Subgroups — C₁₂₇×#2583 #30635535

Order 16256.

544 conjugacy classes of subgroups.

The subgroup lattice has 196 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
2	C₂²×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#398×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#395×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#389×C₁₂₇	C₂	1	4	yes	maximal, complemented
6	#385×C₁₂₇	C₂	1	2	yes	maximal
7	#385×C₁₂₇	C₂	1	2	yes	maximal
8	C₄×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
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	—	2	4	no	—
15	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
24	#95×C₁₂₇	C₂²	1	2	yes	—
25	#95×C₁₂₇	C₂²	1	4	yes	—
26	#95×C₁₂₇	C₂²	1	4	yes	—
27	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂×Q₈×C₁₂₇	—	2	2	no	—
29	C₂×Q₈×C₁₂₇	—	4	2	no	—
30	C₂²×C₄×C₁₂₇	—	2	2	no	—
31	C₂²×C₄×C₁₂₇	—	2	4	no	—
32	C₂²×C₄×C₁₂₇	—	2	2	no	—
33	C₂²×C₄×C₁₂₇	—	4	2	no	—
34	C₂²×C₄×C₁₂₇	—	4	2	no	—
35	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
36	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
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	4	yes	—
42	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
43	C₄⋊C₄×C₁₂₇	—	2	4	no	—
44	C₄⋊C₄×C₁₂₇	—	2	4	no	—
45	C₄⋊C₄×C₁₂₇	—	2	2	no	—
46	C₄⋊C₄×C₁₂₇	—	2	4	no	—
47	C₄⋊C₄×C₁₂₇	—	2	8	no	—
48	C₄⋊C₄×C₁₂₇	—	4	4	no	—
49	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
50	C₂²⋊C₄×C₁₂₇	Q₈	1	8	yes	complemented
51	C₄²×C₁₂₇	—	4	2	no	—
52	C₂³×C₁₂₇	—	2	2	no	—
53	C₂³×C₁₂₇	—	2	2	no	—
54	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
55	C₂³×C₁₂₇	C₂×Q₈	1	4	yes	—
56	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
57	Q₈×C₁₂₇	—	4	4	no	—
58	C₂×C₄×C₁₂₇	—	2	8	no	—
59	C₂×C₄×C₁₂₇	—	2	2	no	—
60	C₂×C₄×C₁₂₇	—	2	4	no	—
61	C₂×C₄×C₁₂₇	—	2	2	no	—
62	C₂×C₄×C₁₂₇	—	2	4	no	—
63	C₂×C₄×C₁₂₇	—	2	2	no	—
64	C₂×C₄×C₁₂₇	—	2	2	no	—
65	C₂×C₄×C₁₂₇	—	2	2	no	—
66	C₂×C₄×C₁₂₇	—	2	2	no	—
67	C₂×C₄×C₁₂₇	—	2	2	no	—
68	C₂×C₄×C₁₂₇	—	2	4	no	—
69	C₂×C₄×C₁₂₇	—	2	4	no	—
70	C₂×C₄×C₁₂₇	—	2	4	no	—
71	C₂×C₄×C₁₂₇	—	4	4	no	—
72	C₂×C₄×C₁₂₇	—	4	4	no	—
73	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₁₂₇	C₂×Q₈	1	8	yes	—
76	C₂×C₄×C₁₂₇	#41	1	4	yes	—
77	C₂²×C₁₂₇	—	2	2	no	—
78	C₂²×C₁₂₇	—	2	2	no	—
79	C₂²×C₁₂₇	—	2	8	no	—
80	C₂²×C₁₂₇	C₄⋊Q₈	1	4	yes	—
81	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
82	C₂²×C₁₂₇	C₂²×Q₈	1	2	yes	—
83	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
84	C₂²×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
85	C₄×C₁₂₇	—	2	4	no	—
86	C₄×C₁₂₇	—	2	2	no	—
87	C₄×C₁₂₇	—	2	2	no	—
88	C₄×C₁₂₇	—	2	8	no	—
89	C₄×C₁₂₇	—	4	2	no	—
90	C₄×C₁₂₇	—	4	2	no	—
91	C₄×C₁₂₇	—	4	2	no	—
92	C₂×C₁₂₇	—	2	4	no	—
93	C₂×C₁₂₇	#525	1	1	yes	characteristic
94	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₁₂₇	#534	1	1	yes	characteristic
96	C₂×C₁₂₇	#554	1	4	yes	—
97	#2583	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₁₂₇	#2583	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×C₂²⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic
100	C₂²×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
101	#398	C₂×C₁₂₇	1	2	yes	complemented
102	#395	C₂×C₁₂₇	1	2	yes	complemented
103	#389	C₂×C₁₂₇	1	4	yes	complemented
104	#385	C₂×C₁₂₇	1	2	yes	—
105	#385	C₂×C₁₂₇	1	2	yes	—
106	C₄×C₂²⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
107	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
108	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
109	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
110	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
111	C₂²⋊Q₈	—	2	4	no	—
112	C₂×C₄⋊C₄	—	2	4	no	—
113	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
114	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
115	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
116	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
118	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
119	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
121	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
122	#95	C₂²×C₁₂₇	1	2	yes	—
123	#95	C₂²×C₁₂₇	1	4	yes	—
124	#95	C₂²×C₁₂₇	1	4	yes	—
125	C₂⁴	C₂³×C₁₂₇	1	1	yes	characteristic
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	—	4	2	no	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
134	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
135	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
136	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
137	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
138	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
139	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
140	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	2	8	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₂²⋊C₄	—	2	4	no	—
148	C₂²⋊C₄	Q₈×C₁₂₇	1	8	yes	complemented
149	C₄²	—	4	2	no	—
150	C₂³	—	2	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
153	C₂³	C₂×Q₈×C₁₂₇	1	4	yes	—
154	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
155	Q₈	—	4	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
172	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
173	C₂×C₄	C₂×Q₈×C₁₂₇	1	8	yes	—
174	C₂×C₄	#41×C₁₂₇	1	4	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	8	no	—
178	C₂²	C₄⋊Q₈×C₁₂₇	1	4	yes	—
179	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
180	C₂²	C₂²×Q₈×C₁₂₇	1	2	yes	—
181	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
182	C₂²	H(5,𝔽₂)×C₁₂₇	1	2	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	2	8	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	2	no	—
190	C₂	—	2	4	no	—
191	C₂	#525×C₁₂₇	1	1	yes	characteristic
192	C₂	C₂×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
193	C₂	#534×C₁₂₇	1	1	yes	characteristic
194	C₂	#554×C₁₂₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₂₇
prime radical	C₂⁴×C₁₂₇	C₂³
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
3. #398×C₁₂₇	C₂	semidirect
4. #395×C₁₂₇	C₂	semidirect
5. #389×C₁₂₇	C₂	semidirect
50. C₂²⋊C₄×C₁₂₇	Q₈	semidirect
97. #2583	C₁₂₇	direct (N × H)
98. C₁₂₇	#2583	direct (N × H)
101. #398	C₂×C₁₂₇	semidirect
102. #395	C₂×C₁₂₇	semidirect
103. #389	C₂×C₁₂₇	semidirect
148. C₂²⋊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
C₂×C₂²⋊Q₈×C₁₂₇	2
C₂²×C₄⋊C₄×C₁₂₇	2
#398×C₁₂₇	2
#395×C₁₂₇	2
#389×C₁₂₇	2
#385×C₁₂₇	2
C₄×C₂²⋊C₄×C₁₂₇	2
#2583	127

Subgroups — C127×#2583 #30635535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2583 #30635535