
48	C₄²×C₁₂₇	—	2	1	no	—
49	C₄²×C₁₂₇	—	2	1	no	—
50	C₄²×C₁₂₇	—	2	1	no	—
51	C₄²×C₁₂₇	—	2	1	no	—
52	C₂³×C₁₂₇	—	2	1	no	—
53	C₂³×C₁₂₇	—	2	1	no	—
54	C₂³×C₁₂₇	—	2	2	no	—
55	C₂³×C₁₂₇	—	2	2	no	—
56	C₂³×C₁₂₇	C₄²	1	2	yes	—
57	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	—	2	2	no	—
59	C₂×C₄×C₁₂₇	—	2	1	no	—
60	C₂×C₄×C₁₂₇	—	2	1	no	—
61	C₂×C₄×C₁₂₇	—	2	2	no	—
62	C₂×C₄×C₁₂₇	—	2	2	no	—
63	C₂×C₄×C₁₂₇	—	2	2	no	—
64	C₂×C₄×C₁₂₇	—	2	2	no	—
65	C₂×C₄×C₁₂₇	—	2	2	no	—
66	C₂×C₄×C₁₂₇	—	2	2	no	—
67	C₂×C₄×C₁₂₇	—	2	2	no	—
68	C₂×C₄×C₁₂₇	—	2	2	no	—
69	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
70	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	complemented
71	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
77	C₈×C₁₂₇	—	2	2	no	—
78	C₈×C₁₂₇	—	2	2	no	—
79	C₈×C₁₂₇	—	2	2	no	—
80	C₈×C₁₂₇	C₂²⋊C₄	1	4	yes	complemented
81	C₂²×C₁₂₇	—	2	2	no	—
82	C₂²×C₁₂₇	—	2	2	no	—
83	C₂²×C₁₂₇	—	2	2	no	—
84	C₂²×C₁₂₇	—	4	2	no	—
85	C₂²×C₁₂₇	—	4	2	no	—
86	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
87	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
89	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
90	C₂²×C₁₂₇	#131	1	2	yes	—
91	C₄×C₁₂₇	—	2	2	no	—
92	C₄×C₁₂₇	—	2	4	no	—
93	C₄×C₁₂₇	—	4	2	no	—
94	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₁₂₇	#117	1	1	yes	characteristic
96	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
97	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
98	C₂×C₁₂₇	—	4	2	no	—
99	C₂×C₁₂₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₁₂₇	C₄×M₁₆	1	1	yes	characteristic
101	C₂×C₁₂₇	#405	1	1	yes	characteristic
102	C₂×C₁₂₇	#435	1	2	yes	—
103	C₂×C₁₂₇	#436	1	2	yes	—
104	#1753	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₁₂₇	#1753	1	1	yes	characteristic, direct product factor, Sylow
106	C₂²×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
107	C₂×C₂²⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic
108	C₂×#97	C₂×C₁₂₇	1	1	yes	characteristic, complemented
109	C₂×C₄×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
110	C₄×C₂²⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
111	#336	C₂×C₁₂₇	1	1	yes	characteristic, complemented
112	#336	C₂×C₁₂₇	1	1	yes	characteristic, complemented
113	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	2	2	no	—
116	C₂×M₁₆	C₄×C₁₂₇	1	4	yes	complemented
117	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
118	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
119	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
120	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
121	C₂×C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
122	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
124	C₂²⋊C₈	C₄×C₁₂₇	1	4	yes	complemented
125	#97	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	#95	C₄×C₁₂₇	1	2	yes	—
128	C₂⁴	C₂×C₄×C₁₂₇	1	1	yes	characteristic
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
134	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
136	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
137	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
138	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
139	M₁₆	—	2	8	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
147	C₂×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
148	C₂×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
149	C₂×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
150	C₂×C₈	D₄×C₁₂₇	1	2	yes	—
151	C₂×C₈	D₄×C₁₂₇	1	2	yes	—
152	C₂²⋊C₄	—	2	4	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₂³	—	2	1	no	—
158	C₂³	—	2	1	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	C₄²×C₁₂₇	1	2	yes	—
162	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₄²×C₁₂₇	1	2	yes	—
175	C₂×C₄	M₁₆×C₁₂₇	1	4	yes	complemented
176	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
178	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
179	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
180	C₂×C₄	#41×C₁₂₇	1	1	yes	characteristic
181	C₂×C₄	#41×C₁₂₇	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	2	2	no	—
184	C₈	—	2	2	no	—
185	C₈	C₂²⋊C₄×C₁₂₇	1	4	yes	complemented
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
192	C₂²	C₄×D₄×C₁₂₇	1	1	yes	characteristic
193	C₂²	C₄×D₄×C₁₂₇	1	1	yes	characteristic
194	C₂²	C₂×M₁₆×C₁₂₇	1	2	yes	—
195	C₂²	#131×C₁₂₇	1	2	yes	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	—	4	2	no	—
199	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
200	C₄	#117×C₁₂₇	1	1	yes	characteristic
201	C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
202	C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
203	C₂	—	4	2	no	—
204	C₂	C₄×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
205	C₂	C₄×M₁₆×C₁₂₇	1	1	yes	characteristic
206	C₂	#405×C₁₂₇	1	1	yes	characteristic
207	C₂	#435×C₁₂₇	1	2	yes	—
208	C₂	#436×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂³
commutator	C₂²	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂³×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
3. C₂×#97×C₁₂₇	C₂	semidirect
4. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
6. #336×C₁₂₇	C₂	semidirect
7. #336×C₁₂₇	C₂	semidirect
11. C₂×M₁₆×C₁₂₇	C₄	semidirect
19. C₂²⋊C₈×C₁₂₇	C₄	semidirect
70. C₂×C₄×C₁₂₇	M₁₆	semidirect
80. C₈×C₁₂₇	C₂²⋊C₄	semidirect
104. #1753	C₁₂₇	direct (N × H)
105. C₁₂₇	#1753	direct (N × H)
108. C₂×#97	C₂×C₁₂₇	semidirect
109. C₂×C₄×C₈	C₂×C₁₂₇	semidirect
111. #336	C₂×C₁₂₇	semidirect
112. #336	C₂×C₁₂₇	semidirect
116. C₂×M₁₆	C₄×C₁₂₇	semidirect
124. C₂²⋊C₈	C₄×C₁₂₇	semidirect
175. C₂×C₄	M₁₆×C₁₂₇	semidirect
185. C₈	C₂²⋊C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₁₂₇	2
C₂×C₂²⋊C₈×C₁₂₇	2
C₂×#97×C₁₂₇	2
C₂×C₄×C₈×C₁₂₇	2
C₄×C₂²⋊C₄×C₁₂₇	2
#336×C₁₂₇	2
#1753	127

Subgroups — C127×#1753 #30634705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#1753 #30634705