Subgroups — C₂×C₁₂₇×#1366 #44575628

Order 32512.

294 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1366	C₂	1	4	yes	maximal, direct product factor
5	C₂×#125×C₁₂₇	C₄	1	2	yes	—
6	C₂×M₃₂×C₁₂₇	—	2	1	no	—
7	C₂×M₃₂×C₁₂₇	—	2	1	no	—
8	#499×C₁₂₇	—	2	1	no	—
9	#499×C₁₂₇	—	2	1	no	—
10	#499×C₁₂₇	C₄	1	2	yes	—
11	#499×C₁₂₇	C₂²	1	2	yes	—
12	C₂×#107×C₁₂₇	—	2	1	no	—
13	C₂×#106×C₁₂₇	—	2	1	no	—
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	#364×C₁₂₇	—	2	1	no	—
16	#364×C₁₂₇	—	2	1	no	—
17	#364×C₁₂₇	C₂²	1	2	yes	—
18	#364×C₁₂₇	C₂²	1	2	yes	—
19	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
21	#125×C₁₂₇	—	2	2	no	—
22	#125×C₁₂₇	—	4	2	no	—
23	#125×C₁₂₇	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₂₇	—	4	2	no	—
26	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₁₂₇	—	2	2	no	—
28	M₃₂×C₁₂₇	—	2	2	no	—
29	M₃₂×C₁₂₇	—	4	1	no	—
30	M₃₂×C₁₂₇	—	4	1	no	—
31	C₂×C₁₆×C₁₂₇	—	4	1	no	—
32	C₂×C₁₆×C₁₂₇	—	4	1	no	—
33	#107×C₁₂₇	—	2	2	no	—
34	#107×C₁₂₇	—	2	2	no	—
35	#106×C₁₂₇	—	2	2	no	—
36	#106×C₁₂₇	—	2	2	no	—
37	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
39	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	—	2	1	no	—
42	C₂²×C₄×C₁₂₇	—	4	2	no	—
43	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
52	C₂×C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₁₂₇	—	2	4	no	—
57	C₄⋊C₄×C₁₂₇	—	4	4	no	—
58	C₄⋊C₄×C₁₂₇	—	4	4	no	—
59	C₄⋊C₄×C₁₂₇	—	4	2	no	—
60	C₄²×C₁₂₇	—	2	1	no	—
61	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
63	C₁₆×C₁₂₇	—	4	2	no	—
64	C₁₆×C₁₂₇	—	4	2	no	—
65	C₂³×C₁₂₇	#100	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	—	2	1	no	—
67	C₂×C₄×C₁₂₇	—	2	1	no	—
68	C₂×C₄×C₁₂₇	—	4	2	no	—
69	C₂×C₄×C₁₂₇	—	4	4	no	—
70	C₂×C₄×C₁₂₇	—	8	4	no	—
71	C₂×C₄×C₁₂₇	#100	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₈×C₁₂₇	#103	1	1	yes	characteristic
78	C₈×C₁₂₇	#103	1	1	yes	characteristic
79	C₈×C₁₂₇	#103	1	1	yes	characteristic
80	C₈×C₁₂₇	#103	1	1	yes	characteristic
81	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
82	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
83	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
84	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
85	C₂²×C₁₂₇	—	2	2	no	—
86	C₂²×C₁₂₇	#333	1	1	yes	characteristic
87	C₂²×C₁₂₇	#333	1	1	yes	characteristic
88	C₂²×C₁₂₇	C₂×#100	1	1	yes	characteristic
89	C₄×C₁₂₇	—	4	1	no	—
90	C₄×C₁₂₇	—	4	1	no	—
91	C₄×C₁₂₇	—	8	4	no	—
92	C₄×C₁₂₇	#333	1	1	yes	characteristic
93	C₄×C₁₂₇	#333	1	1	yes	characteristic
94	C₄×C₁₂₇	C₂×#103	1	1	yes	characteristic
95	C₄×C₁₂₇	C₂×#103	1	1	yes	characteristic
96	C₂×#1366	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂×C₁₂₇	—	2	1	no	—
98	C₂×C₁₂₇	—	2	1	no	—
99	C₂×C₁₂₇	#1366	1	2	yes	direct product factor
100	C₂×C₁₂₇	C₂×#333	1	1	yes	characteristic
101	C₂×#499	C₂×C₁₂₇	1	1	yes	characteristic
102	C₂×#364	C₂×C₁₂₇	1	1	yes	characteristic
103	C₂×#364	C₂×C₁₂₇	1	1	yes	characteristic
104	#1366	C₂×C₁₂₇	1	4	yes	direct product factor
105	C₁₂₇	C₂×#1366	1	1	yes	characteristic, direct product factor, Sylow
106	C₂×#125	C₄×C₁₂₇	1	2	yes	—
107	C₂×M₃₂	—	2	1	no	—
108	C₂×M₃₂	—	2	1	no	—
109	#499	—	2	1	no	—
110	#499	—	2	1	no	—
111	#499	C₄×C₁₂₇	1	2	yes	—
112	#499	C₂²×C₁₂₇	1	2	yes	—
113	C₂×#107	—	2	1	no	—
114	C₂×#106	—	2	1	no	—
115	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
116	#364	—	2	1	no	—
117	#364	—	2	1	no	—
118	#364	C₂²×C₁₂₇	1	2	yes	—
119	#364	C₂²×C₁₂₇	1	2	yes	—
120	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
121	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
122	#125	—	2	2	no	—
123	#125	—	4	2	no	—
124	#125	C₂×C₄×C₁₂₇	1	4	yes	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	4	2	no	—
127	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	M₃₂	—	4	1	no	—
131	M₃₂	—	4	1	no	—
132	C₂×C₁₆	—	4	1	no	—
133	C₂×C₁₆	—	4	1	no	—
134	#107	—	2	2	no	—
135	#107	—	2	2	no	—
136	#106	—	2	2	no	—
137	#106	—	2	2	no	—
138	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
139	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
140	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
141	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
145	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
146	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
147	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
148	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
149	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
150	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
151	C₂×C₈	Q₁₆×C₁₂₇	1	1	yes	characteristic
152	C₂×C₈	Q₁₆×C₁₂₇	1	1	yes	characteristic
153	C₂×C₈	Q₁₆×C₁₂₇	1	1	yes	characteristic
154	C₂×C₈	Q₁₆×C₁₂₇	1	1	yes	characteristic
155	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
156	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄²	—	2	1	no	—
162	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
163	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
164	C₁₆	—	4	2	no	—
165	C₁₆	—	4	2	no	—
166	C₂³	#100×C₁₂₇	1	1	yes	characteristic
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	8	4	no	—
172	C₂×C₄	#100×C₁₂₇	1	1	yes	characteristic
173	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
174	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
175	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
176	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
177	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
178	C₈	#103×C₁₂₇	1	1	yes	characteristic
179	C₈	#103×C₁₂₇	1	1	yes	characteristic
180	C₈	#103×C₁₂₇	1	1	yes	characteristic
181	C₈	#103×C₁₂₇	1	1	yes	characteristic
182	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
183	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
184	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
185	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	#333×C₁₂₇	1	1	yes	characteristic
188	C₂²	#333×C₁₂₇	1	1	yes	characteristic
189	C₂²	C₂×#100×C₁₂₇	1	1	yes	characteristic
190	C₄	—	4	1	no	—
191	C₄	—	4	1	no	—
192	C₄	—	8	4	no	—
193	C₄	#333×C₁₂₇	1	1	yes	characteristic
194	C₄	#333×C₁₂₇	1	1	yes	characteristic
195	C₄	C₂×#103×C₁₂₇	1	1	yes	characteristic
196	C₄	C₂×#103×C₁₂₇	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	C₁₂₇×#1366	1	2	yes	direct product factor
200	C₂	C₂×#333×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#333
commutator	C₄²	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1366	C₂	direct (N × H)
96. C₂×#1366	C₁₂₇	direct (N × H)
99. C₂×C₁₂₇	#1366	direct (N × H)
104. #1366	C₂×C₁₂₇	direct (N × H)
105. C₁₂₇	C₂×#1366	direct (N × H)
199. C₂	C₁₂₇×#1366	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×#499×C₁₂₇	2
C₂×#364×C₁₂₇	2
C₁₂₇×#1366	2
C₂×#1366	127

Subgroups — C2×C127×#1366 #44575628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1366 #44575628