Subgroups — C₁₂₇×#7278 #44570951

Order 32512.

246 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	C₁₂₇×#2150	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1844	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₂₇	—	2	1	no	—
5	#431×C₁₂₇	—	2	1	no	—
6	C₂×#106×C₁₂₇	—	2	1	no	—
7	C₂×#106×C₁₂₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	#384×C₁₂₇	—	2	1	no	—
10	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
11	C₄⋊C₁₆×C₁₂₇	C₄	1	2	yes	complemented
12	#335×C₁₂₇	C₄	1	2	yes	complemented
13	#334×C₁₂₇	—	2	1	no	—
14	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
17	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
18	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
19	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
20	M₃₂×C₁₂₇	—	4	1	no	—
21	M₃₂×C₁₂₇	—	4	1	no	—
22	C₂×C₁₆×C₁₂₇	—	2	2	no	—
23	#106×C₁₂₇	—	4	1	no	—
24	#106×C₁₂₇	—	4	1	no	—
25	#106×C₁₂₇	—	4	2	no	—
26	C₄⋊C₈×C₁₂₇	—	2	2	no	—
27	C₂²⋊C₈×C₁₂₇	—	4	1	no	—
28	C₄×C₈×C₁₂₇	—	4	1	no	—
29	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₂₇	—	4	2	no	—
34	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	—	2	1	no	—
38	C₂×C₈×C₁₂₇	—	2	1	no	—
39	C₂×C₈×C₁₂₇	—	4	2	no	—
40	C₂×C₈×C₁₂₇	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	D₈	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
48	C₄⋊C₄×C₁₂₇	—	4	2	no	—
49	C₄⋊C₄×C₁₂₇	—	4	2	no	—
50	C₄⋊C₄×C₁₂₇	—	8	2	no	—
51	C₄⋊C₄×C₁₂₇	—	8	2	no	—
52	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
55	C₄²×C₁₂₇	Q₁₆	1	1	yes	characteristic
56	C₁₆×C₁₂₇	—	4	2	no	—
57	C₂³×C₁₂₇	#102	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	—	2	2	no	—
59	C₂×C₄×C₁₂₇	—	2	2	no	—
60	C₂×C₄×C₁₂₇	—	8	2	no	—
61	C₂×C₄×C₁₂₇	—	8	2	no	—
62	C₂×C₄×C₁₂₇	—	8	2	no	—
63	C₂×C₄×C₁₂₇	#95	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	#100	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	#101	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	#106	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
73	C₈×C₁₂₇	—	2	1	no	—
74	C₈×C₁₂₇	—	2	1	no	—
75	C₈×C₁₂₇	—	8	2	no	—
76	C₈×C₁₂₇	#102	1	1	yes	characteristic
77	C₈×C₁₂₇	#103	1	1	yes	characteristic
78	C₈×C₁₂₇	#106	1	1	yes	characteristic
79	C₈×C₁₂₇	#107	1	1	yes	characteristic
80	C₂²×C₁₂₇	—	2	1	no	—
81	C₂²×C₁₂₇	—	2	1	no	—
82	C₂²×C₁₂₇	#331	1	1	yes	characteristic
83	C₂²×C₁₂₇	#332	1	1	yes	characteristic
84	C₂²×C₁₂₇	#340	1	1	yes	characteristic
85	C₄×C₁₂₇	—	2	1	no	—
86	C₄×C₁₂₇	—	2	1	no	—
87	C₄×C₁₂₇	—	16	2	no	—
88	C₄×C₁₂₇	#332	1	1	yes	characteristic
89	C₄×C₁₂₇	#333	1	1	yes	characteristic
90	C₄×C₁₂₇	#340	1	1	yes	characteristic
91	C₄×C₁₂₇	#343	1	1	yes	characteristic
92	C₄×C₁₂₇	#360	1	2	yes	—
93	C₄×C₁₂₇	#365	1	2	yes	—
94	#7278	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	#1296	1	1	yes	characteristic
98	C₂×C₁₂₇	#1383	1	1	yes	characteristic
99	C₂×C₁₂₇	#1386	1	1	yes	characteristic
100	#2150	C₂×C₁₂₇	1	1	yes	characteristic
101	#1844	C₂×C₁₂₇	1	1	yes	characteristic
102	#1564	C₂×C₁₂₇	1	1	yes	characteristic
103	C₁₂₇	#7278	1	1	yes	characteristic, direct product factor, Sylow
104	C₂×M₃₂	—	2	1	no	—
105	#431	—	2	1	no	—
106	C₂×#106	—	2	1	no	—
107	C₂×#106	C₄×C₁₂₇	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
109	#384	—	2	1	no	—
110	C₄⋊C₁₆	—	2	1	no	—
111	C₄⋊C₁₆	C₄×C₁₂₇	1	2	yes	complemented
112	#335	C₄×C₁₂₇	1	2	yes	complemented
113	#334	—	2	1	no	—
114	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
115	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	4	1	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
120	M₃₂	—	4	1	no	—
121	M₃₂	—	4	1	no	—
122	C₂×C₁₆	—	2	2	no	—
123	#106	—	4	1	no	—
124	#106	—	4	1	no	—
125	#106	—	4	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	4	1	no	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
130	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
131	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
132	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
135	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
136	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	C₄²×C₁₂₇	1	1	yes	characteristic
141	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
142	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
143	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
144	C₂×C₈	D₈×C₁₂₇	1	1	yes	characteristic
145	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
146	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
147	C₂×C₈	Q₁₆×C₁₂₇	1	1	yes	characteristic
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	8	2	no	—
151	C₄⋊C₄	—	8	2	no	—
152	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
153	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
154	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
155	C₄²	Q₁₆×C₁₂₇	1	1	yes	characteristic
156	C₁₆	—	4	2	no	—
157	C₂³	#102×C₁₂₇	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	#95×C₁₂₇	1	1	yes	characteristic
164	C₂×C₄	#100×C₁₂₇	1	1	yes	characteristic
165	C₂×C₄	#101×C₁₂₇	1	1	yes	characteristic
166	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
167	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
168	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
169	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
171	C₂×C₄	#106×C₁₂₇	1	1	yes	characteristic
172	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	8	2	no	—
176	C₈	#102×C₁₂₇	1	1	yes	characteristic
177	C₈	#103×C₁₂₇	1	1	yes	characteristic
178	C₈	#106×C₁₂₇	1	1	yes	characteristic
179	C₈	#107×C₁₂₇	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	#331×C₁₂₇	1	1	yes	characteristic
183	C₂²	#332×C₁₂₇	1	1	yes	characteristic
184	C₂²	#340×C₁₂₇	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	16	2	no	—
188	C₄	#332×C₁₂₇	1	1	yes	characteristic
189	C₄	#333×C₁₂₇	1	1	yes	characteristic
190	C₄	#340×C₁₂₇	1	1	yes	characteristic
191	C₄	#343×C₁₂₇	1	1	yes	characteristic
192	C₄	#360×C₁₂₇	1	2	yes	—
193	C₄	#365×C₁₂₇	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	C₁₂₇×#1296	1	1	yes	characteristic
197	C₂	C₁₂₇×#1383	1	1	yes	characteristic
198	C₂	C₁₂₇×#1386	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#340
commutator	C₂×C₈	C₄²×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₂₇
prime radical	C₂³×C₁₂₇	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₄⋊C₁₆×C₁₂₇	C₄	semidirect
12. #335×C₁₂₇	C₄	semidirect
94. #7278	C₁₂₇	direct (N × H)
103. C₁₂₇	#7278	direct (N × H)
111. C₄⋊C₁₆	C₄×C₁₂₇	semidirect
112. #335	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2150	2
C₁₂₇×#1844	2
C₁₂₇×#1564	2
#7278	127

Subgroups — C127×#7278 #44570951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7278 #44570951