Subgroups — C₂×#443×C₁₂₇ #30635905

Order 16256.

472 conjugacy classes of subgroups.

The subgroup lattice has 188 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	1	yes	maximal, characteristic
2	C₂×#131×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	#443×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#104×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₁₂₇	C₂²	1	2	yes	—
8	#135×C₁₂₇	C₄	1	4	yes	complemented
9	#135×C₁₂₇	C₂²	1	4	yes	—
10	C₂×Q₁₆×C₁₂₇	C₄	1	1	yes	characteristic, complemented
11	C₂×SD₁₆×C₁₂₇	C₄	1	2	yes	complemented
12	C₂×D₈×C₁₂₇	C₄	1	1	yes	characteristic, complemented
13	#131×C₁₂₇	—	2	4	no	—
14	#131×C₁₂₇	C₂²	1	8	yes	complemented
15	C₂×M₁₆×C₁₂₇	—	2	2	no	—
16	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
17	C₂²×C₈×C₁₂₇	—	2	2	no	—
18	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
21	#108×C₁₂₇	C₂²	1	4	yes	complemented
22	#104×C₁₂₇	C₂²	1	8	yes	complemented
23	C₄×C₈×C₁₂₇	C₂²	1	4	yes	complemented
24	#41×C₁₂₇	—	2	2	no	—
25	#41×C₁₂₇	—	2	2	no	—
26	#41×C₁₂₇	C₂×C₄	1	4	yes	—
27	#41×C₁₂₇	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₁₂₇	—	2	2	no	—
30	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₂₇	—	2	1	no	—
32	C₂²×C₄×C₁₂₇	—	2	2	no	—
33	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
35	SD₁₆×C₁₂₇	C₂×C₄	1	8	yes	complemented
36	D₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
37	M₁₆×C₁₂₇	—	2	2	no	—
38	M₁₆×C₁₂₇	—	2	8	no	—
39	M₁₆×C₁₂₇	C₂³	1	4	yes	—
40	C₂×C₈×C₁₂₇	—	2	2	no	—
41	C₂×C₈×C₁₂₇	—	2	8	no	—
42	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
45	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
47	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
49	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
50	C₄²×C₁₂₇	—	2	1	no	—
51	C₄²×C₁₂₇	C₂³	1	2	yes	—
52	C₂³×C₁₂₇	—	2	2	no	—
53	C₂³×C₁₂₇	#41	1	1	yes	characteristic
54	Q₈×C₁₂₇	—	2	2	no	—
55	Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
56	D₄×C₁₂₇	—	2	4	no	—
57	D₄×C₁₂₇	—	2	4	no	—
58	D₄×C₁₂₇	—	2	2	no	—
59	D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₁₂₇	—	2	4	no	—
61	C₂×C₄×C₁₂₇	—	2	2	no	—
62	C₂×C₄×C₁₂₇	—	2	2	no	—
63	C₂×C₄×C₁₂₇	—	2	4	no	—
64	C₂×C₄×C₁₂₇	—	2	4	no	—
65	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
71	C₈×C₁₂₇	—	2	4	no	—
72	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
73	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
74	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
75	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
76	C₂²×C₁₂₇	—	2	2	no	—
77	C₂²×C₁₂₇	—	2	4	no	—
78	C₂²×C₁₂₇	—	4	4	no	—
79	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
80	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
81	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
82	C₄×C₁₂₇	—	2	4	no	—
83	C₄×C₁₂₇	—	2	4	no	—
84	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
85	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
86	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
87	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₁₂₇	—	2	1	no	—
89	C₂×C₁₂₇	—	2	1	no	—
90	C₂×C₁₂₇	—	4	4	no	—
91	C₂×C₁₂₇	#443	1	2	yes	direct product factor
92	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
93	C₂×#443	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₁₂₇	C₂×#443	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×#135	C₂×C₁₂₇	1	1	yes	characteristic
96	C₂×#131	C₂×C₁₂₇	1	2	yes	complemented
97	#443	C₂×C₁₂₇	1	8	yes	direct product factor
98	C₂×#108	C₂×C₁₂₇	1	1	yes	characteristic, complemented
99	C₂×#104	C₂×C₁₂₇	1	2	yes	complemented
100	C₂×C₄×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
101	C₂×#41	C₂²×C₁₂₇	1	2	yes	—
102	#135	C₄×C₁₂₇	1	4	yes	complemented
103	#135	C₂²×C₁₂₇	1	4	yes	—
104	C₂×Q₁₆	C₄×C₁₂₇	1	1	yes	characteristic, complemented
105	C₂×SD₁₆	C₄×C₁₂₇	1	2	yes	complemented
106	C₂×D₈	C₄×C₁₂₇	1	1	yes	characteristic, complemented
107	#131	—	2	4	no	—
108	#131	C₂²×C₁₂₇	1	8	yes	complemented
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	C₂²×C₁₂₇	1	2	yes	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
113	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
114	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
115	#108	C₂²×C₁₂₇	1	4	yes	complemented
116	#104	C₂²×C₁₂₇	1	8	yes	complemented
117	C₄×C₈	C₂²×C₁₂₇	1	4	yes	complemented
118	#41	—	2	2	no	—
119	#41	—	2	2	no	—
120	#41	C₂×C₄×C₁₂₇	1	4	yes	—
121	#41	C₂³×C₁₂₇	1	4	yes	—
122	C₂×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	C₂×C₄×C₁₂₇	1	2	yes	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
128	Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	complemented
129	SD₁₆	C₂×C₄×C₁₂₇	1	8	yes	complemented
130	D₈	C₂×C₄×C₁₂₇	1	4	yes	complemented
131	M₁₆	—	2	2	no	—
132	M₁₆	—	2	8	no	—
133	M₁₆	C₂³×C₁₂₇	1	4	yes	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	8	no	—
136	C₂×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
137	C₂×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
138	C₂×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
139	C₂×C₈	D₄×C₁₂₇	1	1	yes	characteristic
140	C₂×C₈	D₄×C₁₂₇	1	2	yes	—
141	C₂×C₈	D₄×C₁₂₇	1	1	yes	characteristic
142	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
143	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
144	C₄²	—	2	1	no	—
145	C₄²	C₂³×C₁₂₇	1	2	yes	—
146	C₂³	—	2	2	no	—
147	C₂³	#41×C₁₂₇	1	1	yes	characteristic
148	Q₈	—	2	2	no	—
149	Q₈	C₂²×C₄×C₁₂₇	1	4	yes	—
150	D₄	—	2	4	no	—
151	D₄	—	2	4	no	—
152	D₄	—	2	2	no	—
153	D₄	C₂²×C₄×C₁₂₇	1	4	yes	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
160	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
161	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
162	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
163	C₂×C₄	#41×C₁₂₇	1	1	yes	characteristic
164	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
165	C₈	—	2	4	no	—
166	C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
167	C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
168	C₈	C₂×D₄×C₁₂₇	1	2	yes	—
169	C₈	C₂×D₄×C₁₂₇	1	2	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	4	4	no	—
173	C₂²	C₄×D₄×C₁₂₇	1	1	yes	characteristic
174	C₂²	C₄×D₄×C₁₂₇	1	1	yes	characteristic
175	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	2	4	no	—
178	C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
179	C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
180	C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
181	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	—	4	4	no	—
185	C₂	#443×C₁₂₇	1	2	yes	direct product factor
186	C₂	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₂₇	D₄
commutator	C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₄×D₄
prime radical	C₂×#135×C₁₂₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#131×C₁₂₇	C₂	semidirect
3. #443×C₁₂₇	C₂	direct (N × H)
4. C₂×#108×C₁₂₇	C₂	semidirect
5. C₂×#104×C₁₂₇	C₂	semidirect
6. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
8. #135×C₁₂₇	C₄	semidirect
10. C₂×Q₁₆×C₁₂₇	C₄	semidirect
11. C₂×SD₁₆×C₁₂₇	C₄	semidirect
12. C₂×D₈×C₁₂₇	C₄	semidirect
14. #131×C₁₂₇	C₂²	semidirect
21. #108×C₁₂₇	C₂²	semidirect
22. #104×C₁₂₇	C₂²	semidirect
23. C₄×C₈×C₁₂₇	C₂²	semidirect
34. Q₁₆×C₁₂₇	C₂×C₄	semidirect
35. SD₁₆×C₁₂₇	C₂×C₄	semidirect
36. D₈×C₁₂₇	C₂×C₄	semidirect
91. C₂×C₁₂₇	#443	direct (N × H)
93. C₂×#443	C₁₂₇	direct (N × H)
94. C₁₂₇	C₂×#443	direct (N × H)
96. C₂×#131	C₂×C₁₂₇	semidirect
97. #443	C₂×C₁₂₇	direct (N × H)
98. C₂×#108	C₂×C₁₂₇	semidirect
99. C₂×#104	C₂×C₁₂₇	semidirect
100. C₂×C₄×C₈	C₂×C₁₂₇	semidirect
102. #135	C₄×C₁₂₇	semidirect
104. C₂×Q₁₆	C₄×C₁₂₇	semidirect
105. C₂×SD₁₆	C₄×C₁₂₇	semidirect
106. C₂×D₈	C₄×C₁₂₇	semidirect
108. #131	C₂²×C₁₂₇	semidirect
115. #108	C₂²×C₁₂₇	semidirect
116. #104	C₂²×C₁₂₇	semidirect
117. C₄×C₈	C₂²×C₁₂₇	semidirect
128. Q₁₆	C₂×C₄×C₁₂₇	semidirect
129. SD₁₆	C₂×C₄×C₁₂₇	semidirect
130. D₈	C₂×C₄×C₁₂₇	semidirect
185. C₂	#443×C₁₂₇	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×#135×C₁₂₇	2
C₂×#131×C₁₂₇	2
#443×C₁₂₇	2
C₂×#108×C₁₂₇	2
C₂×#104×C₁₂₇	2
C₂×C₄×C₈×C₁₂₇	2
C₂×#443	127

Subgroups — C2×#443×C127 #30635905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#443×C₁₂₇ #30635905