Subgroups — C₁₂₇×#3061 #30636013

Order 16256.

484 conjugacy classes of subgroups.

The subgroup lattice has 218 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	C₂×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#537×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#468×C₁₂₇	C₂	1	4	yes	maximal, complemented
5	#467×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	#466×C₁₂₇	C₂	1	2	yes	maximal, complemented
7	C₂×#107×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#417×C₁₂₇	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₁₂₇	C₂²	1	2	yes	—
10	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
11	#135×C₁₂₇	—	2	4	no	—
12	C₂×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	#131×C₁₂₇	—	2	4	no	—
16	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
18	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
19	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
21	C₂²⋊Q₈×C₁₂₇	C₂²	1	4	yes	—
22	#121×C₁₂₇	—	2	2	no	—
23	#121×C₁₂₇	—	2	4	no	—
24	#121×C₁₂₇	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
26	#107×C₁₂₇	C₂²	1	2	yes	—
27	#107×C₁₂₇	C₂²	1	2	yes	—
28	#103×C₁₂₇	C₂²	1	4	yes	—
29	#102×C₁₂₇	C₂²	1	4	yes	—
30	#41×C₁₂₇	—	2	2	no	—
31	#41×C₁₂₇	—	2	2	no	—
32	#41×C₁₂₇	—	2	4	no	—
33	#41×C₁₂₇	—	2	4	no	—
34	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
36	C₂×D₄×C₁₂₇	—	2	2	no	—
37	C₂×D₄×C₁₂₇	—	2	4	no	—
38	C₂×D₄×C₁₂₇	D₄	1	2	yes	—
39	C₂×D₄×C₁₂₇	D₄	1	1	yes	characteristic
40	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₂₇	—	2	2	no	—
42	C₂²×C₄×C₁₂₇	—	2	2	no	—
43	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
45	Q₁₆×C₁₂₇	—	2	2	no	—
46	SD₁₆×C₁₂₇	—	2	4	no	—
47	D₈×C₁₂₇	—	2	2	no	—
48	M₁₆×C₁₂₇	—	2	2	no	—
49	M₁₆×C₁₂₇	—	2	4	no	—
50	C₂×C₈×C₁₂₇	—	2	4	no	—
51	C₂×C₈×C₁₂₇	D₄	1	4	yes	complemented
52	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
53	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₁₂₇	—	2	2	no	—
56	C₄⋊C₄×C₁₂₇	—	2	4	no	—
57	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
58	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
59	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
60	C₂³×C₁₂₇	—	2	2	no	—
61	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
62	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
63	Q₈×C₁₂₇	—	2	2	no	—
64	Q₈×C₁₂₇	—	2	2	no	—
65	Q₈×C₁₂₇	—	2	2	no	—
66	D₄×C₁₂₇	—	2	4	no	—
67	D₄×C₁₂₇	—	2	2	no	—
68	D₄×C₁₂₇	—	2	2	no	—
69	D₄×C₁₂₇	—	2	2	no	—
70	D₄×C₁₂₇	—	4	4	no	—
71	D₄×C₁₂₇	—	8	4	no	—
72	C₂×C₄×C₁₂₇	—	2	2	no	—
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	—	2	4	no	—
75	C₂×C₄×C₁₂₇	—	2	2	no	—
76	C₂×C₄×C₁₂₇	—	4	4	no	—
77	C₂×C₄×C₁₂₇	—	8	2	no	—
78	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
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	1	no	—
87	C₂²×C₁₂₇	—	2	1	no	—
88	C₂²×C₁₂₇	—	2	2	no	—
89	C₂²×C₁₂₇	—	2	2	no	—
90	C₂²×C₁₂₇	—	2	2	no	—
91	C₂²×C₁₂₇	—	4	2	no	—
92	C₂²×C₁₂₇	—	4	2	no	—
93	C₂²×C₁₂₇	—	4	2	no	—
94	C₂²×C₁₂₇	#121	1	2	yes	—
95	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₁₂₇	—	2	2	no	—
97	C₄×C₁₂₇	—	4	2	no	—
98	C₄×C₁₂₇	—	4	4	no	—
99	C₄×C₁₂₇	#121	1	2	yes	—
100	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
102	C₂×C₁₂₇	—	2	2	no	—
103	C₂×C₁₂₇	—	4	2	no	—
104	C₂×C₁₂₇	—	8	2	no	—
105	C₂×C₁₂₇	C₂×#121	1	1	yes	characteristic
106	C₂×C₁₂₇	#576	1	1	yes	characteristic
107	C₂×C₁₂₇	#578	1	1	yes	characteristic
108	#3061	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₁₂₇	#3061	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×#135	C₂×C₁₂₇	1	1	yes	characteristic, complemented
111	C₂×#131	C₂×C₁₂₇	1	1	yes	characteristic, complemented
112	#537	C₂×C₁₂₇	1	2	yes	complemented
113	#468	C₂×C₁₂₇	1	4	yes	complemented
114	#467	C₂×C₁₂₇	1	2	yes	complemented
115	#466	C₂×C₁₂₇	1	2	yes	complemented
116	C₂×#107	C₂×C₁₂₇	1	1	yes	characteristic, complemented
117	#417	C₂×C₁₂₇	1	2	yes	complemented
118	C₂×#41	C₂²×C₁₂₇	1	2	yes	—
119	C₂×#41	C₂²×C₁₂₇	1	1	yes	characteristic
120	#135	—	2	4	no	—
121	C₂×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
122	C₂×SD₁₆	C₂²×C₁₂₇	1	2	yes	—
123	C₂×D₈	C₂²×C₁₂₇	1	1	yes	characteristic
124	#131	—	2	4	no	—
125	C₂×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
126	C₂×M₁₆	C₂²×C₁₂₇	1	2	yes	—
127	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
128	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
129	C₂²⋊Q₈	—	2	2	no	—
130	C₂²⋊Q₈	C₂²×C₁₂₇	1	4	yes	—
131	#121	—	2	2	no	—
132	#121	—	2	4	no	—
133	#121	C₂²×C₁₂₇	1	4	yes	—
134	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
135	#107	C₂²×C₁₂₇	1	2	yes	—
136	#107	C₂²×C₁₂₇	1	2	yes	—
137	#103	C₂²×C₁₂₇	1	4	yes	—
138	#102	C₂²×C₁₂₇	1	4	yes	—
139	#41	—	2	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	4	no	—
142	#41	—	2	4	no	—
143	C₂×Q₈	D₄×C₁₂₇	1	1	yes	characteristic
144	C₂×Q₈	C₂³×C₁₂₇	1	2	yes	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	D₄×C₁₂₇	1	2	yes	—
148	C₂×D₄	D₄×C₁₂₇	1	1	yes	characteristic
149	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
153	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
154	Q₁₆	—	2	2	no	—
155	SD₁₆	—	2	4	no	—
156	D₈	—	2	2	no	—
157	M₁₆	—	2	2	no	—
158	M₁₆	—	2	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	D₄×C₁₂₇	1	4	yes	complemented
161	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
162	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
163	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₂³	—	2	2	no	—
170	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
171	C₂³	C₂×D₄×C₁₂₇	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	Q₈	—	2	2	no	—
174	Q₈	—	2	2	no	—
175	D₄	—	2	4	no	—
176	D₄	—	2	2	no	—
177	D₄	—	2	2	no	—
178	D₄	—	2	2	no	—
179	D₄	—	4	4	no	—
180	D₄	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	2	no	—
187	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
188	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
189	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
190	C₂×C₄	#41×C₁₂₇	1	2	yes	—
191	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	C₂×D₄×C₁₂₇	1	2	yes	—
194	C₈	C₂×D₄×C₁₂₇	1	2	yes	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	#121×C₁₂₇	1	2	yes	—
204	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	4	no	—
208	C₄	#121×C₁₂₇	1	2	yes	—
209	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
210	C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
211	C₂	—	2	2	no	—
212	C₂	—	4	2	no	—
213	C₂	—	8	2	no	—
214	C₂	C₂×#121×C₁₂₇	1	1	yes	characteristic
215	C₂	#576×C₁₂₇	1	1	yes	characteristic
216	C₂	#578×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 ◁ C₂×#41×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. C₂×#131×C₁₂₇	C₂	semidirect
3. #537×C₁₂₇	C₂	semidirect
4. #468×C₁₂₇	C₂	semidirect
5. #467×C₁₂₇	C₂	semidirect
6. #466×C₁₂₇	C₂	semidirect
7. C₂×#107×C₁₂₇	C₂	semidirect
8. #417×C₁₂₇	C₂	semidirect
51. C₂×C₈×C₁₂₇	D₄	semidirect
108. #3061	C₁₂₇	direct (N × H)
109. C₁₂₇	#3061	direct (N × H)
110. C₂×#135	C₂×C₁₂₇	semidirect
111. C₂×#131	C₂×C₁₂₇	semidirect
112. #537	C₂×C₁₂₇	semidirect
113. #468	C₂×C₁₂₇	semidirect
114. #467	C₂×C₁₂₇	semidirect
115. #466	C₂×C₁₂₇	semidirect
116. C₂×#107	C₂×C₁₂₇	semidirect
117. #417	C₂×C₁₂₇	semidirect
160. C₂×C₈	D₄×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
C₂×#131×C₁₂₇	2
#537×C₁₂₇	2
#468×C₁₂₇	2
#467×C₁₂₇	2
#466×C₁₂₇	2
C₂×#107×C₁₂₇	2
#417×C₁₂₇	2
#3061	127

Subgroups — C127×#3061 #30636013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#3061 #30636013