Subgroups — C₂×C₁₂₇×#1420 #44577297

Order 32512.

286 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#368×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1420	C₂	1	4	yes	maximal, direct product factor
5	C₂²×Q₁₆×C₁₂₇	—	2	1	no	—
6	C₂×Q₃₂×C₁₂₇	—	2	1	no	—
7	C₂×Q₃₂×C₁₂₇	—	2	1	no	—
8	C₂×Q₃₂×C₁₂₇	—	2	1	no	—
9	C₂×Q₃₂×C₁₂₇	C₄	1	2	yes	—
10	C₂×Q₃₂×C₁₂₇	C₄	1	2	yes	—
11	C₂×Q₃₂×C₁₂₇	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₁₂₇	—	2	1	no	—
14	M₆₄×C₁₂₇	—	2	1	no	—
15	M₆₄×C₁₂₇	C₂²	1	2	yes	—
16	C₂×C₃₂×C₁₂₇	—	2	1	no	—
17	#368×C₁₂₇	—	2	1	no	—
18	#368×C₁₂₇	C₂²	1	2	yes	—
19	C₂²×Q₈×C₁₂₇	—	4	1	no	—
20	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
21	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
22	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
23	C₂×Q₁₆×C₁₂₇	—	4	1	no	—
24	C₂×Q₁₆×C₁₂₇	—	4	1	no	—
25	C₂×Q₁₆×C₁₂₇	—	4	1	no	—
26	C₂×M₁₆×C₁₂₇	—	4	1	no	—
27	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
28	Q₃₂×C₁₂₇	—	2	2	no	—
29	Q₃₂×C₁₂₇	—	2	2	no	—
30	Q₃₂×C₁₂₇	—	2	2	no	—
31	Q₃₂×C₁₂₇	C₂×C₄	1	4	yes	—
32	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
38	#108×C₁₂₇	—	2	2	no	—
39	#108×C₁₂₇	—	4	1	no	—
40	C₃₂×C₁₂₇	—	2	2	no	—
41	C₂×Q₈×C₁₂₇	—	4	2	no	—
42	C₂×Q₈×C₁₂₇	—	4	2	no	—
43	C₂×Q₈×C₁₂₇	—	4	2	no	—
44	C₂×Q₈×C₁₂₇	—	8	1	no	—
45	C₂×Q₈×C₁₂₇	—	8	1	no	—
46	C₂×Q₈×C₁₂₇	—	8	1	no	—
47	C₂²×C₄×C₁₂₇	—	8	1	no	—
48	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
49	Q₁₆×C₁₂₇	—	2	4	no	—
50	Q₁₆×C₁₂₇	—	4	2	no	—
51	Q₁₆×C₁₂₇	—	4	2	no	—
52	Q₁₆×C₁₂₇	—	4	2	no	—
53	M₁₆×C₁₂₇	—	4	2	no	—
54	M₁₆×C₁₂₇	—	8	1	no	—
55	C₂×C₈×C₁₂₇	—	8	1	no	—
56	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₂₇	D₈	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
60	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
61	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
62	C₁₆×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₁₆×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₁₆×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	C₁₆×C₁₂₇	C₂×D₄	1	1	yes	characteristic
66	C₂³×C₁₂₇	SD₃₂	1	1	yes	characteristic
67	Q₈×C₁₂₇	—	4	4	no	—
68	Q₈×C₁₂₇	—	8	2	no	—
69	Q₈×C₁₂₇	—	8	2	no	—
70	Q₈×C₁₂₇	—	8	2	no	—
71	C₂×C₄×C₁₂₇	—	8	2	no	—
72	C₂×C₄×C₁₂₇	—	8	2	no	—
73	C₂×C₄×C₁₂₇	—	8	2	no	—
74	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	SD₃₂	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
80	C₈×C₁₂₇	—	8	2	no	—
81	C₈×C₁₂₇	#102	1	1	yes	characteristic
82	C₈×C₁₂₇	#102	1	1	yes	characteristic
83	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
85	C₂²×C₁₂₇	—	2	2	no	—
86	C₂²×C₁₂₇	#357	1	1	yes	characteristic
87	C₂²×C₁₂₇	#357	1	1	yes	characteristic
88	C₂²×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
89	C₄×C₁₂₇	—	8	4	no	—
90	C₄×C₁₂₇	#357	1	1	yes	characteristic
91	C₄×C₁₂₇	#357	1	1	yes	characteristic
92	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
93	C₄×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
94	C₂×#1420	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	#1420	1	2	yes	direct product factor
98	C₂×C₁₂₇	C₂×#357	1	1	yes	characteristic
99	C₂²×Q₃₂	C₂×C₁₂₇	1	1	yes	characteristic
100	C₂×M₆₄	C₂×C₁₂₇	1	1	yes	characteristic
101	C₂×#368	C₂×C₁₂₇	1	1	yes	characteristic
102	#1420	C₂×C₁₂₇	1	4	yes	direct product factor
103	C₁₂₇	C₂×#1420	1	1	yes	characteristic, direct product factor, Sylow
104	C₂²×Q₁₆	—	2	1	no	—
105	C₂×Q₃₂	—	2	1	no	—
106	C₂×Q₃₂	—	2	1	no	—
107	C₂×Q₃₂	—	2	1	no	—
108	C₂×Q₃₂	C₄×C₁₂₇	1	2	yes	—
109	C₂×Q₃₂	C₄×C₁₂₇	1	2	yes	—
110	C₂×Q₃₂	C₂²×C₁₂₇	1	2	yes	—
111	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
112	C₂×#108	—	2	1	no	—
113	M₆₄	—	2	1	no	—
114	M₆₄	C₂²×C₁₂₇	1	2	yes	—
115	C₂×C₃₂	—	2	1	no	—
116	#368	—	2	1	no	—
117	#368	C₂²×C₁₂₇	1	2	yes	—
118	C₂²×Q₈	—	4	1	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	—	2	2	no	—
122	C₂×Q₁₆	—	4	1	no	—
123	C₂×Q₁₆	—	4	1	no	—
124	C₂×Q₁₆	—	4	1	no	—
125	C₂×M₁₆	—	4	1	no	—
126	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
127	Q₃₂	—	2	2	no	—
128	Q₃₂	—	2	2	no	—
129	Q₃₂	—	2	2	no	—
130	Q₃₂	C₂×C₄×C₁₂₇	1	4	yes	—
131	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
132	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
133	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
134	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
135	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
136	C₂×C₁₆	C₂³×C₁₂₇	1	1	yes	characteristic
137	#108	—	2	2	no	—
138	#108	—	4	1	no	—
139	C₃₂	—	2	2	no	—
140	C₂×Q₈	—	4	2	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	8	1	no	—
144	C₂×Q₈	—	8	1	no	—
145	C₂×Q₈	—	8	1	no	—
146	C₂²×C₄	—	8	1	no	—
147	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
148	Q₁₆	—	2	4	no	—
149	Q₁₆	—	4	2	no	—
150	Q₁₆	—	4	2	no	—
151	Q₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	8	1	no	—
154	C₂×C₈	—	8	1	no	—
155	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
157	C₂×C₈	D₈×C₁₂₇	1	1	yes	characteristic
158	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
159	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
161	C₁₆	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
162	C₁₆	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
163	C₁₆	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
164	C₁₆	C₂×D₄×C₁₂₇	1	1	yes	characteristic
165	C₂³	SD₃₂×C₁₂₇	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	Q₈	—	8	2	no	—
168	Q₈	—	8	2	no	—
169	Q₈	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
174	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
175	C₂×C₄	D₁₆×C₁₂₇	1	1	yes	characteristic
176	C₂×C₄	D₁₆×C₁₂₇	1	1	yes	characteristic
177	C₂×C₄	SD₃₂×C₁₂₇	1	1	yes	characteristic
178	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
179	C₈	—	8	2	no	—
180	C₈	#102×C₁₂₇	1	1	yes	characteristic
181	C₈	#102×C₁₂₇	1	1	yes	characteristic
182	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
183	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	#357×C₁₂₇	1	1	yes	characteristic
186	C₂²	#357×C₁₂₇	1	1	yes	characteristic
187	C₂²	C₂×SD₃₂×C₁₂₇	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	#357×C₁₂₇	1	1	yes	characteristic
190	C₄	#357×C₁₂₇	1	1	yes	characteristic
191	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
192	C₄	C₂×D₁₆×C₁₂₇	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₁₂₇×#1420	1	2	yes	direct product factor
196	C₂	C₂×#357×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#357
commutator	C₁₆	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ C₂²×C₁₆×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1420	C₂	direct (N × H)
94. C₂×#1420	C₁₂₇	direct (N × H)
97. C₂×C₁₂₇	#1420	direct (N × H)
102. #1420	C₂×C₁₂₇	direct (N × H)
103. C₁₂₇	C₂×#1420	direct (N × H)
195. C₂	C₁₂₇×#1420	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×Q₃₂×C₁₂₇	2
C₂×M₆₄×C₁₂₇	2
C₂×#368×C₁₂₇	2
C₁₂₇×#1420	2
C₂×#1420	127

Subgroups — C2×C127×#1420 #44577297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1420 #44577297