Subgroups — C₁₂₇×#10693 #44574366

Order 32512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#336×C₁₂₇	C₂	1	4	yes	maximal
4	C₂³×C₈×C₁₂₇	C₂²	1	4	yes	—
5	C₂²×C₄²×C₁₂₇	C₄	1	4	yes	—
6	C₂²×C₄²×C₁₂₇	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#97×C₁₂₇	—	2	4	no	—
9	C₂×#97×C₁₂₇	—	2	4	no	—
10	C₂×#97×C₁₂₇	C₄	1	8	yes	complemented
11	C₄³×C₁₂₇	—	2	4	no	—
12	#336×C₁₂₇	C₄	1	16	yes	complemented
13	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
14	C₂³×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
15	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
16	C₂²×C₈×C₁₂₇	—	2	4	no	—
17	C₂²×C₈×C₁₂₇	—	2	4	no	—
18	C₂²×C₈×C₁₂₇	—	2	8	no	—
19	C₂²×C₈×C₁₂₇	C₂×C₄	1	16	yes	—
20	C₂²×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
21	C₂×C₄²×C₁₂₇	—	2	8	no	—
22	C₂×C₄²×C₁₂₇	—	2	16	no	—
23	C₂×C₄²×C₁₂₇	—	2	4	no	—
24	C₂×C₄²×C₁₂₇	—	2	4	no	—
25	C₂×C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
28	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
29	C₂×C₄²×C₁₂₇	Q₈	1	2	yes	—
30	#97×C₁₂₇	—	2	16	no	—
31	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	—	2	4	no	—
33	C₂²×C₄×C₁₂₇	—	2	4	no	—
34	C₂²×C₄×C₁₂₇	—	2	8	no	—
35	C₂²×C₄×C₁₂₇	—	2	8	no	—
36	C₂²×C₄×C₁₂₇	C₄²	1	4	yes	—
37	C₂²×C₄×C₁₂₇	C₄²	1	8	yes	—
38	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₁₂₇	M₁₆	1	16	yes	—
41	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
47	C₂×C₈×C₁₂₇	—	2	8	no	—
48	C₂×C₈×C₁₂₇	—	2	8	no	—
49	C₂×C₈×C₁₂₇	—	2	16	no	—
50	C₂×C₈×C₁₂₇	—	2	16	no	—
51	C₂×C₈×C₁₂₇	C₄²	1	16	yes	complemented
52	C₄²×C₁₂₇	—	2	32	no	—
53	C₄²×C₁₂₇	—	2	16	no	—
54	C₄²×C₁₂₇	C₂²⋊C₄	1	8	yes	—
55	C₄²×C₁₂₇	C₄⋊C₄	1	8	yes	—
56	C₂³×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
57	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
58	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂³×C₁₂₇	#117	1	2	yes	—
61	C₂³×C₁₂₇	C₂×M₁₆	1	8	yes	—
62	C₂×C₄×C₁₂₇	—	2	8	no	—
63	C₂×C₄×C₁₂₇	—	2	8	no	—
64	C₂×C₄×C₁₂₇	—	2	8	no	—
65	C₂×C₄×C₁₂₇	—	2	8	no	—
66	C₂×C₄×C₁₂₇	—	2	16	no	—
67	C₂×C₄×C₁₂₇	—	2	8	no	—
68	C₂×C₄×C₁₂₇	—	2	16	no	—
69	C₂×C₄×C₁₂₇	#95	1	8	yes	—
70	C₂×C₄×C₁₂₇	#97	1	16	yes	complemented
71	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
72	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₁₂₇	#117	1	4	yes	—
75	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
76	C₂×C₄×C₁₂₇	C₄×D₄	1	8	yes	—
77	C₂×C₄×C₁₂₇	C₄×Q₈	1	4	yes	—
78	C₈×C₁₂₇	—	2	16	no	—
79	C₂²×C₁₂₇	C₄³	1	1	yes	characteristic
80	C₂²×C₁₂₇	C₂×#95	1	1	yes	characteristic
81	C₂²×C₁₂₇	#376	1	1	yes	characteristic
82	C₂²×C₁₂₇	C₄×C₂²⋊C₄	1	2	yes	—
83	C₂²×C₁₂₇	C₄×C₄⋊C₄	1	2	yes	—
84	C₂²×C₁₂₇	C₂×#97	1	4	yes	—
85	C₂²×C₁₂₇	C₄×M₁₆	1	4	yes	—
86	C₂²×C₁₂₇	C₄×M₁₆	1	4	yes	—
87	C₂²×C₁₂₇	#407	1	4	yes	—
88	C₂²×C₁₂₇	#423	1	4	yes	—
89	C₂²×C₁₂₇	#432	1	8	yes	—
90	C₄×C₁₂₇	—	2	8	no	—
91	C₄×C₁₂₇	—	2	16	no	—
92	C₄×C₁₂₇	C₄×C₂²⋊C₄	1	4	yes	—
93	C₄×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
94	#10693	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₁₂₇	C₄×#95	1	1	yes	characteristic
96	C₂×C₁₂₇	C₄×#97	1	2	yes	—
97	C₂×C₁₂₇	#1728	1	2	yes	—
98	C₂×C₁₂₇	#1745	1	2	yes	—
99	C₂×C₁₂₇	#1749	1	4	yes	—
100	C₂×C₁₂₇	#1767	1	4	yes	—
101	C₂²×#97	C₂×C₁₂₇	1	2	yes	—
102	C₂×C₄³	C₂×C₁₂₇	1	1	yes	characteristic
103	C₂×#336	C₂×C₁₂₇	1	4	yes	—
104	C₁₂₇	#10693	1	1	yes	characteristic, direct product factor, Sylow
105	C₂³×C₈	C₂²×C₁₂₇	1	4	yes	—
106	C₂²×C₄²	C₄×C₁₂₇	1	4	yes	—
107	C₂²×C₄²	C₂²×C₁₂₇	1	2	yes	—
108	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
109	C₂×#97	—	2	4	no	—
110	C₂×#97	—	2	4	no	—
111	C₂×#97	C₄×C₁₂₇	1	8	yes	complemented
112	C₄³	—	2	4	no	—
113	#336	C₄×C₁₂₇	1	16	yes	complemented
114	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
115	C₂³×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
116	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	—	2	4	no	—
119	C₂²×C₈	—	2	8	no	—
120	C₂²×C₈	C₂×C₄×C₁₂₇	1	16	yes	—
121	C₂²×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
122	C₂×C₄²	—	2	8	no	—
123	C₂×C₄²	—	2	16	no	—
124	C₂×C₄²	—	2	4	no	—
125	C₂×C₄²	—	2	4	no	—
126	C₂×C₄²	C₂×C₄×C₁₂₇	1	8	yes	—
127	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
128	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
129	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
130	C₂×C₄²	Q₈×C₁₂₇	1	2	yes	—
131	#97	—	2	16	no	—
132	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	C₄²×C₁₂₇	1	4	yes	—
138	C₂²×C₄	C₄²×C₁₂₇	1	8	yes	—
139	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
140	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
141	C₂²×C₄	M₁₆×C₁₂₇	1	16	yes	—
142	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
143	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
144	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
146	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
147	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	2	16	no	—
151	C₂×C₈	—	2	16	no	—
152	C₂×C₈	C₄²×C₁₂₇	1	16	yes	complemented
153	C₄²	—	2	32	no	—
154	C₄²	—	2	16	no	—
155	C₄²	C₂²⋊C₄×C₁₂₇	1	8	yes	—
156	C₄²	C₄⋊C₄×C₁₂₇	1	8	yes	—
157	C₂³	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
158	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
159	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
160	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
161	C₂³	#117×C₁₂₇	1	2	yes	—
162	C₂³	C₂×M₁₆×C₁₂₇	1	8	yes	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	8	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	16	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	#95×C₁₂₇	1	8	yes	—
171	C₂×C₄	#97×C₁₂₇	1	16	yes	complemented
172	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
173	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
175	C₂×C₄	#117×C₁₂₇	1	4	yes	—
176	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
177	C₂×C₄	C₄×D₄×C₁₂₇	1	8	yes	—
178	C₂×C₄	C₄×Q₈×C₁₂₇	1	4	yes	—
179	C₈	—	2	16	no	—
180	C₂²	C₄³×C₁₂₇	1	1	yes	characteristic
181	C₂²	C₂×#95×C₁₂₇	1	1	yes	characteristic
182	C₂²	#376×C₁₂₇	1	1	yes	characteristic
183	C₂²	C₄×C₂²⋊C₄×C₁₂₇	1	2	yes	—
184	C₂²	C₄×C₄⋊C₄×C₁₂₇	1	2	yes	—
185	C₂²	C₂×#97×C₁₂₇	1	4	yes	—
186	C₂²	C₄×M₁₆×C₁₂₇	1	4	yes	—
187	C₂²	C₄×M₁₆×C₁₂₇	1	4	yes	—
188	C₂²	#407×C₁₂₇	1	4	yes	—
189	C₂²	#423×C₁₂₇	1	4	yes	—
190	C₂²	#432×C₁₂₇	1	8	yes	—
191	C₄	—	2	8	no	—
192	C₄	—	2	16	no	—
193	C₄	C₄×C₂²⋊C₄×C₁₂₇	1	4	yes	—
194	C₄	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
195	C₂	C₄×#95×C₁₂₇	1	1	yes	characteristic
196	C₂	C₄×#97×C₁₂₇	1	2	yes	—
197	C₂	C₁₂₇×#1728	1	2	yes	—
198	C₂	C₁₂₇×#1745	1	2	yes	—
199	C₂	C₁₂₇×#1749	1	4	yes	—
200	C₂	C₁₂₇×#1767	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₂₇	C₂³
commutator	C₂²	C₄³×C₁₂₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₂₇
socle, prime radical	C₂⁴×C₁₂₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂×#97×C₁₂₇	C₄	semidirect
12. #336×C₁₂₇	C₄	semidirect
51. C₂×C₈×C₁₂₇	C₄²	semidirect
70. C₂×C₄×C₁₂₇	#97	semidirect
94. #10693	C₁₂₇	direct (N × H)
104. C₁₂₇	#10693	direct (N × H)
111. C₂×#97	C₄×C₁₂₇	semidirect
113. #336	C₄×C₁₂₇	semidirect
152. C₂×C₈	C₄²×C₁₂₇	semidirect
171. C₂×C₄	#97×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#97×C₁₂₇	2
C₂×C₄³×C₁₂₇	2
C₂×#336×C₁₂₇	2
#10693	127

Subgroups — C127×#10693 #44574366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#10693 #44574366