Subgroups — C₁₂₇×#2799 #30635751

Order 16256.

376 conjugacy classes of subgroups.

The subgroup lattice has 200 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	1	yes	maximal, characteristic
3	#398×C₁₂₇	C₂	1	2	yes	maximal
4	#395×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#391×C₁₂₇	C₂	1	2	yes	maximal
6	#391×C₁₂₇	C₂	1	2	yes	maximal
7	#391×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	#387×C₁₂₇	C₂	1	2	yes	maximal
9	#387×C₁₂₇	C₂	1	2	yes	maximal
10	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
20	#95×C₁₂₇	C₂²	1	2	yes	—
21	#95×C₁₂₇	C₂²	1	2	yes	—
22	#95×C₁₂₇	C₂²	1	2	yes	—
23	#95×C₁₂₇	C₂²	1	2	yes	—
24	#95×C₁₂₇	C₂²	1	2	yes	—
25	#95×C₁₂₇	C₂²	1	2	yes	—
26	#95×C₁₂₇	C₂²	1	2	yes	—
27	#95×C₁₂₇	C₂²	1	2	yes	—
28	#95×C₁₂₇	C₂²	1	2	yes	—
29	#95×C₁₂₇	C₂²	1	1	yes	characteristic
30	#95×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂×Q₈×C₁₂₇	—	2	4	no	—
32	C₂×Q₈×C₁₂₇	—	4	1	no	—
33	C₂×Q₈×C₁₂₇	—	4	1	no	—
34	C₂×Q₈×C₁₂₇	—	4	1	no	—
35	C₂×Q₈×C₁₂₇	Q₈	1	4	yes	complemented
36	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
45	C₄⋊C₄×C₁₂₇	—	2	4	no	—
46	C₄⋊C₄×C₁₂₇	—	2	4	no	—
47	C₄⋊C₄×C₁₂₇	—	4	2	no	—
48	C₄⋊C₄×C₁₂₇	—	4	1	no	—
49	C₄⋊C₄×C₁₂₇	—	4	1	no	—
50	C₄⋊C₄×C₁₂₇	—	4	2	no	—
51	C₄²×C₁₂₇	—	4	2	no	—
52	C₄²×C₁₂₇	—	4	1	no	—
53	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
54	Q₈×C₁₂₇	—	4	4	no	—
55	Q₈×C₁₂₇	—	4	4	no	—
56	C₂×C₄×C₁₂₇	—	2	2	no	—
57	C₂×C₄×C₁₂₇	—	2	1	no	—
58	C₂×C₄×C₁₂₇	—	2	2	no	—
59	C₂×C₄×C₁₂₇	—	2	2	no	—
60	C₂×C₄×C₁₂₇	—	2	2	no	—
61	C₂×C₄×C₁₂₇	—	2	1	no	—
62	C₂×C₄×C₁₂₇	—	2	2	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	2	no	—
69	C₂×C₄×C₁₂₇	—	2	2	no	—
70	C₂×C₄×C₁₂₇	—	2	2	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	2	no	—
75	C₂×C₄×C₁₂₇	—	2	2	no	—
76	C₂×C₄×C₁₂₇	—	2	4	no	—
77	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
78	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
79	C₂×C₄×C₁₂₇	#41	1	4	yes	—
80	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
82	C₂²×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂²×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
84	C₂²×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
85	C₄×C₁₂₇	—	2	2	no	—
86	C₄×C₁₂₇	—	2	4	no	—
87	C₄×C₁₂₇	—	2	4	no	—
88	C₄×C₁₂₇	—	4	2	no	—
89	C₄×C₁₂₇	—	4	2	no	—
90	C₄×C₁₂₇	—	4	2	no	—
91	C₄×C₁₂₇	—	4	1	no	—
92	C₄×C₁₂₇	—	4	1	no	—
93	C₄×C₁₂₇	—	4	2	no	—
94	C₂×C₁₂₇	#542	1	2	yes	—
95	C₂×C₁₂₇	#543	1	1	yes	characteristic
96	C₂×C₁₂₇	#551	1	1	yes	characteristic
97	C₂×C₁₂₇	#557	1	2	yes	—
98	C₂×C₁₂₇	#563	1	1	yes	characteristic
99	#2799	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₁₂₇	#2799	1	1	yes	characteristic, direct product factor, Sylow
101	#400	C₂×C₁₂₇	1	2	yes	—
102	#400	C₂×C₁₂₇	1	1	yes	characteristic
103	#398	C₂×C₁₂₇	1	2	yes	—
104	#395	C₂×C₁₂₇	1	1	yes	characteristic
105	#391	C₂×C₁₂₇	1	2	yes	—
106	#391	C₂×C₁₂₇	1	2	yes	—
107	#391	C₂×C₁₂₇	1	1	yes	characteristic
108	#387	C₂×C₁₂₇	1	2	yes	—
109	#387	C₂×C₁₂₇	1	2	yes	—
110	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
111	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
114	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
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	2	yes	—
118	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
119	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
120	#95	C₂²×C₁₂₇	1	2	yes	—
121	#95	C₂²×C₁₂₇	1	2	yes	—
122	#95	C₂²×C₁₂₇	1	2	yes	—
123	#95	C₂²×C₁₂₇	1	2	yes	—
124	#95	C₂²×C₁₂₇	1	2	yes	—
125	#95	C₂²×C₁₂₇	1	2	yes	—
126	#95	C₂²×C₁₂₇	1	2	yes	—
127	#95	C₂²×C₁₂₇	1	2	yes	—
128	#95	C₂²×C₁₂₇	1	2	yes	—
129	#95	C₂²×C₁₂₇	1	1	yes	characteristic
130	#95	C₂²×C₁₂₇	1	1	yes	characteristic
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	1	no	—
135	C₂×Q₈	Q₈×C₁₂₇	1	4	yes	complemented
136	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
137	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
138	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
139	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
140	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
141	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
144	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	1	no	—
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄²	—	4	2	no	—
152	C₄²	—	4	1	no	—
153	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
154	Q₈	—	4	4	no	—
155	Q₈	—	4	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	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	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
178	C₂×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
179	C₂×C₄	#41×C₁₂₇	1	4	yes	—
180	C₂²	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
181	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
182	C₂²	H(5,𝔽₂)×C₁₂₇	1	1	yes	characteristic
183	C₂²	H(5,𝔽₂)×C₁₂₇	1	1	yes	characteristic
184	C₂²	2₋¹⁺⁴×C₁₂₇	1	2	yes	—
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	2	4	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	4	1	no	—
192	C₄	—	4	1	no	—
193	C₄	—	4	2	no	—
194	C₂	#542×C₁₂₇	1	2	yes	—
195	C₂	#543×C₁₂₇	1	1	yes	characteristic
196	C₂	#551×C₁₂₇	1	1	yes	characteristic
197	C₂	#557×C₁₂₇	1	2	yes	—
198	C₂	#563×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
35. C₂×Q₈×C₁₂₇	Q₈	semidirect
99. #2799	C₁₂₇	direct (N × H)
100. C₁₂₇	#2799	direct (N × H)
135. C₂×Q₈	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
#398×C₁₂₇	2
#395×C₁₂₇	2
#391×C₁₂₇	2
#387×C₁₂₇	2
#2799	127

Subgroups — C127×#2799 #30635751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2799 #30635751