Subgroups — C₄²×C₄⋊C₄×C₁₂₇ #44577421

Order 32512.

2478 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×C₄⋊C₄×C₁₂₇	C₂	1	6	yes	maximal
2	C₂×C₄³×C₁₂₇	C₂	1	2	yes	maximal
3	C₂×C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#95×C₁₂₇	C₂	1	6	yes	maximal
5	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
6	C₂²×C₄²×C₁₂₇	C₂²	1	3	yes	—
7	C₂²×C₄²×C₁₂₇	C₂²	1	12	yes	—
8	C₂²×C₄²×C₁₂₇	C₂²	1	3	yes	—
9	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊C₄×C₁₂₇	C₄	1	48	yes	direct product factor
11	C₂×#95×C₁₂₇	C₂²	1	12	yes	—
12	C₄³×C₁₂₇	—	2	8	no	—
13	C₄³×C₁₂₇	C₄	1	8	yes	complemented
14	C₂³×C₄×C₁₂₇	C₂³	1	8	yes	—
15	C₂³×C₄×C₁₂₇	C₂³	1	3	yes	—
16	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
17	C₂³×C₄×C₁₂₇	C₂³	1	3	yes	—
18	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	48	yes	—
19	C₂×C₄²×C₁₂₇	—	2	24	no	—
20	C₂×C₄²×C₁₂₇	—	2	12	no	—
21	C₂×C₄²×C₁₂₇	—	2	24	no	—
22	C₂×C₄²×C₁₂₇	C₂×C₄	1	12	yes	—
23	C₂×C₄²×C₁₂₇	C₂×C₄	1	48	yes	—
24	C₂×C₄²×C₁₂₇	C₂×C₄	1	12	yes	—
25	C₂×C₄²×C₁₂₇	C₂×C₄	1	24	yes	—
26	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
28	C₂×C₄²×C₁₂₇	Q₈	1	4	yes	—
29	#95×C₁₂₇	C₂×C₄	1	48	yes	—
30	C₂⁴×C₁₂₇	C₂⁴	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	—	2	8	no	—
32	C₂²×C₄×C₁₂₇	—	2	12	no	—
33	C₂²×C₄×C₁₂₇	—	2	24	no	—
34	C₂²×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
35	C₂²×C₄×C₁₂₇	C₂²×C₄	1	48	yes	—
36	C₂²×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
37	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
39	C₂²×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
40	C₂²×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
41	C₂²×C₄×C₁₂₇	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
43	C₂²×C₄×C₁₂₇	#41	1	12	yes	—
44	C₄⋊C₄×C₁₂₇	C₄²	1	64	yes	direct product factor
45	C₄²×C₁₂₇	—	2	24	no	—
46	C₄²×C₁₂₇	—	2	96	no	—
47	C₄²×C₁₂₇	C₄²	1	48	yes	complemented
48	C₄²×C₁₂₇	C₄⋊C₄	1	16	yes	direct product factor
49	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
50	C₂³×C₁₂₇	C₂³×C₄	1	6	yes	—
51	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
52	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
53	C₂³×C₁₂₇	C₂×#41	1	6	yes	—
54	C₂×C₄×C₁₂₇	—	2	12	no	—
55	C₂×C₄×C₁₂₇	—	2	48	no	—
56	C₂×C₄×C₁₂₇	—	2	24	no	—
57	C₂×C₄×C₁₂₇	—	2	48	no	—
58	C₂×C₄×C₁₂₇	C₂×C₄²	1	12	yes	—
59	C₂×C₄×C₁₂₇	C₂×C₄²	1	32	yes	—
60	C₂×C₄×C₁₂₇	C₂×C₄²	1	12	yes	—
61	C₂×C₄×C₁₂₇	C₂×C₄²	1	16	yes	—
62	C₂×C₄×C₁₂₇	C₂×C₄²	1	12	yes	—
63	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	12	yes	—
64	C₂×C₄×C₁₂₇	#117	1	12	yes	—
65	C₂×C₄×C₁₂₇	C₄×D₄	1	24	yes	—
66	C₂×C₄×C₁₂₇	C₄×Q₈	1	24	yes	—
67	C₂²×C₁₂₇	C₂²×C₄²	1	3	yes	—
68	C₂²×C₁₂₇	C₂²×C₄²	1	4	yes	—
69	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₁₂₇	C₂×#117	1	3	yes	—
71	C₂²×C₁₂₇	C₂×C₄×D₄	1	6	yes	—
72	C₂²×C₁₂₇	C₂×C₄×Q₈	1	6	yes	—
73	C₂²×C₁₂₇	C₄×#41	1	12	yes	—
74	C₄×C₁₂₇	—	2	12	no	—
75	C₄×C₁₂₇	—	2	32	no	—
76	C₄×C₁₂₇	C₄³	1	8	yes	complemented
77	C₄×C₁₂₇	C₄×C₄⋊C₄	1	24	yes	direct product factor
78	C₄²×C₄⋊C₄	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×C₁₂₇	C₂×C₄³	1	1	yes	characteristic
80	C₂×C₁₂₇	C₂×C₄×C₄⋊C₄	1	3	yes	—
81	C₂×C₁₂₇	C₄×#117	1	3	yes	—
82	C₂×C₁₂₇	C₄²×D₄	1	4	yes	—
83	C₂×C₁₂₇	C₄²×Q₈	1	4	yes	—
84	C₂×C₄×C₄⋊C₄	C₂×C₁₂₇	1	6	yes	—
85	C₂×C₄³	C₂×C₁₂₇	1	2	yes	—
86	C₂×C₄³	C₂×C₁₂₇	1	1	yes	characteristic
87	C₄×#95	C₂×C₁₂₇	1	6	yes	—
88	C₁₂₇	C₄²×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
90	C₂²×C₄²	C₂²×C₁₂₇	1	3	yes	—
91	C₂²×C₄²	C₂²×C₁₂₇	1	12	yes	—
92	C₂²×C₄²	C₂²×C₁₂₇	1	3	yes	—
93	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
94	C₄×C₄⋊C₄	C₄×C₁₂₇	1	48	yes	direct product factor
95	C₂×#95	C₂²×C₁₂₇	1	12	yes	—
96	C₄³	—	2	8	no	—
97	C₄³	C₄×C₁₂₇	1	8	yes	complemented
98	C₂³×C₄	C₂³×C₁₂₇	1	8	yes	—
99	C₂³×C₄	C₂³×C₁₂₇	1	3	yes	—
100	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
101	C₂³×C₄	C₂³×C₁₂₇	1	3	yes	—
102	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	48	yes	—
103	C₂×C₄²	—	2	24	no	—
104	C₂×C₄²	—	2	12	no	—
105	C₂×C₄²	—	2	24	no	—
106	C₂×C₄²	C₂×C₄×C₁₂₇	1	12	yes	—
107	C₂×C₄²	C₂×C₄×C₁₂₇	1	48	yes	—
108	C₂×C₄²	C₂×C₄×C₁₂₇	1	12	yes	—
109	C₂×C₄²	C₂×C₄×C₁₂₇	1	24	yes	—
110	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
111	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
112	C₂×C₄²	Q₈×C₁₂₇	1	4	yes	—
113	#95	C₂×C₄×C₁₂₇	1	48	yes	—
114	C₂⁴	C₂⁴×C₁₂₇	1	1	yes	characteristic
115	C₂²×C₄	—	2	8	no	—
116	C₂²×C₄	—	2	12	no	—
117	C₂²×C₄	—	2	24	no	—
118	C₂²×C₄	C₂²×C₄×C₁₂₇	1	12	yes	—
119	C₂²×C₄	C₂²×C₄×C₁₂₇	1	48	yes	—
120	C₂²×C₄	C₂²×C₄×C₁₂₇	1	6	yes	—
121	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
122	C₂²×C₄	C₂²×C₄×C₁₂₇	1	12	yes	—
123	C₂²×C₄	C₂²×C₄×C₁₂₇	1	6	yes	—
124	C₂²×C₄	C₂²×C₄×C₁₂₇	1	12	yes	—
125	C₂²×C₄	C₂×D₄×C₁₂₇	1	6	yes	—
126	C₂²×C₄	C₂×Q₈×C₁₂₇	1	6	yes	—
127	C₂²×C₄	#41×C₁₂₇	1	12	yes	—
128	C₄⋊C₄	C₄²×C₁₂₇	1	64	yes	direct product factor
129	C₄²	—	2	24	no	—
130	C₄²	—	2	96	no	—
131	C₄²	C₄²×C₁₂₇	1	48	yes	complemented
132	C₄²	C₄⋊C₄×C₁₂₇	1	16	yes	direct product factor
133	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
134	C₂³	C₂³×C₄×C₁₂₇	1	6	yes	—
135	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
136	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
137	C₂³	C₂×#41×C₁₂₇	1	6	yes	—
138	C₂×C₄	—	2	12	no	—
139	C₂×C₄	—	2	48	no	—
140	C₂×C₄	—	2	24	no	—
141	C₂×C₄	—	2	48	no	—
142	C₂×C₄	C₂×C₄²×C₁₂₇	1	12	yes	—
143	C₂×C₄	C₂×C₄²×C₁₂₇	1	32	yes	—
144	C₂×C₄	C₂×C₄²×C₁₂₇	1	12	yes	—
145	C₂×C₄	C₂×C₄²×C₁₂₇	1	16	yes	—
146	C₂×C₄	C₂×C₄²×C₁₂₇	1	12	yes	—
147	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	12	yes	—
148	C₂×C₄	#117×C₁₂₇	1	12	yes	—
149	C₂×C₄	C₄×D₄×C₁₂₇	1	24	yes	—
150	C₂×C₄	C₄×Q₈×C₁₂₇	1	24	yes	—
151	C₂²	C₂²×C₄²×C₁₂₇	1	3	yes	—
152	C₂²	C₂²×C₄²×C₁₂₇	1	4	yes	—
153	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
154	C₂²	C₂×#117×C₁₂₇	1	3	yes	—
155	C₂²	C₂×C₄×D₄×C₁₂₇	1	6	yes	—
156	C₂²	C₂×C₄×Q₈×C₁₂₇	1	6	yes	—
157	C₂²	C₄×#41×C₁₂₇	1	12	yes	—
158	C₄	—	2	12	no	—
159	C₄	—	2	32	no	—
160	C₄	C₄³×C₁₂₇	1	8	yes	complemented
161	C₄	C₄×C₄⋊C₄×C₁₂₇	1	24	yes	direct product factor
162	C₂	C₂×C₄³×C₁₂₇	1	1	yes	characteristic
163	C₂	C₂×C₄×C₄⋊C₄×C₁₂₇	1	3	yes	—
164	C₂	C₄×#117×C₁₂₇	1	3	yes	—
165	C₂	C₄²×D₄×C₁₂₇	1	4	yes	—
166	C₂	C₄²×Q₈×C₁₂₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
10. C₄×C₄⋊C₄×C₁₂₇	C₄	direct (N × H)
13. C₄³×C₁₂₇	C₄	semidirect
44. C₄⋊C₄×C₁₂₇	C₄²	direct (N × H)
47. C₄²×C₁₂₇	C₄²	semidirect
48. C₄²×C₁₂₇	C₄⋊C₄	direct (N × H)
76. C₄×C₁₂₇	C₄³	semidirect
77. C₄×C₁₂₇	C₄×C₄⋊C₄	direct (N × H)
78. C₄²×C₄⋊C₄	C₁₂₇	direct (N × H)
88. C₁₂₇	C₄²×C₄⋊C₄	direct (N × H)
94. C₄×C₄⋊C₄	C₄×C₁₂₇	direct (N × H)
97. C₄³	C₄×C₁₂₇	semidirect
128. C₄⋊C₄	C₄²×C₁₂₇	direct (N × H)
131. C₄²	C₄²×C₁₂₇	semidirect
132. C₄²	C₄⋊C₄×C₁₂₇	direct (N × H)
160. C₄	C₄³×C₁₂₇	semidirect
161. C₄	C₄×C₄⋊C₄×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×C₄⋊C₄×C₁₂₇	2
C₂×C₄³×C₁₂₇	2
C₄×#95×C₁₂₇	2
C₄²×C₄⋊C₄	127

Subgroups — C42×C4⋊C4×C127 #44577421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₄⋊C₄×C₁₂₇ #44577421