Subgroups — C₁₂₇×#12671 #44576344

Order 32512.

310 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#368×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1772	C₂	1	2	yes	maximal
5	C₁₂₇×#1382	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
9	#427×C₁₂₇	C₂²	1	2	yes	—
10	#405×C₁₂₇	—	2	2	no	—
11	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	#368×C₁₂₇	C₄	1	4	yes	complemented
13	#367×C₁₂₇	C₄	1	2	yes	complemented
14	#366×C₁₂₇	C₄	1	2	yes	complemented
15	#346×C₁₂₇	C₄	1	4	yes	complemented
16	#339×C₁₂₇	—	2	2	no	—
17	C₂×M₁₆×C₁₂₇	—	2	2	no	—
18	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
20	#117×C₁₂₇	—	2	2	no	—
21	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₂₇	—	2	1	no	—
23	C₂×C₁₆×C₁₂₇	—	2	1	no	—
24	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
29	#108×C₁₂₇	—	2	2	no	—
30	#108×C₁₂₇	C₂×C₄	1	4	yes	—
31	#107×C₁₂₇	C₂×C₄	1	4	yes	—
32	#106×C₁₂₇	—	2	2	no	—
33	#97×C₁₂₇	—	4	2	no	—
34	C₄×C₈×C₁₂₇	—	4	2	no	—
35	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₂₇	Q₈	1	2	yes	—
37	C₂²×C₄×C₁₂₇	—	2	1	no	—
38	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	M₁₆×C₁₂₇	—	2	4	no	—
40	M₁₆×C₁₂₇	—	4	2	no	—
41	C₂×C₈×C₁₂₇	—	4	2	no	—
42	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
44	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
45	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
51	C₄⋊C₄×C₁₂₇	—	2	4	no	—
52	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
53	C₄²×C₁₂₇	—	2	1	no	—
54	C₄²×C₁₂₇	—	4	2	no	—
55	C₄²×C₁₂₇	C₄⋊C₄	1	2	yes	—
56	C₁₆×C₁₂₇	—	2	2	no	—
57	C₁₆×C₁₂₇	C₄²	1	2	yes	—
58	C₁₆×C₁₂₇	C₄²	1	2	yes	—
59	C₂³×C₁₂₇	#135	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	—	2	2	no	—
61	C₂×C₄×C₁₂₇	—	4	2	no	—
62	C₂×C₄×C₁₂₇	—	4	4	no	—
63	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
69	C₈×C₁₂₇	—	4	4	no	—
70	C₈×C₁₂₇	#107	1	4	yes	—
71	C₈×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
72	C₈×C₁₂₇	#117	1	1	yes	characteristic
73	C₈×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
74	C₈×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
75	C₂²×C₁₂₇	—	2	2	no	—
76	C₂²×C₁₂₇	#427	1	1	yes	characteristic
77	C₂²×C₁₂₇	C₄×D₈	1	1	yes	characteristic
78	C₂²×C₁₂₇	C₄×Q₁₆	1	1	yes	characteristic
79	C₄×C₁₂₇	—	4	2	no	—
80	C₄×C₁₂₇	—	8	4	no	—
81	C₄×C₁₂₇	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₄×C₁₂₇	C₂×#107	1	1	yes	characteristic
83	C₄×C₁₂₇	C₄×D₈	1	1	yes	characteristic
84	C₄×C₁₂₇	C₄×Q₁₆	1	1	yes	characteristic
85	#12671	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂×C₁₂₇	—	2	1	no	—
87	C₂×C₁₂₇	—	2	1	no	—
88	C₂×C₁₂₇	C₄×#107	1	1	yes	characteristic
89	C₂×C₁₂₇	#2179	1	2	yes	—
90	C₂×#368	C₂×C₁₂₇	1	1	yes	characteristic
91	#2158	C₂×C₁₂₇	1	1	yes	characteristic
92	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic
93	#1772	C₂×C₁₂₇	1	2	yes	—
94	#1382	C₂×C₁₂₇	1	2	yes	—
95	C₁₂₇	#12671	1	1	yes	characteristic, direct product factor, Sylow
96	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
97	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
98	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
99	#427	C₂²×C₁₂₇	1	2	yes	—
100	#405	—	2	2	no	—
101	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
102	#368	C₄×C₁₂₇	1	4	yes	complemented
103	#367	C₄×C₁₂₇	1	2	yes	complemented
104	#366	C₄×C₁₂₇	1	2	yes	complemented
105	#346	C₄×C₁₂₇	1	4	yes	complemented
106	#339	—	2	2	no	—
107	C₂×M₁₆	—	2	2	no	—
108	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
109	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
110	#117	—	2	2	no	—
111	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
115	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
116	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
117	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
118	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
119	#108	—	2	2	no	—
120	#108	C₂×C₄×C₁₂₇	1	4	yes	—
121	#107	C₂×C₄×C₁₂₇	1	4	yes	—
122	#106	—	2	2	no	—
123	#97	—	4	2	no	—
124	C₄×C₈	—	4	2	no	—
125	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
126	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
129	M₁₆	—	2	4	no	—
130	M₁₆	—	4	2	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
133	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
134	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
135	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
136	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
137	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
138	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
139	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
140	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
141	C₄⋊C₄	—	2	4	no	—
142	C₂²⋊C₄	—	4	2	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	4	2	no	—
145	C₄²	C₄⋊C₄×C₁₂₇	1	2	yes	—
146	C₁₆	—	2	2	no	—
147	C₁₆	C₄²×C₁₂₇	1	2	yes	—
148	C₁₆	C₄²×C₁₂₇	1	2	yes	—
149	C₂³	#135×C₁₂₇	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
154	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
155	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
156	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
157	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
158	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
159	C₈	—	4	4	no	—
160	C₈	#107×C₁₂₇	1	4	yes	—
161	C₈	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
162	C₈	#117×C₁₂₇	1	1	yes	characteristic
163	C₈	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
164	C₈	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
165	C₂²	—	2	2	no	—
166	C₂²	#427×C₁₂₇	1	1	yes	characteristic
167	C₂²	C₄×D₈×C₁₂₇	1	1	yes	characteristic
168	C₂²	C₄×Q₁₆×C₁₂₇	1	1	yes	characteristic
169	C₄	—	4	2	no	—
170	C₄	—	8	4	no	—
171	C₄	C₄×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
172	C₄	C₂×#107×C₁₂₇	1	1	yes	characteristic
173	C₄	C₄×D₈×C₁₂₇	1	1	yes	characteristic
174	C₄	C₄×Q₁₆×C₁₂₇	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	C₄×#107×C₁₂₇	1	1	yes	characteristic
178	C₂	C₁₂₇×#2179	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₈
commutator	C₈	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₄×D₈
prime radical	C₂³×C₁₂₇	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
12. #368×C₁₂₇	C₄	semidirect
13. #367×C₁₂₇	C₄	semidirect
14. #366×C₁₂₇	C₄	semidirect
15. #346×C₁₂₇	C₄	semidirect
85. #12671	C₁₂₇	direct (N × H)
95. C₁₂₇	#12671	direct (N × H)
102. #368	C₄×C₁₂₇	semidirect
103. #367	C₄×C₁₂₇	semidirect
104. #366	C₄×C₁₂₇	semidirect
105. #346	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×#368×C₁₂₇	2
C₁₂₇×#2158	2
C₂×#346×C₁₂₇	2
C₁₂₇×#1772	2
C₁₂₇×#1382	2
#12671	127

Subgroups — C127×#12671 #44576344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12671 #44576344