Subgroups — C₂×C₄×#95×C₁₂₇ #44577418

Order 32512.

4462 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₁₂₇	C₂	1	3	yes	maximal
2	C₂²×#95×C₁₂₇	C₂	1	4	yes	maximal
3	C₄×#95×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
4	C₂⁴×C₄×C₁₂₇	C₂²	1	6	yes	—
5	C₂⁴×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₁₂₇	—	2	24	no	—
7	C₂²×C₄²×C₁₂₇	C₄	1	24	yes	complemented
8	C₂²×C₄²×C₁₂₇	C₂²	1	12	yes	—
9	C₂×#95×C₁₂₇	C₄	1	32	yes	direct product factor
10	C₂×#95×C₁₂₇	C₂²	1	16	yes	—
11	C₂⁵×C₁₂₇	C₂³	1	1	yes	characteristic
12	C₂³×C₄×C₁₂₇	—	2	24	no	—
13	C₂³×C₄×C₁₂₇	—	2	24	no	—
14	C₂³×C₄×C₁₂₇	C₂×C₄	1	24	yes	—
15	C₂³×C₄×C₁₂₇	C₂×C₄	1	48	yes	—
16	C₂³×C₄×C₁₂₇	C₂×C₄	1	12	yes	—
17	C₂³×C₄×C₁₂₇	D₄	1	12	yes	—
18	C₂³×C₄×C₁₂₇	Q₈	1	4	yes	—
19	C₂³×C₄×C₁₂₇	C₂³	1	12	yes	—
20	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
21	C₂×C₄²×C₁₂₇	—	2	96	no	—
22	C₂×C₄²×C₁₂₇	—	2	48	no	—
23	C₂×C₄²×C₁₂₇	C₂×C₄	1	48	yes	complemented
24	#95×C₁₂₇	C₂×C₄	1	64	yes	direct product factor
25	C₂⁴×C₁₂₇	C₂²×C₄	1	6	yes	—
26	C₂⁴×C₁₂₇	C₂²×C₄	1	8	yes	—
27	C₂⁴×C₁₂₇	C₂×D₄	1	6	yes	—
28	C₂⁴×C₁₂₇	C₂×Q₈	1	2	yes	—
29	C₂⁴×C₁₂₇	#41	1	8	yes	—
30	C₂⁴×C₁₂₇	C₂⁴	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	—	2	48	no	—
32	C₂²×C₄×C₁₂₇	—	2	48	no	—
33	C₂²×C₄×C₁₂₇	—	2	192	no	—
34	C₂²×C₄×C₁₂₇	—	2	24	no	—
35	C₂²×C₄×C₁₂₇	—	2	24	no	—
36	C₂²×C₄×C₁₂₇	C₄²	1	16	yes	—
37	C₂²×C₄×C₁₂₇	C₄²	1	96	yes	complemented
38	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	48	yes	—
39	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	48	yes	—
40	C₂²×C₄×C₁₂₇	C₂²×C₄	1	48	yes	—
41	C₂²×C₄×C₁₂₇	C₂²×C₄	1	24	yes	—
42	C₂²×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
43	C₂²×C₄×C₁₂₇	C₂×D₄	1	12	yes	—
44	C₂²×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
45	C₄²×C₁₂₇	—	2	96	no	—
46	C₂³×C₁₂₇	C₂×C₄²	1	4	yes	—
47	C₂³×C₁₂₇	C₂×C₄²	1	24	yes	—
48	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	12	yes	—
49	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	12	yes	—
50	C₂³×C₁₂₇	#117	1	24	yes	—
51	C₂³×C₁₂₇	C₄×D₄	1	48	yes	—
52	C₂³×C₁₂₇	C₄×Q₈	1	16	yes	—
53	C₂³×C₁₂₇	C₂³×C₄	1	3	yes	—
54	C₂³×C₁₂₇	C₂³×C₄	1	4	yes	—
55	C₂³×C₁₂₇	C₂²×D₄	1	3	yes	—
56	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
57	C₂³×C₁₂₇	C₂×#41	1	4	yes	—
58	C₂×C₄×C₁₂₇	—	2	24	no	—
59	C₂×C₄×C₁₂₇	—	2	24	no	—
60	C₂×C₄×C₁₂₇	—	2	96	no	—
61	C₂×C₄×C₁₂₇	—	2	192	no	—
62	C₂×C₄×C₁₂₇	#95	1	64	yes	direct product factor
63	C₂×C₄×C₁₂₇	C₂×C₄²	1	8	yes	—
64	C₂×C₄×C₁₂₇	C₂×C₄²	1	48	yes	complemented
65	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	24	yes	—
66	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	24	yes	—
67	C₂²×C₁₂₇	C₄³	1	8	yes	—
68	C₂²×C₁₂₇	C₂×#95	1	8	yes	—
69	C₂²×C₁₂₇	#376	1	8	yes	—
70	C₂²×C₁₂₇	C₄×C₂²⋊C₄	1	48	yes	—
71	C₂²×C₁₂₇	C₄×C₄⋊C₄	1	48	yes	—
72	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
73	C₂²×C₁₂₇	C₂²×C₄²	1	6	yes	—
74	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	3	yes	—
75	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	3	yes	—
76	C₂²×C₁₂₇	C₂×#117	1	6	yes	—
77	C₂²×C₁₂₇	C₂×C₄×D₄	1	12	yes	—
78	C₂²×C₁₂₇	C₂×C₄×Q₈	1	4	yes	—
79	C₄×C₁₂₇	—	2	48	no	—
80	C₄×C₁₂₇	C₂×#95	1	16	yes	direct product factor
81	C₂×C₄×#95	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₁₂₇	C₄×#95	1	16	yes	direct product factor
83	C₂×C₁₂₇	C₂×C₄³	1	1	yes	characteristic
84	C₂×C₁₂₇	C₂²×#95	1	1	yes	characteristic
85	C₂×C₁₂₇	C₂×#376	1	1	yes	characteristic
86	C₂×C₁₂₇	C₂×C₄×C₂²⋊C₄	1	6	yes	—
87	C₂×C₁₂₇	C₂×C₄×C₄⋊C₄	1	6	yes	—
88	C₂³×C₄²	C₂×C₁₂₇	1	3	yes	—
89	C₂²×#95	C₂×C₁₂₇	1	4	yes	—
90	C₄×#95	C₂×C₁₂₇	1	8	yes	direct product factor
91	C₁₂₇	C₂×C₄×#95	1	1	yes	characteristic, direct product factor, Sylow
92	C₂⁴×C₄	C₂²×C₁₂₇	1	6	yes	—
93	C₂⁴×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
94	C₂²×C₄²	—	2	24	no	—
95	C₂²×C₄²	C₄×C₁₂₇	1	24	yes	complemented
96	C₂²×C₄²	C₂²×C₁₂₇	1	12	yes	—
97	C₂×#95	C₄×C₁₂₇	1	32	yes	direct product factor
98	C₂×#95	C₂²×C₁₂₇	1	16	yes	—
99	C₂⁵	C₂³×C₁₂₇	1	1	yes	characteristic
100	C₂³×C₄	—	2	24	no	—
101	C₂³×C₄	—	2	24	no	—
102	C₂³×C₄	C₂×C₄×C₁₂₇	1	48	yes	—
103	C₂³×C₄	C₂×C₄×C₁₂₇	1	24	yes	—
104	C₂³×C₄	C₂×C₄×C₁₂₇	1	12	yes	—
105	C₂³×C₄	D₄×C₁₂₇	1	12	yes	—
106	C₂³×C₄	Q₈×C₁₂₇	1	4	yes	—
107	C₂³×C₄	C₂³×C₁₂₇	1	12	yes	—
108	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
109	C₂×C₄²	—	2	96	no	—
110	C₂×C₄²	—	2	48	no	—
111	C₂×C₄²	C₂×C₄×C₁₂₇	1	48	yes	complemented
112	#95	C₂×C₄×C₁₂₇	1	64	yes	direct product factor
113	C₂⁴	C₂²×C₄×C₁₂₇	1	6	yes	—
114	C₂⁴	C₂²×C₄×C₁₂₇	1	8	yes	—
115	C₂⁴	C₂×D₄×C₁₂₇	1	6	yes	—
116	C₂⁴	C₂×Q₈×C₁₂₇	1	2	yes	—
117	C₂⁴	#41×C₁₂₇	1	8	yes	—
118	C₂⁴	C₂⁴×C₁₂₇	1	1	yes	characteristic
119	C₂²×C₄	—	2	48	no	—
120	C₂²×C₄	—	2	48	no	—
121	C₂²×C₄	—	2	192	no	—
122	C₂²×C₄	—	2	24	no	—
123	C₂²×C₄	—	2	24	no	—
124	C₂²×C₄	C₄²×C₁₂₇	1	16	yes	—
125	C₂²×C₄	C₄²×C₁₂₇	1	96	yes	complemented
126	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	48	yes	—
127	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	48	yes	—
128	C₂²×C₄	C₂²×C₄×C₁₂₇	1	48	yes	—
129	C₂²×C₄	C₂²×C₄×C₁₂₇	1	24	yes	—
130	C₂²×C₄	C₂²×C₄×C₁₂₇	1	12	yes	—
131	C₂²×C₄	C₂×D₄×C₁₂₇	1	12	yes	—
132	C₂²×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
133	C₄²	—	2	96	no	—
134	C₂³	C₂×C₄²×C₁₂₇	1	4	yes	—
135	C₂³	C₂×C₄²×C₁₂₇	1	24	yes	—
136	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	12	yes	—
137	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	12	yes	—
138	C₂³	#117×C₁₂₇	1	24	yes	—
139	C₂³	C₄×D₄×C₁₂₇	1	48	yes	—
140	C₂³	C₄×Q₈×C₁₂₇	1	16	yes	—
141	C₂³	C₂³×C₄×C₁₂₇	1	3	yes	—
142	C₂³	C₂³×C₄×C₁₂₇	1	4	yes	—
143	C₂³	C₂²×D₄×C₁₂₇	1	3	yes	—
144	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
145	C₂³	C₂×#41×C₁₂₇	1	4	yes	—
146	C₂×C₄	—	2	24	no	—
147	C₂×C₄	—	2	24	no	—
148	C₂×C₄	—	2	96	no	—
149	C₂×C₄	—	2	192	no	—
150	C₂×C₄	#95×C₁₂₇	1	64	yes	direct product factor
151	C₂×C₄	C₂×C₄²×C₁₂₇	1	8	yes	—
152	C₂×C₄	C₂×C₄²×C₁₂₇	1	48	yes	complemented
153	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	24	yes	—
154	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	24	yes	—
155	C₂²	C₄³×C₁₂₇	1	8	yes	—
156	C₂²	C₂×#95×C₁₂₇	1	8	yes	—
157	C₂²	#376×C₁₂₇	1	8	yes	—
158	C₂²	C₄×C₂²⋊C₄×C₁₂₇	1	48	yes	—
159	C₂²	C₄×C₄⋊C₄×C₁₂₇	1	48	yes	—
160	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
161	C₂²	C₂²×C₄²×C₁₂₇	1	6	yes	—
162	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	3	yes	—
163	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	3	yes	—
164	C₂²	C₂×#117×C₁₂₇	1	6	yes	—
165	C₂²	C₂×C₄×D₄×C₁₂₇	1	12	yes	—
166	C₂²	C₂×C₄×Q₈×C₁₂₇	1	4	yes	—
167	C₄	—	2	48	no	—
168	C₄	C₂×#95×C₁₂₇	1	16	yes	direct product factor
169	C₂	C₄×#95×C₁₂₇	1	16	yes	direct product factor
170	C₂	C₂×C₄³×C₁₂₇	1	1	yes	characteristic
171	C₂	C₂²×#95×C₁₂₇	1	1	yes	characteristic
172	C₂	C₂×#376×C₁₂₇	1	1	yes	characteristic
173	C₂	C₂×C₄×C₂²⋊C₄×C₁₂₇	1	6	yes	—
174	C₂	C₂×C₄×C₄⋊C₄×C₁₂₇	1	6	yes	—

	subgroup	G/H
center	C₂⁴×C₄×C₁₂₇	C₂²
commutator	C₂	C₂×C₄³×C₁₂₇
Frattini, cosocle	C₂⁴	C₂⁴×C₁₂₇
socle, prime radical	C₂⁵×C₁₂₇	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×#95×C₁₂₇	C₂	direct (N × H)
7. C₂²×C₄²×C₁₂₇	C₄	semidirect
9. C₂×#95×C₁₂₇	C₄	direct (N × H)
23. C₂×C₄²×C₁₂₇	C₂×C₄	semidirect
24. #95×C₁₂₇	C₂×C₄	direct (N × H)
37. C₂²×C₄×C₁₂₇	C₄²	semidirect
62. C₂×C₄×C₁₂₇	#95	direct (N × H)
64. C₂×C₄×C₁₂₇	C₂×C₄²	semidirect
80. C₄×C₁₂₇	C₂×#95	direct (N × H)
81. C₂×C₄×#95	C₁₂₇	direct (N × H)
82. C₂×C₁₂₇	C₄×#95	direct (N × H)
90. C₄×#95	C₂×C₁₂₇	direct (N × H)
91. C₁₂₇	C₂×C₄×#95	direct (N × H)
95. C₂²×C₄²	C₄×C₁₂₇	semidirect
97. C₂×#95	C₄×C₁₂₇	direct (N × H)
111. C₂×C₄²	C₂×C₄×C₁₂₇	semidirect
112. #95	C₂×C₄×C₁₂₇	direct (N × H)
125. C₂²×C₄	C₄²×C₁₂₇	semidirect
150. C₂×C₄	#95×C₁₂₇	direct (N × H)
152. C₂×C₄	C₂×C₄²×C₁₂₇	semidirect
168. C₄	C₂×#95×C₁₂₇	direct (N × H)
169. C₂	C₄×#95×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×C₄²×C₁₂₇	2
C₂²×#95×C₁₂₇	2
C₄×#95×C₁₂₇	2
C₂×C₄×#95	127

Subgroups — C2×C4×#95×C127 #44577418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#95×C₁₂₇ #44577418