Subgroups — C₂²×#338×C₁₂₇ #44584012

Order 32512.

2110 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₁₂₇	C₂	1	3	yes	maximal
2	C₂×#338×C₁₂₇	C₂	1	12	yes	maximal, direct product factor
3	C₂⁴×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
4	C₂²×M₁₆×C₁₂₇	—	2	6	no	—
5	C₂²×M₁₆×C₁₂₇	—	2	18	no	—
6	C₂²×M₁₆×C₁₂₇	C₄	1	18	yes	—
7	C₂²×M₁₆×C₁₂₇	C₂²	1	18	yes	—
8	C₂³×C₈×C₁₂₇	C₄	1	6	yes	—
9	#338×C₁₂₇	C₂²	1	16	yes	direct product factor
10	C₂⁵×C₁₂₇	Q₈	1	1	yes	characteristic
11	C₂³×C₄×C₁₂₇	C₂×C₄	1	9	yes	—
12	C₂³×C₄×C₁₂₇	C₂×C₄	1	3	yes	—
13	C₂³×C₄×C₁₂₇	D₄	1	3	yes	—
14	C₂³×C₄×C₁₂₇	D₄	1	9	yes	—
15	C₂³×C₄×C₁₂₇	Q₈	1	3	yes	—
16	C₂³×C₄×C₁₂₇	C₂³	1	3	yes	—
17	C₂×M₁₆×C₁₂₇	—	2	36	no	—
18	C₂×M₁₆×C₁₂₇	—	2	36	no	—
19	C₂×M₁₆×C₁₂₇	—	2	36	no	—
20	C₂×M₁₆×C₁₂₇	—	2	36	no	—
21	C₂×M₁₆×C₁₂₇	C₂×C₄	1	36	yes	—
22	C₂×M₁₆×C₁₂₇	C₂³	1	12	yes	—
23	C₂²×C₈×C₁₂₇	—	2	6	no	—
24	C₂²×C₈×C₁₂₇	—	2	18	no	—
25	C₂²×C₈×C₁₂₇	C₂×C₄	1	36	yes	—
26	C₂⁴×C₁₂₇	—	4	4	no	—
27	C₂⁴×C₁₂₇	C₄⋊C₄	1	3	yes	—
28	C₂⁴×C₁₂₇	C₄⋊C₄	1	9	yes	—
29	C₂⁴×C₁₂₇	C₂×Q₈	1	3	yes	—
30	C₂²×C₄×C₁₂₇	C₄²	1	6	yes	—
31	C₂²×C₄×C₁₂₇	C₄²	1	9	yes	—
32	C₂²×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	3	yes	—
34	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	9	yes	—
35	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	18	yes	—
36	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	18	yes	—
37	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	9	yes	—
38	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	18	yes	—
39	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	9	yes	—
40	C₂²×C₄×C₁₂₇	C₂²×C₄	1	9	yes	—
41	C₂²×C₄×C₁₂₇	C₂²×C₄	1	9	yes	—
42	C₂²×C₄×C₁₂₇	C₂×D₄	1	9	yes	—
43	C₂²×C₄×C₁₂₇	C₂×D₄	1	9	yes	—
44	C₂²×C₄×C₁₂₇	C₂×Q₈	1	3	yes	—
45	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
46	M₁₆×C₁₂₇	—	2	72	no	—
47	M₁₆×C₁₂₇	—	2	24	no	—
48	C₂×C₈×C₁₂₇	—	2	36	no	—
49	C₂×C₈×C₁₂₇	—	2	36	no	—
50	C₂×C₈×C₁₂₇	C₂²×C₄	1	24	yes	—
51	C₂³×C₁₂₇	—	2	12	no	—
52	C₂³×C₁₂₇	—	4	18	no	—
53	C₂³×C₁₂₇	—	4	6	no	—
54	C₂³×C₁₂₇	#95	1	9	yes	—
55	C₂³×C₁₂₇	#95	1	6	yes	—
56	C₂³×C₁₂₇	#95	1	1	yes	characteristic
57	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	9	yes	—
58	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	9	yes	—
59	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	#95	1	18	yes	—
61	C₂×C₄×C₁₂₇	#95	1	18	yes	—
62	C₂×C₄×C₁₂₇	#95	1	9	yes	—
63	C₂×C₄×C₁₂₇	#95	1	3	yes	—
64	C₂×C₄×C₁₂₇	C₂×C₄²	1	9	yes	—
65	C₂×C₄×C₁₂₇	C₂×C₄²	1	3	yes	—
66	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	9	yes	—
67	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	9	yes	—
68	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	18	yes	—
69	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	9	yes	—
70	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	9	yes	—
71	C₂×C₄×C₁₂₇	C₂³×C₄	1	3	yes	—
72	C₂×C₄×C₁₂₇	C₂²×D₄	1	3	yes	—
73	C₈×C₁₂₇	—	2	24	no	—
74	C₂²×C₁₂₇	—	2	18	no	—
75	C₂²×C₁₂₇	—	2	18	no	—
76	C₂²×C₁₂₇	—	4	6	no	—
77	C₂²×C₁₂₇	—	4	9	no	—
78	C₂²×C₁₂₇	—	4	1	no	—
79	C₂²×C₁₂₇	#338	1	4	yes	direct product factor
80	C₂²×C₁₂₇	C₂×#95	1	9	yes	—
81	C₂²×C₁₂₇	C₂×#95	1	3	yes	—
82	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	3	yes	—
83	C₄×C₁₂₇	C₂×#95	1	9	yes	—
84	C₄×C₁₂₇	C₂×#95	1	3	yes	—
85	C₄×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
86	C₄×C₁₂₇	C₂²×C₂²⋊C₄	1	3	yes	—
87	C₂²×#338	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₁₂₇	—	2	3	no	—
89	C₂×C₁₂₇	—	2	9	no	—
90	C₂×C₁₂₇	C₂×#338	1	6	yes	direct product factor
91	C₂×C₁₂₇	C₂²×#95	1	1	yes	characteristic
92	C₂³×M₁₆	C₂×C₁₂₇	1	3	yes	—
93	C₂×#338	C₂×C₁₂₇	1	12	yes	direct product factor
94	C₁₂₇	C₂²×#338	1	1	yes	characteristic, direct product factor, Sylow
95	C₂⁴×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
96	C₂²×M₁₆	—	2	6	no	—
97	C₂²×M₁₆	—	2	18	no	—
98	C₂²×M₁₆	C₄×C₁₂₇	1	18	yes	—
99	C₂²×M₁₆	C₂²×C₁₂₇	1	18	yes	—
100	C₂³×C₈	C₄×C₁₂₇	1	6	yes	—
101	#338	C₂²×C₁₂₇	1	16	yes	direct product factor
102	C₂⁵	Q₈×C₁₂₇	1	1	yes	characteristic
103	C₂³×C₄	C₂×C₄×C₁₂₇	1	3	yes	—
104	C₂³×C₄	C₂×C₄×C₁₂₇	1	9	yes	—
105	C₂³×C₄	D₄×C₁₂₇	1	3	yes	—
106	C₂³×C₄	D₄×C₁₂₇	1	9	yes	—
107	C₂³×C₄	Q₈×C₁₂₇	1	3	yes	—
108	C₂³×C₄	C₂³×C₁₂₇	1	3	yes	—
109	C₂×M₁₆	—	2	36	no	—
110	C₂×M₁₆	—	2	36	no	—
111	C₂×M₁₆	—	2	36	no	—
112	C₂×M₁₆	—	2	36	no	—
113	C₂×M₁₆	C₂×C₄×C₁₂₇	1	36	yes	—
114	C₂×M₁₆	C₂³×C₁₂₇	1	12	yes	—
115	C₂²×C₈	—	2	18	no	—
116	C₂²×C₈	—	2	6	no	—
117	C₂²×C₈	C₂×C₄×C₁₂₇	1	36	yes	—
118	C₂⁴	—	4	4	no	—
119	C₂⁴	C₄⋊C₄×C₁₂₇	1	3	yes	—
120	C₂⁴	C₄⋊C₄×C₁₂₇	1	9	yes	—
121	C₂⁴	C₂×Q₈×C₁₂₇	1	3	yes	—
122	C₂²×C₄	C₄²×C₁₂₇	1	9	yes	—
123	C₂²×C₄	C₄²×C₁₂₇	1	6	yes	—
124	C₂²×C₄	C₄²×C₁₂₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	9	yes	—
126	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	3	yes	—
127	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	18	yes	—
128	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	18	yes	—
129	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	9	yes	—
130	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	9	yes	—
131	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	18	yes	—
132	C₂²×C₄	C₂²×C₄×C₁₂₇	1	9	yes	—
133	C₂²×C₄	C₂²×C₄×C₁₂₇	1	9	yes	—
134	C₂²×C₄	C₂×D₄×C₁₂₇	1	9	yes	—
135	C₂²×C₄	C₂×D₄×C₁₂₇	1	9	yes	—
136	C₂²×C₄	C₂×Q₈×C₁₂₇	1	3	yes	—
137	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
138	M₁₆	—	2	72	no	—
139	M₁₆	—	2	24	no	—
140	C₂×C₈	—	2	36	no	—
141	C₂×C₈	—	2	36	no	—
142	C₂×C₈	C₂²×C₄×C₁₂₇	1	24	yes	—
143	C₂³	—	2	12	no	—
144	C₂³	—	4	18	no	—
145	C₂³	—	4	6	no	—
146	C₂³	#95×C₁₂₇	1	9	yes	—
147	C₂³	#95×C₁₂₇	1	6	yes	—
148	C₂³	#95×C₁₂₇	1	1	yes	characteristic
149	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	9	yes	—
150	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	9	yes	—
151	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
152	C₂×C₄	#95×C₁₂₇	1	18	yes	—
153	C₂×C₄	#95×C₁₂₇	1	18	yes	—
154	C₂×C₄	#95×C₁₂₇	1	9	yes	—
155	C₂×C₄	#95×C₁₂₇	1	3	yes	—
156	C₂×C₄	C₂×C₄²×C₁₂₇	1	9	yes	—
157	C₂×C₄	C₂×C₄²×C₁₂₇	1	3	yes	—
158	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	9	yes	—
159	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	9	yes	—
160	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	18	yes	—
161	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	9	yes	—
162	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	9	yes	—
163	C₂×C₄	C₂³×C₄×C₁₂₇	1	3	yes	—
164	C₂×C₄	C₂²×D₄×C₁₂₇	1	3	yes	—
165	C₈	—	2	24	no	—
166	C₂²	—	2	18	no	—
167	C₂²	—	2	18	no	—
168	C₂²	—	4	6	no	—
169	C₂²	—	4	9	no	—
170	C₂²	—	4	1	no	—
171	C₂²	#338×C₁₂₇	1	4	yes	direct product factor
172	C₂²	C₂×#95×C₁₂₇	1	9	yes	—
173	C₂²	C₂×#95×C₁₂₇	1	3	yes	—
174	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	3	yes	—
175	C₄	C₂×#95×C₁₂₇	1	3	yes	—
176	C₄	C₂×#95×C₁₂₇	1	9	yes	—
177	C₄	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
178	C₄	C₂²×C₂²⋊C₄×C₁₂₇	1	3	yes	—
179	C₂	—	2	3	no	—
180	C₂	—	2	9	no	—
181	C₂	C₂×#338×C₁₂₇	1	6	yes	direct product factor
182	C₂	C₂²×#95×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₄²
commutator	C₄	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	#95
prime radical	C₂⁵×C₁₂₇	Q₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#338×C₁₂₇	C₂	direct (N × H)
9. #338×C₁₂₇	C₂²	direct (N × H)
79. C₂²×C₁₂₇	#338	direct (N × H)
87. C₂²×#338	C₁₂₇	direct (N × H)
90. C₂×C₁₂₇	C₂×#338	direct (N × H)
93. C₂×#338	C₂×C₁₂₇	direct (N × H)
94. C₁₂₇	C₂²×#338	direct (N × H)
101. #338	C₂²×C₁₂₇	direct (N × H)
171. C₂²	#338×C₁₂₇	direct (N × H)
181. C₂	C₂×#338×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
C₂³×M₁₆×C₁₂₇	2
C₂×#338×C₁₂₇	2
C₂²×#338	127

Subgroups — C22×#338×C127 #44584012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#338×C₁₂₇ #44584012