Subgroups — C₁₂₇×#2561 #30635513

Order 16256.

672 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#397×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#393×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#388×C₁₂₇	C₂	1	4	yes	maximal, complemented
6	#381×C₁₂₇	C₂	1	2	yes	maximal, complemented
7	#379×C₁₂₇	C₂	1	2	yes	maximal
8	C₄×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
11	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²⋊Q₈×C₁₂₇	—	2	4	no	—
14	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	complemented
15	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₂²⋊C₄×C₁₂₇	—	2	4	no	—
17	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	—
22	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	—
23	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	complemented
24	#95×C₁₂₇	C₂²	1	2	yes	complemented
25	#95×C₁₂₇	C₂²	1	4	yes	complemented
26	#95×C₁₂₇	C₂²	1	4	yes	complemented
27	C₂⁴×C₁₂₇	—	2	2	no	—
28	C₂⁴×C₁₂₇	—	2	4	no	—
29	C₂⁴×C₁₂₇	—	2	4	no	—
30	C₂⁴×C₁₂₇	—	4	2	no	—
31	C₂⁴×C₁₂₇	C₂³	1	2	yes	—
32	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₁₂₇	—	2	2	no	—
34	C₂×Q₈×C₁₂₇	—	4	2	no	—
35	C₂²×C₄×C₁₂₇	—	2	2	no	—
36	C₂²×C₄×C₁₂₇	—	2	4	no	—
37	C₂²×C₄×C₁₂₇	—	4	2	no	—
38	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₁₂₇	—	2	2	no	—
46	C₄⋊C₄×C₁₂₇	—	2	4	no	—
47	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
48	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
49	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
50	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
51	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
52	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
53	C₄²×C₁₂₇	—	4	2	no	—
54	C₂³×C₁₂₇	—	2	2	no	—
55	C₂³×C₁₂₇	—	2	2	no	—
56	C₂³×C₁₂₇	—	2	8	no	—
57	C₂³×C₁₂₇	—	2	2	no	—
58	C₂³×C₁₂₇	—	2	2	no	—
59	C₂³×C₁₂₇	—	2	4	no	—
60	C₂³×C₁₂₇	—	2	4	no	—
61	C₂³×C₁₂₇	—	4	16	no	—
62	C₂³×C₁₂₇	—	4	4	no	—
63	C₂³×C₁₂₇	—	4	4	no	—
64	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
65	C₂³×C₁₂₇	#41	1	4	yes	—
66	C₂³×C₁₂₇	#41	1	4	yes	—
67	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
68	Q₈×C₁₂₇	—	4	4	no	—
69	C₂×C₄×C₁₂₇	—	2	2	no	—
70	C₂×C₄×C₁₂₇	—	2	2	no	—
71	C₂×C₄×C₁₂₇	—	2	2	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	4	no	—
76	C₂×C₄×C₁₂₇	—	2	4	no	—
77	C₂×C₄×C₁₂₇	—	2	4	no	—
78	C₂×C₄×C₁₂₇	—	2	4	no	—
79	C₂×C₄×C₁₂₇	—	2	4	no	—
80	C₂×C₄×C₁₂₇	—	4	4	no	—
81	C₂×C₄×C₁₂₇	—	4	4	no	—
82	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
84	C₂²×C₁₂₇	—	2	2	no	—
85	C₂²×C₁₂₇	—	2	2	no	—
86	C₂²×C₁₂₇	—	2	8	no	—
87	C₂²×C₁₂₇	—	2	4	no	—
88	C₂²×C₁₂₇	—	2	8	no	—
89	C₂²×C₁₂₇	—	4	16	no	—
90	C₂²×C₁₂₇	—	4	2	no	—
91	C₂²×C₁₂₇	—	4	2	no	—
92	C₂²×C₁₂₇	—	4	4	no	—
93	C₂²×C₁₂₇	#124	1	4	yes	—
94	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
95	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
96	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
97	C₂²×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
98	C₄×C₁₂₇	—	2	2	no	—
99	C₄×C₁₂₇	—	2	2	no	—
100	C₄×C₁₂₇	—	4	2	no	—
101	C₄×C₁₂₇	—	4	2	no	—
102	C₄×C₁₂₇	—	4	2	no	—
103	C₄×C₁₂₇	—	4	4	no	—
104	C₂×C₁₂₇	—	2	4	no	—
105	C₂×C₁₂₇	—	4	4	no	—
106	C₂×C₁₂₇	#525	1	1	yes	characteristic
107	C₂×C₁₂₇	C₂×#124	1	1	yes	characteristic
108	C₂×C₁₂₇	#534	1	1	yes	characteristic
109	C₂×C₁₂₇	#551	1	4	yes	—
110	#2561	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₁₂₇	#2561	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×C₂²⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
113	C₂²×C₂²⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
114	#397	C₂×C₁₂₇	1	2	yes	complemented
115	#393	C₂×C₁₂₇	1	2	yes	complemented
116	#388	C₂×C₁₂₇	1	4	yes	complemented
117	#381	C₂×C₁₂₇	1	2	yes	complemented
118	#379	C₂×C₁₂₇	1	2	yes	—
119	C₄×C₂²⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
120	C₂⁵	C₂²×C₁₂₇	1	1	yes	characteristic
121	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
122	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
123	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
124	C₂²⋊Q₈	—	2	4	no	—
125	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	complemented
126	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
127	C₂×C₂²⋊C₄	—	2	4	no	—
128	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	4	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	4	yes	—
134	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	complemented
135	#95	C₂²×C₁₂₇	1	2	yes	complemented
136	#95	C₂²×C₁₂₇	1	4	yes	complemented
137	#95	C₂²×C₁₂₇	1	4	yes	complemented
138	C₂⁴	—	2	2	no	—
139	C₂⁴	—	2	4	no	—
140	C₂⁴	—	2	4	no	—
141	C₂⁴	—	4	2	no	—
142	C₂⁴	C₂³×C₁₂₇	1	2	yes	—
143	C₂⁴	C₂³×C₁₂₇	1	1	yes	characteristic
144	C₂×Q₈	—	2	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
150	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
151	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
152	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
153	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
154	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	2	8	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	C₄²	—	4	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	8	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	4	no	—
171	C₂³	—	2	4	no	—
172	C₂³	—	4	16	no	—
173	C₂³	—	4	4	no	—
174	C₂³	—	4	4	no	—
175	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
176	C₂³	#41×C₁₂₇	1	4	yes	—
177	C₂³	#41×C₁₂₇	1	4	yes	—
178	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
194	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	8	no	—
200	C₂²	—	4	16	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	4	no	—
204	C₂²	#124×C₁₂₇	1	4	yes	—
205	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
206	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
207	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
208	C₂²	H(5,𝔽₂)×C₁₂₇	1	2	yes	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	4	no	—
215	C₂	—	2	4	no	—
216	C₂	—	4	4	no	—
217	C₂	#525×C₁₂₇	1	1	yes	characteristic
218	C₂	C₂×#124×C₁₂₇	1	1	yes	characteristic
219	C₂	#534×C₁₂₇	1	1	yes	characteristic
220	C₂	#551×C₁₂₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₂₇
prime radical	C₂⁵×C₁₂₇	C₂²
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×C₂²⋊Q₈×C₁₂₇	C₂	semidirect
3. #397×C₁₂₇	C₂	semidirect
4. #393×C₁₂₇	C₂	semidirect
5. #388×C₁₂₇	C₂	semidirect
6. #381×C₁₂₇	C₂	semidirect
8. C₄×C₂²⋊C₄×C₁₂₇	C₂	semidirect
10. C₂²×Q₈×C₁₂₇	C₂²	semidirect
14. C₂×C₄⋊C₄×C₁₂₇	C₂²	semidirect
15. C₂×C₄⋊C₄×C₁₂₇	C₂²	semidirect
23. C₂×C₄²×C₁₂₇	C₂²	semidirect
24. #95×C₁₂₇	C₂²	semidirect
25. #95×C₁₂₇	C₂²	semidirect
26. #95×C₁₂₇	C₂²	semidirect
110. #2561	C₁₂₇	direct (N × H)
111. C₁₂₇	#2561	direct (N × H)
112. C₂×C₂²⋊Q₈	C₂×C₁₂₇	semidirect
114. #397	C₂×C₁₂₇	semidirect
115. #393	C₂×C₁₂₇	semidirect
116. #388	C₂×C₁₂₇	semidirect
117. #381	C₂×C₁₂₇	semidirect
119. C₄×C₂²⋊C₄	C₂×C₁₂₇	semidirect
121. C₂²×Q₈	C₂²×C₁₂₇	semidirect
125. C₂×C₄⋊C₄	C₂²×C₁₂₇	semidirect
126. C₂×C₄⋊C₄	C₂²×C₁₂₇	semidirect
134. C₂×C₄²	C₂²×C₁₂₇	semidirect
135. #95	C₂²×C₁₂₇	semidirect
136. #95	C₂²×C₁₂₇	semidirect
137. #95	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×C₂²⋊Q₈×C₁₂₇	2
C₂²×C₂²⋊C₄×C₁₂₇	2
#397×C₁₂₇	2
#393×C₁₂₇	2
#388×C₁₂₇	2
#381×C₁₂₇	2
#379×C₁₂₇	2
C₄×C₂²⋊C₄×C₁₂₇	2
#2561	127

Subgroups — C127×#2561 #30635513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2561 #30635513