Subgroups — C₁₂₇×#13171 #44576844

Order 32512.

274 conjugacy classes of subgroups.

The subgroup lattice has 192 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	C₂×#366×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2108	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1952	C₂	1	2	yes	maximal
5	C₁₂₇×#1387	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
9	#428×C₁₂₇	—	2	2	no	—
10	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
11	#405×C₁₂₇	C₂²	1	1	yes	characteristic
12	#368×C₁₂₇	C₄	1	4	yes	complemented
13	#366×C₁₂₇	—	2	1	no	—
14	#366×C₁₂₇	—	2	1	no	—
15	#346×C₁₂₇	—	2	1	no	—
16	C₄×C₁₆×C₁₂₇	—	2	1	no	—
17	#343×C₁₂₇	—	2	2	no	—
18	C₂×M₁₆×C₁₂₇	—	2	2	no	—
19	C₂×M₁₆×C₁₂₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
21	#117×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
23	M₃₂×C₁₂₇	—	2	1	no	—
24	M₃₂×C₁₂₇	Q₈	1	2	yes	—
25	C₂×C₁₆×C₁₂₇	—	2	1	no	—
26	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₂₇	Q₈	1	2	yes	—
30	#108×C₁₂₇	—	2	2	no	—
31	#108×C₁₂₇	C₂×C₄	1	4	yes	—
32	#107×C₁₂₇	—	2	2	no	—
33	#107×C₁₂₇	—	2	2	no	—
34	#107×C₁₂₇	—	4	2	no	—
35	#106×C₁₂₇	—	4	2	no	—
36	#97×C₁₂₇	—	2	1	no	—
37	C₄×C₈×C₁₂₇	—	2	1	no	—
38	C₂²×C₄×C₁₂₇	—	4	2	no	—
39	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	M₁₆×C₁₂₇	—	2	1	no	—
41	M₁₆×C₁₂₇	—	2	4	no	—
42	M₁₆×C₁₂₇	—	4	2	no	—
43	M₁₆×C₁₂₇	Q₁₆	1	2	yes	—
44	C₂×C₈×C₁₂₇	—	2	1	no	—
45	C₂×C₈×C₁₂₇	—	4	2	no	—
46	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
52	C₄⋊C₄×C₁₂₇	—	4	2	no	—
53	C₄⋊C₄×C₁₂₇	—	4	2	no	—
54	C₄⋊C₄×C₁₂₇	D₈	1	1	yes	characteristic
55	C₄⋊C₄×C₁₂₇	D₈	1	1	yes	characteristic
56	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
57	C₄²×C₁₂₇	—	2	1	no	—
58	C₁₆×C₁₂₇	—	2	2	no	—
59	C₁₆×C₁₂₇	C₄⋊C₄	1	2	yes	—
60	C₁₆×C₁₂₇	C₄⋊C₄	1	2	yes	—
61	C₂³×C₁₂₇	#135	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	—	2	1	no	—
63	C₂×C₄×C₁₂₇	—	2	1	no	—
64	C₂×C₄×C₁₂₇	—	8	2	no	—
65	C₂×C₄×C₁₂₇	—	16	2	no	—
66	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
72	C₈×C₁₂₇	—	2	2	no	—
73	C₈×C₁₂₇	—	4	4	no	—
74	C₈×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
76	C₈×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
77	C₈×C₁₂₇	#125	1	1	yes	characteristic
78	C₂²×C₁₂₇	—	2	1	no	—
79	C₂²×C₁₂₇	—	2	1	no	—
80	C₂²×C₁₂₇	C₄×Q₁₆	1	1	yes	characteristic
81	C₂²×C₁₂₇	#466	1	1	yes	characteristic
82	C₂²×C₁₂₇	#499	1	1	yes	characteristic
83	C₄×C₁₂₇	—	4	1	no	—
84	C₄×C₁₂₇	—	4	1	no	—
85	C₄×C₁₂₇	—	16	2	no	—
86	C₄×C₁₂₇	#389	1	1	yes	characteristic
87	C₄×C₁₂₇	C₄×D₈	1	1	yes	characteristic
88	C₄×C₁₂₇	#467	1	1	yes	characteristic
89	C₄×C₁₂₇	#500	1	1	yes	characteristic
90	#13171	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₁₂₇	—	2	1	no	—
92	C₂×C₁₂₇	—	2	1	no	—
93	C₂×C₁₂₇	#1942	1	1	yes	characteristic
94	C₂×C₁₂₇	#2221	1	1	yes	characteristic
95	C₂×C₁₂₇	#2222	1	1	yes	characteristic
96	C₂×#368	C₂×C₁₂₇	1	1	yes	characteristic
97	C₂×#366	C₂×C₁₂₇	1	1	yes	characteristic
98	#2108	C₂×C₁₂₇	1	1	yes	characteristic
99	#1952	C₂×C₁₂₇	1	2	yes	—
100	#1387	C₂×C₁₂₇	1	2	yes	—
101	C₁₂₇	#13171	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
103	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
104	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
105	#428	—	2	2	no	—
106	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
107	#405	C₂²×C₁₂₇	1	1	yes	characteristic
108	#368	C₄×C₁₂₇	1	4	yes	complemented
109	#366	—	2	1	no	—
110	#366	—	2	1	no	—
111	#346	—	2	1	no	—
112	C₄×C₁₆	—	2	1	no	—
113	#343	—	2	2	no	—
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	D₄×C₁₂₇	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
117	#117	D₄×C₁₂₇	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	2	2	no	—
119	M₃₂	—	2	1	no	—
120	M₃₂	Q₈×C₁₂₇	1	2	yes	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
123	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
124	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
125	C₂×C₁₆	Q₈×C₁₂₇	1	2	yes	—
126	#108	—	2	2	no	—
127	#108	C₂×C₄×C₁₂₇	1	4	yes	—
128	#107	—	2	2	no	—
129	#107	—	2	2	no	—
130	#107	—	4	2	no	—
131	#106	—	4	2	no	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
136	M₁₆	—	2	1	no	—
137	M₁₆	—	2	4	no	—
138	M₁₆	—	4	2	no	—
139	M₁₆	Q₁₆×C₁₂₇	1	2	yes	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
143	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
144	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
145	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
146	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
147	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	D₈×C₁₂₇	1	1	yes	characteristic
151	C₄⋊C₄	D₈×C₁₂₇	1	1	yes	characteristic
152	C₂²⋊C₄	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₁₆	—	2	2	no	—
155	C₁₆	C₄⋊C₄×C₁₂₇	1	2	yes	—
156	C₁₆	C₄⋊C₄×C₁₂₇	1	2	yes	—
157	C₂³	#135×C₁₂₇	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	16	2	no	—
162	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
163	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
164	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
165	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
166	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
167	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	4	4	no	—
170	C₈	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
171	C₈	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
172	C₈	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
173	C₈	#125×C₁₂₇	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×Q₁₆×C₁₂₇	1	1	yes	characteristic
177	C₂²	#466×C₁₂₇	1	1	yes	characteristic
178	C₂²	#499×C₁₂₇	1	1	yes	characteristic
179	C₄	—	4	1	no	—
180	C₄	—	4	1	no	—
181	C₄	—	16	2	no	—
182	C₄	#389×C₁₂₇	1	1	yes	characteristic
183	C₄	C₄×D₈×C₁₂₇	1	1	yes	characteristic
184	C₄	#467×C₁₂₇	1	1	yes	characteristic
185	C₄	#500×C₁₂₇	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	C₁₂₇×#1942	1	1	yes	characteristic
189	C₂	C₁₂₇×#2221	1	1	yes	characteristic
190	C₂	C₁₂₇×#2222	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#466
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #405×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. #368×C₁₂₇	C₄	semidirect
90. #13171	C₁₂₇	direct (N × H)
101. C₁₂₇	#13171	direct (N × H)
108. #368	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
C₂×#366×C₁₂₇	2
C₁₂₇×#2108	2
C₁₂₇×#1952	2
C₁₂₇×#1387	2
#13171	127

Subgroups — C127×#13171 #44576844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13171 #44576844