Subgroups — C₁₂₇×#13167 #44576840

Order 32512.

318 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#366×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1942	C₂	1	2	yes	maximal
5	C₁₂₇×#1380	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
9	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	#389×C₁₂₇	—	2	2	no	—
12	#367×C₁₂₇	C₄	1	4	yes	complemented
13	#366×C₁₂₇	—	2	2	no	—
14	#346×C₁₂₇	—	2	2	no	—
15	#340×C₁₂₇	—	2	2	no	—
16	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₂₇	—	4	2	no	—
21	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₂₇	—	2	1	no	—
23	C₂×C₁₆×C₁₂₇	—	2	1	no	—
24	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
26	C₂×C₁₆×C₁₂₇	Q₈	1	2	yes	—
27	C₂×C₁₆×C₁₂₇	Q₈	1	2	yes	—
28	#107×C₁₂₇	—	2	2	no	—
29	#107×C₁₂₇	—	2	2	no	—
30	#107×C₁₂₇	—	2	2	no	—
31	#107×C₁₂₇	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₁₂₇	—	2	1	no	—
33	C₄×C₈×C₁₂₇	—	2	1	no	—
34	#95×C₁₂₇	—	4	2	no	—
35	C₂²×C₄×C₁₂₇	—	4	2	no	—
36	C₂²×C₄×C₁₂₇	—	4	2	no	—
37	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	—	2	1	no	—
41	C₂×C₈×C₁₂₇	—	2	1	no	—
42	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
43	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₁₂₇	—	2	4	no	—
50	C₄⋊C₄×C₁₂₇	—	4	2	no	—
51	C₄⋊C₄×C₁₂₇	—	4	2	no	—
52	C₄⋊C₄×C₁₂₇	—	4	2	no	—
53	C₄⋊C₄×C₁₂₇	—	8	4	no	—
54	C₄²×C₁₂₇	—	2	1	no	—
55	C₄²×C₁₂₇	—	2	1	no	—
56	C₁₆×C₁₂₇	—	2	2	no	—
57	C₁₆×C₁₂₇	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	—	2	1	no	—
60	C₂×C₄×C₁₂₇	—	2	1	no	—
61	C₂×C₄×C₁₂₇	—	2	2	no	—
62	C₂×C₄×C₁₂₇	—	2	2	no	—
63	C₂×C₄×C₁₂₇	—	4	4	no	—
64	C₂×C₄×C₁₂₇	—	8	2	no	—
65	C₂×C₄×C₁₂₇	—	8	2	no	—
66	C₂×C₄×C₁₂₇	—	8	2	no	—
67	C₂×C₄×C₁₂₇	—	8	2	no	—
68	C₂×C₄×C₁₂₇	—	8	2	no	—
69	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
75	C₈×C₁₂₇	—	2	2	no	—
76	C₈×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₈×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
78	C₈×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
79	C₈×C₁₂₇	#125	1	1	yes	characteristic
80	C₂²×C₁₂₇	C₄×D₈	1	1	yes	characteristic
81	C₂²×C₁₂₇	#466	1	1	yes	characteristic
82	C₂²×C₁₂₇	#500	1	1	yes	characteristic
83	C₂²×C₁₂₇	#509	1	2	yes	—
84	C₂²×C₁₂₇	#510	1	2	yes	—
85	C₄×C₁₂₇	—	4	1	no	—
86	C₄×C₁₂₇	—	4	1	no	—
87	C₄×C₁₂₇	—	8	4	no	—
88	C₄×C₁₂₇	—	16	2	no	—
89	C₄×C₁₂₇	#389	1	1	yes	characteristic
90	C₄×C₁₂₇	C₄×Q₁₆	1	1	yes	characteristic
91	C₄×C₁₂₇	#467	1	1	yes	characteristic
92	C₄×C₁₂₇	#499	1	1	yes	characteristic
93	#13167	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×C₁₂₇	#1942	1	1	yes	characteristic
95	C₂×C₁₂₇	#2175	1	2	yes	—
96	C₂×C₁₂₇	#2218	1	1	yes	characteristic
97	C₂×C₁₂₇	#2219	1	1	yes	characteristic
98	C₂×C₁₂₇	#2255	1	2	yes	—
99	C₂×#367	C₂×C₁₂₇	1	1	yes	characteristic
100	C₂×#366	C₂×C₁₂₇	1	1	yes	characteristic
101	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic
102	#1942	C₂×C₁₂₇	1	2	yes	—
103	#1380	C₂×C₁₂₇	1	2	yes	—
104	C₁₂₇	#13167	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
106	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
107	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
108	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
110	#389	—	2	2	no	—
111	#367	C₄×C₁₂₇	1	4	yes	complemented
112	#366	—	2	2	no	—
113	#346	—	2	2	no	—
114	#340	—	2	2	no	—
115	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	4	2	no	—
120	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
124	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
125	C₂×C₁₆	Q₈×C₁₂₇	1	2	yes	—
126	C₂×C₁₆	Q₈×C₁₂₇	1	2	yes	—
127	#107	—	2	2	no	—
128	#107	—	2	2	no	—
129	#107	—	2	2	no	—
130	#107	C₂×C₄×C₁₂₇	1	4	yes	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	#95	—	4	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
137	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
142	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
143	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
144	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
145	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
146	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
147	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	8	4	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₁₆	—	2	2	no	—
156	C₁₆	C₄⋊C₄×C₁₂₇	1	4	yes	complemented
157	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
169	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
171	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
172	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
173	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
176	C₈	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
177	C₈	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
178	C₈	#125×C₁₂₇	1	1	yes	characteristic
179	C₂²	C₄×D₈×C₁₂₇	1	1	yes	characteristic
180	C₂²	#466×C₁₂₇	1	1	yes	characteristic
181	C₂²	#500×C₁₂₇	1	1	yes	characteristic
182	C₂²	#509×C₁₂₇	1	2	yes	—
183	C₂²	#510×C₁₂₇	1	2	yes	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	8	4	no	—
187	C₄	—	16	2	no	—
188	C₄	#389×C₁₂₇	1	1	yes	characteristic
189	C₄	C₄×Q₁₆×C₁₂₇	1	1	yes	characteristic
190	C₄	#467×C₁₂₇	1	1	yes	characteristic
191	C₄	#499×C₁₂₇	1	1	yes	characteristic
192	C₂	C₁₂₇×#1942	1	1	yes	characteristic
193	C₂	C₁₂₇×#2175	1	2	yes	—
194	C₂	C₁₂₇×#2218	1	1	yes	characteristic
195	C₂	C₁₂₇×#2219	1	1	yes	characteristic
196	C₂	C₁₂₇×#2255	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. #367×C₁₂₇	C₄	semidirect
57. C₁₆×C₁₂₇	C₄⋊C₄	semidirect
93. #13167	C₁₂₇	direct (N × H)
104. C₁₂₇	#13167	direct (N × H)
111. #367	C₄×C₁₂₇	semidirect
156. C₁₆	C₄⋊C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#367×C₁₂₇	2
C₂×#366×C₁₂₇	2
C₂×#346×C₁₂₇	2
C₁₂₇×#1942	2
C₁₂₇×#1380	2
#13167	127

Subgroups — C127×#13167 #44576840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13167 #44576840