Subgroups — C₁₂₇×#3050 #30636002

Order 16256.

520 conjugacy classes of subgroups.

The subgroup lattice has 214 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, complemented
2	C₂³×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#525×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#467×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#466×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	#465×C₁₂₇	C₂	1	4	yes	maximal, complemented
7	#427×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#416×C₁₂₇	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₁₂₇	C₂²	1	2	yes	—
10	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
11	#135×C₁₂₇	—	2	4	no	—
12	C₂×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
13	C₂×SD₁₆×C₁₂₇	C₂²	1	2	yes	complemented
14	C₂×D₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₈×C₁₂₇	—	2	4	no	—
16	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
18	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
19	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	#123×C₁₂₇	—	2	2	no	—
21	C₂²⋊Q₈×C₁₂₇	C₂²	1	4	yes	—
22	#121×C₁₂₇	C₂²	1	4	yes	—
23	#120×C₁₂₇	—	2	2	no	—
24	C₄×D₄×C₁₂₇	—	2	4	no	—
25	#117×C₁₂₇	C₂²	1	2	yes	—
26	#107×C₁₂₇	C₂²	1	2	yes	complemented
27	#106×C₁₂₇	C₂²	1	2	yes	complemented
28	#103×C₁₂₇	C₂²	1	4	yes	complemented
29	#102×C₁₂₇	C₂²	1	4	yes	complemented
30	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
31	#41×C₁₂₇	—	2	4	no	—
32	#41×C₁₂₇	—	2	4	no	—
33	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
34	C₂×D₄×C₁₂₇	—	2	4	no	—
35	C₂×D₄×C₁₂₇	—	2	2	no	—
36	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₂₇	—	2	2	no	—
38	C₂²×C₄×C₁₂₇	—	2	2	no	—
39	C₂²×C₄×C₁₂₇	—	2	2	no	—
40	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
42	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
44	Q₁₆×C₁₂₇	—	2	2	no	—
45	SD₁₆×C₁₂₇	—	2	4	no	—
46	D₈×C₁₂₇	—	2	2	no	—
47	C₂×C₈×C₁₂₇	—	2	2	no	—
48	C₂×C₈×C₁₂₇	—	2	4	no	—
49	C₂×C₈×C₁₂₇	—	2	4	no	—
50	C₂×C₈×C₁₂₇	D₄	1	4	yes	complemented
51	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
52	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₁₂₇	—	2	4	no	—
55	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
56	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
57	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
58	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
59	C₄²×C₁₂₇	—	2	2	no	—
60	C₂³×C₁₂₇	—	2	2	no	—
61	C₂³×C₁₂₇	—	2	4	no	—
62	C₂³×C₁₂₇	—	2	2	no	—
63	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
64	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
65	Q₈×C₁₂₇	—	2	2	no	—
66	Q₈×C₁₂₇	—	2	2	no	—
67	D₄×C₁₂₇	—	2	2	no	—
68	D₄×C₁₂₇	—	2	2	no	—
69	D₄×C₁₂₇	—	4	4	no	—
70	D₄×C₁₂₇	—	4	8	no	—
71	C₂×C₄×C₁₂₇	—	2	2	no	—
72	C₂×C₄×C₁₂₇	—	2	4	no	—
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	—	2	2	no	—
75	C₂×C₄×C₁₂₇	—	2	4	no	—
76	C₂×C₄×C₁₂₇	—	2	2	no	—
77	C₂×C₄×C₁₂₇	—	4	4	no	—
78	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	#41	1	2	yes	—
82	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
83	C₈×C₁₂₇	—	2	2	no	—
84	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
85	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
86	C₂²×C₁₂₇	—	2	8	no	—
87	C₂²×C₁₂₇	—	2	2	no	—
88	C₂²×C₁₂₇	—	2	2	no	—
89	C₂²×C₁₂₇	—	2	2	no	—
90	C₂²×C₁₂₇	—	4	2	no	—
91	C₂²×C₁₂₇	—	4	4	no	—
92	C₂²×C₁₂₇	#121	1	2	yes	—
93	C₂²×C₁₂₇	#135	1	4	yes	complemented
94	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
95	C₄×C₁₂₇	—	2	2	no	—
96	C₄×C₁₂₇	—	4	2	no	—
97	C₄×C₁₂₇	—	4	4	no	—
98	C₄×C₁₂₇	#121	1	2	yes	—
99	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
100	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
101	C₂×C₁₂₇	—	2	2	no	—
102	C₂×C₁₂₇	—	2	4	no	—
103	C₂×C₁₂₇	—	8	2	no	—
104	C₂×C₁₂₇	C₂×#121	1	1	yes	characteristic
105	C₂×C₁₂₇	C₂×#135	1	2	yes	—
106	#3050	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₁₂₇	#3050	1	1	yes	characteristic, direct product factor, Sylow
108	C₂×#135	C₂×C₁₂₇	1	1	yes	characteristic, complemented
109	C₂³×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
110	#525	C₂×C₁₂₇	1	2	yes	complemented
111	#467	C₂×C₁₂₇	1	2	yes	complemented
112	#466	C₂×C₁₂₇	1	2	yes	complemented
113	#465	C₂×C₁₂₇	1	4	yes	complemented
114	#427	C₂×C₁₂₇	1	1	yes	characteristic, complemented
115	#416	C₂×C₁₂₇	1	2	yes	complemented
116	C₂×#41	C₂²×C₁₂₇	1	2	yes	—
117	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
118	#135	—	2	4	no	—
119	C₂×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
120	C₂×SD₁₆	C₂²×C₁₂₇	1	2	yes	complemented
121	C₂×D₈	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
122	C₂²×C₈	—	2	4	no	—
123	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
124	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
125	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
126	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
127	#123	—	2	2	no	—
128	C₂²⋊Q₈	C₂²×C₁₂₇	1	4	yes	—
129	#121	C₂²×C₁₂₇	1	4	yes	—
130	#120	—	2	2	no	—
131	C₄×D₄	—	2	4	no	—
132	#117	C₂²×C₁₂₇	1	2	yes	—
133	#107	C₂²×C₁₂₇	1	2	yes	complemented
134	#106	C₂²×C₁₂₇	1	2	yes	complemented
135	#103	C₂²×C₁₂₇	1	4	yes	complemented
136	#102	C₂²×C₁₂₇	1	4	yes	complemented
137	C₂⁴	D₄×C₁₂₇	1	1	yes	characteristic
138	#41	—	2	4	no	—
139	#41	—	2	4	no	—
140	C₂×Q₈	C₂³×C₁₂₇	1	2	yes	—
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
148	C₂²×C₄	D₄×C₁₂₇	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
150	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
151	Q₁₆	—	2	2	no	—
152	SD₁₆	—	2	4	no	—
153	D₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	D₄×C₁₂₇	1	4	yes	complemented
158	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
159	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
160	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	—	2	4	no	—
165	C₂²⋊C₄	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	4	no	—
169	C₂³	—	2	2	no	—
170	C₂³	C₂×D₄×C₁₂₇	1	1	yes	characteristic
171	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
172	Q₈	—	2	2	no	—
173	Q₈	—	2	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	4	4	no	—
177	D₄	—	4	8	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
186	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
187	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
188	C₂×C₄	#41×C₁₂₇	1	2	yes	—
189	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	C₂×D₄×C₁₂₇	1	2	yes	—
192	C₈	C₂×D₄×C₁₂₇	1	2	yes	—
193	C₂²	—	2	8	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	4	no	—
199	C₂²	#121×C₁₂₇	1	2	yes	—
200	C₂²	#135×C₁₂₇	1	4	yes	complemented
201	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	4	no	—
205	C₄	#121×C₁₂₇	1	2	yes	—
206	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
207	C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	—	2	4	no	—
210	C₂	—	8	2	no	—
211	C₂	C₂×#121×C₁₂₇	1	1	yes	characteristic
212	C₂	C₂×#135×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂³×C₄×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. C₂³×C₈×C₁₂₇	C₂	semidirect
3. #525×C₁₂₇	C₂	semidirect
4. #467×C₁₂₇	C₂	semidirect
5. #466×C₁₂₇	C₂	semidirect
6. #465×C₁₂₇	C₂	semidirect
7. #427×C₁₂₇	C₂	semidirect
8. #416×C₁₂₇	C₂	semidirect
12. C₂×Q₁₆×C₁₂₇	C₂²	semidirect
13. C₂×SD₁₆×C₁₂₇	C₂²	semidirect
14. C₂×D₈×C₁₂₇	C₂²	semidirect
26. #107×C₁₂₇	C₂²	semidirect
27. #106×C₁₂₇	C₂²	semidirect
28. #103×C₁₂₇	C₂²	semidirect
29. #102×C₁₂₇	C₂²	semidirect
50. C₂×C₈×C₁₂₇	D₄	semidirect
93. C₂²×C₁₂₇	#135	semidirect
106. #3050	C₁₂₇	direct (N × H)
107. C₁₂₇	#3050	direct (N × H)
108. C₂×#135	C₂×C₁₂₇	semidirect
109. C₂³×C₈	C₂×C₁₂₇	semidirect
110. #525	C₂×C₁₂₇	semidirect
111. #467	C₂×C₁₂₇	semidirect
112. #466	C₂×C₁₂₇	semidirect
113. #465	C₂×C₁₂₇	semidirect
114. #427	C₂×C₁₂₇	semidirect
115. #416	C₂×C₁₂₇	semidirect
119. C₂×Q₁₆	C₂²×C₁₂₇	semidirect
120. C₂×SD₁₆	C₂²×C₁₂₇	semidirect
121. C₂×D₈	C₂²×C₁₂₇	semidirect
133. #107	C₂²×C₁₂₇	semidirect
134. #106	C₂²×C₁₂₇	semidirect
135. #103	C₂²×C₁₂₇	semidirect
136. #102	C₂²×C₁₂₇	semidirect
157. C₂×C₈	D₄×C₁₂₇	semidirect
200. C₂²	#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
C₂³×C₈×C₁₂₇	2
#525×C₁₂₇	2
#467×C₁₂₇	2
#466×C₁₂₇	2
#465×C₁₂₇	2
#427×C₁₂₇	2
#416×C₁₂₇	2
#3050	127

Subgroups — C127×#3050 #30636002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#3050 #30636002