Subgroups — C₁₂₇×#19603 #44583276

Order 32512.

922 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	C₁₂₇×#3169	C₂	1	4	yes	maximal, complemented
2	C₁₂₇×#3147	C₂	1	2	yes	maximal, complemented
3	C₁₂₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1709	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1706	C₂	1	4	yes	maximal, complemented
7	C₂×C₈²×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₁₂₇	—	2	4	no	—
9	#532×C₁₂₇	C₂²	1	4	yes	—
10	#500×C₁₂₇	C₂²	1	4	yes	—
11	#495×C₁₂₇	—	2	4	no	—
12	#494×C₁₂₇	C₂²	1	2	yes	—
13	#493×C₁₂₇	C₂²	1	2	yes	—
14	#493×C₁₂₇	C₂²	1	4	yes	—
15	#492×C₁₂₇	C₂²	1	4	yes	—
16	#486×C₁₂₇	C₂²	1	8	yes	—
17	#466×C₁₂₇	—	2	8	no	—
18	C₄×D₈×C₁₂₇	—	2	8	no	—
19	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
21	C₈²×C₁₂₇	C₂²	1	2	yes	—
22	C₈²×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₈²×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×#41×C₁₂₇	—	2	4	no	—
25	#135×C₁₂₇	—	4	16	no	—
26	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
27	C₂×SD₁₆×C₁₂₇	—	2	8	no	—
28	C₂×D₈×C₁₂₇	—	2	4	no	—
29	C₂×D₈×C₁₂₇	—	2	8	no	—
30	C₂²×C₈×C₁₂₇	D₄	1	4	yes	—
31	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
32	C₄⋊Q₈×C₁₂₇	C₂³	1	4	yes	—
33	#127×C₁₂₇	C₂³	1	4	yes	—
34	#124×C₁₂₇	—	2	4	no	—
35	#121×C₁₂₇	—	2	8	no	—
36	C₄×D₄×C₁₂₇	—	2	8	no	—
37	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
38	#107×C₁₂₇	—	2	8	no	—
39	#102×C₁₂₇	—	2	16	no	—
40	C₄×C₈×C₁₂₇	D₄	1	4	yes	—
41	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
42	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
43	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
44	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
46	#41×C₁₂₇	—	4	16	no	—
47	C₂×Q₈×C₁₂₇	—	2	4	no	—
48	C₂×D₄×C₁₂₇	—	2	8	no	—
49	C₂×D₄×C₁₂₇	—	2	4	no	—
50	C₂×D₄×C₁₂₇	—	4	4	no	—
51	C₂²×C₄×C₁₂₇	—	4	4	no	—
52	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	Q₁₆×C₁₂₇	—	4	8	no	—
55	SD₁₆×C₁₂₇	—	4	16	no	—
56	D₈×C₁₂₇	—	4	8	no	—
57	D₈×C₁₂₇	—	4	16	no	—
58	C₂×C₈×C₁₂₇	—	2	8	no	—
59	C₂×C₈×C₁₂₇	D₈	1	8	yes	complemented
60	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
61	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
64	C₄⋊C₄×C₁₂₇	—	2	8	no	—
65	C₂²⋊C₄×C₁₂₇	—	4	8	no	—
66	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
68	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
69	C₂³×C₁₂₇	—	4	4	no	—
70	C₂³×C₁₂₇	#127	1	1	yes	characteristic
71	Q₈×C₁₂₇	—	4	8	no	—
72	D₄×C₁₂₇	—	4	8	no	—
73	D₄×C₁₂₇	—	4	16	no	—
74	D₄×C₁₂₇	—	8	8	no	—
75	C₂×C₄×C₁₂₇	—	2	4	no	—
76	C₂×C₄×C₁₂₇	—	4	4	no	—
77	C₂×C₄×C₁₂₇	—	8	8	no	—
78	C₂×C₄×C₁₂₇	#127	1	2	yes	—
79	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂×D₈	1	4	yes	—
81	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
83	C₈×C₁₂₇	—	2	4	no	—
84	C₈×C₁₂₇	C₂×D₈	1	4	yes	—
85	C₈×C₁₂₇	C₂×D₈	1	4	yes	—
86	C₈×C₁₂₇	#135	1	8	yes	complemented
87	C₂²×C₁₂₇	—	2	2	no	—
88	C₂²×C₁₂₇	—	8	4	no	—
89	C₂²×C₁₂₇	—	8	8	no	—
90	C₂²×C₁₂₇	#493	1	2	yes	—
91	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
92	C₄×C₁₂₇	—	2	2	no	—
93	C₄×C₁₂₇	—	8	4	no	—
94	C₄×C₁₂₇	#493	1	2	yes	—
95	C₄×C₁₂₇	C₂²×D₈	1	2	yes	—
96	C₄×C₁₂₇	C₂×#135	1	4	yes	—
97	#19603	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₁₂₇	—	2	2	no	—
99	C₂×C₁₂₇	—	16	4	no	—
100	C₂×C₁₂₇	C₂×#493	1	1	yes	characteristic
101	C₂×C₁₂₇	#3147	1	2	yes	—
102	#3169	C₂×C₁₂₇	1	4	yes	complemented
103	#3147	C₂×C₁₂₇	1	2	yes	complemented
104	#1713	C₂×C₁₂₇	1	1	yes	characteristic, complemented
105	#1712	C₂×C₁₂₇	1	1	yes	characteristic, complemented
106	#1709	C₂×C₁₂₇	1	2	yes	complemented
107	#1706	C₂×C₁₂₇	1	4	yes	complemented
108	C₂×C₈²	C₂×C₁₂₇	1	1	yes	characteristic, complemented
109	C₁₂₇	#19603	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×#135	—	2	4	no	—
111	#532	C₂²×C₁₂₇	1	4	yes	—
112	#500	C₂²×C₁₂₇	1	4	yes	—
113	#495	—	2	4	no	—
114	#494	C₂²×C₁₂₇	1	2	yes	—
115	#493	C₂²×C₁₂₇	1	2	yes	—
116	#493	C₂²×C₁₂₇	1	4	yes	—
117	#492	C₂²×C₁₂₇	1	4	yes	—
118	#486	C₂²×C₁₂₇	1	8	yes	—
119	#466	—	2	8	no	—
120	C₄×D₈	—	2	8	no	—
121	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
123	C₈²	C₂²×C₁₂₇	1	2	yes	—
124	C₈²	C₂²×C₁₂₇	1	1	yes	characteristic
125	C₈²	C₂²×C₁₂₇	1	1	yes	characteristic
126	C₂×#41	—	2	4	no	—
127	#135	—	4	16	no	—
128	C₂×Q₁₆	—	2	4	no	—
129	C₂×SD₁₆	—	2	8	no	—
130	C₂×D₈	—	2	4	no	—
131	C₂×D₈	—	2	8	no	—
132	C₂²×C₈	D₄×C₁₂₇	1	4	yes	—
133	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
134	C₄⋊Q₈	C₂³×C₁₂₇	1	4	yes	—
135	#127	C₂³×C₁₂₇	1	4	yes	—
136	#124	—	2	4	no	—
137	#121	—	2	8	no	—
138	C₄×D₄	—	2	8	no	—
139	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
140	#107	—	2	8	no	—
141	#102	—	2	16	no	—
142	C₄×C₈	D₄×C₁₂₇	1	4	yes	—
143	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
144	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
145	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
146	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
147	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
148	#41	—	4	16	no	—
149	C₂×Q₈	—	2	4	no	—
150	C₂×D₄	—	2	8	no	—
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	4	4	no	—
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
155	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
156	Q₁₆	—	4	8	no	—
157	SD₁₆	—	4	16	no	—
158	D₈	—	4	8	no	—
159	D₈	—	4	16	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	D₈×C₁₂₇	1	8	yes	complemented
162	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
163	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
164	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
166	C₄⋊C₄	—	2	8	no	—
167	C₂²⋊C₄	—	4	8	no	—
168	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
169	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
170	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
171	C₂³	—	4	4	no	—
172	C₂³	#127×C₁₂₇	1	1	yes	characteristic
173	Q₈	—	4	8	no	—
174	D₄	—	4	8	no	—
175	D₄	—	4	16	no	—
176	D₄	—	8	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	8	no	—
180	C₂×C₄	#127×C₁₂₇	1	2	yes	—
181	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
182	C₂×C₄	C₂×D₈×C₁₂₇	1	4	yes	—
183	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
184	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
185	C₈	—	2	4	no	—
186	C₈	C₂×D₈×C₁₂₇	1	4	yes	—
187	C₈	C₂×D₈×C₁₂₇	1	4	yes	—
188	C₈	#135×C₁₂₇	1	8	yes	complemented
189	C₂²	—	2	2	no	—
190	C₂²	—	8	4	no	—
191	C₂²	—	8	8	no	—
192	C₂²	#493×C₁₂₇	1	2	yes	—
193	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	8	4	no	—
196	C₄	#493×C₁₂₇	1	2	yes	—
197	C₄	C₂²×D₈×C₁₂₇	1	2	yes	—
198	C₄	C₂×#135×C₁₂₇	1	4	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	16	4	no	—
201	C₂	C₂×#493×C₁₂₇	1	1	yes	characteristic
202	C₂	C₁₂₇×#3147	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₂₇×#3169	C₂	semidirect
2. C₁₂₇×#3147	C₂	semidirect
3. C₁₂₇×#1713	C₂	semidirect
4. C₁₂₇×#1712	C₂	semidirect
5. C₁₂₇×#1709	C₂	semidirect
6. C₁₂₇×#1706	C₂	semidirect
7. C₂×C₈²×C₁₂₇	C₂	semidirect
59. C₂×C₈×C₁₂₇	D₈	semidirect
86. C₈×C₁₂₇	#135	semidirect
97. #19603	C₁₂₇	direct (N × H)
102. #3169	C₂×C₁₂₇	semidirect
103. #3147	C₂×C₁₂₇	semidirect
104. #1713	C₂×C₁₂₇	semidirect
105. #1712	C₂×C₁₂₇	semidirect
106. #1709	C₂×C₁₂₇	semidirect
107. #1706	C₂×C₁₂₇	semidirect
108. C₂×C₈²	C₂×C₁₂₇	semidirect
109. C₁₂₇	#19603	direct (N × H)
161. C₂×C₈	D₈×C₁₂₇	semidirect
188. C₈	#135×C₁₂₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₂₇×#3169	2
C₁₂₇×#3147	2
C₁₂₇×#1713	2
C₁₂₇×#1712	2
C₁₂₇×#1709	2
C₁₂₇×#1706	2
C₂×C₈²×C₁₂₇	2
#19603	127

Subgroups — C127×#19603 #44583276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19603 #44583276