Subgroups — C₂×C₁₂₇×#1373 #44575683

Order 32512.

230 conjugacy classes of subgroups.

The subgroup lattice has 162 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×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1373	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	C₄×C₁₆×C₁₂₇	C₄	1	2	yes	—
11	C₄×C₁₆×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	D₈	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	D₈	1	1	yes	characteristic
38	C₂×C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
39	C₂×C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₁₂₇	—	2	1	no	—
43	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
45	C₁₆×C₁₂₇	—	2	2	no	—
46	C₁₆×C₁₂₇	—	4	4	no	—
47	C₁₆×C₁₂₇	C₂×C₈	1	4	yes	—
48	C₂³×C₁₂₇	#108	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	—	2	4	no	—
50	C₂×C₄×C₁₂₇	—	2	1	no	—
51	C₂×C₄×C₁₂₇	—	2	1	no	—
52	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	#108	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₈×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
59	C₈×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₁₂₇	#107	1	1	yes	characteristic
61	C₈×C₁₂₇	#107	1	1	yes	characteristic
62	C₈×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
63	C₈×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₁₂₇	C₂×D₈	1	1	yes	characteristic
65	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂²×C₁₂₇	—	2	2	no	—
67	C₂²×C₁₂₇	#335	1	1	yes	characteristic
68	C₂²×C₁₂₇	#335	1	1	yes	characteristic
69	C₂²×C₁₂₇	C₂×#108	1	1	yes	characteristic
70	C₄×C₁₂₇	—	2	2	no	—
71	C₄×C₁₂₇	—	2	2	no	—
72	C₄×C₁₂₇	#335	1	1	yes	characteristic
73	C₄×C₁₂₇	#335	1	1	yes	characteristic
74	C₄×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₄×C₁₂₇	C₂×#107	1	1	yes	characteristic
76	C₂×#1373	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₁₂₇	—	2	1	no	—
78	C₂×C₁₂₇	—	2	1	no	—
79	C₂×C₁₂₇	#1373	1	2	yes	direct product factor
80	C₂×C₁₂₇	C₂×#335	1	1	yes	characteristic
81	C₂×#364	C₂×C₁₂₇	1	2	yes	—
82	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
83	#1373	C₂×C₁₂₇	1	4	yes	direct product factor
84	C₁₂₇	C₂×#1373	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×M₃₂	—	2	2	no	—
86	C₂²×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
87	C₂²×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
89	#364	—	2	2	no	—
90	#364	C₂²×C₁₂₇	1	4	yes	—
91	C₄×C₁₆	C₄×C₁₂₇	1	2	yes	—
92	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
93	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
94	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
95	C₂×C₄²	Q₈×C₁₂₇	1	1	yes	characteristic
96	M₃₂	—	2	4	no	—
97	M₃₂	—	4	2	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	4	2	no	—
101	C₂×C₁₆	C₈×C₁₂₇	1	2	yes	—
102	C₂×C₁₆	C₈×C₁₂₇	1	2	yes	—
103	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
104	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
105	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
106	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
107	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
108	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
111	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
112	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
113	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
114	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
115	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
116	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
117	C₂×C₈	D₈×C₁₂₇	1	1	yes	characteristic
118	C₂×C₈	D₈×C₁₂₇	1	1	yes	characteristic
119	C₂×C₈	Q₁₆×C₁₂₇	1	1	yes	characteristic
120	C₂×C₈	Q₁₆×C₁₂₇	1	1	yes	characteristic
121	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
122	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
123	C₄²	—	2	1	no	—
124	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
125	C₄²	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
126	C₁₆	—	2	2	no	—
127	C₁₆	—	4	4	no	—
128	C₁₆	C₂×C₈×C₁₂₇	1	4	yes	—
129	C₂³	#108×C₁₂₇	1	1	yes	characteristic
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
134	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
135	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
136	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
137	C₂×C₄	#108×C₁₂₇	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
139	C₈	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
140	C₈	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
141	C₈	#107×C₁₂₇	1	1	yes	characteristic
142	C₈	#107×C₁₂₇	1	1	yes	characteristic
143	C₈	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
144	C₈	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
145	C₈	C₂×D₈×C₁₂₇	1	1	yes	characteristic
146	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
147	C₂²	—	2	2	no	—
148	C₂²	#335×C₁₂₇	1	1	yes	characteristic
149	C₂²	#335×C₁₂₇	1	1	yes	characteristic
150	C₂²	C₂×#108×C₁₂₇	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	#335×C₁₂₇	1	1	yes	characteristic
154	C₄	#335×C₁₂₇	1	1	yes	characteristic
155	C₄	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
156	C₄	C₂×#107×C₁₂₇	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	C₁₂₇×#1373	1	2	yes	direct product factor
160	C₂	C₂×#335×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₂₇	D₈
commutator	C₈	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	#335
prime radical	C₂³×C₁₂₇	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×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×#1373	C₂	direct (N × H)
76. C₂×#1373	C₁₂₇	direct (N × H)
79. C₂×C₁₂₇	#1373	direct (N × H)
83. #1373	C₂×C₁₂₇	direct (N × H)
84. C₁₂₇	C₂×#1373	direct (N × H)
159. C₂	C₁₂₇×#1373	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×C₄×C₁₆×C₁₂₇	2
C₁₂₇×#1373	2
C₂×#1373	127

Subgroups — C2×C127×#1373 #44575683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1373 #44575683