Subgroups — C₁₂₇×#2855 #30635807

Order 16256.

564 conjugacy classes of subgroups.

The subgroup lattice has 206 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×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#391×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#389×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#387×C₁₂₇	C₂	1	4	yes	maximal, complemented
6	#385×C₁₂₇	C₂	1	2	yes	maximal
7	#381×C₁₂₇	C₂	1	2	yes	maximal
8	#380×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	C₂²	1	2	yes	—
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	4	yes	—
18	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₂₇	—	2	4	no	—
21	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
23	#95×C₁₂₇	C₂²	1	2	yes	—
24	#95×C₁₂₇	C₂²	1	4	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	—	2	4	no	—
30	C₂²×C₄×C₁₂₇	—	2	2	no	—
31	C₂²×C₄×C₁₂₇	—	2	2	no	—
32	C₂²×C₄×C₁₂₇	—	2	2	no	—
33	C₂²×C₄×C₁₂₇	—	2	4	no	—
34	C₂²×C₄×C₁₂₇	—	2	4	no	—
35	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
36	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	complemented
37	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
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	1	yes	characteristic
44	C₄⋊C₄×C₁₂₇	—	2	2	no	—
45	C₄⋊C₄×C₁₂₇	—	2	4	no	—
46	C₄⋊C₄×C₁₂₇	—	2	8	no	—
47	C₄⋊C₄×C₁₂₇	—	2	4	no	—
48	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
49	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
50	C₄²×C₁₂₇	—	2	4	no	—
51	C₄²×C₁₂₇	—	2	4	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	2	yes	—
56	C₂³×C₁₂₇	#41	1	2	yes	—
57	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
58	Q₈×C₁₂₇	—	4	4	no	—
59	C₂×C₄×C₁₂₇	—	2	8	no	—
60	C₂×C₄×C₁₂₇	—	2	8	no	—
61	C₂×C₄×C₁₂₇	—	2	2	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	8	no	—
67	C₂×C₄×C₁₂₇	—	2	2	no	—
68	C₂×C₄×C₁₂₇	—	2	2	no	—
69	C₂×C₄×C₁₂₇	—	2	4	no	—
70	C₂×C₄×C₁₂₇	—	2	4	no	—
71	C₂×C₄×C₁₂₇	—	2	4	no	—
72	C₂×C₄×C₁₂₇	—	2	4	no	—
73	C₂×C₄×C₁₂₇	—	2	4	no	—
74	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
77	C₂×C₄×C₁₂₇	#41	1	4	yes	—
78	C₂×C₄×C₁₂₇	#41	1	4	yes	—
79	C₂²×C₁₂₇	—	2	2	no	—
80	C₂²×C₁₂₇	—	2	2	no	—
81	C₂²×C₁₂₇	—	2	4	no	—
82	C₂²×C₁₂₇	—	2	4	no	—
83	C₂²×C₁₂₇	C₂²⋊Q₈	1	4	yes	—
84	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
87	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
88	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
89	C₄×C₁₂₇	—	2	4	no	—
90	C₄×C₁₂₇	—	2	2	no	—
91	C₄×C₁₂₇	—	2	4	no	—
92	C₄×C₁₂₇	—	4	2	no	—
93	C₄×C₁₂₇	—	4	2	no	—
94	C₄×C₁₂₇	—	4	4	no	—
95	C₄×C₁₂₇	#123	1	4	yes	—
96	C₂×C₁₂₇	—	2	4	no	—
97	C₂×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₁₂₇	C₂×#123	1	1	yes	characteristic
99	C₂×C₁₂₇	#525	1	1	yes	characteristic
100	C₂×C₁₂₇	#529	1	2	yes	—
101	C₂×C₁₂₇	#533	1	2	yes	—
102	#2855	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₁₂₇	#2855	1	1	yes	characteristic, direct product factor, Sylow
104	C₂×C₂²⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic
105	C₂²×C₄²	C₂×C₁₂₇	1	1	yes	characteristic
106	#391	C₂×C₁₂₇	1	2	yes	complemented
107	#389	C₂×C₁₂₇	1	2	yes	complemented
108	#387	C₂×C₁₂₇	1	4	yes	complemented
109	#385	C₂×C₁₂₇	1	2	yes	—
110	#381	C₂×C₁₂₇	1	2	yes	—
111	#380	C₂×C₁₂₇	1	1	yes	characteristic
112	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
113	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
114	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
115	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
116	C₂²⋊Q₈	—	2	4	no	—
117	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
121	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
122	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
123	C₂×C₄²	—	2	4	no	—
124	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
125	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
126	#95	C₂²×C₁₂₇	1	2	yes	—
127	#95	C₂²×C₁₂₇	1	4	yes	—
128	#95	C₂²×C₁₂₇	1	4	yes	—
129	#95	C₂²×C₁₂₇	1	4	yes	—
130	C₂⁴	C₂³×C₁₂₇	1	1	yes	characteristic
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	2	4	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
139	C₂²×C₄	Q₈×C₁₂₇	1	4	yes	complemented
140	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
141	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
142	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
143	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
145	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
146	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	8	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₂²⋊C₄	—	2	4	no	—
152	C₂²⋊C₄	—	2	4	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	2	4	no	—
155	C₂³	—	2	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
158	C₂³	C₂×Q₈×C₁₂₇	1	2	yes	—
159	C₂³	#41×C₁₂₇	1	2	yes	—
160	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
161	Q₈	—	4	4	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	2	8	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	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
178	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
179	C₂×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
180	C₂×C₄	#41×C₁₂₇	1	4	yes	—
181	C₂×C₄	#41×C₁₂₇	1	4	yes	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	4	no	—
185	C₂²	—	2	4	no	—
186	C₂²	C₂²⋊Q₈×C₁₂₇	1	4	yes	—
187	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
188	C₂²	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
189	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
190	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
191	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
192	C₄	—	2	4	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	4	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	4	no	—
198	C₄	#123×C₁₂₇	1	4	yes	—
199	C₂	—	2	4	no	—
200	C₂	C₂×C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
201	C₂	C₂×#123×C₁₂₇	1	1	yes	characteristic
202	C₂	#525×C₁₂₇	1	1	yes	characteristic
203	C₂	#529×C₁₂₇	1	2	yes	—
204	C₂	#533×C₁₂₇	1	2	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. #391×C₁₂₇	C₂	semidirect
4. #389×C₁₂₇	C₂	semidirect
5. #387×C₁₂₇	C₂	semidirect
36. C₂²×C₄×C₁₂₇	Q₈	semidirect
102. #2855	C₁₂₇	direct (N × H)
103. C₁₂₇	#2855	direct (N × H)
106. #391	C₂×C₁₂₇	semidirect
107. #389	C₂×C₁₂₇	semidirect
108. #387	C₂×C₁₂₇	semidirect
139. 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×C₄²×C₁₂₇	2
#391×C₁₂₇	2
#389×C₁₂₇	2
#387×C₁₂₇	2
#385×C₁₂₇	2
#381×C₁₂₇	2
#380×C₁₂₇	2
#2855	127

Subgroups — C127×#2855 #30635807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2855 #30635807