Subgroups — C₂×#472×C₁₂₇ #30636017

Order 16256.

488 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×AGL(1,ℤ/8)×C₁₂₇	C₂	1	2	yes	maximal, complemented
2	C₂²×D₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#472×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#100×C₁₂₇	C₂	1	2	yes	maximal, complemented
7	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₁₂₇	C₂²	1	2	yes	—
9	AGL(1,ℤ/8)×C₁₂₇	—	2	4	no	—
10	AGL(1,ℤ/8)×C₁₂₇	C₂²	1	8	yes	complemented
11	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
12	C₂×D₈×C₁₂₇	—	2	2	no	—
13	C₂×D₈×C₁₂₇	—	2	1	no	—
14	C₂×D₈×C₁₂₇	—	2	1	no	—
15	C₂×D₈×C₁₂₇	—	2	2	no	—
16	C₂×D₈×C₁₂₇	C₂²	1	4	yes	complemented
17	#131×C₁₂₇	—	2	2	no	—
18	#131×C₁₂₇	C₂²	1	4	yes	complemented
19	C₂×M₁₆×C₁₂₇	—	2	1	no	—
20	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
22	C₂²×C₈×C₁₂₇	—	2	1	no	—
23	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	#108×C₁₂₇	C₂²	1	4	yes	complemented
25	#100×C₁₂₇	C₂²	1	8	yes	complemented
26	C₂⁴×C₁₂₇	—	2	2	no	—
27	#41×C₁₂₇	—	2	1	no	—
28	#41×C₁₂₇	—	2	1	no	—
29	#41×C₁₂₇	D₄	1	2	yes	—
30	#41×C₁₂₇	C₂³	1	2	yes	—
31	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂×D₄×C₁₂₇	—	2	1	no	—
33	C₂×D₄×C₁₂₇	—	2	2	no	—
34	C₂×D₄×C₁₂₇	—	2	2	no	—
35	C₂×D₄×C₁₂₇	—	2	2	no	—
36	C₂×D₄×C₁₂₇	—	2	2	no	—
37	C₂×D₄×C₁₂₇	—	2	2	no	—
38	C₂×D₄×C₁₂₇	D₄	1	1	yes	characteristic
39	C₂×D₄×C₁₂₇	C₂³	1	4	yes	—
40	C₂²×C₄×C₁₂₇	—	2	1	no	—
41	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	SD₁₆×C₁₂₇	—	2	8	no	—
43	D₈×C₁₂₇	—	2	8	no	—
44	D₈×C₁₂₇	—	2	4	no	—
45	D₈×C₁₂₇	—	2	4	no	—
46	M₁₆×C₁₂₇	—	2	1	no	—
47	M₁₆×C₁₂₇	—	2	2	no	—
48	M₁₆×C₁₂₇	—	2	4	no	—
49	M₁₆×C₁₂₇	C₂³	1	2	yes	—
50	M₁₆×C₁₂₇	C₂³	1	4	yes	—
51	C₂×C₈×C₁₂₇	—	2	1	no	—
52	C₂×C₈×C₁₂₇	—	2	2	no	—
53	C₂×C₈×C₁₂₇	—	2	4	no	—
54	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
57	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
58	C₂³×C₁₂₇	—	2	1	no	—
59	C₂³×C₁₂₇	—	2	4	no	—
60	C₂³×C₁₂₇	—	4	2	no	—
61	C₂³×C₁₂₇	—	4	2	no	—
62	C₂³×C₁₂₇	—	8	4	no	—
63	C₂³×C₁₂₇	#41	1	1	yes	characteristic
64	Q₈×C₁₂₇	—	2	1	no	—
65	Q₈×C₁₂₇	C₂×D₄	1	2	yes	—
66	D₄×C₁₂₇	—	2	2	no	—
67	D₄×C₁₂₇	—	2	2	no	—
68	D₄×C₁₂₇	—	2	1	no	—
69	D₄×C₁₂₇	—	2	4	no	—
70	D₄×C₁₂₇	—	2	4	no	—
71	D₄×C₁₂₇	—	4	2	no	—
72	D₄×C₁₂₇	—	4	2	no	—
73	D₄×C₁₂₇	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₁₂₇	—	2	1	no	—
75	C₂×C₄×C₁₂₇	—	2	2	no	—
76	C₂×C₄×C₁₂₇	—	2	2	no	—
77	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
83	C₈×C₁₂₇	—	2	2	no	—
84	C₈×C₁₂₇	—	2	4	no	—
85	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
86	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
87	C₂²×C₁₂₇	—	2	2	no	—
88	C₂²×C₁₂₇	—	2	2	no	—
89	C₂²×C₁₂₇	—	4	2	no	—
90	C₂²×C₁₂₇	—	4	4	no	—
91	C₂²×C₁₂₇	—	8	4	no	—
92	C₂²×C₁₂₇	—	8	4	no	—
93	C₂²×C₁₂₇	—	8	4	no	—
94	C₂²×C₁₂₇	#121	1	1	yes	characteristic
95	C₂²×C₁₂₇	#121	1	1	yes	characteristic
96	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
97	C₄×C₁₂₇	—	2	2	no	—
98	C₄×C₁₂₇	#121	1	1	yes	characteristic
99	C₄×C₁₂₇	#121	1	1	yes	characteristic
100	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₁₂₇	—	2	1	no	—
103	C₂×C₁₂₇	—	2	1	no	—
104	C₂×C₁₂₇	—	4	2	no	—
105	C₂×C₁₂₇	—	8	4	no	—
106	C₂×C₁₂₇	#472	1	2	yes	direct product factor
107	C₂×C₁₂₇	C₂×#121	1	1	yes	characteristic
108	C₂×#472	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₁₂₇	C₂×#472	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×AGL(1,ℤ/8)	C₂×C₁₂₇	1	2	yes	complemented
111	C₂²×D₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
112	C₂×#131	C₂×C₁₂₇	1	1	yes	characteristic, complemented
113	#472	C₂×C₁₂₇	1	8	yes	direct product factor
114	C₂×#108	C₂×C₁₂₇	1	1	yes	characteristic, complemented
115	C₂×#100	C₂×C₁₂₇	1	2	yes	complemented
116	C₂×#41	C₂²×C₁₂₇	1	1	yes	characteristic
117	C₂²×D₄	C₂²×C₁₂₇	1	2	yes	—
118	AGL(1,ℤ/8)	—	2	4	no	—
119	AGL(1,ℤ/8)	C₂²×C₁₂₇	1	8	yes	complemented
120	C₂×SD₁₆	—	2	2	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×D₈	—	2	1	no	—
123	C₂×D₈	—	2	1	no	—
124	C₂×D₈	—	2	2	no	—
125	C₂×D₈	C₂²×C₁₂₇	1	4	yes	complemented
126	#131	—	2	2	no	—
127	#131	C₂²×C₁₂₇	1	4	yes	complemented
128	C₂×M₁₆	—	2	1	no	—
129	C₂×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
130	C₂×M₁₆	C₂²×C₁₂₇	1	2	yes	—
131	C₂²×C₈	—	2	1	no	—
132	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
133	#108	C₂²×C₁₂₇	1	4	yes	complemented
134	#100	C₂²×C₁₂₇	1	8	yes	complemented
135	C₂⁴	—	2	2	no	—
136	#41	—	2	1	no	—
137	#41	—	2	1	no	—
138	#41	D₄×C₁₂₇	1	2	yes	—
139	#41	C₂³×C₁₂₇	1	2	yes	—
140	C₂×Q₈	D₄×C₁₂₇	1	1	yes	characteristic
141	C₂×D₄	—	2	1	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	D₄×C₁₂₇	1	1	yes	characteristic
148	C₂×D₄	C₂³×C₁₂₇	1	4	yes	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
151	SD₁₆	—	2	8	no	—
152	D₈	—	2	8	no	—
153	D₈	—	2	4	no	—
154	D₈	—	2	4	no	—
155	M₁₆	—	2	1	no	—
156	M₁₆	—	2	2	no	—
157	M₁₆	—	2	4	no	—
158	M₁₆	C₂³×C₁₂₇	1	2	yes	—
159	M₁₆	C₂³×C₁₂₇	1	4	yes	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	D₄×C₁₂₇	1	1	yes	characteristic
164	C₂×C₈	D₄×C₁₂₇	1	1	yes	characteristic
165	C₂×C₈	D₄×C₁₂₇	1	2	yes	—
166	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	4	no	—
169	C₂³	—	4	2	no	—
170	C₂³	—	4	2	no	—
171	C₂³	—	8	4	no	—
172	C₂³	#41×C₁₂₇	1	1	yes	characteristic
173	Q₈	—	2	1	no	—
174	Q₈	C₂×D₄×C₁₂₇	1	2	yes	—
175	D₄	—	2	2	no	—
176	D₄	—	2	2	no	—
177	D₄	—	2	1	no	—
178	D₄	—	2	4	no	—
179	D₄	—	2	4	no	—
180	D₄	—	4	2	no	—
181	D₄	—	4	2	no	—
182	D₄	C₂×D₄×C₁₂₇	1	2	yes	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
187	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
188	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
189	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
190	C₂×C₄	#41×C₁₂₇	1	1	yes	characteristic
191	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	4	no	—
194	C₈	C₂×D₄×C₁₂₇	1	2	yes	—
195	C₈	C₂×D₄×C₁₂₇	1	2	yes	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	—	4	4	no	—
200	C₂²	—	8	4	no	—
201	C₂²	—	8	4	no	—
202	C₂²	—	8	4	no	—
203	C₂²	#121×C₁₂₇	1	1	yes	characteristic
204	C₂²	#121×C₁₂₇	1	1	yes	characteristic
205	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	#121×C₁₂₇	1	1	yes	characteristic
208	C₄	#121×C₁₂₇	1	1	yes	characteristic
209	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
210	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	—	4	2	no	—
214	C₂	—	8	4	no	—
215	C₂	#472×C₁₂₇	1	2	yes	direct product factor
216	C₂	C₂×#121×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×AGL(1,ℤ/8)×C₁₂₇	C₂	semidirect
2. C₂²×D₈×C₁₂₇	C₂	semidirect
3. C₂×#131×C₁₂₇	C₂	semidirect
4. #472×C₁₂₇	C₂	direct (N × H)
5. C₂×#108×C₁₂₇	C₂	semidirect
6. C₂×#100×C₁₂₇	C₂	semidirect
10. AGL(1,ℤ/8)×C₁₂₇	C₂²	semidirect
16. C₂×D₈×C₁₂₇	C₂²	semidirect
18. #131×C₁₂₇	C₂²	semidirect
24. #108×C₁₂₇	C₂²	semidirect
25. #100×C₁₂₇	C₂²	semidirect
106. C₂×C₁₂₇	#472	direct (N × H)
108. C₂×#472	C₁₂₇	direct (N × H)
109. C₁₂₇	C₂×#472	direct (N × H)
110. C₂×AGL(1,ℤ/8)	C₂×C₁₂₇	semidirect
111. C₂²×D₈	C₂×C₁₂₇	semidirect
112. C₂×#131	C₂×C₁₂₇	semidirect
113. #472	C₂×C₁₂₇	direct (N × H)
114. C₂×#108	C₂×C₁₂₇	semidirect
115. C₂×#100	C₂×C₁₂₇	semidirect
119. AGL(1,ℤ/8)	C₂²×C₁₂₇	semidirect
125. C₂×D₈	C₂²×C₁₂₇	semidirect
127. #131	C₂²×C₁₂₇	semidirect
133. #108	C₂²×C₁₂₇	semidirect
134. #100	C₂²×C₁₂₇	semidirect
215. C₂	#472×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×AGL(1,ℤ/8)×C₁₂₇	2
C₂²×D₈×C₁₂₇	2
C₂×#131×C₁₂₇	2
#472×C₁₂₇	2
C₂×#108×C₁₂₇	2
C₂×#100×C₁₂₇	2
C₂×#472	127

Subgroups — C2×#472×C127 #30636017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#472×C₁₂₇ #30636017