Subgroups — C₁₂₇×#9623 #44573296

Order 32512.

298 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	C₁₂₇×#1950	C₂	1	2	yes	maximal
2	C₁₂₇×#1566	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₈×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1301	C₂	1	2	yes	maximal
6	#431×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×#108×C₁₂₇	—	2	2	no	—
8	#423×C₁₂₇	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
10	C₄×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	#341×C₁₂₇	—	2	2	no	—
13	#334×C₁₂₇	—	2	1	no	—
14	#334×C₁₂₇	—	2	1	no	—
15	#334×C₁₂₇	C₄	1	2	yes	—
16	#334×C₁₂₇	C₄	1	2	yes	—
17	#322×C₁₂₇	—	2	1	no	—
18	C₈²×C₁₂₇	—	2	1	no	—
19	C₂×M₁₆×C₁₂₇	—	2	2	no	—
20	C₂×M₁₆×C₁₂₇	D₄	1	2	yes	—
21	C₂²×C₈×C₁₂₇	—	2	2	no	—
22	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
25	#108×C₁₂₇	—	4	4	no	—
26	C₄⋊C₈×C₁₂₇	—	2	2	no	—
27	C₄⋊C₈×C₁₂₇	—	2	2	no	—
28	C₄⋊C₈×C₁₂₇	—	2	1	no	—
29	C₄⋊C₈×C₁₂₇	—	2	2	no	—
30	C₄⋊C₈×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂²⋊C₈×C₁₂₇	Q₈	1	2	yes	—
32	#97×C₁₂₇	—	2	1	no	—
33	C₄×C₈×C₁₂₇	—	2	1	no	—
34	C₄×C₈×C₁₂₇	—	2	1	no	—
35	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
38	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₁₂₇	—	2	2	no	—
43	M₁₆×C₁₂₇	—	4	2	no	—
44	M₁₆×C₁₂₇	—	4	2	no	—
45	M₁₆×C₁₂₇	SD₁₆	1	4	yes	—
46	C₂×C₈×C₁₂₇	—	2	2	no	—
47	C₂×C₈×C₁₂₇	—	2	2	no	—
48	C₂×C₈×C₁₂₇	—	2	4	no	—
49	C₂×C₈×C₁₂₇	—	2	2	no	—
50	C₂×C₈×C₁₂₇	—	4	1	no	—
51	C₂×C₈×C₁₂₇	—	4	2	no	—
52	C₂×C₈×C₁₂₇	—	4	2	no	—
53	C₂×C₈×C₁₂₇	—	4	2	no	—
54	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₁₂₇	#41	1	1	yes	characteristic
61	C₄²×C₁₂₇	#41	1	1	yes	characteristic
62	C₂³×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	—	2	1	no	—
64	C₂×C₄×C₁₂₇	—	2	1	no	—
65	C₂×C₄×C₁₂₇	—	4	1	no	—
66	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	#125	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₁₂₇	#135	1	2	yes	—
74	C₈×C₁₂₇	—	2	1	no	—
75	C₈×C₁₂₇	—	2	1	no	—
76	C₈×C₁₂₇	—	2	1	no	—
77	C₈×C₁₂₇	—	2	4	no	—
78	C₈×C₁₂₇	—	4	2	no	—
79	C₈×C₁₂₇	—	4	4	no	—
80	C₈×C₁₂₇	—	8	2	no	—
81	C₈×C₁₂₇	#108	1	2	yes	complemented
82	C₂²×C₁₂₇	—	2	1	no	—
83	C₂²×C₁₂₇	—	2	1	no	—
84	C₂²×C₁₂₇	#389	1	1	yes	characteristic
85	C₂²×C₁₂₇	#498	1	1	yes	characteristic
86	C₂²×C₁₂₇	#499	1	1	yes	characteristic
87	C₄×C₁₂₇	—	2	1	no	—
88	C₄×C₁₂₇	—	2	1	no	—
89	C₄×C₁₂₇	—	2	1	no	—
90	C₄×C₁₂₇	C₂×#108	1	1	yes	characteristic
91	C₄×C₁₂₇	#430	1	1	yes	characteristic
92	C₄×C₁₂₇	C₄×SD₁₆	1	2	yes	—
93	C₄×C₁₂₇	#465	1	2	yes	—
94	#9623	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₁₂₇	—	2	1	no	—
96	C₂×C₁₂₇	—	2	1	no	—
97	C₂×C₁₂₇	#1941	1	1	yes	characteristic
98	C₂×C₁₂₇	#1950	1	1	yes	characteristic
99	C₂×C₁₂₇	#1953	1	1	yes	characteristic
100	#1950	C₂×C₁₂₇	1	2	yes	—
101	#1566	C₂×C₁₂₇	1	1	yes	characteristic
102	C₂×#334	C₂×C₁₂₇	1	1	yes	characteristic
103	C₈×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
104	#1301	C₂×C₁₂₇	1	2	yes	—
105	C₁₂₇	#9623	1	1	yes	characteristic, direct product factor, Sylow
106	#431	C₂²×C₁₂₇	1	1	yes	characteristic
107	C₂×#108	—	2	2	no	—
108	#423	C₂²×C₁₂₇	1	2	yes	—
109	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
110	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
112	#341	—	2	2	no	—
113	#334	—	2	1	no	—
114	#334	—	2	1	no	—
115	#334	C₄×C₁₂₇	1	2	yes	—
116	#334	C₄×C₁₂₇	1	2	yes	—
117	#322	—	2	1	no	—
118	C₈²	—	2	1	no	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	D₄×C₁₂₇	1	2	yes	—
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
123	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
124	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
125	#108	—	4	4	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	1	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	C₂×C₄×C₁₂₇	1	4	yes	—
131	C₂²⋊C₈	Q₈×C₁₂₇	1	2	yes	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	—	2	1	no	—
135	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
136	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
137	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
138	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
140	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
141	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	4	2	no	—
145	M₁₆	SD₁₆×C₁₂₇	1	4	yes	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
155	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
156	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
157	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
158	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
159	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
160	C₄²	#41×C₁₂₇	1	1	yes	characteristic
161	C₄²	#41×C₁₂₇	1	1	yes	characteristic
162	C₂³	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
167	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
168	C₂×C₄	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
169	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
171	C₂×C₄	#125×C₁₂₇	1	1	yes	characteristic
172	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
173	C₂×C₄	#135×C₁₂₇	1	2	yes	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	4	no	—
178	C₈	—	4	2	no	—
179	C₈	—	4	4	no	—
180	C₈	—	8	2	no	—
181	C₈	#108×C₁₂₇	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#389×C₁₂₇	1	1	yes	characteristic
185	C₂²	#498×C₁₂₇	1	1	yes	characteristic
186	C₂²	#499×C₁₂₇	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	C₂×#108×C₁₂₇	1	1	yes	characteristic
191	C₄	#430×C₁₂₇	1	1	yes	characteristic
192	C₄	C₄×SD₁₆×C₁₂₇	1	2	yes	—
193	C₄	#465×C₁₂₇	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	C₁₂₇×#1941	1	1	yes	characteristic
197	C₂	C₁₂₇×#1950	1	1	yes	characteristic
198	C₂	C₁₂₇×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#121
commutator	C₄²	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	#389
prime radical	C₂³×C₁₂₇	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
81. C₈×C₁₂₇	#108	semidirect
94. #9623	C₁₂₇	direct (N × H)
105. C₁₂₇	#9623	direct (N × H)
181. C₈	#108×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#1950	2
C₁₂₇×#1566	2
C₂×#334×C₁₂₇	2
C₈×M₁₆×C₁₂₇	2
C₁₂₇×#1301	2
#9623	127

Subgroups — C127×#9623 #44573296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#9623 #44573296