Subgroups — C₁₂₇×#20467 #44584140

Order 32512.

930 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	C₂×#432×C₁₂₇	C₂	1	2	yes	maximal
2	C₄×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1768	C₂	1	4	yes	maximal, complemented
4	C₁₂₇×#1750	C₂	1	4	yes	maximal
5	C₁₂₇×#1745	C₂	1	2	yes	maximal, complemented
6	C₄×#97×C₁₂₇	C₂	1	2	yes	maximal, complemented
7	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
9	#432×C₁₂₇	C₄	1	16	yes	complemented
10	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
11	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	—
12	C₂×#97×C₁₂₇	C₂²	1	4	yes	—
13	C₂×#97×C₁₂₇	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄×C₁₂₇	C₄	1	4	yes	—
16	C₄×C₂²⋊C₄×C₁₂₇	C₄	1	4	yes	—
17	#376×C₁₂₇	C₂²	1	2	yes	—
18	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
20	#336×C₁₂₇	C₂²	1	8	yes	—
21	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₁₂₇	C₂³	1	8	yes	—
24	#117×C₁₂₇	—	2	8	no	—
25	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
26	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
32	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
35	C₄⋊C₈×C₁₂₇	C₂×C₄	1	16	yes	—
36	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	16	yes	—
37	#97×C₁₂₇	—	2	8	no	—
38	#97×C₁₂₇	C₂×C₄	1	16	yes	—
39	C₄×C₈×C₁₂₇	—	2	8	no	—
40	#95×C₁₂₇	C₂×C₄	1	8	yes	—
41	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	—	2	4	no	—
43	C₂²×C₄×C₁₂₇	—	4	4	no	—
44	C₂²×C₄×C₁₂₇	C₄²	1	8	yes	—
45	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₁₂₇	—	2	8	no	—
53	C₂×C₈×C₁₂₇	—	2	8	no	—
54	C₂×C₈×C₁₂₇	C₂²×C₄	1	16	yes	—
55	C₄⋊C₄×C₁₂₇	—	2	8	no	—
56	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
57	C₄²×C₁₂₇	—	2	4	no	—
58	C₄²×C₁₂₇	—	4	4	no	—
59	C₄²×C₁₂₇	—	4	4	no	—
60	C₄²×C₁₂₇	C₄²	1	8	yes	—
61	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
62	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
63	C₄²×C₁₂₇	#41	1	8	yes	—
64	C₂³×C₁₂₇	—	2	2	no	—
65	C₂³×C₁₂₇	—	2	4	no	—
66	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
67	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	—	2	2	no	—
69	C₂×C₄×C₁₂₇	—	2	4	no	—
70	C₂×C₄×C₁₂₇	—	2	8	no	—
71	C₂×C₄×C₁₂₇	—	2	4	no	—
72	C₂×C₄×C₁₂₇	—	2	8	no	—
73	C₂×C₄×C₁₂₇	—	2	4	no	—
74	C₂×C₄×C₁₂₇	—	2	8	no	—
75	C₂×C₄×C₁₂₇	—	4	8	no	—
76	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
77	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
78	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
79	C₂×C₄×C₁₂₇	#117	1	8	yes	—
80	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
81	C₂×C₄×C₁₂₇	C₂×#41	1	4	yes	—
82	C₈×C₁₂₇	—	2	16	no	—
83	C₂²×C₁₂₇	—	2	4	no	—
84	C₂²×C₁₂₇	—	2	2	no	—
85	C₂²×C₁₂₇	—	4	4	no	—
86	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
87	C₂²×C₁₂₇	C₂×#117	1	2	yes	—
88	C₂²×C₁₂₇	#568	1	4	yes	—
89	C₄×C₁₂₇	—	2	4	no	—
90	C₄×C₁₂₇	—	2	4	no	—
91	C₄×C₁₂₇	—	2	8	no	—
92	C₄×C₁₂₇	—	2	2	no	—
93	C₄×C₁₂₇	—	4	4	no	—
94	C₄×C₁₂₇	C₄×#41	1	4	yes	—
95	#20467	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₁₂₇	—	4	2	no	—
97	C₂×C₁₂₇	C₄×#117	1	1	yes	characteristic
98	C₂×C₁₂₇	#2873	1	2	yes	—
99	C₂×C₁₂₇	#2923	1	4	yes	—
100	C₂×#432	C₂×C₁₂₇	1	2	yes	—
101	C₄×#117	C₂×C₁₂₇	1	1	yes	characteristic
102	#1768	C₂×C₁₂₇	1	4	yes	complemented
103	#1750	C₂×C₁₂₇	1	4	yes	—
104	#1745	C₂×C₁₂₇	1	2	yes	complemented
105	C₄×#97	C₂×C₁₂₇	1	2	yes	complemented
106	C₁₂₇	#20467	1	1	yes	characteristic, direct product factor, Sylow
107	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
108	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
109	#432	C₄×C₁₂₇	1	16	yes	complemented
110	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	—
111	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	4	yes	—
112	C₂×#97	C₂²×C₁₂₇	1	4	yes	—
113	C₂×#97	C₂²×C₁₂₇	1	4	yes	—
114	C₂×C₄×C₈	C₂²×C₁₂₇	1	4	yes	—
115	C₄×C₄⋊C₄	C₄×C₁₂₇	1	4	yes	—
116	C₄×C₂²⋊C₄	C₄×C₁₂₇	1	4	yes	—
117	#376	C₂²×C₁₂₇	1	2	yes	—
118	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
119	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
120	#336	C₂²×C₁₂₇	1	8	yes	—
121	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
122	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
123	C₂²×C₈	C₂³×C₁₂₇	1	8	yes	—
124	#117	—	2	8	no	—
125	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
126	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
127	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
128	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
129	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
130	C₂×C₄²	C₂×C₄×C₁₂₇	1	8	yes	—
131	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
132	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
133	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
134	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
135	C₄⋊C₈	C₂×C₄×C₁₂₇	1	16	yes	—
136	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	16	yes	—
137	#97	—	2	8	no	—
138	#97	C₂×C₄×C₁₂₇	1	16	yes	—
139	C₄×C₈	—	2	8	no	—
140	#95	C₂×C₄×C₁₂₇	1	8	yes	—
141	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	4	4	no	—
144	C₂²×C₄	C₄²×C₁₂₇	1	8	yes	—
145	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
146	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
147	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
148	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
150	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
151	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	2	8	no	—
154	C₂×C₈	C₂²×C₄×C₁₂₇	1	16	yes	—
155	C₄⋊C₄	—	2	8	no	—
156	C₂²⋊C₄	—	2	8	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	4	4	no	—
159	C₄²	—	4	4	no	—
160	C₄²	C₄²×C₁₂₇	1	8	yes	—
161	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
162	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
163	C₄²	#41×C₁₂₇	1	8	yes	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	4	no	—
166	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
167	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	4	8	no	—
176	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
177	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
178	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
179	C₂×C₄	#117×C₁₂₇	1	8	yes	—
180	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
181	C₂×C₄	C₂×#41×C₁₂₇	1	4	yes	—
182	C₈	—	2	16	no	—
183	C₂²	—	2	4	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	4	no	—
186	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
187	C₂²	C₂×#117×C₁₂₇	1	2	yes	—
188	C₂²	#568×C₁₂₇	1	4	yes	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	2	8	no	—
192	C₄	—	2	2	no	—
193	C₄	—	4	4	no	—
194	C₄	C₄×#41×C₁₂₇	1	4	yes	—
195	C₂	—	4	2	no	—
196	C₂	C₄×#117×C₁₂₇	1	1	yes	characteristic
197	C₂	C₁₂₇×#2873	1	2	yes	—
198	C₂	C₁₂₇×#2923	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂³×C₄
prime radical	C₂⁴×C₁₂₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1768	C₂	semidirect
5. C₁₂₇×#1745	C₂	semidirect
6. C₄×#97×C₁₂₇	C₂	semidirect
9. #432×C₁₂₇	C₄	semidirect
95. #20467	C₁₂₇	direct (N × H)
102. #1768	C₂×C₁₂₇	semidirect
104. #1745	C₂×C₁₂₇	semidirect
105. C₄×#97	C₂×C₁₂₇	semidirect
106. C₁₂₇	#20467	direct (N × H)
109. #432	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₁₂₇	2
C₄×#117×C₁₂₇	2
C₁₂₇×#1768	2
C₁₂₇×#1750	2
C₁₂₇×#1745	2
C₄×#97×C₁₂₇	2
#20467	127

Subgroups — C127×#20467 #44584140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20467 #44584140