Subgroups — C₄²×Q₁₆×C₁₂₇ #44584172

Order 32512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₁₂₇	C₂	1	3	yes	maximal
2	C₄²×Q₈×C₁₂₇	C₂	1	2	yes	maximal
3	C₄×#107×C₁₂₇	C₂	1	3	yes	maximal
4	C₄×#103×C₁₂₇	C₂	1	6	yes	maximal
5	C₄²×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₂₇	C₂²	1	6	yes	—
8	C₄×Q₁₆×C₁₂₇	C₄	1	24	yes	direct product factor
9	C₂×#107×C₁₂₇	C₂²	1	3	yes	—
10	C₂×#103×C₁₂₇	C₂²	1	6	yes	—
11	C₂×C₄×C₈×C₁₂₇	C₂²	1	3	yes	—
12	C₂×C₄×C₈×C₁₂₇	C₂²	1	3	yes	—
13	C₄×C₄⋊C₄×C₁₂₇	—	2	6	no	—
14	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	6	yes	—
15	C₄³×C₁₂₇	—	2	2	no	—
16	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
17	#340×C₁₂₇	C₂²	1	6	yes	—
18	C₂²×Q₈×C₁₂₇	C₂³	1	2	yes	—
19	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	12	yes	—
20	C₂²×C₈×C₁₂₇	C₂³	1	3	yes	—
21	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₁₂₇	—	2	12	no	—
23	C₄×Q₈×C₁₂₇	C₂×C₄	1	24	yes	—
24	C₂×C₄⋊C₄×C₁₂₇	—	2	6	no	—
25	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	6	yes	—
26	C₂×C₄²×C₁₂₇	—	2	6	no	—
27	C₂×C₄²×C₁₂₇	—	2	6	no	—
28	C₂×C₄²×C₁₂₇	D₄	1	3	yes	—
29	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₂₇	C₂³	1	3	yes	—
31	#107×C₁₂₇	C₂×C₄	1	12	yes	—
32	#103×C₁₂₇	C₂×C₄	1	24	yes	—
33	C₄×C₈×C₁₂₇	—	2	6	no	—
34	C₄×C₈×C₁₂₇	C₂×C₄	1	12	yes	—
35	#95×C₁₂₇	—	2	6	no	—
36	C₂×Q₈×C₁₂₇	—	2	6	no	—
37	C₂×Q₈×C₁₂₇	C₂²×C₄	1	12	yes	—
38	C₂²×C₄×C₁₂₇	—	2	2	no	—
39	C₂²×C₄×C₁₂₇	—	2	6	no	—
40	C₂²×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
43	Q₁₆×C₁₂₇	C₄²	1	16	yes	direct product factor
44	C₂×C₈×C₁₂₇	—	2	6	no	—
45	C₂×C₈×C₁₂₇	C₂²×C₄	1	6	yes	—
46	C₂×C₈×C₁₂₇	C₂²×C₄	1	6	yes	—
47	C₄⋊C₄×C₁₂₇	—	2	24	no	—
48	C₄⋊C₄×C₁₂₇	—	2	6	no	—
49	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
50	C₄²×C₁₂₇	—	2	24	no	—
51	C₄²×C₁₂₇	—	2	6	no	—
52	C₄²×C₁₂₇	—	4	12	no	—
53	C₄²×C₁₂₇	Q₁₆	1	4	yes	direct product factor
54	C₄²×C₁₂₇	C₂²×C₄	1	6	yes	—
55	C₄²×C₁₂₇	#41	1	6	yes	—
56	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
57	Q₈×C₁₂₇	—	2	12	no	—
58	Q₈×C₁₂₇	C₂×C₄²	1	8	yes	—
59	C₂×C₄×C₁₂₇	—	2	12	no	—
60	C₂×C₄×C₁₂₇	—	2	6	no	—
61	C₂×C₄×C₁₂₇	—	2	12	no	—
62	C₂×C₄×C₁₂₇	—	4	12	no	—
63	C₂×C₄×C₁₂₇	C₄×D₄	1	6	yes	—
64	C₂×C₄×C₁₂₇	C₄×D₄	1	6	yes	—
65	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	6	yes	—
66	C₂×C₄×C₁₂₇	#135	1	6	yes	—
67	C₂×C₄×C₁₂₇	C₂³×C₄	1	3	yes	—
68	C₂×C₄×C₁₂₇	C₂×#41	1	3	yes	—
69	C₈×C₁₂₇	—	2	6	no	—
70	C₈×C₁₂₇	C₂×C₄²	1	4	yes	—
71	C₂²×C₁₂₇	C₂×C₄×D₄	1	3	yes	—
72	C₂²×C₁₂₇	C₂²×Q₁₆	1	1	yes	characteristic
73	C₂²×C₁₂₇	C₂×#135	1	3	yes	—
74	C₄×C₁₂₇	—	2	6	no	—
75	C₄×C₁₂₇	—	2	8	no	—
76	C₄×C₁₂₇	—	4	12	no	—
77	C₄×C₁₂₇	C₄×Q₁₆	1	12	yes	direct product factor
78	C₄×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
79	C₄×C₁₂₇	C₄×#41	1	3	yes	—
80	C₄²×Q₁₆	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×C₁₂₇	C₄²×D₄	1	1	yes	characteristic
82	C₂×C₁₂₇	C₂×C₄×Q₁₆	1	3	yes	—
83	C₂×C₁₂₇	C₄×#135	1	3	yes	—
84	C₂×C₄×Q₁₆	C₂×C₁₂₇	1	3	yes	—
85	C₄²×Q₈	C₂×C₁₂₇	1	2	yes	—
86	C₄×#107	C₂×C₁₂₇	1	3	yes	—
87	C₄×#103	C₂×C₁₂₇	1	6	yes	—
88	C₄²×C₈	C₂×C₁₂₇	1	1	yes	characteristic
89	C₁₂₇	C₄²×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
91	C₂×C₄×Q₈	C₂²×C₁₂₇	1	6	yes	—
92	C₄×Q₁₆	C₄×C₁₂₇	1	24	yes	direct product factor
93	C₂×#107	C₂²×C₁₂₇	1	3	yes	—
94	C₂×#103	C₂²×C₁₂₇	1	6	yes	—
95	C₂×C₄×C₈	C₂²×C₁₂₇	1	3	yes	—
96	C₂×C₄×C₈	C₂²×C₁₂₇	1	3	yes	—
97	C₄×C₄⋊C₄	—	2	6	no	—
98	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	6	yes	—
99	C₄³	—	2	2	no	—
100	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
101	#340	C₂²×C₁₂₇	1	6	yes	—
102	C₂²×Q₈	C₂³×C₁₂₇	1	2	yes	—
103	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	12	yes	—
104	C₂²×C₈	C₂³×C₁₂₇	1	3	yes	—
105	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
106	C₄×Q₈	—	2	12	no	—
107	C₄×Q₈	C₂×C₄×C₁₂₇	1	24	yes	—
108	C₂×C₄⋊C₄	—	2	6	no	—
109	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	6	yes	—
110	C₂×C₄²	—	2	6	no	—
111	C₂×C₄²	—	2	6	no	—
112	C₂×C₄²	D₄×C₁₂₇	1	3	yes	—
113	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
114	C₂×C₄²	C₂³×C₁₂₇	1	3	yes	—
115	#107	C₂×C₄×C₁₂₇	1	12	yes	—
116	#103	C₂×C₄×C₁₂₇	1	24	yes	—
117	C₄×C₈	—	2	6	no	—
118	C₄×C₈	C₂×C₄×C₁₂₇	1	12	yes	—
119	#95	—	2	6	no	—
120	C₂×Q₈	—	2	6	no	—
121	C₂×Q₈	C₂²×C₄×C₁₂₇	1	12	yes	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	6	no	—
124	C₂²×C₄	C₂×D₄×C₁₂₇	1	3	yes	—
125	C₂²×C₄	C₂×D₄×C₁₂₇	1	3	yes	—
126	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
127	Q₁₆	C₄²×C₁₂₇	1	16	yes	direct product factor
128	C₂×C₈	—	2	6	no	—
129	C₂×C₈	C₂²×C₄×C₁₂₇	1	6	yes	—
130	C₂×C₈	C₂²×C₄×C₁₂₇	1	6	yes	—
131	C₄⋊C₄	—	2	24	no	—
132	C₄⋊C₄	—	2	6	no	—
133	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	12	yes	—
134	C₄²	—	2	24	no	—
135	C₄²	—	2	6	no	—
136	C₄²	—	4	12	no	—
137	C₄²	Q₁₆×C₁₂₇	1	4	yes	direct product factor
138	C₄²	C₂²×C₄×C₁₂₇	1	6	yes	—
139	C₄²	#41×C₁₂₇	1	6	yes	—
140	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
141	Q₈	—	2	12	no	—
142	Q₈	C₂×C₄²×C₁₂₇	1	8	yes	—
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	—	2	6	no	—
145	C₂×C₄	—	2	12	no	—
146	C₂×C₄	—	4	12	no	—
147	C₂×C₄	C₄×D₄×C₁₂₇	1	6	yes	—
148	C₂×C₄	C₄×D₄×C₁₂₇	1	6	yes	—
149	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	6	yes	—
150	C₂×C₄	#135×C₁₂₇	1	6	yes	—
151	C₂×C₄	C₂³×C₄×C₁₂₇	1	3	yes	—
152	C₂×C₄	C₂×#41×C₁₂₇	1	3	yes	—
153	C₈	—	2	6	no	—
154	C₈	C₂×C₄²×C₁₂₇	1	4	yes	—
155	C₂²	C₂×C₄×D₄×C₁₂₇	1	3	yes	—
156	C₂²	C₂²×Q₁₆×C₁₂₇	1	1	yes	characteristic
157	C₂²	C₂×#135×C₁₂₇	1	3	yes	—
158	C₄	—	2	6	no	—
159	C₄	—	2	8	no	—
160	C₄	—	4	12	no	—
161	C₄	C₄×Q₁₆×C₁₂₇	1	12	yes	direct product factor
162	C₄	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
163	C₄	C₄×#41×C₁₂₇	1	3	yes	—
164	C₂	C₄²×D₄×C₁₂₇	1	1	yes	characteristic
165	C₂	C₂×C₄×Q₁₆×C₁₂₇	1	3	yes	—
166	C₂	C₄×#135×C₁₂₇	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₂₇	D₄
commutator	C₄	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle, prime radical	C₂³×C₁₂₇	C₂²×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₂₇ ◁ C₄³×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₄×Q₁₆×C₁₂₇	C₄	direct (N × H)
43. Q₁₆×C₁₂₇	C₄²	direct (N × H)
53. C₄²×C₁₂₇	Q₁₆	direct (N × H)
77. C₄×C₁₂₇	C₄×Q₁₆	direct (N × H)
80. C₄²×Q₁₆	C₁₂₇	direct (N × H)
89. C₁₂₇	C₄²×Q₁₆	direct (N × H)
92. C₄×Q₁₆	C₄×C₁₂₇	direct (N × H)
127. Q₁₆	C₄²×C₁₂₇	direct (N × H)
137. C₄²	Q₁₆×C₁₂₇	direct (N × H)
161. C₄	C₄×Q₁₆×C₁₂₇	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×Q₁₆×C₁₂₇	2
C₄²×Q₈×C₁₂₇	2
C₄×#107×C₁₂₇	2
C₄×#103×C₁₂₇	2
C₄²×C₈×C₁₂₇	2
C₄²×Q₁₆	127

Subgroups — C42×Q16×C127 #44584172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×Q₁₆×C₁₂₇ #44584172