Subgroups — C₁₂₇×#12657 #44576330

Order 32512.

334 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2150	C₂	1	2	yes	maximal
2	C₂×#347×C₁₂₇	C₂	1	2	yes	maximal
3	C₁₂₇×#1726	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1375	C₂	1	2	yes	maximal
5	C₂×M₃₂×C₁₂₇	C₂²	1	4	yes	—
6	C₂×#97×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	#376×C₁₂₇	C₄	1	1	yes	characteristic
13	C₄⋊C₁₆×C₁₂₇	C₄	1	8	yes	complemented
14	#347×C₁₂₇	—	2	4	no	—
15	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	M₃₂×C₁₂₇	—	2	4	no	—
23	M₃₂×C₁₂₇	—	2	4	no	—
24	C₂×C₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
25	#97×C₁₂₇	—	2	2	no	—
26	#97×C₁₂₇	—	2	2	no	—
27	#97×C₁₂₇	D₄	1	2	yes	—
28	#97×C₁₂₇	Q₈	1	2	yes	—
29	C₄×C₈×C₁₂₇	—	2	2	no	—
30	C₄×C₈×C₁₂₇	—	2	2	no	—
31	C₄×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
32	#95×C₁₂₇	—	2	2	no	—
33	C₂²×C₄×C₁₂₇	—	2	1	no	—
34	C₂²×C₄×C₁₂₇	—	2	1	no	—
35	C₂²×C₄×C₁₂₇	M₁₆	1	2	yes	—
36	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	—	2	1	no	—
38	C₂×C₈×C₁₂₇	—	2	1	no	—
39	C₂×C₈×C₁₂₇	—	2	1	no	—
40	C₂×C₈×C₁₂₇	—	2	1	no	—
41	C₂×C₈×C₁₂₇	—	2	1	no	—
42	C₂×C₈×C₁₂₇	—	2	1	no	—
43	C₂×C₈×C₁₂₇	C₄²	1	2	yes	—
44	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
46	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	#41	1	2	yes	—
50	C₄²×C₁₂₇	—	2	1	no	—
51	C₄²×C₁₂₇	—	2	1	no	—
52	C₄²×C₁₂₇	C₄²	1	2	yes	—
53	C₄²×C₁₂₇	C₄⋊C₄	1	2	yes	—
54	C₄²×C₁₂₇	M₁₆	1	2	yes	—
55	C₄²×C₁₂₇	M₁₆	1	2	yes	—
56	C₁₆×C₁₂₇	—	2	8	no	—
57	C₂³×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	—	2	2	no	—
59	C₂×C₄×C₁₂₇	—	2	2	no	—
60	C₂×C₄×C₁₂₇	—	4	2	no	—
61	C₂×C₄×C₁₂₇	—	4	2	no	—
62	C₂×C₄×C₁₂₇	—	4	2	no	—
63	C₂×C₄×C₁₂₇	#97	1	4	yes	—
64	C₂×C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	#117	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₁₂₇	—	2	2	no	—
71	C₈×C₁₂₇	—	2	2	no	—
72	C₈×C₁₂₇	—	2	2	no	—
73	C₈×C₁₂₇	C₄×D₄	1	2	yes	—
74	C₈×C₁₂₇	C₄×Q₈	1	2	yes	—
75	C₂²×C₁₂₇	—	2	2	no	—
76	C₂²×C₁₂₇	C₂×#97	1	1	yes	characteristic
77	C₂²×C₁₂₇	#423	1	1	yes	characteristic
78	C₂²×C₁₂₇	#432	1	1	yes	characteristic
79	C₄×C₁₂₇	—	2	4	no	—
80	C₄×C₁₂₇	—	4	2	no	—
81	C₄×C₁₂₇	—	4	2	no	—
82	C₄×C₁₂₇	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₄×C₁₂₇	C₄×M₁₆	1	1	yes	characteristic
84	C₄×C₁₂₇	#423	1	1	yes	characteristic
85	C₄×C₁₂₇	#432	1	1	yes	characteristic
86	#12657	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂×C₁₂₇	—	2	1	no	—
88	C₂×C₁₂₇	—	2	1	no	—
89	C₂×C₁₂₇	#1767	1	1	yes	characteristic
90	C₂×C₁₂₇	#2110	1	2	yes	—
91	#2150	C₂×C₁₂₇	1	2	yes	—
92	C₂×#347	C₂×C₁₂₇	1	2	yes	—
93	#1726	C₂×C₁₂₇	1	1	yes	characteristic
94	#1375	C₂×C₁₂₇	1	2	yes	—
95	C₁₂₇	#12657	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×M₃₂	C₂²×C₁₂₇	1	4	yes	—
97	C₂×#97	C₄×C₁₂₇	1	1	yes	characteristic
98	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
99	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
100	C₂×C₄×C₈	C₄×C₁₂₇	1	1	yes	characteristic
101	C₂×C₄×C₈	C₄×C₁₂₇	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
103	#376	C₄×C₁₂₇	1	1	yes	characteristic
104	C₄⋊C₁₆	C₄×C₁₂₇	1	8	yes	complemented
105	#347	—	2	4	no	—
106	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
109	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
112	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
113	M₃₂	—	2	4	no	—
114	M₃₂	—	2	4	no	—
115	C₂×C₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
116	#97	—	2	2	no	—
117	#97	—	2	2	no	—
118	#97	D₄×C₁₂₇	1	2	yes	—
119	#97	Q₈×C₁₂₇	1	2	yes	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
123	#95	—	2	2	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	M₁₆×C₁₂₇	1	2	yes	—
127	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	C₄²×C₁₂₇	1	2	yes	—
135	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
136	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
137	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
138	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
139	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
140	C₂×C₈	#41×C₁₂₇	1	2	yes	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	C₄²×C₁₂₇	1	2	yes	—
144	C₄²	C₄⋊C₄×C₁₂₇	1	2	yes	—
145	C₄²	M₁₆×C₁₂₇	1	2	yes	—
146	C₄²	M₁₆×C₁₂₇	1	2	yes	—
147	C₁₆	—	2	8	no	—
148	C₂³	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	#97×C₁₂₇	1	4	yes	—
155	C₂×C₄	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
157	C₂×C₄	#117×C₁₂₇	1	1	yes	characteristic
158	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
159	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
160	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	2	2	no	—
164	C₈	C₄×D₄×C₁₂₇	1	2	yes	—
165	C₈	C₄×Q₈×C₁₂₇	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	C₂×#97×C₁₂₇	1	1	yes	characteristic
168	C₂²	#423×C₁₂₇	1	1	yes	characteristic
169	C₂²	#432×C₁₂₇	1	1	yes	characteristic
170	C₄	—	2	4	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	C₄×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
174	C₄	C₄×M₁₆×C₁₂₇	1	1	yes	characteristic
175	C₄	#423×C₁₂₇	1	1	yes	characteristic
176	C₄	#432×C₁₂₇	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₁₂₇×#1767	1	1	yes	characteristic
180	C₂	C₁₂₇×#2110	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#97
prime radical	C₂³×C₁₂₇	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄⋊C₁₆×C₁₂₇	C₄	semidirect
86. #12657	C₁₂₇	direct (N × H)
95. C₁₂₇	#12657	direct (N × H)
104. C₄⋊C₁₆	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×#2150	2
C₂×#347×C₁₂₇	2
C₁₂₇×#1726	2
C₁₂₇×#1375	2
#12657	127

Subgroups — C127×#12657 #44576330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12657 #44576330