Subgroups — C₁₂₇×#2612 #30635564

Order 16256.

388 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	#400×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#398×C₁₂₇	C₂	1	2	yes	maximal
3	#395×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#391×C₁₂₇	C₂	1	2	yes	maximal
5	#389×C₁₂₇	C₂	1	2	yes	maximal
6	#387×C₁₂₇	C₂	1	2	yes	maximal
7	#387×C₁₂₇	C₂	1	2	yes	maximal
8	#382×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	#376×C₁₂₇	C₂	1	2	yes	maximal
10	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
21	#95×C₁₂₇	C₂²	1	2	yes	—
22	#95×C₁₂₇	C₂²	1	2	yes	—
23	#95×C₁₂₇	C₂²	1	2	yes	—
24	#95×C₁₂₇	C₂²	1	2	yes	—
25	#95×C₁₂₇	C₂²	1	2	yes	—
26	#95×C₁₂₇	C₂²	1	2	yes	—
27	#95×C₁₂₇	C₂²	1	1	yes	characteristic
28	#95×C₁₂₇	C₂²	1	1	yes	characteristic
29	#95×C₁₂₇	C₂²	1	2	yes	—
30	#95×C₁₂₇	C₂²	1	2	yes	—
31	C₂×Q₈×C₁₂₇	—	2	4	no	—
32	C₂×Q₈×C₁₂₇	—	4	1	no	—
33	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
42	C₄⋊C₄×C₁₂₇	—	2	4	no	—
43	C₄⋊C₄×C₁₂₇	—	2	4	no	—
44	C₄⋊C₄×C₁₂₇	—	2	4	no	—
45	C₄⋊C₄×C₁₂₇	—	4	1	no	—
46	C₄⋊C₄×C₁₂₇	—	4	1	no	—
47	C₄⋊C₄×C₁₂₇	—	4	2	no	—
48	C₄⋊C₄×C₁₂₇	—	4	1	no	—
49	C₄²×C₁₂₇	—	2	4	no	—
50	C₄²×C₁₂₇	—	2	4	no	—
51	C₄²×C₁₂₇	Q₈	1	4	yes	complemented
52	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
53	Q₈×C₁₂₇	—	4	4	no	—
54	C₂×C₄×C₁₂₇	—	2	2	no	—
55	C₂×C₄×C₁₂₇	—	2	2	no	—
56	C₂×C₄×C₁₂₇	—	2	2	no	—
57	C₂×C₄×C₁₂₇	—	2	2	no	—
58	C₂×C₄×C₁₂₇	—	2	1	no	—
59	C₂×C₄×C₁₂₇	—	2	2	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	—	2	2	no	—
64	C₂×C₄×C₁₂₇	—	2	2	no	—
65	C₂×C₄×C₁₂₇	—	2	2	no	—
66	C₂×C₄×C₁₂₇	—	2	2	no	—
67	C₂×C₄×C₁₂₇	—	2	2	no	—
68	C₂×C₄×C₁₂₇	—	2	2	no	—
69	C₂×C₄×C₁₂₇	—	2	2	no	—
70	C₂×C₄×C₁₂₇	—	2	1	no	—
71	C₂×C₄×C₁₂₇	—	2	2	no	—
72	C₂×C₄×C₁₂₇	—	2	2	no	—
73	C₂×C₄×C₁₂₇	—	2	1	no	—
74	C₂×C₄×C₁₂₇	—	2	2	no	—
75	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
76	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₁₂₇	#41	1	4	yes	—
78	C₂×C₄×C₁₂₇	#41	1	4	yes	—
79	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
81	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
82	C₂²×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂²×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
84	C₄×C₁₂₇	—	2	4	no	—
85	C₄×C₁₂₇	—	2	2	no	—
86	C₄×C₁₂₇	—	2	4	no	—
87	C₄×C₁₂₇	—	2	4	no	—
88	C₄×C₁₂₇	—	4	1	no	—
89	C₄×C₁₂₇	—	4	1	no	—
90	C₄×C₁₂₇	—	4	2	no	—
91	C₄×C₁₂₇	—	4	2	no	—
92	C₄×C₁₂₇	—	4	2	no	—
93	C₂×C₁₂₇	#533	1	2	yes	—
94	C₂×C₁₂₇	#544	1	1	yes	characteristic
95	C₂×C₁₂₇	#551	1	1	yes	characteristic
96	C₂×C₁₂₇	#552	1	2	yes	—
97	C₂×C₁₂₇	#555	1	1	yes	characteristic
98	#2612	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₁₂₇	#2612	1	1	yes	characteristic, direct product factor, Sylow
100	#400	C₂×C₁₂₇	1	1	yes	characteristic
101	#398	C₂×C₁₂₇	1	2	yes	—
102	#395	C₂×C₁₂₇	1	1	yes	characteristic
103	#391	C₂×C₁₂₇	1	2	yes	—
104	#389	C₂×C₁₂₇	1	2	yes	—
105	#387	C₂×C₁₂₇	1	2	yes	—
106	#387	C₂×C₁₂₇	1	2	yes	—
107	#382	C₂×C₁₂₇	1	1	yes	characteristic
108	#376	C₂×C₁₂₇	1	2	yes	—
109	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
112	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
117	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
118	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
119	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
120	#95	C₂²×C₁₂₇	1	2	yes	—
121	#95	C₂²×C₁₂₇	1	2	yes	—
122	#95	C₂²×C₁₂₇	1	2	yes	—
123	#95	C₂²×C₁₂₇	1	2	yes	—
124	#95	C₂²×C₁₂₇	1	2	yes	—
125	#95	C₂²×C₁₂₇	1	2	yes	—
126	#95	C₂²×C₁₂₇	1	1	yes	characteristic
127	#95	C₂²×C₁₂₇	1	1	yes	characteristic
128	#95	C₂²×C₁₂₇	1	2	yes	—
129	#95	C₂²×C₁₂₇	1	2	yes	—
130	C₂×Q₈	—	2	4	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
133	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
135	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
137	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
138	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
139	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
140	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	Q₈×C₁₂₇	1	4	yes	complemented
151	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
152	Q₈	—	4	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
175	C₂×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
176	C₂×C₄	#41×C₁₂₇	1	4	yes	—
177	C₂×C₄	#41×C₁₂₇	1	4	yes	—
178	C₂²	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
179	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
180	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
181	C₂²	H(5,𝔽₂)×C₁₂₇	1	1	yes	characteristic
182	C₂²	2₋¹⁺⁴×C₁₂₇	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	4	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	4	2	no	—
192	C₂	#533×C₁₂₇	1	2	yes	—
193	C₂	#544×C₁₂₇	1	1	yes	characteristic
194	C₂	#551×C₁₂₇	1	1	yes	characteristic
195	C₂	#552×C₁₂₇	1	2	yes	—
196	C₂	#555×C₁₂₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
51. C₄²×C₁₂₇	Q₈	semidirect
98. #2612	C₁₂₇	direct (N × H)
99. C₁₂₇	#2612	direct (N × H)
150. C₄²	Q₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#400×C₁₂₇	2
#398×C₁₂₇	2
#395×C₁₂₇	2
#391×C₁₂₇	2
#389×C₁₂₇	2
#387×C₁₂₇	2
#382×C₁₂₇	2
#376×C₁₂₇	2
#2612	127

Subgroups — C127×#2612 #30635564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2612 #30635564