Subgroups — C₂×C₁₂₇×#1400 #44577228

Order 32512.

306 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₁₂₇×#1400	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂×M₃₂×C₁₂₇	—	2	2	no	—
6	C₂×M₃₂×C₁₂₇	—	2	2	no	—
7	C₂×M₃₂×C₁₂₇	C₄	1	2	yes	—
8	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
9	C₂²×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
10	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	M₆₄×C₁₂₇	—	2	2	no	—
12	M₆₄×C₁₂₇	C₂²	1	4	yes	complemented
13	C₂×C₃₂×C₁₂₇	—	2	2	no	—
14	C₂³×C₄×C₁₂₇	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	—	2	1	no	—
16	C₂²×C₈×C₁₂₇	—	2	1	no	—
17	C₂²×C₈×C₁₂₇	—	2	1	no	—
18	C₂²×C₈×C₁₂₇	—	2	1	no	—
19	C₂²×C₈×C₁₂₇	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₂₇	C₈	1	2	yes	—
21	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	M₃₂×C₁₂₇	—	2	4	no	—
24	M₃₂×C₁₂₇	—	2	4	no	—
25	C₂×C₁₆×C₁₂₇	—	2	1	no	—
26	C₂×C₁₆×C₁₂₇	—	2	1	no	—
27	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
30	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
31	C₂×C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₃₂×C₁₂₇	—	2	4	no	—
33	C₂⁴×C₁₂₇	C₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₁₂₇	—	2	1	no	—
35	C₂²×C₄×C₁₂₇	—	2	1	no	—
36	C₂²×C₄×C₁₂₇	—	2	1	no	—
37	C₂²×C₄×C₁₂₇	—	2	1	no	—
38	C₂²×C₄×C₁₂₇	C₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	C₁₆	1	2	yes	—
40	C₂²×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₁₂₇	—	2	1	no	—
43	C₂×C₈×C₁₂₇	—	2	1	no	—
44	C₂×C₈×C₁₂₇	—	2	2	no	—
45	C₂×C₈×C₁₂₇	—	2	2	no	—
46	C₂×C₈×C₁₂₇	—	2	2	no	—
47	C₂×C₈×C₁₂₇	—	2	2	no	—
48	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
55	C₁₆×C₁₂₇	—	2	2	no	—
56	C₁₆×C₁₂₇	C₂²⋊C₄	1	2	yes	—
57	C₁₆×C₁₂₇	C₂×D₄	1	2	yes	—
58	C₂³×C₁₂₇	—	2	1	no	—
59	C₂³×C₁₂₇	—	2	1	no	—
60	C₂³×C₁₂₇	—	4	2	no	—
61	C₂³×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
62	C₂³×C₁₂₇	C₂×C₁₆	1	2	yes	—
63	C₂×C₄×C₁₂₇	—	2	1	no	—
64	C₂×C₄×C₁₂₇	—	2	1	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	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	C₂×C₁₆	1	2	yes	—
72	C₂×C₄×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	M₃₂	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	M₃₂	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
76	C₈×C₁₂₇	—	2	2	no	—
77	C₈×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
78	C₈×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
79	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₈×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
81	C₂²×C₁₂₇	—	2	2	no	—
82	C₂²×C₁₂₇	—	2	2	no	—
83	C₂²×C₁₂₇	—	4	2	no	—
84	C₂²×C₁₂₇	—	4	2	no	—
85	C₂²×C₁₂₇	—	4	2	no	—
86	C₂²×C₁₂₇	C₂²⋊C₁₆	1	1	yes	characteristic
87	C₂²×C₁₂₇	C₂²⋊C₁₆	1	1	yes	characteristic
88	C₂²×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
89	C₄×C₁₂₇	—	2	2	no	—
90	C₄×C₁₂₇	C₂²⋊C₁₆	1	1	yes	characteristic
91	C₄×C₁₂₇	C₂²⋊C₁₆	1	1	yes	characteristic
92	C₄×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
93	C₄×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
94	C₂×#1400	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₁₂₇	—	2	1	no	—
96	C₂×C₁₂₇	—	2	1	no	—
97	C₂×C₁₂₇	—	4	2	no	—
98	C₂×C₁₂₇	#1400	1	2	yes	direct product factor
99	C₂×C₁₂₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
100	C₂²×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
101	C₂×M₆₄	C₂×C₁₂₇	1	2	yes	complemented
102	#1400	C₂×C₁₂₇	1	4	yes	direct product factor
103	C₁₂₇	C₂×#1400	1	1	yes	characteristic, direct product factor, Sylow
104	C₂³×C₈	C₄×C₁₂₇	1	1	yes	characteristic
105	C₂×M₃₂	—	2	2	no	—
106	C₂×M₃₂	—	2	2	no	—
107	C₂×M₃₂	C₄×C₁₂₇	1	2	yes	—
108	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
109	C₂²×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
110	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
111	M₆₄	—	2	2	no	—
112	M₆₄	C₂²×C₁₂₇	1	4	yes	complemented
113	C₂×C₃₂	—	2	2	no	—
114	C₂³×C₄	C₈×C₁₂₇	1	1	yes	characteristic
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	C₈×C₁₂₇	1	1	yes	characteristic
120	C₂²×C₈	C₈×C₁₂₇	1	2	yes	—
121	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
122	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
123	M₃₂	—	2	4	no	—
124	M₃₂	—	2	4	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
128	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
129	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
130	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
131	C₂×C₁₆	C₂³×C₁₂₇	1	1	yes	characteristic
132	C₃₂	—	2	4	no	—
133	C₂⁴	C₁₆×C₁₂₇	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	C₁₆×C₁₂₇	1	1	yes	characteristic
139	C₂²×C₄	C₁₆×C₁₂₇	1	2	yes	—
140	C₂²×C₄	C₂×C₈×C₁₂₇	1	2	yes	—
141	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
149	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
150	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
151	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
152	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
153	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
154	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
155	C₁₆	—	2	2	no	—
156	C₁₆	C₂²⋊C₄×C₁₂₇	1	2	yes	—
157	C₁₆	C₂×D₄×C₁₂₇	1	2	yes	—
158	C₂³	—	2	1	no	—
159	C₂³	—	2	1	no	—
160	C₂³	—	4	2	no	—
161	C₂³	C₂×C₁₆×C₁₂₇	1	1	yes	characteristic
162	C₂³	C₂×C₁₆×C₁₂₇	1	2	yes	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
171	C₂×C₄	C₂×C₁₆×C₁₂₇	1	2	yes	—
172	C₂×C₄	C₂×C₁₆×C₁₂₇	1	1	yes	characteristic
173	C₂×C₄	M₃₂×C₁₂₇	1	1	yes	characteristic
174	C₂×C₄	M₃₂×C₁₂₇	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
178	C₈	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
179	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
180	C₈	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂²⋊C₁₆×C₁₂₇	1	1	yes	characteristic
187	C₂²	C₂²⋊C₁₆×C₁₂₇	1	1	yes	characteristic
188	C₂²	C₂²×C₁₆×C₁₂₇	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	C₂²⋊C₁₆×C₁₂₇	1	1	yes	characteristic
191	C₄	C₂²⋊C₁₆×C₁₂₇	1	1	yes	characteristic
192	C₄	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
193	C₄	C₂×M₃₂×C₁₂₇	1	1	yes	characteristic
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	—	4	2	no	—
197	C₂	C₁₂₇×#1400	1	2	yes	direct product factor
198	C₂	C₂×C₂²⋊C₁₆×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₁₆×C₁₂₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂²⋊C₁₆
prime radical	C₂⁴×C₁₂₇	C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂²×C₁₆×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×M₆₄×C₁₂₇	C₂	semidirect
3. C₁₂₇×#1400	C₂	direct (N × H)
12. M₆₄×C₁₂₇	C₂²	semidirect
94. C₂×#1400	C₁₂₇	direct (N × H)
98. C₂×C₁₂₇	#1400	direct (N × H)
101. C₂×M₆₄	C₂×C₁₂₇	semidirect
102. #1400	C₂×C₁₂₇	direct (N × H)
103. C₁₂₇	C₂×#1400	direct (N × H)
112. M₆₄	C₂²×C₁₂₇	semidirect
197. C₂	C₁₂₇×#1400	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₁₂₇	2
C₂×M₆₄×C₁₂₇	2
C₁₂₇×#1400	2
C₂×#1400	127

Subgroups — C2×C127×#1400 #44577228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1400 #44577228