Subgroups — C₁₂₇×#2957 #30635909

Order 16256.

452 conjugacy classes of subgroups.

The subgroup lattice has 204 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	#568×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	#444×C₁₂₇	C₂	1	4	yes	maximal, complemented
4	#443×C₁₂₇	C₂	1	4	yes	maximal, complemented
5	C₂×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#421×C₁₂₇	C₂	1	2	yes	maximal, complemented
7	#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₁₂₇	C₂²	1	2	yes	—
9	#135×C₁₂₇	C₄	1	2	yes	complemented
10	#135×C₁₂₇	C₄	1	2	yes	complemented
11	#135×C₁₂₇	C₂²	1	2	yes	—
12	#135×C₁₂₇	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₁₂₇	C₄	1	1	yes	characteristic, complemented
14	C₂×SD₁₆×C₁₂₇	C₄	1	2	yes	complemented
15	C₂×D₈×C₁₂₇	C₄	1	1	yes	characteristic, complemented
16	#131×C₁₂₇	—	2	4	no	—
17	#131×C₁₂₇	C₂²	1	8	yes	complemented
18	C₂×M₁₆×C₁₂₇	—	2	4	no	—
19	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
20	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	#117×C₁₂₇	C₂²	1	1	yes	characteristic
23	#108×C₁₂₇	C₂²	1	2	yes	complemented
24	#108×C₁₂₇	C₂²	1	2	yes	complemented
25	#104×C₁₂₇	C₂²	1	8	yes	complemented
26	#97×C₁₂₇	C₂²	1	2	yes	complemented
27	C₄×C₈×C₁₂₇	C₂²	1	2	yes	complemented
28	#41×C₁₂₇	—	2	2	no	—
29	#41×C₁₂₇	—	2	2	no	—
30	#41×C₁₂₇	C₂×C₄	1	4	yes	—
31	#41×C₁₂₇	C₂³	1	4	yes	—
32	C₂×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₁₂₇	—	2	2	no	—
34	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₂₇	—	2	2	no	—
36	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
38	Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
39	SD₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
40	SD₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
41	D₈×C₁₂₇	C₂×C₄	1	2	yes	—
42	D₈×C₁₂₇	C₂×C₄	1	2	yes	—
43	M₁₆×C₁₂₇	—	2	2	no	—
44	M₁₆×C₁₂₇	—	2	8	no	—
45	M₁₆×C₁₂₇	D₄	1	2	yes	—
46	M₁₆×C₁₂₇	D₄	1	2	yes	—
47	M₁₆×C₁₂₇	C₂³	1	4	yes	—
48	C₂×C₈×C₁₂₇	—	2	2	no	—
49	C₂×C₈×C₁₂₇	—	2	8	no	—
50	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
57	C₄⋊C₄×C₁₂₇	—	2	1	no	—
58	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
59	C₄²×C₁₂₇	C₂³	1	2	yes	—
60	C₂³×C₁₂₇	—	2	2	no	—
61	C₂³×C₁₂₇	#41	1	1	yes	characteristic
62	Q₈×C₁₂₇	—	2	2	no	—
63	Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
64	D₄×C₁₂₇	—	2	4	no	—
65	D₄×C₁₂₇	—	2	4	no	—
66	D₄×C₁₂₇	—	2	2	no	—
67	D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₁₂₇	—	2	2	no	—
69	C₂×C₄×C₁₂₇	—	2	4	no	—
70	C₂×C₄×C₁₂₇	—	2	4	no	—
71	C₂×C₄×C₁₂₇	—	2	2	no	—
72	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
78	C₈×C₁₂₇	—	2	4	no	—
79	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
82	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₈×C₁₂₇	C₂×D₄	1	4	yes	—
84	C₂²×C₁₂₇	—	2	4	no	—
85	C₂²×C₁₂₇	—	4	1	no	—
86	C₂²×C₁₂₇	—	8	2	no	—
87	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
89	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
90	C₄×C₁₂₇	—	2	4	no	—
91	C₄×C₁₂₇	—	4	2	no	—
92	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
93	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
94	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
95	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₁₂₇	—	2	1	no	—
97	C₂×C₁₂₇	—	2	1	no	—
98	C₂×C₁₂₇	—	2	1	no	—
99	C₂×C₁₂₇	—	4	4	no	—
100	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
101	#2957	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₁₂₇	#2957	1	1	yes	characteristic, direct product factor, Sylow
103	C₂×#135	C₂×C₁₂₇	1	1	yes	characteristic
104	#568	C₂×C₁₂₇	1	2	yes	complemented
105	#444	C₂×C₁₂₇	1	4	yes	complemented
106	#443	C₂×C₁₂₇	1	4	yes	complemented
107	C₂×#108	C₂×C₁₂₇	1	1	yes	characteristic, complemented
108	#421	C₂×C₁₂₇	1	2	yes	complemented
109	#405	C₂×C₁₂₇	1	1	yes	characteristic, complemented
110	C₂×#41	C₂²×C₁₂₇	1	2	yes	—
111	#135	C₄×C₁₂₇	1	2	yes	complemented
112	#135	C₄×C₁₂₇	1	2	yes	complemented
113	#135	C₂²×C₁₂₇	1	2	yes	—
114	#135	C₂²×C₁₂₇	1	2	yes	—
115	C₂×Q₁₆	C₄×C₁₂₇	1	1	yes	characteristic, complemented
116	C₂×SD₁₆	C₄×C₁₂₇	1	2	yes	complemented
117	C₂×D₈	C₄×C₁₂₇	1	1	yes	characteristic, complemented
118	#131	—	2	4	no	—
119	#131	C₂²×C₁₂₇	1	8	yes	complemented
120	C₂×M₁₆	—	2	4	no	—
121	C₂×M₁₆	C₂²×C₁₂₇	1	2	yes	—
122	C₂×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
124	#117	C₂²×C₁₂₇	1	1	yes	characteristic
125	#108	C₂²×C₁₂₇	1	2	yes	complemented
126	#108	C₂²×C₁₂₇	1	2	yes	complemented
127	#104	C₂²×C₁₂₇	1	8	yes	complemented
128	#97	C₂²×C₁₂₇	1	2	yes	complemented
129	C₄×C₈	C₂²×C₁₂₇	1	2	yes	complemented
130	#41	—	2	2	no	—
131	#41	—	2	2	no	—
132	#41	C₂×C₄×C₁₂₇	1	4	yes	—
133	#41	C₂³×C₁₂₇	1	4	yes	—
134	C₂×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
135	C₂×D₄	—	2	2	no	—
136	C₂×D₄	C₂×C₄×C₁₂₇	1	2	yes	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
139	Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
140	Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
141	SD₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
142	SD₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
143	D₈	C₂×C₄×C₁₂₇	1	2	yes	—
144	D₈	C₂×C₄×C₁₂₇	1	2	yes	—
145	M₁₆	—	2	2	no	—
146	M₁₆	—	2	8	no	—
147	M₁₆	D₄×C₁₂₇	1	2	yes	—
148	M₁₆	D₄×C₁₂₇	1	2	yes	—
149	M₁₆	C₂³×C₁₂₇	1	4	yes	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
153	C₂×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
154	C₂×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
155	C₂×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
156	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
157	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
158	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
159	C₄⋊C₄	—	2	1	no	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₄²	C₂³×C₁₂₇	1	2	yes	—
162	C₂³	—	2	2	no	—
163	C₂³	#41×C₁₂₇	1	1	yes	characteristic
164	Q₈	—	2	2	no	—
165	Q₈	C₂²×C₄×C₁₂₇	1	4	yes	—
166	D₄	—	2	4	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	2	no	—
169	D₄	C₂²×C₄×C₁₂₇	1	4	yes	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
176	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
177	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
178	C₂×C₄	#41×C₁₂₇	1	1	yes	characteristic
179	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
182	C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
183	C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
184	C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
185	C₈	C₂×D₄×C₁₂₇	1	4	yes	—
186	C₂²	—	2	4	no	—
187	C₂²	—	4	1	no	—
188	C₂²	—	8	2	no	—
189	C₂²	C₄×D₄×C₁₂₇	1	1	yes	characteristic
190	C₂²	C₄×D₄×C₁₂₇	1	1	yes	characteristic
191	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	4	2	no	—
194	C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
195	C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
196	C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
197	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	4	no	—
202	C₂	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₂₇	C₂²×D₄
commutator	C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂×C₁₂₇	C₂×C₄×D₄
prime radical	C₂×#135×C₁₂₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂×M₁₆×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #568×C₁₂₇	C₂	semidirect
3. #444×C₁₂₇	C₂	semidirect
4. #443×C₁₂₇	C₂	semidirect
5. C₂×#108×C₁₂₇	C₂	semidirect
6. #421×C₁₂₇	C₂	semidirect
7. #405×C₁₂₇	C₂	semidirect
9. #135×C₁₂₇	C₄	semidirect
10. #135×C₁₂₇	C₄	semidirect
13. C₂×Q₁₆×C₁₂₇	C₄	semidirect
14. C₂×SD₁₆×C₁₂₇	C₄	semidirect
15. C₂×D₈×C₁₂₇	C₄	semidirect
17. #131×C₁₂₇	C₂²	semidirect
23. #108×C₁₂₇	C₂²	semidirect
24. #108×C₁₂₇	C₂²	semidirect
25. #104×C₁₂₇	C₂²	semidirect
26. #97×C₁₂₇	C₂²	semidirect
27. C₄×C₈×C₁₂₇	C₂²	semidirect
101. #2957	C₁₂₇	direct (N × H)
102. C₁₂₇	#2957	direct (N × H)
104. #568	C₂×C₁₂₇	semidirect
105. #444	C₂×C₁₂₇	semidirect
106. #443	C₂×C₁₂₇	semidirect
107. C₂×#108	C₂×C₁₂₇	semidirect
108. #421	C₂×C₁₂₇	semidirect
109. #405	C₂×C₁₂₇	semidirect
111. #135	C₄×C₁₂₇	semidirect
112. #135	C₄×C₁₂₇	semidirect
115. C₂×Q₁₆	C₄×C₁₂₇	semidirect
116. C₂×SD₁₆	C₄×C₁₂₇	semidirect
117. C₂×D₈	C₄×C₁₂₇	semidirect
119. #131	C₂²×C₁₂₇	semidirect
125. #108	C₂²×C₁₂₇	semidirect
126. #108	C₂²×C₁₂₇	semidirect
127. #104	C₂²×C₁₂₇	semidirect
128. #97	C₂²×C₁₂₇	semidirect
129. C₄×C₈	C₂²×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
#568×C₁₂₇	2
#444×C₁₂₇	2
#443×C₁₂₇	2
C₂×#108×C₁₂₇	2
#421×C₁₂₇	2
#405×C₁₂₇	2
#2957	127

Subgroups — C127×#2957 #30635909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2957 #30635909