Subgroups — C₁₂₇×#13170 #44576843

Order 32512.

278 conjugacy classes of subgroups.

The subgroup lattice has 178 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×#1945	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	#424×C₁₂₇	—	2	2	no	—
11	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	#368×C₁₂₇	—	2	2	no	—
13	#367×C₁₂₇	C₄	1	2	yes	complemented
14	#366×C₁₂₇	C₄	1	2	yes	complemented
15	#346×C₁₂₇	—	2	2	no	—
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	1	yes	characteristic
25	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
27	C₂×C₁₆×C₁₂₇	Q₈	1	2	yes	—
28	C₂×C₁₆×C₁₂₇	Q₈	1	2	yes	—
29	#108×C₁₂₇	—	2	2	no	—
30	#108×C₁₂₇	—	2	2	no	—
31	#107×C₁₂₇	C₂×C₄	1	4	yes	—
32	#106×C₁₂₇	—	2	2	no	—
33	C₄⋊C₈×C₁₂₇	—	4	4	no	—
34	C₄×C₈×C₁₂₇	—	2	1	no	—
35	C₄×C₈×C₁₂₇	—	2	1	no	—
36	C₂²×C₄×C₁₂₇	—	2	1	no	—
37	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₁₂₇	—	4	2	no	—
39	M₁₆×C₁₂₇	—	4	2	no	—
40	C₂×C₈×C₁₂₇	—	2	1	no	—
41	C₂×C₈×C₁₂₇	—	2	1	no	—
42	C₂×C₈×C₁₂₇	—	4	2	no	—
43	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
44	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₁₂₇	—	2	4	no	—
51	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
52	C₄²×C₁₂₇	—	2	1	no	—
53	C₄²×C₁₂₇	—	4	2	no	—
54	C₄²×C₁₂₇	Q₁₆	1	2	yes	—
55	C₁₆×C₁₂₇	—	2	2	no	—
56	C₁₆×C₁₂₇	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₁₂₇	C₄⋊C₄	1	2	yes	—
58	C₂³×C₁₂₇	#135	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	—	2	2	no	—
60	C₂×C₄×C₁₂₇	—	4	2	no	—
61	C₂×C₄×C₁₂₇	—	4	4	no	—
62	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
68	C₈×C₁₂₇	—	2	2	no	—
69	C₈×C₁₂₇	—	8	2	no	—
70	C₈×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₈×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
72	C₈×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
73	C₈×C₁₂₇	#125	1	1	yes	characteristic
74	C₂²×C₁₂₇	—	2	2	no	—
75	C₂²×C₁₂₇	C₄×D₈	1	1	yes	characteristic
76	C₂²×C₁₂₇	#467	1	1	yes	characteristic
77	C₂²×C₁₂₇	#499	1	1	yes	characteristic
78	C₄×C₁₂₇	—	4	2	no	—
79	C₄×C₁₂₇	—	8	4	no	—
80	C₄×C₁₂₇	#389	1	1	yes	characteristic
81	C₄×C₁₂₇	C₄×Q₁₆	1	1	yes	characteristic
82	C₄×C₁₂₇	#466	1	1	yes	characteristic
83	C₄×C₁₂₇	#500	1	1	yes	characteristic
84	#13170	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂×C₁₂₇	—	2	1	no	—
86	C₂×C₁₂₇	—	2	1	no	—
87	C₂×C₁₂₇	#1942	1	1	yes	characteristic
88	C₂×C₁₂₇	#2179	1	2	yes	—
89	C₂×#368	C₂×C₁₂₇	1	1	yes	characteristic
90	#2158	C₂×C₁₂₇	1	1	yes	characteristic
91	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic
92	#1945	C₂×C₁₂₇	1	2	yes	—
93	#1382	C₂×C₁₂₇	1	2	yes	—
94	C₁₂₇	#13170	1	1	yes	characteristic, direct product factor, Sylow
95	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
96	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
97	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
98	#427	C₂²×C₁₂₇	1	2	yes	—
99	#424	—	2	2	no	—
100	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
101	#368	—	2	2	no	—
102	#367	C₄×C₁₂₇	1	2	yes	complemented
103	#366	C₄×C₁₂₇	1	2	yes	complemented
104	#346	—	2	2	no	—
105	#339	—	2	2	no	—
106	C₂×M₁₆	—	2	2	no	—
107	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
108	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
109	#117	—	2	2	no	—
110	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
114	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
115	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
116	C₂×C₁₆	Q₈×C₁₂₇	1	2	yes	—
117	C₂×C₁₆	Q₈×C₁₂₇	1	2	yes	—
118	#108	—	2	2	no	—
119	#108	—	2	2	no	—
120	#107	C₂×C₄×C₁₂₇	1	4	yes	—
121	#106	—	2	2	no	—
122	C₄⋊C₈	—	4	4	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
127	M₁₆	—	4	2	no	—
128	M₁₆	—	4	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
133	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
134	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
135	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
136	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
137	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
138	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
139	C₄⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	4	2	no	—
143	C₄²	Q₁₆×C₁₂₇	1	2	yes	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₄⋊C₄×C₁₂₇	1	2	yes	—
146	C₁₆	C₄⋊C₄×C₁₂₇	1	2	yes	—
147	C₂³	#135×C₁₂₇	1	1	yes	characteristic
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
152	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
153	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
154	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
155	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
156	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	8	2	no	—
159	C₈	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
160	C₈	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
161	C₈	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
162	C₈	#125×C₁₂₇	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	C₄×D₈×C₁₂₇	1	1	yes	characteristic
165	C₂²	#467×C₁₂₇	1	1	yes	characteristic
166	C₂²	#499×C₁₂₇	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	—	8	4	no	—
169	C₄	#389×C₁₂₇	1	1	yes	characteristic
170	C₄	C₄×Q₁₆×C₁₂₇	1	1	yes	characteristic
171	C₄	#466×C₁₂₇	1	1	yes	characteristic
172	C₄	#500×C₁₂₇	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	C₁₂₇×#1942	1	1	yes	characteristic
176	C₂	C₁₂₇×#2179	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₄×D₈
prime radical	C₂³×C₁₂₇	#135
LCS	G ▷ C₂×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
13. #367×C₁₂₇	C₄	semidirect
14. #366×C₁₂₇	C₄	semidirect
84. #13170	C₁₂₇	direct (N × H)
94. C₁₂₇	#13170	direct (N × H)
102. #367	C₄×C₁₂₇	semidirect
103. #366	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×#1945	2
C₁₂₇×#1382	2
#13170	127

Subgroups — C127×#13170 #44576843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13170 #44576843