Subgroups — C₂×C₁₂₇×#1369 #44575674

Order 32512.

222 conjugacy classes of subgroups.

The subgroup lattice has 154 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	2	yes	maximal
2	C₂×#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1369	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₃₂×C₁₂₇	—	2	2	no	—
5	C₂²×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	#364×C₁₂₇	—	2	2	no	—
9	#364×C₁₂₇	C₂²	1	4	yes	—
10	#346×C₁₂₇	C₄	1	2	yes	—
11	#346×C₁₂₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₂₇	Q₈	1	1	yes	characteristic
15	M₃₂×C₁₂₇	—	2	4	no	—
16	M₃₂×C₁₂₇	—	4	2	no	—
17	C₂×C₁₆×C₁₂₇	—	2	1	no	—
18	C₂×C₁₆×C₁₂₇	—	2	1	no	—
19	C₂×C₁₆×C₁₂₇	—	4	2	no	—
20	C₂×C₁₆×C₁₂₇	C₈	1	2	yes	—
21	C₂×C₁₆×C₁₂₇	C₈	1	2	yes	—
22	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	—	2	1	no	—
29	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
35	C₂×C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
37	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
38	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₁₂₇	—	2	1	no	—
41	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
43	C₁₆×C₁₂₇	—	2	2	no	—
44	C₁₆×C₁₂₇	—	4	4	no	—
45	C₁₆×C₁₂₇	C₂×C₈	1	4	yes	—
46	C₂³×C₁₂₇	#108	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	—	2	1	no	—
48	C₂×C₄×C₁₂₇	—	2	1	no	—
49	C₂×C₄×C₁₂₇	—	4	2	no	—
50	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	#106	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	#106	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	#108	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₈×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
57	C₈×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
58	C₈×C₁₂₇	#106	1	2	yes	—
59	C₈×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₁₂₇	C₂×SD₁₆	1	2	yes	—
62	C₂²×C₁₂₇	—	2	2	no	—
63	C₂²×C₁₂₇	#334	1	1	yes	characteristic
64	C₂²×C₁₂₇	#334	1	1	yes	characteristic
65	C₂²×C₁₂₇	C₂×#108	1	1	yes	characteristic
66	C₄×C₁₂₇	—	4	1	no	—
67	C₄×C₁₂₇	—	4	1	no	—
68	C₄×C₁₂₇	#334	1	1	yes	characteristic
69	C₄×C₁₂₇	#334	1	1	yes	characteristic
70	C₄×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₁₂₇	C₂×#106	1	1	yes	characteristic
72	C₂×#1369	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂×C₁₂₇	—	2	1	no	—
74	C₂×C₁₂₇	—	2	1	no	—
75	C₂×C₁₂₇	#1369	1	2	yes	direct product factor
76	C₂×C₁₂₇	C₂×#334	1	1	yes	characteristic
77	C₂×#364	C₂×C₁₂₇	1	2	yes	—
78	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic
79	#1369	C₂×C₁₂₇	1	4	yes	direct product factor
80	C₁₂₇	C₂×#1369	1	1	yes	characteristic, direct product factor, Sylow
81	C₂×M₃₂	—	2	2	no	—
82	C₂²×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
83	C₂²×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
85	#364	—	2	2	no	—
86	#364	C₂²×C₁₂₇	1	4	yes	—
87	#346	C₄×C₁₂₇	1	2	yes	—
88	#346	C₂²×C₁₂₇	1	2	yes	—
89	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
90	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
91	C₂×C₄²	Q₈×C₁₂₇	1	1	yes	characteristic
92	M₃₂	—	2	4	no	—
93	M₃₂	—	4	2	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	—	4	2	no	—
97	C₂×C₁₆	C₈×C₁₂₇	1	2	yes	—
98	C₂×C₁₆	C₈×C₁₂₇	1	2	yes	—
99	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
100	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
101	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
102	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
103	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
104	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
107	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
108	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
109	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
110	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
111	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
112	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
113	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
114	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
115	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
116	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
119	C₄²	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
120	C₁₆	—	2	2	no	—
121	C₁₆	—	4	4	no	—
122	C₁₆	C₂×C₈×C₁₂₇	1	4	yes	—
123	C₂³	#108×C₁₂₇	1	1	yes	characteristic
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
128	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
129	C₂×C₄	#106×C₁₂₇	1	1	yes	characteristic
130	C₂×C₄	#106×C₁₂₇	1	1	yes	characteristic
131	C₂×C₄	#108×C₁₂₇	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
133	C₈	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
134	C₈	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
135	C₈	#106×C₁₂₇	1	2	yes	—
136	C₈	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
137	C₈	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
138	C₈	C₂×SD₁₆×C₁₂₇	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	#334×C₁₂₇	1	1	yes	characteristic
141	C₂²	#334×C₁₂₇	1	1	yes	characteristic
142	C₂²	C₂×#108×C₁₂₇	1	1	yes	characteristic
143	C₄	—	4	1	no	—
144	C₄	—	4	1	no	—
145	C₄	#334×C₁₂₇	1	1	yes	characteristic
146	C₄	#334×C₁₂₇	1	1	yes	characteristic
147	C₄	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
148	C₄	C₂×#106×C₁₂₇	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	C₁₂₇×#1369	1	2	yes	direct product factor
152	C₂	C₂×#334×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#106
commutator	C₈	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	#334
prime radical	C₂³×C₁₂₇	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1369	C₂	direct (N × H)
72. C₂×#1369	C₁₂₇	direct (N × H)
75. C₂×C₁₂₇	#1369	direct (N × H)
79. #1369	C₂×C₁₂₇	direct (N × H)
80. C₁₂₇	C₂×#1369	direct (N × H)
151. C₂	C₁₂₇×#1369	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×#364×C₁₂₇	2
C₂×#346×C₁₂₇	2
C₁₂₇×#1369	2
C₂×#1369	127

Subgroups — C2×C127×#1369 #44575674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1369 #44575674