Subgroups — C₁₂₇×#12005 #44575678

Order 32512.

242 conjugacy classes of subgroups.

The subgroup lattice has 170 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	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1450	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1370	C₂	1	4	yes	maximal, complemented
5	C₂×M₃₂×C₁₂₇	—	2	1	no	—
6	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
7	#431×C₁₂₇	C₄	1	1	yes	characteristic
8	#431×C₁₂₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
11	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
12	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
14	#322×C₁₂₇	C₄	1	2	yes	—
15	C₈²×C₁₂₇	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	M₃₂×C₁₂₇	—	4	2	no	—
20	C₂×C₁₆×C₁₂₇	—	2	2	no	—
21	C₂×C₁₆×C₁₂₇	—	2	2	no	—
22	C₂×C₁₆×C₁₂₇	—	4	2	no	—
23	C₄⋊C₈×C₁₂₇	—	2	1	no	—
24	C₄⋊C₈×C₁₂₇	C₈	1	2	yes	—
25	C₂²⋊C₈×C₁₂₇	—	2	1	no	—
26	C₂²⋊C₈×C₁₂₇	C₈	1	2	yes	—
27	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₁₂₇	—	2	2	no	—
37	C₂×C₈×C₁₂₇	—	2	2	no	—
38	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
42	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
43	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
48	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₁₆×C₁₂₇	—	4	2	no	—
50	C₁₆×C₁₂₇	—	4	2	no	—
51	C₂³×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	—	2	2	no	—
53	C₂×C₄×C₁₂₇	—	4	1	no	—
54	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	#107	1	2	yes	—
58	C₂×C₄×C₁₂₇	#108	1	2	yes	—
59	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
62	C₈×C₁₂₇	—	2	1	no	—
63	C₈×C₁₂₇	—	2	1	no	—
64	C₈×C₁₂₇	—	2	4	no	—
65	C₈×C₁₂₇	—	4	2	no	—
66	C₈×C₁₂₇	#107	1	2	yes	—
67	C₈×C₁₂₇	C₂×D₈	1	1	yes	characteristic
68	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₂²×C₁₂₇	—	2	1	no	—
70	C₂²×C₁₂₇	—	2	1	no	—
71	C₂²×C₁₂₇	#335	1	2	yes	—
72	C₂²×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₁₂₇	—	2	1	no	—
74	C₄×C₁₂₇	—	2	1	no	—
75	C₄×C₁₂₇	—	2	1	no	—
76	C₄×C₁₂₇	#335	1	2	yes	—
77	C₄×C₁₂₇	C₂×#107	1	1	yes	characteristic
78	C₄×C₁₂₇	C₂×#108	1	1	yes	characteristic
79	C₄×C₁₂₇	#504	1	2	yes	—
80	#12005	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×C₁₂₇	—	2	2	no	—
82	C₂×C₁₂₇	C₂×#335	1	1	yes	characteristic
83	C₂×C₁₂₇	#2151	1	1	yes	characteristic
84	C₂×C₁₂₇	#2153	1	1	yes	characteristic
85	#2150	C₂×C₁₂₇	1	1	yes	characteristic
86	C₂×C₄⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
87	#1450	C₂×C₁₂₇	1	1	yes	characteristic
88	#1370	C₂×C₁₂₇	1	4	yes	complemented
89	C₁₂₇	#12005	1	1	yes	characteristic, direct product factor, Sylow
90	C₂×M₃₂	—	2	1	no	—
91	C₂²×C₁₆	—	2	1	no	—
92	#431	C₄×C₁₂₇	1	1	yes	characteristic
93	#431	C₄×C₁₂₇	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
95	C₄⋊C₁₆	—	2	1	no	—
96	C₄⋊C₁₆	—	2	1	no	—
97	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
98	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
99	#322	C₄×C₁₂₇	1	2	yes	—
100	C₈²	C₂²×C₁₂₇	1	2	yes	—
101	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
102	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
103	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
104	M₃₂	—	4	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	—	4	2	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	C₈×C₁₂₇	1	2	yes	—
110	C₂²⋊C₈	—	2	1	no	—
111	C₂²⋊C₈	C₈×C₁₂₇	1	2	yes	—
112	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
113	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
114	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
115	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
116	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
117	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
118	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
119	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
120	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
124	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
125	C₂×C₈	C₂×C₈×C₁₂₇	1	4	yes	—
126	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
127	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
128	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
129	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
130	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
131	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
132	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
133	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
134	C₁₆	—	4	2	no	—
135	C₁₆	—	4	2	no	—
136	C₂³	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
140	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
141	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
142	C₂×C₄	#107×C₁₂₇	1	2	yes	—
143	C₂×C₄	#108×C₁₂₇	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
145	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
146	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
147	C₈	—	2	4	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	4	2	no	—
151	C₈	#107×C₁₂₇	1	2	yes	—
152	C₈	C₂×D₈×C₁₂₇	1	1	yes	characteristic
153	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#335×C₁₂₇	1	2	yes	—
157	C₂²	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#335×C₁₂₇	1	2	yes	—
162	C₄	C₂×#107×C₁₂₇	1	1	yes	characteristic
163	C₄	C₂×#108×C₁₂₇	1	1	yes	characteristic
164	C₄	#504×C₁₂₇	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	C₂×#335×C₁₂₇	1	1	yes	characteristic
167	C₂	C₁₂₇×#2151	1	1	yes	characteristic
168	C₂	C₁₂₇×#2153	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
4. C₁₂₇×#1370	C₂	semidirect
80. #12005	C₁₂₇	direct (N × H)
88. #1370	C₂×C₁₂₇	semidirect
89. C₁₂₇	#12005	direct (N × H)

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×C₄⋊C₁₆×C₁₂₇	2
C₁₂₇×#1450	2
C₁₂₇×#1370	2
#12005	127

Subgroups — C127×#12005 #44575678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12005 #44575678