Subgroups — C₁₂₇×#2511 #30635463

Order 16256.

388 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400×C₁₂₇	C₂	1	2	yes	maximal
2	#400×C₁₂₇	C₂	1	2	yes	maximal
3	#391×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#391×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#391×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#389×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	#387×C₁₂₇	C₂	1	2	yes	maximal
8	#387×C₁₂₇	C₂	1	2	yes	maximal
9	#384×C₁₂₇	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
25	#95×C₁₂₇	C₂²	1	1	yes	characteristic
26	#95×C₁₂₇	C₂²	1	2	yes	—
27	#95×C₁₂₇	C₂²	1	1	yes	characteristic
28	#95×C₁₂₇	C₂²	1	2	yes	—
29	#95×C₁₂₇	C₂²	1	2	yes	—
30	#95×C₁₂₇	C₂²	1	2	yes	—
31	#95×C₁₂₇	C₂²	1	2	yes	—
32	#95×C₁₂₇	C₂²	1	2	yes	—
33	#95×C₁₂₇	C₂²	1	2	yes	—
34	#95×C₁₂₇	C₂²	1	2	yes	—
35	C₂×Q₈×C₁₂₇	—	2	2	no	—
36	C₂×Q₈×C₁₂₇	—	2	2	no	—
37	C₂×Q₈×C₁₂₇	—	2	2	no	—
38	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
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	4	no	—
51	C₄⋊C₄×C₁₂₇	—	2	2	no	—
52	C₄⋊C₄×C₁₂₇	—	2	2	no	—
53	C₄⋊C₄×C₁₂₇	—	2	2	no	—
54	C₄⋊C₄×C₁₂₇	—	4	2	no	—
55	C₄⋊C₄×C₁₂₇	—	4	2	no	—
56	C₄⋊C₄×C₁₂₇	Q₈	1	4	yes	complemented
57	C₄²×C₁₂₇	—	2	2	no	—
58	C₄²×C₁₂₇	—	2	2	no	—
59	C₄²×C₁₂₇	—	2	2	no	—
60	C₄²×C₁₂₇	—	4	2	no	—
61	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
62	Q₈×C₁₂₇	—	4	4	no	—
63	C₂×C₄×C₁₂₇	—	2	1	no	—
64	C₂×C₄×C₁₂₇	—	2	1	no	—
65	C₂×C₄×C₁₂₇	—	2	1	no	—
66	C₂×C₄×C₁₂₇	—	2	1	no	—
67	C₂×C₄×C₁₂₇	—	2	1	no	—
68	C₂×C₄×C₁₂₇	—	2	1	no	—
69	C₂×C₄×C₁₂₇	—	2	1	no	—
70	C₂×C₄×C₁₂₇	—	2	1	no	—
71	C₂×C₄×C₁₂₇	—	2	1	no	—
72	C₂×C₄×C₁₂₇	—	2	1	no	—
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	—	2	1	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	2	no	—
79	C₂×C₄×C₁₂₇	—	2	2	no	—
80	C₂×C₄×C₁₂₇	—	2	2	no	—
81	C₂×C₄×C₁₂₇	—	2	2	no	—
82	C₂×C₄×C₁₂₇	—	2	2	no	—
83	C₂×C₄×C₁₂₇	—	2	2	no	—
84	C₂×C₄×C₁₂₇	—	2	1	no	—
85	C₂×C₄×C₁₂₇	—	2	2	no	—
86	C₂×C₄×C₁₂₇	—	2	1	no	—
87	C₂×C₄×C₁₂₇	—	2	2	no	—
88	C₂×C₄×C₁₂₇	—	2	1	no	—
89	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
90	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
92	C₂×C₄×C₁₂₇	#41	1	2	yes	—
93	C₂×C₄×C₁₂₇	#41	1	2	yes	—
94	C₂×C₄×C₁₂₇	#41	1	2	yes	—
95	C₂×C₄×C₁₂₇	#41	1	2	yes	—
96	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
98	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
99	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
100	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
101	C₂²×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂²×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₁₂₇	—	2	2	no	—
104	C₄×C₁₂₇	—	2	2	no	—
105	C₄×C₁₂₇	—	2	2	no	—
106	C₄×C₁₂₇	—	2	2	no	—
107	C₄×C₁₂₇	—	4	2	no	—
108	C₄×C₁₂₇	—	4	1	no	—
109	C₄×C₁₂₇	—	4	2	no	—
110	C₄×C₁₂₇	—	4	1	no	—
111	C₄×C₁₂₇	—	4	2	no	—
112	C₄×C₁₂₇	—	4	2	no	—
113	C₄×C₁₂₇	#126	1	4	yes	—
114	C₂×C₁₂₇	C₂×#126	1	1	yes	characteristic
115	C₂×C₁₂₇	#533	1	1	yes	characteristic
116	C₂×C₁₂₇	#542	1	1	yes	characteristic
117	C₂×C₁₂₇	#552	1	1	yes	characteristic
118	C₂×C₁₂₇	#554	1	1	yes	characteristic
119	C₂×C₁₂₇	#557	1	1	yes	characteristic
120	C₂×C₁₂₇	#559	1	1	yes	characteristic
121	#2511	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₁₂₇	#2511	1	1	yes	characteristic, direct product factor, Sylow
123	#400	C₂×C₁₂₇	1	2	yes	—
124	#400	C₂×C₁₂₇	1	2	yes	—
125	#391	C₂×C₁₂₇	1	1	yes	characteristic
126	#391	C₂×C₁₂₇	1	1	yes	characteristic
127	#391	C₂×C₁₂₇	1	1	yes	characteristic
128	#389	C₂×C₁₂₇	1	1	yes	characteristic
129	#387	C₂×C₁₂₇	1	2	yes	—
130	#387	C₂×C₁₂₇	1	2	yes	—
131	#384	C₂×C₁₂₇	1	1	yes	characteristic
132	C₄×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
133	C₄×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
134	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
141	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
142	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
143	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
144	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
145	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
146	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
147	#95	C₂²×C₁₂₇	1	1	yes	characteristic
148	#95	C₂²×C₁₂₇	1	2	yes	—
149	#95	C₂²×C₁₂₇	1	1	yes	characteristic
150	#95	C₂²×C₁₂₇	1	2	yes	—
151	#95	C₂²×C₁₂₇	1	2	yes	—
152	#95	C₂²×C₁₂₇	1	2	yes	—
153	#95	C₂²×C₁₂₇	1	2	yes	—
154	#95	C₂²×C₁₂₇	1	2	yes	—
155	#95	C₂²×C₁₂₇	1	2	yes	—
156	#95	C₂²×C₁₂₇	1	2	yes	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
161	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
164	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
165	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
166	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
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	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	Q₈×C₁₂₇	1	4	yes	complemented
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	4	2	no	—
183	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
184	Q₈	—	4	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
212	C₂×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
213	C₂×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
214	C₂×C₄	#41×C₁₂₇	1	2	yes	—
215	C₂×C₄	#41×C₁₂₇	1	2	yes	—
216	C₂×C₄	#41×C₁₂₇	1	2	yes	—
217	C₂×C₄	#41×C₁₂₇	1	2	yes	—
218	C₂²	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
219	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
220	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
221	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
222	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
223	C₂²	H(5,𝔽₂)×C₁₂₇	1	1	yes	characteristic
224	C₂²	2₋¹⁺⁴×C₁₂₇	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	#126×C₁₂₇	1	4	yes	—
236	C₂	C₂×#126×C₁₂₇	1	1	yes	characteristic
237	C₂	#533×C₁₂₇	1	1	yes	characteristic
238	C₂	#542×C₁₂₇	1	1	yes	characteristic
239	C₂	#552×C₁₂₇	1	1	yes	characteristic
240	C₂	#554×C₁₂₇	1	1	yes	characteristic
241	C₂	#557×C₁₂₇	1	1	yes	characteristic
242	C₂	#559×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	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
56. C₄⋊C₄×C₁₂₇	Q₈	semidirect
121. #2511	C₁₂₇	direct (N × H)
122. C₁₂₇	#2511	direct (N × H)
178. C₄⋊C₄	Q₈×C₁₂₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#400×C₁₂₇	2
#391×C₁₂₇	2
#389×C₁₂₇	2
#387×C₁₂₇	2
#384×C₁₂₇	2
C₄×C₄⋊C₄×C₁₂₇	2
#2511	127

Subgroups — C127×#2511 #30635463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2511 #30635463