Subgroups — C₂×C₁₂₇×#1709 #44583270

Order 32512.

982 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#493×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#486×C₁₂₇	C₂	1	4	yes	maximal, complemented
4	C₁₂₇×#1709	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈²×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₁₂₇	—	2	2	no	—
7	C₂²×D₈×C₁₂₇	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
9	C₂×#127×C₁₂₇	C₂²	1	2	yes	—
10	#494×C₁₂₇	—	2	2	no	—
11	#494×C₁₂₇	C₂²	1	4	yes	complemented
12	#493×C₁₂₇	—	2	2	no	—
13	#493×C₁₂₇	C₂²	1	4	yes	—
14	#486×C₁₂₇	—	2	8	no	—
15	#486×C₁₂₇	C₂²	1	16	yes	complemented
16	C₂×#102×C₁₂₇	—	2	8	no	—
17	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₈²×C₁₂₇	C₂²	1	4	yes	complemented
20	C₂²×Q₈×C₁₂₇	—	2	2	no	—
21	C₂²×D₄×C₁₂₇	—	2	2	no	—
22	C₂²×D₄×C₁₂₇	—	2	4	no	—
23	C₂×Q₁₆×C₁₂₇	—	2	8	no	—
24	C₂×Q₁₆×C₁₂₇	—	4	8	no	—
25	C₂×D₈×C₁₂₇	—	2	8	no	—
26	C₂×D₈×C₁₂₇	—	4	8	no	—
27	C₂²×C₈×C₁₂₇	D₄	1	4	yes	—
28	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
29	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
30	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
31	C₄⋊Q₈×C₁₂₇	C₂³	1	4	yes	—
32	#127×C₁₂₇	—	2	4	no	—
33	#127×C₁₂₇	—	2	2	no	—
34	#127×C₁₂₇	C₂³	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
36	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
37	#102×C₁₂₇	—	2	32	no	—
38	C₄×C₈×C₁₂₇	D₄	1	4	yes	—
39	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
40	C₄×C₈×C₁₂₇	C₂³	1	4	yes	—
41	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
42	C₂⁴×C₁₂₇	—	4	2	no	—
43	C₂×Q₈×C₁₂₇	—	2	4	no	—
44	C₂×Q₈×C₁₂₇	—	4	2	no	—
45	C₂×Q₈×C₁₂₇	—	4	8	no	—
46	C₂×D₄×C₁₂₇	—	2	8	no	—
47	C₂×D₄×C₁₂₇	—	2	4	no	—
48	C₂×D₄×C₁₂₇	—	4	2	no	—
49	C₂×D₄×C₁₂₇	—	4	4	no	—
50	C₂×D₄×C₁₂₇	—	4	8	no	—
51	C₂×D₄×C₁₂₇	—	4	16	no	—
52	C₂²×C₄×C₁₂₇	—	4	2	no	—
53	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
55	Q₁₆×C₁₂₇	—	4	16	no	—
56	D₈×C₁₂₇	—	4	16	no	—
57	C₂×C₈×C₁₂₇	—	2	8	no	—
58	C₂×C₈×C₁₂₇	D₈	1	8	yes	—
59	C₂×C₈×C₁₂₇	C₂×D₄	1	8	yes	—
60	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
61	C₄⋊C₄×C₁₂₇	—	2	8	no	—
62	C₄⋊C₄×C₁₂₇	—	4	4	no	—
63	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
64	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
66	C₂³×C₁₂₇	—	4	4	no	—
67	C₂³×C₁₂₇	—	8	4	no	—
68	C₂³×C₁₂₇	—	8	8	no	—
69	C₂³×C₁₂₇	#127	1	1	yes	characteristic
70	Q₈×C₁₂₇	—	4	8	no	—
71	Q₈×C₁₂₇	—	8	4	no	—
72	D₄×C₁₂₇	—	4	8	no	—
73	D₄×C₁₂₇	—	4	16	no	—
74	D₄×C₁₂₇	—	8	4	no	—
75	D₄×C₁₂₇	—	8	8	no	—
76	C₂×C₄×C₁₂₇	—	4	4	no	—
77	C₂×C₄×C₁₂₇	—	8	4	no	—
78	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	#127	1	2	yes	—
80	C₂×C₄×C₁₂₇	C₂×D₈	1	4	yes	—
81	C₂×C₄×C₁₂₇	#135	1	8	yes	—
82	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
84	C₈×C₁₂₇	—	2	8	no	—
85	C₈×C₁₂₇	C₂×D₈	1	8	yes	—
86	C₂²×C₁₂₇	—	8	4	no	—
87	C₂²×C₁₂₇	—	8	8	no	—
88	C₂²×C₁₂₇	—	16	8	no	—
89	C₂²×C₁₂₇	#493	1	2	yes	—
90	C₂²×C₁₂₇	#495	1	4	yes	—
91	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
92	C₄×C₁₂₇	—	8	4	no	—
93	C₄×C₁₂₇	#493	1	2	yes	—
94	C₄×C₁₂₇	#495	1	4	yes	—
95	C₄×C₁₂₇	C₂²×D₈	1	2	yes	—
96	C₄×C₁₂₇	C₂×#135	1	4	yes	—
97	C₂×#1709	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₁₂₇	—	16	4	no	—
99	C₂×C₁₂₇	#1709	1	4	yes	direct product factor
100	C₂×C₁₂₇	C₂×#493	1	1	yes	characteristic
101	C₂×C₁₂₇	C₂×#495	1	2	yes	—
102	C₂×#494	C₂×C₁₂₇	1	1	yes	characteristic, complemented
103	C₂×#493	C₂×C₁₂₇	1	1	yes	characteristic
104	C₂×#486	C₂×C₁₂₇	1	4	yes	complemented
105	#1709	C₂×C₁₂₇	1	8	yes	direct product factor
106	C₂×C₈²	C₂×C₁₂₇	1	1	yes	characteristic, complemented
107	C₁₂₇	C₂×#1709	1	1	yes	characteristic, direct product factor, Sylow
108	C₂²×Q₁₆	—	2	2	no	—
109	C₂²×D₈	—	2	2	no	—
110	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
111	C₂×#127	C₂²×C₁₂₇	1	2	yes	—
112	#494	—	2	2	no	—
113	#494	C₂²×C₁₂₇	1	4	yes	complemented
114	#493	—	2	2	no	—
115	#493	C₂²×C₁₂₇	1	4	yes	—
116	#486	—	2	8	no	—
117	#486	C₂²×C₁₂₇	1	16	yes	complemented
118	C₂×#102	—	2	8	no	—
119	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
120	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
121	C₈²	C₂²×C₁₂₇	1	4	yes	complemented
122	C₂²×Q₈	—	2	2	no	—
123	C₂²×D₄	—	2	2	no	—
124	C₂²×D₄	—	2	4	no	—
125	C₂×Q₁₆	—	2	8	no	—
126	C₂×Q₁₆	—	4	8	no	—
127	C₂×D₈	—	2	8	no	—
128	C₂×D₈	—	4	8	no	—
129	C₂²×C₈	D₄×C₁₂₇	1	4	yes	—
130	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
131	C₄⋊Q₈	—	2	4	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	C₂³×C₁₂₇	1	4	yes	—
134	#127	—	2	4	no	—
135	#127	—	2	2	no	—
136	#127	C₂³×C₁₂₇	1	4	yes	—
137	C₂×C₄⋊C₄	—	2	4	no	—
138	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
139	#102	—	2	32	no	—
140	C₄×C₈	D₄×C₁₂₇	1	4	yes	—
141	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
142	C₄×C₈	C₂³×C₁₂₇	1	4	yes	—
143	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
144	C₂⁴	—	4	2	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	8	no	—
148	C₂×D₄	—	2	8	no	—
149	C₂×D₄	—	2	4	no	—
150	C₂×D₄	—	4	2	no	—
151	C₂×D₄	—	4	4	no	—
152	C₂×D₄	—	4	8	no	—
153	C₂×D₄	—	4	16	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
157	Q₁₆	—	4	16	no	—
158	D₈	—	4	16	no	—
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	D₈×C₁₂₇	1	8	yes	—
161	C₂×C₈	C₂×D₄×C₁₂₇	1	8	yes	—
162	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
163	C₄⋊C₄	—	2	8	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
166	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
167	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
168	C₂³	—	4	4	no	—
169	C₂³	—	8	4	no	—
170	C₂³	—	8	8	no	—
171	C₂³	#127×C₁₂₇	1	1	yes	characteristic
172	Q₈	—	4	8	no	—
173	Q₈	—	8	4	no	—
174	D₄	—	4	8	no	—
175	D₄	—	4	16	no	—
176	D₄	—	8	4	no	—
177	D₄	—	8	8	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
181	C₂×C₄	#127×C₁₂₇	1	2	yes	—
182	C₂×C₄	C₂×D₈×C₁₂₇	1	4	yes	—
183	C₂×C₄	#135×C₁₂₇	1	8	yes	—
184	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
185	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
186	C₈	—	2	8	no	—
187	C₈	C₂×D₈×C₁₂₇	1	8	yes	—
188	C₂²	—	8	4	no	—
189	C₂²	—	8	8	no	—
190	C₂²	—	16	8	no	—
191	C₂²	#493×C₁₂₇	1	2	yes	—
192	C₂²	#495×C₁₂₇	1	4	yes	—
193	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
194	C₄	—	8	4	no	—
195	C₄	#493×C₁₂₇	1	2	yes	—
196	C₄	#495×C₁₂₇	1	4	yes	—
197	C₄	C₂²×D₈×C₁₂₇	1	2	yes	—
198	C₄	C₂×#135×C₁₂₇	1	4	yes	—
199	C₂	—	16	4	no	—
200	C₂	C₁₂₇×#1709	1	4	yes	direct product factor
201	C₂	C₂×#493×C₁₂₇	1	1	yes	characteristic
202	C₂	C₂×#495×C₁₂₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₂×#493×C₁₂₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#494×C₁₂₇	C₂	semidirect
3. C₂×#486×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1709	C₂	direct (N × H)
5. C₂×C₈²×C₁₂₇	C₂	semidirect
11. #494×C₁₂₇	C₂²	semidirect
15. #486×C₁₂₇	C₂²	semidirect
19. C₈²×C₁₂₇	C₂²	semidirect
97. C₂×#1709	C₁₂₇	direct (N × H)
99. C₂×C₁₂₇	#1709	direct (N × H)
102. C₂×#494	C₂×C₁₂₇	semidirect
104. C₂×#486	C₂×C₁₂₇	semidirect
105. #1709	C₂×C₁₂₇	direct (N × H)
106. C₂×C₈²	C₂×C₁₂₇	semidirect
107. C₁₂₇	C₂×#1709	direct (N × H)
113. #494	C₂²×C₁₂₇	semidirect
117. #486	C₂²×C₁₂₇	semidirect
121. C₈²	C₂²×C₁₂₇	semidirect
200. C₂	C₁₂₇×#1709	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×#494×C₁₂₇	2
C₂×#493×C₁₂₇	2
C₂×#486×C₁₂₇	2
C₁₂₇×#1709	2
C₂×C₈²×C₁₂₇	2
C₂×#1709	127

Subgroups — C2×C127×#1709 #44583270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1709 #44583270