Subgroups — C₂×#108×D₁₂₇ #30647331

Order 16256.

480 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₂₇	C₂	1	2	yes	maximal
2	C₂²×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#16861145	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16861144	C₂	1	2	yes	maximal, complemented
5	#108×D₁₂₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#14523232	C₂²	1	2	yes	—
9	C₂×#14523231	—	2	2	no	—
10	M₁₆×D₁₂₇	—	2	4	no	—
11	M₁₆×D₁₂₇	C₂²	1	8	yes	—
12	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₁₂₇	—	2	2	no	—
14	C₂×C₈×D₁₂₇	C₄	1	4	yes	—
15	C₂×C₈×D₁₂₇	C₄	1	2	yes	—
16	C₂×C₈×D₁₂₇	C₄	1	2	yes	—
17	C₂×C₈×D₁₂₇	C₂²	1	4	yes	—
18	#16861145	C₂²	1	4	yes	complemented
19	#16861144	C₂²	1	8	yes	complemented
20	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
21	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	#108×C₁₂₇	C₂²	1	4	yes	complemented
23	C₂³×D₁₂₇	Q₈	1	1	yes	characteristic
24	C₂²×Dic₁₂₇	Q₈	1	1	yes	characteristic
25	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
29	C₂×C₄×D₁₂₇	D₄	1	2	yes	—
30	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
31	C₂×C₄×D₁₂₇	Q₈	1	2	yes	—
32	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
33	#14523232	—	2	2	no	—
34	#14523232	C₂³	1	4	yes	—
35	#14523231	—	2	8	no	—
36	C₂×C₁₂₇⋊C₈	—	2	2	no	—
37	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
38	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
40	C₈×D₁₂₇	—	2	8	no	—
41	C₈×D₁₂₇	C₂×C₄	1	8	yes	—
42	C₈×D₁₂₇	C₂×C₄	1	8	yes	—
43	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
44	M₁₆×C₁₂₇	—	2	2	no	—
45	M₁₆×C₁₂₇	C₂³	1	4	yes	—
46	C₂×C₈×C₁₂₇	—	2	2	no	—
47	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
48	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
49	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
50	C₂²×D₁₂₇	—	2	4	no	—
51	C₂²×D₁₂₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×D₁₂₇	C₄⋊C₄	1	2	yes	—
53	C₂²×D₁₂₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×D₁₂₇	C₂×Q₈	1	2	yes	—
55	C₂×Dic₁₂₇	C₄⋊C₄	1	2	yes	—
56	C₂×Dic₁₂₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂×Dic₁₂₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×Dic₁₂₇	C₂×Q₈	1	2	yes	—
59	C₄×D₁₂₇	C₄⋊C₄	1	2	yes	—
60	C₄×D₁₂₇	C₄⋊C₄	1	2	yes	—
61	C₄×D₁₂₇	C₄⋊C₄	1	2	yes	—
62	C₄×D₁₂₇	C₄⋊C₄	1	2	yes	—
63	C₄×D₁₂₇	C₂²×C₄	1	2	yes	—
64	C₄×D₁₂₇	C₂²×C₄	1	2	yes	—
65	C₄×D₁₂₇	C₂×D₄	1	2	yes	—
66	C₄×D₁₂₇	C₂×D₄	1	2	yes	—
67	C₁₂₇⋊C₈	—	2	4	no	—
68	C₁₂₇⋊C₈	C₂²×C₄	1	4	yes	—
69	C₂³×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
76	C₈×C₁₂₇	—	2	4	no	—
77	C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
78	C₂²×C₁₂₇	—	2	2	no	—
79	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
82	C₂×D₁₂₇	—	2	2	no	—
83	C₂×D₁₂₇	—	2	4	no	—
84	C₂×D₁₂₇	—	2	4	no	—
85	C₂×D₁₂₇	#108	1	4	yes	direct product factor
86	C₂×D₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×D₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
91	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
92	Dic₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
93	Dic₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₁₂₇	—	2	1	no	—
95	C₂×C₁₂₇	—	2	1	no	—
96	C₂×C₁₂₇	C₂×#108	1	2	yes	complemented
97	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	D₁₂₇	—	2	2	no	—
99	D₁₂₇	C₂×#108	1	4	yes	direct product factor
100	C₂²×#108	—	127	1	no	maximal, Sylow
101	C₁₂₇	C₂²×#108	1	1	yes	characteristic, complemented, Sylow
102	C₂²×M₁₆	—	127	2	no	—
103	C₂³×C₈	—	127	1	no	—
104	C₂×#108	—	127	2	no	—
105	C₂×#108	—	127	1	no	—
106	C₂×#108	—	127	8	no	—
107	C₂×#108	D₁₂₇	1	1	yes	characteristic, direct product factor
108	C₂³×C₄	—	127	1	no	—
109	C₂×M₁₆	—	127	2	no	—
110	C₂×M₁₆	—	127	8	no	—
111	C₂×M₁₆	—	254	2	no	—
112	C₂×M₁₆	—	254	4	no	—
113	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
114	C₂²×C₈	—	127	1	no	—
115	C₂²×C₈	—	127	2	no	—
116	C₂²×C₈	—	127	2	no	—
117	C₂²×C₈	—	127	4	no	—
118	C₂²×C₈	—	127	4	no	—
119	C₂²×C₈	—	254	2	no	—
120	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
121	#108	—	127	8	no	—
122	#108	—	127	4	no	—
123	#108	C₂×D₁₂₇	1	4	yes	direct product factor
124	C₂⁴	—	127	1	no	—
125	C₂²×C₄	—	127	1	no	—
126	C₂²×C₄	—	127	1	no	—
127	C₂²×C₄	—	127	2	no	—
128	C₂²×C₄	—	127	1	no	—
129	C₂²×C₄	—	127	2	no	—
130	C₂²×C₄	—	127	1	no	—
131	C₂²×C₄	—	127	1	no	—
132	C₂²×C₄	—	127	2	no	—
133	C₂²×C₄	—	127	2	no	—
134	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
135	M₁₆	—	2	2	no	—
136	M₁₆	—	127	4	no	—
137	M₁₆	—	254	8	no	—
138	M₁₆	—	254	2	no	—
139	M₁₆	C₂²×D₁₂₇	1	4	yes	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	127	2	no	—
142	C₂×C₈	—	127	2	no	—
143	C₂×C₈	—	127	2	no	—
144	C₂×C₈	—	127	8	no	—
145	C₂×C₈	—	127	8	no	—
146	C₂×C₈	—	254	8	no	—
147	C₂×C₈	—	254	2	no	—
148	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
149	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
150	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
151	C₂³	—	127	1	no	—
152	C₂³	—	127	2	no	—
153	C₂³	—	127	2	no	—
154	C₂³	—	127	1	no	—
155	C₂³	—	254	4	no	—
156	C₂³	Q₈×D₁₂₇	1	1	yes	characteristic
157	C₂×C₄	—	127	2	no	—
158	C₂×C₄	—	127	2	no	—
159	C₂×C₄	—	127	1	no	—
160	C₂×C₄	—	127	1	no	—
161	C₂×C₄	—	127	2	no	—
162	C₂×C₄	—	127	2	no	—
163	C₂×C₄	—	127	2	no	—
164	C₂×C₄	—	127	2	no	—
165	C₂×C₄	—	127	2	no	—
166	C₂×C₄	—	127	2	no	—
167	C₂×C₄	—	127	2	no	—
168	C₂×C₄	—	127	2	no	—
169	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
171	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
172	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
173	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
174	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
175	C₈	—	2	4	no	—
176	C₈	—	127	4	no	—
177	C₈	—	254	4	no	—
178	C₈	C₂×C₄×D₁₂₇	1	4	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	—	127	2	no	—
181	C₂²	—	127	4	no	—
182	C₂²	—	127	2	no	—
183	C₂²	—	254	4	no	—
184	C₂²	—	254	4	no	—
185	C₂²	—	254	2	no	—
186	C₂²	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
187	C₂²	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
188	C₂²	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
189	C₄	—	127	2	no	—
190	C₄	—	127	2	no	—
191	C₄	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
192	C₄	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
193	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
194	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	—	127	4	no	—
198	C₂	—	254	2	no	—
199	C₂	#108×D₁₂₇	1	2	yes	direct product factor
200	C₂	C₂×C₄⋊C₄×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#108×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₄⋊C₄
prime radical	C₂³×D₁₂₇	Q₈
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#108	D₁₂₇
hypocenter	C₁₂₇	C₂²×#108
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
3. C₂×#16861145	C₂	semidirect
4. C₂×#16861144	C₂	semidirect
5. #108×D₁₂₇	C₂	direct (N × H)
6. C₂×#108×C₁₂₇	C₂	semidirect
18. #16861145	C₂²	semidirect
19. #16861144	C₂²	semidirect
22. #108×C₁₂₇	C₂²	semidirect
85. C₂×D₁₂₇	#108	direct (N × H)
96. C₂×C₁₂₇	C₂×#108	semidirect
99. D₁₂₇	C₂×#108	direct (N × H)
101. C₁₂₇	C₂²×#108	semidirect
107. C₂×#108	D₁₂₇	direct (N × H)
123. #108	C₂×D₁₂₇	direct (N × H)
199. C₂	#108×D₁₂₇	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×M₁₆×D₁₂₇	2
C₂²×C₈×D₁₂₇	2
C₂×#16861145	2
C₂×#16861144	2
#108×D₁₂₇	2
C₂×#108×C₁₂₇	2
C₂²×#108	127

Subgroups — C2×#108×D127 #30647331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#108×D₁₂₇ #30647331