Subgroups — C₁₂₇×#3287 #30636239

Order 16256.

504 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135×C₁₂₇	C₂	1	2	yes	maximal, complemented
2	#550×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	#493×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#466×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#460×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	#449×C₁₂₇	C₂	1	4	yes	maximal, complemented
7	C₄×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₁₂₇	C₂²	1	4	yes	—
10	#135×C₁₂₇	—	2	8	no	—
11	C₂×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆×C₁₂₇	C₂²	1	4	yes	—
13	C₂×D₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂×D₈×C₁₂₇	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
16	#127×C₁₂₇	—	2	2	no	—
17	#127×C₁₂₇	C₂²	1	2	yes	—
18	#121×C₁₂₇	—	2	4	no	—
19	#121×C₁₂₇	C₂²	1	4	yes	—
20	C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
21	C₄×D₄×C₁₂₇	—	2	2	no	—
22	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
23	#107×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
25	#103×C₁₂₇	C₂²	1	2	yes	—
26	#102×C₁₂₇	C₂²	1	4	yes	—
27	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
28	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
29	#41×C₁₂₇	—	2	8	no	—
30	#41×C₁₂₇	—	4	4	no	—
31	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
32	C₂×D₄×C₁₂₇	—	2	4	no	—
33	C₂×D₄×C₁₂₇	—	2	4	no	—
34	C₂×D₄×C₁₂₇	—	2	2	no	—
35	C₂×D₄×C₁₂₇	—	2	2	no	—
36	C₂×D₄×C₁₂₇	D₄	1	1	yes	characteristic
37	C₂×D₄×C₁₂₇	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₂₇	—	2	4	no	—
39	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
40	Q₁₆×C₁₂₇	D₄	1	4	yes	complemented
41	SD₁₆×C₁₂₇	—	2	8	no	—
42	D₈×C₁₂₇	—	2	4	no	—
43	D₈×C₁₂₇	—	4	2	no	—
44	C₂×C₈×C₁₂₇	—	2	4	no	—
45	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
47	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₁₂₇	—	2	2	no	—
49	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
51	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
52	C₂²⋊C₄×C₁₂₇	D₄	1	2	yes	—
53	C₄²×C₁₂₇	—	2	2	no	—
54	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
55	C₂³×C₁₂₇	—	2	4	no	—
56	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
57	Q₈×C₁₂₇	—	2	2	no	—
58	Q₈×C₁₂₇	C₂×D₄	1	4	yes	—
59	D₄×C₁₂₇	—	2	2	no	—
60	D₄×C₁₂₇	—	2	4	no	—
61	D₄×C₁₂₇	—	2	4	no	—
62	D₄×C₁₂₇	—	4	8	no	—
63	D₄×C₁₂₇	—	4	4	no	—
64	D₄×C₁₂₇	—	8	2	no	—
65	D₄×C₁₂₇	—	8	4	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	—	4	8	no	—
71	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
75	C₈×C₁₂₇	—	2	2	no	—
76	C₈×C₁₂₇	—	2	1	no	—
77	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
78	C₂²×C₁₂₇	—	2	2	no	—
79	C₂²×C₁₂₇	—	2	2	no	—
80	C₂²×C₁₂₇	—	2	2	no	—
81	C₂²×C₁₂₇	—	4	4	no	—
82	C₂²×C₁₂₇	—	4	4	no	—
83	C₂²×C₁₂₇	—	4	4	no	—
84	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
85	C₄×C₁₂₇	—	2	2	no	—
86	C₄×C₁₂₇	—	2	4	no	—
87	C₄×C₁₂₇	—	2	1	no	—
88	C₄×C₁₂₇	—	4	2	no	—
89	C₄×C₁₂₇	#135	1	2	yes	complemented
90	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
91	C₄×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂×C₁₂₇	—	4	2	no	—
93	C₂×C₁₂₇	—	8	4	no	—
94	C₂×C₁₂₇	D₄²	1	1	yes	characteristic
95	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
96	C₂×C₁₂₇	#576	1	1	yes	characteristic
97	#3287	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₁₂₇	#3287	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×#135	C₂×C₁₂₇	1	2	yes	complemented
100	#550	C₂×C₁₂₇	1	2	yes	complemented
101	#493	C₂×C₁₂₇	1	1	yes	characteristic, complemented
102	#466	C₂×C₁₂₇	1	2	yes	complemented
103	#460	C₂×C₁₂₇	1	2	yes	complemented
104	#449	C₂×C₁₂₇	1	4	yes	complemented
105	C₄×Q₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
106	C₈×D₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
107	C₂×#41	C₂²×C₁₂₇	1	4	yes	—
108	#135	—	2	8	no	—
109	C₂×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
110	C₂×SD₁₆	C₂²×C₁₂₇	1	4	yes	—
111	C₂×D₈	C₂²×C₁₂₇	1	2	yes	—
112	C₂×D₈	C₂²×C₁₂₇	1	2	yes	—
113	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
114	#127	—	2	2	no	—
115	#127	C₂²×C₁₂₇	1	2	yes	—
116	#121	—	2	4	no	—
117	#121	C₂²×C₁₂₇	1	4	yes	—
118	C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
119	C₄×D₄	—	2	2	no	—
120	C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
121	#107	C₂²×C₁₂₇	1	1	yes	characteristic
122	C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
123	#103	C₂²×C₁₂₇	1	2	yes	—
124	#102	C₂²×C₁₂₇	1	4	yes	—
125	C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
126	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
127	#41	—	2	8	no	—
128	#41	—	4	4	no	—
129	C₂×Q₈	C₂³×C₁₂₇	1	2	yes	—
130	C₂×D₄	—	2	4	no	—
131	C₂×D₄	—	2	4	no	—
132	C₂×D₄	—	2	2	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂×D₄	D₄×C₁₂₇	1	1	yes	characteristic
135	C₂×D₄	C₂³×C₁₂₇	1	4	yes	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
138	Q₁₆	D₄×C₁₂₇	1	4	yes	complemented
139	SD₁₆	—	2	8	no	—
140	D₈	—	2	4	no	—
141	D₈	—	4	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
144	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
145	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	D₄×C₁₂₇	1	1	yes	characteristic
148	C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
149	C₂²⋊C₄	—	2	4	no	—
150	C₂²⋊C₄	D₄×C₁₂₇	1	2	yes	—
151	C₄²	—	2	2	no	—
152	C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
153	C₂³	—	2	4	no	—
154	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
155	Q₈	—	2	2	no	—
156	Q₈	C₂×D₄×C₁₂₇	1	4	yes	—
157	D₄	—	2	2	no	—
158	D₄	—	2	4	no	—
159	D₄	—	2	4	no	—
160	D₄	—	4	8	no	—
161	D₄	—	4	4	no	—
162	D₄	—	8	2	no	—
163	D₄	—	8	4	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	—	4	8	no	—
169	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
171	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
172	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	1	no	—
175	C₈	C₂×D₄×C₁₂₇	1	2	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	4	no	—
180	C₂²	—	4	4	no	—
181	C₂²	—	4	4	no	—
182	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	4	no	—
185	C₄	—	2	1	no	—
186	C₄	—	4	2	no	—
187	C₄	#135×C₁₂₇	1	2	yes	complemented
188	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
189	C₄	H(5,𝔽₂)×C₁₂₇	1	1	yes	characteristic
190	C₂	—	4	2	no	—
191	C₂	—	8	4	no	—
192	C₂	D₄²×C₁₂₇	1	1	yes	characteristic
193	C₂	C₂×#135×C₁₂₇	1	1	yes	characteristic
194	C₂	#576×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄×D₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#135×C₁₂₇	C₂	semidirect
2. #550×C₁₂₇	C₂	semidirect
3. #493×C₁₂₇	C₂	semidirect
4. #466×C₁₂₇	C₂	semidirect
5. #460×C₁₂₇	C₂	semidirect
6. #449×C₁₂₇	C₂	semidirect
7. C₄×Q₁₆×C₁₂₇	C₂	semidirect
8. C₈×D₄×C₁₂₇	C₂	semidirect
40. Q₁₆×C₁₂₇	D₄	semidirect
89. C₄×C₁₂₇	#135	semidirect
97. #3287	C₁₂₇	direct (N × H)
98. C₁₂₇	#3287	direct (N × H)
99. C₂×#135	C₂×C₁₂₇	semidirect
100. #550	C₂×C₁₂₇	semidirect
101. #493	C₂×C₁₂₇	semidirect
102. #466	C₂×C₁₂₇	semidirect
103. #460	C₂×C₁₂₇	semidirect
104. #449	C₂×C₁₂₇	semidirect
105. C₄×Q₁₆	C₂×C₁₂₇	semidirect
106. C₈×D₄	C₂×C₁₂₇	semidirect
138. Q₁₆	D₄×C₁₂₇	semidirect
187. C₄	#135×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135×C₁₂₇	2
#550×C₁₂₇	2
#493×C₁₂₇	2
#466×C₁₂₇	2
#460×C₁₂₇	2
#449×C₁₂₇	2
C₄×Q₁₆×C₁₂₇	2
C₈×D₄×C₁₂₇	2
#3287	127

Subgroups — C127×#3287 #30636239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#3287 #30636239