Subgroups — C₁₂₇×#21819 #44585492

Order 32512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₁₂₇	C₂	1	2	yes	maximal
3	C₁₂₇×#1984	C₂	1	8	yes	maximal, complemented
4	C₁₂₇×#1818	C₂	1	4	yes	maximal
5	#533×C₁₂₇	—	2	8	no	—
6	C₂×C₄⋊Q₈×C₁₂₇	C₄	1	2	yes	—
7	C₂×#125×C₁₂₇	C₄	1	2	yes	—
8	C₂×C₂²⋊Q₈×C₁₂₇	C₄	1	4	yes	—
9	C₂×C₄×Q₈×C₁₂₇	C₂²	1	4	yes	—
10	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	#431×C₁₂₇	—	2	8	no	—
13	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	4	yes	—
16	#336×C₁₂₇	C₂²	1	16	yes	—
17	C₂²×Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
18	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₁₂₇	C₂³	1	8	yes	—
21	C₄⋊Q₈×C₁₂₇	—	2	8	no	—
22	#125×C₁₂₇	—	2	8	no	—
23	C₂²⋊Q₈×C₁₂₇	—	2	16	no	—
24	C₄×Q₈×C₁₂₇	—	2	8	no	—
25	C₄×Q₈×C₁₂₇	—	2	8	no	—
26	#117×C₁₂₇	—	2	8	no	—
27	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
29	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₂₇	D₄	1	8	yes	—
34	C₂×C₄²×C₁₂₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
36	C₄⋊C₈×C₁₂₇	—	2	8	no	—
37	C₂²⋊C₈×C₁₂₇	—	2	8	no	—
38	C₄×C₈×C₁₂₇	—	2	8	no	—
39	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×Q₈×C₁₂₇	—	2	8	no	—
41	C₂×Q₈×C₁₂₇	—	2	16	no	—
42	C₂²×C₄×C₁₂₇	—	2	8	no	—
43	C₂²×C₄×C₁₂₇	—	2	4	no	—
44	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	8	yes	—
45	C₂²×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
46	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
50	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	—	2	8	no	—
52	C₂×C₈×C₁₂₇	—	2	16	no	—
53	C₄⋊C₄×C₁₂₇	—	2	8	no	—
54	C₄⋊C₄×C₁₂₇	—	2	16	no	—
55	C₄⋊C₄×C₁₂₇	—	2	8	no	—
56	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
57	C₄²×C₁₂₇	—	2	8	no	—
58	C₄²×C₁₂₇	C₂²⋊C₄	1	4	yes	—
59	C₄²×C₁₂₇	C₂²⋊C₄	1	4	yes	—
60	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
61	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
62	C₂³×C₁₂₇	—	2	2	no	—
63	C₂³×C₁₂₇	—	2	4	no	—
64	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
65	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
66	Q₈×C₁₂₇	—	4	16	no	—
67	C₂×C₄×C₁₂₇	—	2	2	no	—
68	C₂×C₄×C₁₂₇	—	2	4	no	—
69	C₂×C₄×C₁₂₇	—	2	8	no	—
70	C₂×C₄×C₁₂₇	—	2	16	no	—
71	C₂×C₄×C₁₂₇	—	2	8	no	—
72	C₂×C₄×C₁₂₇	—	2	16	no	—
73	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
74	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
75	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
76	C₂×C₄×C₁₂₇	#131	1	8	yes	—
77	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
78	C₂×C₄×C₁₂₇	2₋¹⁺⁴	1	4	yes	—
79	C₈×C₁₂₇	—	4	8	no	—
80	C₂²×C₁₂₇	—	2	2	no	—
81	C₂²×C₁₂₇	—	2	4	no	—
82	C₂²×C₁₂₇	—	4	4	no	—
83	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₁₂₇	#519	1	2	yes	—
85	C₂²×C₁₂₇	C₂×#131	1	4	yes	—
86	C₄×C₁₂₇	—	2	2	no	—
87	C₄×C₁₂₇	—	2	4	no	—
88	C₄×C₁₂₇	—	4	8	no	—
89	C₄×C₁₂₇	#408	1	8	yes	—
90	C₄×C₁₂₇	#536	1	4	yes	—
91	#21819	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₁₂₇	—	4	2	no	—
93	C₂×C₁₂₇	#2310	1	1	yes	characteristic
94	C₂×C₁₂₇	C₂×#408	1	2	yes	—
95	C₂×C₁₂₇	#2986	1	4	yes	—
96	C₂×#533	C₂×C₁₂₇	1	1	yes	characteristic
97	C₂×#431	C₂×C₁₂₇	1	2	yes	—
98	#1984	C₂×C₁₂₇	1	8	yes	complemented
99	#1818	C₂×C₁₂₇	1	4	yes	—
100	C₁₂₇	#21819	1	1	yes	characteristic, direct product factor, Sylow
101	#533	—	2	8	no	—
102	C₂×C₄⋊Q₈	C₄×C₁₂₇	1	2	yes	—
103	C₂×#125	C₄×C₁₂₇	1	2	yes	—
104	C₂×C₂²⋊Q₈	C₄×C₁₂₇	1	4	yes	—
105	C₂×C₄×Q₈	C₂²×C₁₂₇	1	4	yes	—
106	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
107	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
108	#431	—	2	8	no	—
109	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	—
110	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	4	yes	—
111	C₂×C₄×C₈	C₂²×C₁₂₇	1	4	yes	—
112	#336	C₂²×C₁₂₇	1	16	yes	—
113	C₂²×Q₈	C₂×C₄×C₁₂₇	1	4	yes	—
114	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
115	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₁₂₇	1	8	yes	—
117	C₄⋊Q₈	—	2	8	no	—
118	#125	—	2	8	no	—
119	C₂²⋊Q₈	—	2	16	no	—
120	C₄×Q₈	—	2	8	no	—
121	C₄×Q₈	—	2	8	no	—
122	#117	—	2	8	no	—
123	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
124	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
125	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
126	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
127	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
128	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
129	C₂×C₄²	D₄×C₁₂₇	1	8	yes	—
130	C₂×C₄²	C₂³×C₁₂₇	1	4	yes	—
131	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
132	C₄⋊C₈	—	2	8	no	—
133	C₂²⋊C₈	—	2	8	no	—
134	C₄×C₈	—	2	8	no	—
135	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
136	C₂×Q₈	—	2	8	no	—
137	C₂×Q₈	—	2	16	no	—
138	C₂²×C₄	—	2	8	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	8	yes	—
141	C₂²×C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
142	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
144	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
145	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
146	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	—	2	16	no	—
149	C₄⋊C₄	—	2	8	no	—
150	C₄⋊C₄	—	2	16	no	—
151	C₄⋊C₄	—	2	8	no	—
152	C₂²⋊C₄	—	2	8	no	—
153	C₄²	—	2	8	no	—
154	C₄²	C₂²⋊C₄×C₁₂₇	1	4	yes	—
155	C₄²	C₂²⋊C₄×C₁₂₇	1	4	yes	—
156	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
157	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
158	C₂³	—	2	2	no	—
159	C₂³	—	2	4	no	—
160	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
161	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
162	Q₈	—	4	16	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	16	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	16	no	—
169	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
170	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
171	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
172	C₂×C₄	#131×C₁₂₇	1	8	yes	—
173	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
174	C₂×C₄	2₋¹⁺⁴×C₁₂₇	1	4	yes	—
175	C₈	—	4	8	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	4	4	no	—
179	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
180	C₂²	#519×C₁₂₇	1	2	yes	—
181	C₂²	C₂×#131×C₁₂₇	1	4	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	8	no	—
185	C₄	#408×C₁₂₇	1	8	yes	—
186	C₄	#536×C₁₂₇	1	4	yes	—
187	C₂	—	4	2	no	—
188	C₂	C₁₂₇×#2310	1	1	yes	characteristic
189	C₂	C₂×#408×C₁₂₇	1	2	yes	—
190	C₂	C₁₂₇×#2986	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1984	C₂	semidirect
91. #21819	C₁₂₇	direct (N × H)
98. #1984	C₂×C₁₂₇	semidirect
100. C₁₂₇	#21819	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×#533×C₁₂₇	2
C₂×#431×C₁₂₇	2
C₁₂₇×#1984	2
C₁₂₇×#1818	2
#21819	127

Subgroups — C127×#21819 #44585492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21819 #44585492