Subgroups — C₂×C₁₂₇×#1372 #44575680

Order 32512.

262 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×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#335×C₁₂₇	C₂	1	2	yes	maximal
3	C₁₂₇×#1372	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₈×C₁₂₇	—	2	2	no	—
7	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₁₂₇	C₄	1	2	yes	—
9	C₄×C₁₆×C₁₂₇	C₂²	1	2	yes	—
10	#335×C₁₂₇	—	2	2	no	—
11	#335×C₁₂₇	C₂²	1	4	yes	—
12	C₂²×C₈×C₁₂₇	—	4	2	no	—
13	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
15	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₁₂₇	—	2	2	no	—
17	C₂×C₁₆×C₁₂₇	C₈	1	4	yes	complemented
18	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₄⋊C₈×C₁₂₇	—	2	4	no	—
21	C₄⋊C₈×C₁₂₇	—	4	2	no	—
22	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
29	C₂×C₈×C₁₂₇	—	2	2	no	—
30	C₂×C₈×C₁₂₇	—	4	4	no	—
31	C₂×C₈×C₁₂₇	—	8	4	no	—
32	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
36	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
40	C₄²×C₁₂₇	Q₁₆	1	1	yes	characteristic
41	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₁₂₇	—	2	2	no	—
43	C₁₆×C₁₂₇	C₂×C₈	1	4	yes	complemented
44	C₂³×C₁₂₇	#107	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	—	2	2	no	—
46	C₂×C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
47	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	#108	1	2	yes	—
51	C₂×C₄×C₁₂₇	SD₃₂	1	4	yes	—
52	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
55	C₈×C₁₂₇	—	2	1	no	—
56	C₈×C₁₂₇	—	2	1	no	—
57	C₈×C₁₂₇	—	8	4	no	—
58	C₈×C₁₂₇	C₄⋊C₈	1	2	yes	—
59	C₈×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
61	C₂²×C₁₂₇	#335	1	2	yes	—
62	C₂²×C₁₂₇	#367	1	2	yes	—
63	C₂²×C₁₂₇	#368	1	2	yes	—
64	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
65	C₄×C₁₂₇	—	2	1	no	—
66	C₄×C₁₂₇	—	2	1	no	—
67	C₄×C₁₂₇	#335	1	2	yes	—
68	C₄×C₁₂₇	#367	1	2	yes	—
69	C₄×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₄×C₁₂₇	C₂×#108	1	1	yes	characteristic
71	C₄×C₁₂₇	C₂×SD₃₂	1	2	yes	—
72	C₂×#1372	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂×C₁₂₇	#1372	1	4	yes	direct product factor
74	C₂×C₁₂₇	C₂×#335	1	1	yes	characteristic
75	C₂×C₁₂₇	C₂×#367	1	1	yes	characteristic
76	C₂×C₁₂₇	C₂×#368	1	1	yes	characteristic
77	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
78	C₂×#335	C₂×C₁₂₇	1	2	yes	—
79	#1372	C₂×C₁₂₇	1	4	yes	direct product factor
80	C₁₂₇	C₂×#1372	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
82	C₂²×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
83	C₂×C₄⋊C₈	—	2	2	no	—
84	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
85	C₄×C₁₆	C₄×C₁₂₇	1	2	yes	—
86	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
87	#335	—	2	2	no	—
88	#335	C₂²×C₁₂₇	1	4	yes	—
89	C₂²×C₈	—	4	2	no	—
90	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
91	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
92	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	C₈×C₁₂₇	1	4	yes	complemented
95	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
96	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
97	C₄⋊C₈	—	2	4	no	—
98	C₄⋊C₈	—	4	2	no	—
99	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
100	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
101	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
102	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
103	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
104	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
105	C₂²×C₄	Q₁₆×C₁₂₇	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	4	4	no	—
108	C₂×C₈	—	8	4	no	—
109	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
110	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
111	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
112	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
113	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
114	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
115	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
116	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
117	C₄²	Q₁₆×C₁₂₇	1	1	yes	characteristic
118	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
119	C₁₆	—	2	2	no	—
120	C₁₆	C₂×C₈×C₁₂₇	1	4	yes	complemented
121	C₂³	#107×C₁₂₇	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	C₄⋊C₈×C₁₂₇	1	2	yes	—
124	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
125	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
126	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
127	C₂×C₄	#108×C₁₂₇	1	2	yes	—
128	C₂×C₄	SD₃₂×C₁₂₇	1	4	yes	—
129	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
130	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
131	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	8	4	no	—
135	C₈	C₄⋊C₈×C₁₂₇	1	2	yes	—
136	C₈	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
137	C₈	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
138	C₂²	#335×C₁₂₇	1	2	yes	—
139	C₂²	#367×C₁₂₇	1	2	yes	—
140	C₂²	#368×C₁₂₇	1	2	yes	—
141	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	#335×C₁₂₇	1	2	yes	—
145	C₄	#367×C₁₂₇	1	2	yes	—
146	C₄	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
147	C₄	C₂×#108×C₁₂₇	1	1	yes	characteristic
148	C₄	C₂×SD₃₂×C₁₂₇	1	2	yes	—
149	C₂	C₁₂₇×#1372	1	4	yes	direct product factor
150	C₂	C₂×#335×C₁₂₇	1	1	yes	characteristic
151	C₂	C₂×#367×C₁₂₇	1	1	yes	characteristic
152	C₂	C₂×#368×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	D₈
commutator	C₈	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle, prime radical	C₂³×C₁₂₇	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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
3. C₁₂₇×#1372	C₂	direct (N × H)
17. C₂×C₁₆×C₁₂₇	C₈	semidirect
43. C₁₆×C₁₂₇	C₂×C₈	semidirect
72. C₂×#1372	C₁₂₇	direct (N × H)
73. C₂×C₁₂₇	#1372	direct (N × H)
79. #1372	C₂×C₁₂₇	direct (N × H)
80. C₁₂₇	C₂×#1372	direct (N × H)
94. C₂×C₁₆	C₈×C₁₂₇	semidirect
120. C₁₆	C₂×C₈×C₁₂₇	semidirect
149. C₂	C₁₂₇×#1372	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×C₄×C₁₆×C₁₂₇	2
C₂×#335×C₁₂₇	2
C₁₂₇×#1372	2
C₂×#1372	127

Subgroups — C2×C127×#1372 #44575680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1372 #44575680