Subgroups — C₁₂₇×#2535 #30635487

Order 16256.

732 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#121×C₁₂₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#397×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#392×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	#386×C₁₂₇	C₂	1	4	yes	maximal, complemented
7	#382×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#380×C₁₂₇	C₂	1	2	yes	maximal, complemented
9	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₁₂₇	C₂²	1	4	yes	—
11	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
14	#121×C₁₂₇	—	2	8	no	—
15	C₄×D₄×C₁₂₇	—	2	4	no	—
16	C₄×D₄×C₁₂₇	—	2	4	no	—
17	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	—
23	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	—
24	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
25	#95×C₁₂₇	C₂²	1	4	yes	—
26	#95×C₁₂₇	C₂²	1	2	yes	—
27	#95×C₁₂₇	C₂²	1	2	yes	—
28	C₂⁴×C₁₂₇	C₂³	1	2	yes	—
29	C₂⁴×C₁₂₇	C₂³	1	2	yes	—
30	C₂×D₄×C₁₂₇	—	2	4	no	—
31	C₂×D₄×C₁₂₇	—	2	8	no	—
32	C₂×D₄×C₁₂₇	—	2	8	no	—
33	C₂×D₄×C₁₂₇	—	2	2	no	—
34	C₂×D₄×C₁₂₇	—	2	2	no	—
35	C₂×D₄×C₁₂₇	—	2	2	no	—
36	C₂×D₄×C₁₂₇	—	4	4	no	—
37	C₂×D₄×C₁₂₇	D₄	1	4	yes	complemented
38	C₂²×C₄×C₁₂₇	—	2	4	no	—
39	C₂²×C₄×C₁₂₇	—	2	4	no	—
40	C₂²×C₄×C₁₂₇	—	2	4	no	—
41	C₂²×C₄×C₁₂₇	—	2	4	no	—
42	C₂²×C₄×C₁₂₇	—	4	4	no	—
43	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
44	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₁₂₇	—	2	2	no	—
50	C₄⋊C₄×C₁₂₇	—	2	2	no	—
51	C₄⋊C₄×C₁₂₇	—	2	4	no	—
52	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
53	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
54	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
55	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
56	C₄²×C₁₂₇	D₄	1	4	yes	complemented
57	C₂³×C₁₂₇	—	2	2	no	—
58	C₂³×C₁₂₇	—	2	2	no	—
59	C₂³×C₁₂₇	—	2	2	no	—
60	C₂³×C₁₂₇	—	2	2	no	—
61	C₂³×C₁₂₇	—	2	2	no	—
62	C₂³×C₁₂₇	—	2	2	no	—
63	C₂³×C₁₂₇	—	2	4	no	—
64	C₂³×C₁₂₇	—	2	2	no	—
65	C₂³×C₁₂₇	—	2	4	no	—
66	C₂³×C₁₂₇	—	2	2	no	—
67	C₂³×C₁₂₇	C₂×D₄	1	4	yes	—
68	C₂³×C₁₂₇	#41	1	4	yes	—
69	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
70	D₄×C₁₂₇	—	2	8	no	—
71	D₄×C₁₂₇	—	2	8	no	—
72	D₄×C₁₂₇	—	4	16	no	—
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	—	2	2	no	—
75	C₂×C₄×C₁₂₇	—	2	2	no	—
76	C₂×C₄×C₁₂₇	—	2	2	no	—
77	C₂×C₄×C₁₂₇	—	2	2	no	—
78	C₂×C₄×C₁₂₇	—	2	4	no	—
79	C₂×C₄×C₁₂₇	—	2	2	no	—
80	C₂×C₄×C₁₂₇	—	2	4	no	—
81	C₂×C₄×C₁₂₇	—	2	4	no	—
82	C₂×C₄×C₁₂₇	—	2	4	no	—
83	C₂×C₄×C₁₂₇	—	4	8	no	—
84	C₂×C₄×C₁₂₇	—	4	8	no	—
85	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₂₇	#41	1	2	yes	—
90	C₂²×C₁₂₇	—	2	4	no	—
91	C₂²×C₁₂₇	—	2	4	no	—
92	C₂²×C₁₂₇	—	2	4	no	—
93	C₂²×C₁₂₇	—	2	8	no	—
94	C₂²×C₁₂₇	—	2	4	no	—
95	C₂²×C₁₂₇	—	4	2	no	—
96	C₂²×C₁₂₇	—	4	2	no	—
97	C₂²×C₁₂₇	—	4	2	no	—
98	C₂²×C₁₂₇	—	4	2	no	—
99	C₂²×C₁₂₇	—	4	2	no	—
100	C₂²×C₁₂₇	—	4	2	no	—
101	C₂²×C₁₂₇	—	4	4	no	—
102	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
103	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
104	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
105	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
106	C₂²×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
107	C₄×C₁₂₇	—	2	2	no	—
108	C₄×C₁₂₇	—	2	2	no	—
109	C₄×C₁₂₇	—	4	2	no	—
110	C₄×C₁₂₇	—	4	2	no	—
111	C₄×C₁₂₇	—	4	4	no	—
112	C₄×C₁₂₇	#121	1	4	yes	—
113	C₂×C₁₂₇	—	4	4	no	—
114	C₂×C₁₂₇	—	4	4	no	—
115	C₂×C₁₂₇	C₂×#121	1	1	yes	characteristic
116	C₂×C₁₂₇	#525	1	1	yes	characteristic
117	C₂×C₁₂₇	#535	1	1	yes	characteristic
118	C₂×C₁₂₇	#546	1	2	yes	—
119	C₂×C₁₂₇	#553	1	2	yes	—
120	#2535	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₁₂₇	#2535	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×#121	C₂×C₁₂₇	1	2	yes	complemented
123	C₂×C₄×D₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
124	C₂×C₄×D₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
125	#397	C₂×C₁₂₇	1	2	yes	complemented
126	#392	C₂×C₁₂₇	1	2	yes	complemented
127	#386	C₂×C₁₂₇	1	4	yes	complemented
128	#382	C₂×C₁₂₇	1	1	yes	characteristic, complemented
129	#380	C₂×C₁₂₇	1	2	yes	complemented
130	C₂²×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
131	C₂²×D₄	C₂²×C₁₂₇	1	4	yes	—
132	C₂²×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
133	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
134	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
135	#121	—	2	8	no	—
136	C₄×D₄	—	2	4	no	—
137	C₄×D₄	—	2	4	no	—
138	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	4	yes	—
144	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	4	yes	—
145	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
146	#95	C₂²×C₁₂₇	1	4	yes	—
147	#95	C₂²×C₁₂₇	1	2	yes	—
148	#95	C₂²×C₁₂₇	1	2	yes	—
149	C₂⁴	C₂³×C₁₂₇	1	2	yes	—
150	C₂⁴	C₂³×C₁₂₇	1	2	yes	—
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	2	8	no	—
153	C₂×D₄	—	2	8	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	2	2	no	—
157	C₂×D₄	—	4	4	no	—
158	C₂×D₄	D₄×C₁₂₇	1	4	yes	complemented
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
165	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
166	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
167	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	2	8	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₂²⋊C₄	—	4	4	no	—
177	C₄²	D₄×C₁₂₇	1	4	yes	complemented
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	4	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	4	no	—
187	C₂³	—	2	2	no	—
188	C₂³	C₂×D₄×C₁₂₇	1	4	yes	—
189	C₂³	#41×C₁₂₇	1	4	yes	—
190	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
191	D₄	—	2	8	no	—
192	D₄	—	2	8	no	—
193	D₄	—	4	16	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
208	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
209	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
210	C₂×C₄	#41×C₁₂₇	1	2	yes	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	4	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	4	no	—
223	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
224	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
225	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
226	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
227	C₂²	H(5,𝔽₂)×C₁₂₇	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	4	no	—
233	C₄	#121×C₁₂₇	1	4	yes	—
234	C₂	—	4	4	no	—
235	C₂	—	4	4	no	—
236	C₂	C₂×#121×C₁₂₇	1	1	yes	characteristic
237	C₂	#525×C₁₂₇	1	1	yes	characteristic
238	C₂	#535×C₁₂₇	1	1	yes	characteristic
239	C₂	#546×C₁₂₇	1	2	yes	—
240	C₂	#553×C₁₂₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₂₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#121×C₁₂₇	C₂	semidirect
2. C₂×C₄×D₄×C₁₂₇	C₂	semidirect
3. C₂×C₄×D₄×C₁₂₇	C₂	semidirect
4. #397×C₁₂₇	C₂	semidirect
5. #392×C₁₂₇	C₂	semidirect
6. #386×C₁₂₇	C₂	semidirect
7. #382×C₁₂₇	C₂	semidirect
8. #380×C₁₂₇	C₂	semidirect
37. C₂×D₄×C₁₂₇	D₄	semidirect
56. C₄²×C₁₂₇	D₄	semidirect
120. #2535	C₁₂₇	direct (N × H)
121. C₁₂₇	#2535	direct (N × H)
122. C₂×#121	C₂×C₁₂₇	semidirect
123. C₂×C₄×D₄	C₂×C₁₂₇	semidirect
124. C₂×C₄×D₄	C₂×C₁₂₇	semidirect
125. #397	C₂×C₁₂₇	semidirect
126. #392	C₂×C₁₂₇	semidirect
127. #386	C₂×C₁₂₇	semidirect
128. #382	C₂×C₁₂₇	semidirect
129. #380	C₂×C₁₂₇	semidirect
158. C₂×D₄	D₄×C₁₂₇	semidirect
177. C₄²	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×#121×C₁₂₇	2
C₂×C₄×D₄×C₁₂₇	2
#397×C₁₂₇	2
#392×C₁₂₇	2
#386×C₁₂₇	2
#382×C₁₂₇	2
#380×C₁₂₇	2
#2535	127

Subgroups — C127×#2535 #30635487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2535 #30635487