Subgroups — C₁₂₇×#13161 #44576834

Order 32512.

298 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2151	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1772	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1379	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×#108×C₁₂₇	C₄	1	1	yes	characteristic
10	#427×C₁₂₇	C₄	1	1	yes	characteristic
11	#405×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	#364×C₁₂₇	—	2	2	no	—
14	C₄⋊C₁₆×C₁₂₇	—	2	4	no	—
15	C₄×C₁₆×C₁₂₇	—	2	2	no	—
16	#339×C₁₂₇	C₄	1	2	yes	—
17	C₂×M₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
20	#117×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	M₃₂×C₁₂₇	—	2	2	no	—
23	M₃₂×C₁₂₇	—	2	2	no	—
24	C₂×C₁₆×C₁₂₇	—	2	1	no	—
25	C₂×C₁₆×C₁₂₇	—	2	1	no	—
26	C₂×C₁₆×C₁₂₇	—	2	1	no	—
27	C₂×C₁₆×C₁₂₇	—	2	1	no	—
28	C₂×C₁₆×C₁₂₇	—	2	2	no	—
29	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
30	C₂×C₁₆×C₁₂₇	Q₈	1	2	yes	—
31	#108×C₁₂₇	—	2	2	no	—
32	#107×C₁₂₇	C₈	1	2	yes	—
33	#106×C₁₂₇	C₈	1	2	yes	—
34	#97×C₁₂₇	—	2	2	no	—
35	C₄×C₈×C₁₂₇	—	2	2	no	—
36	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
37	C₄×C₈×C₁₂₇	Q₈	1	2	yes	—
38	C₂²×C₄×C₁₂₇	—	2	1	no	—
39	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₁₂₇	—	2	2	no	—
41	M₁₆×C₁₂₇	—	2	2	no	—
42	C₂×C₈×C₁₂₇	—	2	2	no	—
43	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
47	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
52	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₁₂₇	C₂×C₈	1	4	yes	—
54	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
55	C₄²×C₁₂₇	—	2	2	no	—
56	C₄²×C₁₂₇	—	2	1	no	—
57	C₄²×C₁₂₇	C₄⋊C₄	1	2	yes	—
58	C₁₆×C₁₂₇	—	2	2	no	—
59	C₁₆×C₁₂₇	—	2	2	no	—
60	C₁₆×C₁₂₇	—	4	2	no	—
61	C₂³×C₁₂₇	#131	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	—	2	4	no	—
63	C₂×C₄×C₁₂₇	—	2	2	no	—
64	C₂×C₄×C₁₂₇	—	2	4	no	—
65	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	#131	1	1	yes	characteristic
71	C₈×C₁₂₇	—	4	2	no	—
72	C₈×C₁₂₇	C₄⋊C₈	1	2	yes	—
73	C₈×C₁₂₇	C₄⋊C₈	1	2	yes	—
74	C₈×C₁₂₇	#121	1	1	yes	characteristic
75	C₈×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₁₂₇	#125	1	1	yes	characteristic
77	C₈×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
78	C₂²×C₁₂₇	—	2	2	no	—
79	C₂²×C₁₂₇	#424	1	1	yes	characteristic
80	C₂²×C₁₂₇	C₈×D₄	1	1	yes	characteristic
81	C₂²×C₁₂₇	#446	1	1	yes	characteristic
82	C₄×C₁₂₇	—	2	4	no	—
83	C₄×C₁₂₇	—	4	4	no	—
84	C₄×C₁₂₇	#389	1	1	yes	characteristic
85	C₄×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₄×C₁₂₇	#436	1	1	yes	characteristic
87	C₄×C₁₂₇	C₈×Q₈	1	1	yes	characteristic
88	#13161	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×C₁₂₇	—	2	1	no	—
90	C₂×C₁₂₇	—	2	1	no	—
91	C₂×C₁₂₇	#1939	1	1	yes	characteristic
92	C₂×C₁₂₇	#2170	1	2	yes	—
93	C₂×#364	C₂×C₁₂₇	1	1	yes	characteristic
94	#2151	C₂×C₁₂₇	1	2	yes	—
95	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
96	#1772	C₂×C₁₂₇	1	1	yes	characteristic
97	#1379	C₂×C₁₂₇	1	2	yes	—
98	C₁₂₇	#13161	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
100	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
101	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
102	C₂×#108	C₄×C₁₂₇	1	1	yes	characteristic
103	#427	C₄×C₁₂₇	1	1	yes	characteristic
104	#405	C₂²×C₁₂₇	1	2	yes	—
105	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
106	#364	—	2	2	no	—
107	C₄⋊C₁₆	—	2	4	no	—
108	C₄×C₁₆	—	2	2	no	—
109	#339	C₄×C₁₂₇	1	2	yes	—
110	C₂×M₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
111	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
112	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
113	#117	C₂×C₄×C₁₂₇	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	—	2	2	no	—
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
123	C₂×C₁₆	Q₈×C₁₂₇	1	2	yes	—
124	#108	—	2	2	no	—
125	#107	C₈×C₁₂₇	1	2	yes	—
126	#106	C₈×C₁₂₇	1	2	yes	—
127	#97	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
130	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
137	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
138	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
139	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
140	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
141	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
142	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
143	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
144	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
145	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
146	C₄⋊C₄	C₂×C₈×C₁₂₇	1	4	yes	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	C₄⋊C₄×C₁₂₇	1	2	yes	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	2	2	no	—
153	C₁₆	—	4	2	no	—
154	C₂³	#131×C₁₂₇	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
159	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
160	C₂×C₄	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
162	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
163	C₂×C₄	#131×C₁₂₇	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₈	C₄⋊C₈×C₁₂₇	1	2	yes	—
166	C₈	C₄⋊C₈×C₁₂₇	1	2	yes	—
167	C₈	#121×C₁₂₇	1	1	yes	characteristic
168	C₈	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
169	C₈	#125×C₁₂₇	1	1	yes	characteristic
170	C₈	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	#424×C₁₂₇	1	1	yes	characteristic
173	C₂²	C₈×D₄×C₁₂₇	1	1	yes	characteristic
174	C₂²	#446×C₁₂₇	1	1	yes	characteristic
175	C₄	—	2	4	no	—
176	C₄	—	4	4	no	—
177	C₄	#389×C₁₂₇	1	1	yes	characteristic
178	C₄	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
179	C₄	#436×C₁₂₇	1	1	yes	characteristic
180	C₄	C₈×Q₈×C₁₂₇	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	C₁₂₇×#1939	1	1	yes	characteristic
184	C₂	C₁₂₇×#2170	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₈×D₄
prime radical	C₂³×C₁₂₇	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂²×C₁₆×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
88. #13161	C₁₂₇	direct (N × H)
98. C₁₂₇	#13161	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#364×C₁₂₇	2
C₁₂₇×#2151	2
C₂×C₄×C₁₆×C₁₂₇	2
C₁₂₇×#1772	2
C₁₂₇×#1379	2
#13161	127

Subgroups — C127×#13161 #44576834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13161 #44576834