Subgroups — C₁₂₇×#7296 #44570969

Order 32512.

242 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2162	C₂	1	1	yes	maximal, characteristic
2	C₄×#107×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1567	C₂	1	1	yes	maximal, characteristic
4	C₂×#107×C₁₂₇	—	2	1	no	—
5	#423×C₁₂₇	—	2	1	no	—
6	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₄×C₄⋊C₄×C₁₂₇	—	2	1	no	—
8	C₄⋊C₁₆×C₁₂₇	C₄	1	2	yes	complemented
9	C₂²⋊C₁₆×C₁₂₇	—	2	1	no	—
10	C₄×C₁₆×C₁₂₇	—	2	1	no	—
11	#340×C₁₂₇	C₄	1	2	yes	—
12	#335×C₁₂₇	—	2	1	no	—
13	#335×C₁₂₇	C₄	1	2	yes	complemented
14	C₂×M₁₆×C₁₂₇	—	4	1	no	—
15	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
18	C₂×C₄²×C₁₂₇	—	4	1	no	—
19	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₂₇	—	2	2	no	—
21	#107×C₁₂₇	—	2	4	no	—
22	C₄⋊C₈×C₁₂₇	—	2	2	no	—
23	C₄⋊C₈×C₁₂₇	—	4	1	no	—
24	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
28	#95×C₁₂₇	—	4	1	no	—
29	C₂²×C₄×C₁₂₇	—	4	2	no	—
30	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
33	M₁₆×C₁₂₇	—	8	1	no	—
34	M₁₆×C₁₂₇	—	8	1	no	—
35	C₂×C₈×C₁₂₇	—	2	2	no	—
36	C₂×C₈×C₁₂₇	—	4	1	no	—
37	C₂×C₈×C₁₂₇	—	4	2	no	—
38	C₂×C₈×C₁₂₇	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
42	C₄⋊C₄×C₁₂₇	—	2	4	no	—
43	C₄⋊C₄×C₁₂₇	—	4	2	no	—
44	C₄²×C₁₂₇	—	8	1	no	—
45	C₄²×C₁₂₇	—	8	1	no	—
46	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
49	C₄²×C₁₂₇	Q₁₆	1	1	yes	characteristic
50	C₁₆×C₁₂₇	—	4	2	no	—
51	C₂³×C₁₂₇	#102	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	—	2	1	no	—
53	C₂×C₄×C₁₂₇	—	2	1	no	—
54	C₂×C₄×C₁₂₇	—	4	1	no	—
55	C₂×C₄×C₁₂₇	—	4	4	no	—
56	C₂×C₄×C₁₂₇	—	8	2	no	—
57	C₂×C₄×C₁₂₇	—	8	2	no	—
58	C₂×C₄×C₁₂₇	#95	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	#106	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	SD₃₂	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	SD₃₂	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	Q₃₂	1	1	yes	characteristic
68	C₈×C₁₂₇	—	2	1	no	—
69	C₈×C₁₂₇	—	2	1	no	—
70	C₈×C₁₂₇	—	2	1	no	—
71	C₈×C₁₂₇	—	8	2	no	—
72	C₈×C₁₂₇	#104	1	2	yes	—
73	C₂²×C₁₂₇	—	2	1	no	—
74	C₂²×C₁₂₇	—	2	1	no	—
75	C₂²×C₁₂₇	#340	1	1	yes	characteristic
76	C₂²×C₁₂₇	#357	1	1	yes	characteristic
77	C₂²×C₁₂₇	#358	1	1	yes	characteristic
78	C₄×C₁₂₇	—	2	1	no	—
79	C₄×C₁₂₇	—	2	1	no	—
80	C₄×C₁₂₇	—	2	1	no	—
81	C₄×C₁₂₇	—	8	4	no	—
82	C₄×C₁₂₇	#339	1	1	yes	characteristic
83	C₄×C₁₂₇	#344	1	1	yes	characteristic
84	C₄×C₁₂₇	#357	1	1	yes	characteristic
85	C₄×C₁₂₇	#358	1	1	yes	characteristic
86	C₄×C₁₂₇	#366	1	1	yes	characteristic
87	C₄×C₁₂₇	#367	1	1	yes	characteristic
88	#7296	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×C₁₂₇	—	2	1	no	—
90	C₂×C₁₂₇	—	2	1	no	—
91	C₂×C₁₂₇	#1303	1	1	yes	characteristic
92	C₂×C₁₂₇	#1380	1	1	yes	characteristic
93	C₂×C₁₂₇	#1389	1	1	yes	characteristic
94	#2162	C₂×C₁₂₇	1	1	yes	characteristic
95	C₄×#107	C₂×C₁₂₇	1	1	yes	characteristic
96	#1567	C₂×C₁₂₇	1	1	yes	characteristic
97	C₁₂₇	#7296	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×#107	—	2	1	no	—
99	#423	—	2	1	no	—
100	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
101	C₄×C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₁₆	C₄×C₁₂₇	1	2	yes	complemented
103	C₂²⋊C₁₆	—	2	1	no	—
104	C₄×C₁₆	—	2	1	no	—
105	#340	C₄×C₁₂₇	1	2	yes	—
106	#335	—	2	1	no	—
107	#335	C₄×C₁₂₇	1	2	yes	complemented
108	C₂×M₁₆	—	4	1	no	—
109	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
110	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄²	—	4	1	no	—
113	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
114	C₂×C₁₆	—	2	2	no	—
115	#107	—	2	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	4	1	no	—
118	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
119	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
120	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
121	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
122	#95	—	4	1	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
125	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
126	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
127	M₁₆	—	8	1	no	—
128	M₁₆	—	8	1	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	C₄²×C₁₂₇	1	1	yes	characteristic
133	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
134	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
135	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄²	—	8	1	no	—
139	C₄²	—	8	1	no	—
140	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
141	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
142	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
143	C₄²	Q₁₆×C₁₂₇	1	1	yes	characteristic
144	C₁₆	—	4	2	no	—
145	C₂³	#102×C₁₂₇	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	#95×C₁₂₇	1	1	yes	characteristic
153	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
154	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
155	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
156	C₂×C₄	#106×C₁₂₇	1	1	yes	characteristic
157	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
158	C₂×C₄	D₁₆×C₁₂₇	1	1	yes	characteristic
159	C₂×C₄	SD₃₂×C₁₂₇	1	1	yes	characteristic
160	C₂×C₄	SD₃₂×C₁₂₇	1	1	yes	characteristic
161	C₂×C₄	Q₃₂×C₁₂₇	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	8	2	no	—
166	C₈	#104×C₁₂₇	1	2	yes	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	#340×C₁₂₇	1	1	yes	characteristic
170	C₂²	#357×C₁₂₇	1	1	yes	characteristic
171	C₂²	#358×C₁₂₇	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	8	4	no	—
176	C₄	#339×C₁₂₇	1	1	yes	characteristic
177	C₄	#344×C₁₂₇	1	1	yes	characteristic
178	C₄	#357×C₁₂₇	1	1	yes	characteristic
179	C₄	#358×C₁₂₇	1	1	yes	characteristic
180	C₄	#366×C₁₂₇	1	1	yes	characteristic
181	C₄	#367×C₁₂₇	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	C₁₂₇×#1303	1	1	yes	characteristic
185	C₂	C₁₂₇×#1380	1	1	yes	characteristic
186	C₂	C₁₂₇×#1389	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#102
commutator	C₂×C₈	C₄²×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₂₇
socle	C₂²×C₁₂₇	#340
prime radical	C₂³×C₁₂₇	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
8. C₄⋊C₁₆×C₁₂₇	C₄	semidirect
13. #335×C₁₂₇	C₄	semidirect
88. #7296	C₁₂₇	direct (N × H)
97. C₁₂₇	#7296	direct (N × H)
102. C₄⋊C₁₆	C₄×C₁₂₇	semidirect
107. #335	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2162	2
C₄×#107×C₁₂₇	2
C₁₂₇×#1567	2
#7296	127

Subgroups — C127×#7296 #44570969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7296 #44570969