Subgroups — C₁₂₇×#7180 #44570853

Order 32512.

240 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×#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1772	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₂₇	—	2	1	no	—
5	#431×C₁₂₇	—	2	1	no	—
6	C₂×#108×C₁₂₇	C₄	1	1	yes	characteristic
7	#427×C₁₂₇	C₄	1	1	yes	characteristic
8	#405×C₁₂₇	—	2	1	no	—
9	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	#364×C₁₂₇	—	2	1	no	—
11	#364×C₁₂₇	C₄	1	2	yes	complemented
12	#339×C₁₂₇	—	2	1	no	—
13	#322×C₁₂₇	—	2	1	no	—
14	C₈²×C₁₂₇	C₄	1	2	yes	complemented
15	C₂×M₁₆×C₁₂₇	—	2	1	no	—
16	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
18	#117×C₁₂₇	—	2	1	no	—
19	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
20	M₃₂×C₁₂₇	—	2	2	no	—
21	M₃₂×C₁₂₇	—	4	1	no	—
22	C₂×C₁₆×C₁₂₇	—	4	1	no	—
23	#108×C₁₂₇	—	2	1	no	—
24	#108×C₁₂₇	C₈	1	2	yes	complemented
25	#107×C₁₂₇	—	2	1	no	—
26	#106×C₁₂₇	C₈	1	2	yes	complemented
27	C₄⋊C₈×C₁₂₇	—	2	2	no	—
28	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
29	#97×C₁₂₇	—	4	1	no	—
30	C₄×C₈×C₁₂₇	—	2	2	no	—
31	C₄×C₈×C₁₂₇	—	4	1	no	—
32	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
35	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	—	2	1	no	—
37	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
38	M₁₆×C₁₂₇	—	2	2	no	—
39	M₁₆×C₁₂₇	—	4	1	no	—
40	C₂×C₈×C₁₂₇	—	2	4	no	—
41	C₂×C₈×C₁₂₇	—	4	1	no	—
42	C₂×C₈×C₁₂₇	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	D₈	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
54	C₄⋊C₄×C₁₂₇	—	2	2	no	—
55	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
56	C₄²×C₁₂₇	—	2	1	no	—
57	C₄²×C₁₂₇	—	4	1	no	—
58	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆×C₁₂₇	—	4	2	no	—
61	C₂³×C₁₂₇	#104	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	—	2	1	no	—
63	C₂×C₄×C₁₂₇	—	2	1	no	—
64	C₂×C₄×C₁₂₇	—	4	2	no	—
65	C₂×C₄×C₁₂₇	—	4	2	no	—
66	C₂×C₄×C₁₂₇	#95	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	#104	1	1	yes	characteristic
72	C₈×C₁₂₇	—	4	4	no	—
73	C₈×C₁₂₇	—	4	2	no	—
74	C₈×C₁₂₇	C₄×C₈	1	1	yes	characteristic
75	C₈×C₁₂₇	#97	1	1	yes	characteristic
76	C₈×C₁₂₇	#102	1	1	yes	characteristic
77	C₈×C₁₂₇	#103	1	1	yes	characteristic
78	C₈×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
79	C₈×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
80	C₈×C₁₂₇	#106	1	1	yes	characteristic
81	C₈×C₁₂₇	#107	1	1	yes	characteristic
82	C₂²×C₁₂₇	—	4	1	no	—
83	C₂²×C₁₂₇	D₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₁₂₇	Q₈⋊C₈	1	1	yes	characteristic
85	C₂²×C₁₂₇	#339	1	1	yes	characteristic
86	C₄×C₁₂₇	—	2	2	no	—
87	C₄×C₁₂₇	—	4	1	no	—
88	C₄×C₁₂₇	—	8	2	no	—
89	C₄×C₁₂₇	D₄⋊C₈	1	1	yes	characteristic
90	C₄×C₁₂₇	Q₈⋊C₈	1	1	yes	characteristic
91	C₄×C₁₂₇	#336	1	1	yes	characteristic
92	C₄×C₁₂₇	#340	1	1	yes	characteristic
93	#7180	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×C₁₂₇	—	2	1	no	—
95	C₂×C₁₂₇	—	2	1	no	—
96	C₂×C₁₂₇	—	2	1	no	—
97	C₂×C₁₂₇	#1279	1	1	yes	characteristic
98	C₂×#364	C₂×C₁₂₇	1	1	yes	characteristic
99	#1772	C₂×C₁₂₇	1	1	yes	characteristic
100	#1450	C₂×C₁₂₇	1	1	yes	characteristic
101	C₁₂₇	#7180	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×M₃₂	—	2	1	no	—
103	#431	—	2	1	no	—
104	C₂×#108	C₄×C₁₂₇	1	1	yes	characteristic
105	#427	C₄×C₁₂₇	1	1	yes	characteristic
106	#405	—	2	1	no	—
107	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
108	#364	—	2	1	no	—
109	#364	C₄×C₁₂₇	1	2	yes	complemented
110	#339	—	2	1	no	—
111	#322	—	2	1	no	—
112	C₈²	C₄×C₁₂₇	1	2	yes	complemented
113	C₂×M₁₆	—	2	1	no	—
114	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
115	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
116	#117	—	2	1	no	—
117	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
118	M₃₂	—	2	2	no	—
119	M₃₂	—	4	1	no	—
120	C₂×C₁₆	—	4	1	no	—
121	#108	—	2	1	no	—
122	#108	C₈×C₁₂₇	1	2	yes	complemented
123	#107	—	2	1	no	—
124	#106	C₈×C₁₂₇	1	2	yes	complemented
125	C₄⋊C₈	—	2	2	no	—
126	C₂²⋊C₈	—	2	2	no	—
127	#97	—	4	1	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	4	1	no	—
130	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
131	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
132	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
133	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	4	1	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	C₄²×C₁₂₇	1	1	yes	characteristic
141	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
142	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
143	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
144	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
145	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
146	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
147	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
148	C₂×C₈	D₈×C₁₂₇	1	1	yes	characteristic
149	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
150	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
151	C₂×C₈	Q₁₆×C₁₂₇	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	4	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	4	1	no	—
156	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
157	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
158	C₁₆	—	4	2	no	—
159	C₂³	#104×C₁₂₇	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	#95×C₁₂₇	1	1	yes	characteristic
165	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
166	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
167	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
168	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
169	C₂×C₄	#104×C₁₂₇	1	1	yes	characteristic
170	C₈	—	4	4	no	—
171	C₈	—	4	2	no	—
172	C₈	C₄×C₈×C₁₂₇	1	1	yes	characteristic
173	C₈	#97×C₁₂₇	1	1	yes	characteristic
174	C₈	#102×C₁₂₇	1	1	yes	characteristic
175	C₈	#103×C₁₂₇	1	1	yes	characteristic
176	C₈	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
177	C₈	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
178	C₈	#106×C₁₂₇	1	1	yes	characteristic
179	C₈	#107×C₁₂₇	1	1	yes	characteristic
180	C₂²	—	4	1	no	—
181	C₂²	D₄⋊C₈×C₁₂₇	1	1	yes	characteristic
182	C₂²	Q₈⋊C₈×C₁₂₇	1	1	yes	characteristic
183	C₂²	#339×C₁₂₇	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	8	2	no	—
187	C₄	D₄⋊C₈×C₁₂₇	1	1	yes	characteristic
188	C₄	Q₈⋊C₈×C₁₂₇	1	1	yes	characteristic
189	C₄	#336×C₁₂₇	1	1	yes	characteristic
190	C₄	#340×C₁₂₇	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	C₁₂₇×#1279	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₂₇	#106
commutator	C₈	C₄×C₈×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₂₇
socle	C₂×C₁₂₇	#1279
prime radical	C₂³×C₁₂₇	#104
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. #364×C₁₂₇	C₄	semidirect
14. C₈²×C₁₂₇	C₄	semidirect
24. #108×C₁₂₇	C₈	semidirect
26. #106×C₁₂₇	C₈	semidirect
93. #7180	C₁₂₇	direct (N × H)
101. C₁₂₇	#7180	direct (N × H)
109. #364	C₄×C₁₂₇	semidirect
112. C₈²	C₄×C₁₂₇	semidirect
122. #108	C₈×C₁₂₇	semidirect
124. #106	C₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#364×C₁₂₇	2
C₁₂₇×#1772	2
C₁₂₇×#1450	2
#7180	127

Subgroups — C127×#7180 #44570853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7180 #44570853