Subgroups — C₁₂₇×#12004 #44575677

Order 32512.

270 conjugacy classes of subgroups.

The subgroup lattice has 166 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	2	yes	maximal
2	C₂×C₈²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1370	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₂₇	—	2	2	no	—
5	C₂×C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
9	C₄⋊C₁₆×C₁₂₇	C₂²	1	4	yes	—
10	C₈²×C₁₂₇	C₄	1	2	yes	—
11	C₈²×C₁₂₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₂₇	—	2	1	no	—
13	C₂²×C₈×C₁₂₇	C₈	1	2	yes	—
14	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	M₃₂×C₁₂₇	—	4	4	no	—
18	C₂×C₁₆×C₁₂₇	—	2	4	no	—
19	C₄×C₈×C₁₂₇	—	2	1	no	—
20	C₄×C₈×C₁₂₇	C₈	1	1	yes	characteristic
21	C₄×C₈×C₁₂₇	C₈	1	1	yes	characteristic
22	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
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	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₁₂₇	—	2	4	no	—
32	C₂×C₈×C₁₂₇	—	2	2	no	—
33	C₂×C₈×C₁₂₇	—	2	2	no	—
34	C₂×C₈×C₁₂₇	—	2	4	no	—
35	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
40	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
41	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
46	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₁₆×C₁₂₇	—	4	4	no	—
48	C₂³×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	—	2	2	no	—
50	C₂×C₄×C₁₂₇	—	2	2	no	—
51	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
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	2	yes	—
55	C₂×C₄×C₁₂₇	#108	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
59	C₈×C₁₂₇	—	2	4	no	—
60	C₈×C₁₂₇	—	2	1	no	—
61	C₈×C₁₂₇	—	2	1	no	—
62	C₈×C₁₂₇	—	2	2	no	—
63	C₈×C₁₂₇	#107	1	2	yes	—
64	C₈×C₁₂₇	M₃₂	1	4	yes	complemented
65	C₈×C₁₂₇	C₂×D₈	1	1	yes	characteristic
66	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₁₂₇	—	2	1	no	—
68	C₂²×C₁₂₇	—	2	1	no	—
69	C₂²×C₁₂₇	#335	1	2	yes	—
70	C₂²×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₁₂₇	—	2	1	no	—
72	C₄×C₁₂₇	—	2	1	no	—
73	C₄×C₁₂₇	#335	1	2	yes	—
74	C₄×C₁₂₇	#364	1	2	yes	—
75	C₄×C₁₂₇	C₂×#107	1	1	yes	characteristic
76	C₄×C₁₂₇	C₂×#108	1	1	yes	characteristic
77	C₄×C₁₂₇	C₂×M₃₂	1	2	yes	—
78	#12004	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×C₁₂₇	—	2	2	no	—
80	C₂×C₁₂₇	C₂×#335	1	1	yes	characteristic
81	C₂×C₁₂₇	#2150	1	1	yes	characteristic
82	C₂×C₁₂₇	C₂×#364	1	1	yes	characteristic
83	#2150	C₂×C₁₂₇	1	2	yes	—
84	C₂×C₈²	C₂×C₁₂₇	1	1	yes	characteristic
85	#1370	C₂×C₁₂₇	1	4	yes	complemented
86	C₁₂₇	#12004	1	1	yes	characteristic, direct product factor, Sylow
87	C₂×M₃₂	—	2	2	no	—
88	C₂×C₄×C₈	C₄×C₁₂₇	1	1	yes	characteristic
89	C₂×C₄×C₈	C₄×C₁₂₇	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
91	C₄⋊C₁₆	—	2	2	no	—
92	C₄⋊C₁₆	C₂²×C₁₂₇	1	4	yes	—
93	C₈²	C₄×C₁₂₇	1	2	yes	—
94	C₈²	C₂²×C₁₂₇	1	2	yes	—
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	C₈×C₁₂₇	1	2	yes	—
97	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
98	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
99	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
100	M₃₂	—	4	4	no	—
101	C₂×C₁₆	—	2	4	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	C₈×C₁₂₇	1	1	yes	characteristic
104	C₄×C₈	C₈×C₁₂₇	1	1	yes	characteristic
105	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
106	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
107	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
108	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
109	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
110	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
111	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
113	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
119	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
120	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
121	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
122	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
123	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
124	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
125	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
126	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
127	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
128	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
129	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
130	C₁₆	—	4	4	no	—
131	C₂³	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
135	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
136	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
137	C₂×C₄	#107×C₁₂₇	1	2	yes	—
138	C₂×C₄	#108×C₁₂₇	1	2	yes	—
139	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
140	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
141	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	#107×C₁₂₇	1	2	yes	—
147	C₈	M₃₂×C₁₂₇	1	4	yes	complemented
148	C₈	C₂×D₈×C₁₂₇	1	1	yes	characteristic
149	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#335×C₁₂₇	1	2	yes	—
153	C₂²	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#335×C₁₂₇	1	2	yes	—
157	C₄	#364×C₁₂₇	1	2	yes	—
158	C₄	C₂×#107×C₁₂₇	1	1	yes	characteristic
159	C₄	C₂×#108×C₁₂₇	1	1	yes	characteristic
160	C₄	C₂×M₃₂×C₁₂₇	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	C₂×#335×C₁₂₇	1	1	yes	characteristic
163	C₂	C₁₂₇×#2150	1	1	yes	characteristic
164	C₂	C₂×#364×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₂₇	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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×#1370	C₂	semidirect
64. C₈×C₁₂₇	M₃₂	semidirect
78. #12004	C₁₂₇	direct (N × H)
85. #1370	C₂×C₁₂₇	semidirect
86. C₁₂₇	#12004	direct (N × H)
147. C₈	M₃₂×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
C₂×C₈²×C₁₂₇	2
C₁₂₇×#1370	2
#12004	127

Subgroups — C127×#12004 #44575677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12004 #44575677