Subgroups — C₁₂₇×#7271 #44570944

Order 32512.

222 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1846	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₂₇	—	2	1	no	—
5	C₂×M₃₂×C₁₂₇	—	2	1	no	—
6	C₂×#107×C₁₂₇	—	2	1	no	—
7	C₂×#106×C₁₂₇	—	2	1	no	—
8	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	#382×C₁₂₇	C₄	1	2	yes	—
10	#364×C₁₂₇	—	2	1	no	—
11	#364×C₁₂₇	—	2	1	no	—
12	#364×C₁₂₇	C₄	1	2	yes	complemented
13	#364×C₁₂₇	C₄	1	2	yes	complemented
14	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
17	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	M₃₂×C₁₂₇	—	2	2	no	—
19	M₃₂×C₁₂₇	—	2	2	no	—
20	M₃₂×C₁₂₇	—	4	1	no	—
21	M₃₂×C₁₂₇	—	4	1	no	—
22	C₂×C₁₆×C₁₂₇	—	4	1	no	—
23	C₂×C₁₆×C₁₂₇	—	4	1	no	—
24	#107×C₁₂₇	—	4	1	no	—
25	#107×C₁₂₇	—	4	1	no	—
26	#106×C₁₂₇	—	4	1	no	—
27	#106×C₁₂₇	—	4	1	no	—
28	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
32	#95×C₁₂₇	—	4	2	no	—
33	C₂²×C₄×C₁₂₇	—	2	1	no	—
34	C₂²×C₄×C₁₂₇	—	4	2	no	—
35	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	D₈	1	1	yes	characteristic
41	C₂×C₈×C₁₂₇	D₈	1	1	yes	characteristic
42	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
43	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	Q₁₆	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	C₄²×C₁₂₇	—	2	1	no	—
51	C₄²×C₁₂₇	C₄²	1	1	yes	characteristic
52	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
53	C₁₆×C₁₂₇	—	4	2	no	—
54	C₁₆×C₁₂₇	—	4	2	no	—
55	C₂³×C₁₂₇	#100	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	—	2	1	no	—
57	C₂×C₄×C₁₂₇	—	2	1	no	—
58	C₂×C₄×C₁₂₇	—	4	2	no	—
59	C₂×C₄×C₁₂₇	—	8	4	no	—
60	C₂×C₄×C₁₂₇	—	8	2	no	—
61	C₂×C₄×C₁₂₇	#95	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	#101	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
67	C₈×C₁₂₇	#102	1	1	yes	characteristic
68	C₈×C₁₂₇	#102	1	1	yes	characteristic
69	C₈×C₁₂₇	#103	1	1	yes	characteristic
70	C₈×C₁₂₇	#103	1	1	yes	characteristic
71	C₈×C₁₂₇	#106	1	1	yes	characteristic
72	C₈×C₁₂₇	#106	1	1	yes	characteristic
73	C₈×C₁₂₇	#107	1	1	yes	characteristic
74	C₈×C₁₂₇	#107	1	1	yes	characteristic
75	C₂²×C₁₂₇	—	2	2	no	—
76	C₂²×C₁₂₇	#331	1	1	yes	characteristic
77	C₂²×C₁₂₇	#333	1	1	yes	characteristic
78	C₂²×C₁₂₇	#343	1	1	yes	characteristic
79	C₄×C₁₂₇	—	4	1	no	—
80	C₄×C₁₂₇	—	4	1	no	—
81	C₄×C₁₂₇	—	16	2	no	—
82	C₄×C₁₂₇	#332	1	1	yes	characteristic
83	C₄×C₁₂₇	#332	1	1	yes	characteristic
84	C₄×C₁₂₇	#340	1	1	yes	characteristic
85	C₄×C₁₂₇	#340	1	1	yes	characteristic
86	#7271	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂×C₁₂₇	—	2	1	no	—
88	C₂×C₁₂₇	—	2	1	no	—
89	C₂×C₁₂₇	#1296	1	1	yes	characteristic
90	C₂×C₁₂₇	#1366	1	2	yes	—
91	C₂×#364	C₂×C₁₂₇	1	1	yes	characteristic
92	C₂×#364	C₂×C₁₂₇	1	1	yes	characteristic
93	#1846	C₂×C₁₂₇	1	1	yes	characteristic
94	C₁₂₇	#7271	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×M₃₂	—	2	1	no	—
96	C₂×M₃₂	—	2	1	no	—
97	C₂×#107	—	2	1	no	—
98	C₂×#106	—	2	1	no	—
99	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
100	#382	C₄×C₁₂₇	1	2	yes	—
101	#364	—	2	1	no	—
102	#364	—	2	1	no	—
103	#364	C₄×C₁₂₇	1	2	yes	complemented
104	#364	C₄×C₁₂₇	1	2	yes	complemented
105	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
106	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
109	M₃₂	—	2	2	no	—
110	M₃₂	—	2	2	no	—
111	M₃₂	—	4	1	no	—
112	M₃₂	—	4	1	no	—
113	C₂×C₁₆	—	4	1	no	—
114	C₂×C₁₆	—	4	1	no	—
115	#107	—	4	1	no	—
116	#107	—	4	1	no	—
117	#106	—	4	1	no	—
118	#106	—	4	1	no	—
119	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
120	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
121	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
122	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
123	#95	—	4	2	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	4	2	no	—
126	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
127	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
128	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
129	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
130	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
131	C₂×C₈	D₈×C₁₂₇	1	1	yes	characteristic
132	C₂×C₈	D₈×C₁₂₇	1	1	yes	characteristic
133	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
134	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
135	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
136	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
137	C₂×C₈	Q₁₆×C₁₂₇	1	1	yes	characteristic
138	C₂×C₈	Q₁₆×C₁₂₇	1	1	yes	characteristic
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄²	—	2	1	no	—
142	C₄²	C₄²×C₁₂₇	1	1	yes	characteristic
143	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
144	C₁₆	—	4	2	no	—
145	C₁₆	—	4	2	no	—
146	C₂³	#100×C₁₂₇	1	1	yes	characteristic
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	#95×C₁₂₇	1	1	yes	characteristic
153	C₂×C₄	#101×C₁₂₇	1	1	yes	characteristic
154	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
155	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
156	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
157	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
158	C₈	#102×C₁₂₇	1	1	yes	characteristic
159	C₈	#102×C₁₂₇	1	1	yes	characteristic
160	C₈	#103×C₁₂₇	1	1	yes	characteristic
161	C₈	#103×C₁₂₇	1	1	yes	characteristic
162	C₈	#106×C₁₂₇	1	1	yes	characteristic
163	C₈	#106×C₁₂₇	1	1	yes	characteristic
164	C₈	#107×C₁₂₇	1	1	yes	characteristic
165	C₈	#107×C₁₂₇	1	1	yes	characteristic
166	C₂²	—	2	2	no	—
167	C₂²	#331×C₁₂₇	1	1	yes	characteristic
168	C₂²	#333×C₁₂₇	1	1	yes	characteristic
169	C₂²	#343×C₁₂₇	1	1	yes	characteristic
170	C₄	—	4	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	16	2	no	—
173	C₄	#332×C₁₂₇	1	1	yes	characteristic
174	C₄	#332×C₁₂₇	1	1	yes	characteristic
175	C₄	#340×C₁₂₇	1	1	yes	characteristic
176	C₄	#340×C₁₂₇	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₁₂₇×#1296	1	1	yes	characteristic
180	C₂	C₁₂₇×#1366	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#333
commutator	C₄²	C₄²×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×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
12. #364×C₁₂₇	C₄	semidirect
13. #364×C₁₂₇	C₄	semidirect
86. #7271	C₁₂₇	direct (N × H)
94. C₁₂₇	#7271	direct (N × H)
103. #364	C₄×C₁₂₇	semidirect
104. #364	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#364×C₁₂₇	2
C₁₂₇×#1846	2
#7271	127

Subgroups — C127×#7271 #44570944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7271 #44570944