Subgroups — C₁₂₇×#13188 #44576861

Order 32512.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2155	C₂	1	2	yes	maximal
2	C₂×#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1848	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1772	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1388	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
7	C₂×#108×C₁₂₇	—	2	1	no	—
8	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
9	C₂×#108×C₁₂₇	C₂²	1	1	yes	characteristic
10	#427×C₁₂₇	C₂²	1	1	yes	characteristic
11	#423×C₁₂₇	—	2	1	no	—
12	#405×C₁₂₇	—	2	1	no	—
13	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	#365×C₁₂₇	—	2	4	no	—
15	#364×C₁₂₇	C₄	1	2	yes	complemented
16	#364×C₁₂₇	C₄	1	2	yes	complemented
17	#339×C₁₂₇	—	2	1	no	—
18	C₂×M₁₆×C₁₂₇	—	2	1	no	—
19	C₂×M₁₆×C₁₂₇	—	2	2	no	—
20	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
22	#117×C₁₂₇	—	2	1	no	—
23	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
24	M₃₂×C₁₂₇	—	2	2	no	—
25	M₃₂×C₁₂₇	C₂×C₄	1	4	yes	—
26	C₂×C₁₆×C₁₂₇	—	2	2	no	—
27	#108×C₁₂₇	—	2	2	no	—
28	#108×C₁₂₇	—	2	1	no	—
29	#108×C₁₂₇	—	2	2	no	—
30	#108×C₁₂₇	—	4	1	no	—
31	#108×C₁₂₇	—	4	1	no	—
32	#108×C₁₂₇	Q₈	1	2	yes	—
33	#107×C₁₂₇	—	2	1	no	—
34	#106×C₁₂₇	D₄	1	2	yes	—
35	C₄⋊C₈×C₁₂₇	—	4	2	no	—
36	#97×C₁₂₇	—	4	1	no	—
37	C₄×C₈×C₁₂₇	—	4	1	no	—
38	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	—	2	1	no	—
43	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₁₂₇	—	2	2	no	—
45	M₁₆×C₁₂₇	—	4	1	no	—
46	M₁₆×C₁₂₇	—	4	2	no	—
47	M₁₆×C₁₂₇	—	4	2	no	—
48	C₂×C₈×C₁₂₇	—	4	1	no	—
49	C₂×C₈×C₁₂₇	—	4	2	no	—
50	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
61	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
62	C₄⋊C₄×C₁₂₇	—	2	2	no	—
63	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
64	C₄²×C₁₂₇	—	2	1	no	—
65	C₄²×C₁₂₇	—	4	1	no	—
66	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
68	C₁₆×C₁₂₇	—	2	4	no	—
69	C₂³×C₁₂₇	#137	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	—	2	1	no	—
71	C₂×C₄×C₁₂₇	—	2	1	no	—
72	C₂×C₄×C₁₂₇	—	4	2	no	—
73	C₂×C₄×C₁₂₇	—	4	2	no	—
74	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₈×C₁₂₇	—	4	2	no	—
81	C₈×C₁₂₇	—	8	2	no	—
82	C₈×C₁₂₇	#106	1	1	yes	characteristic
83	C₈×C₁₂₇	#106	1	1	yes	characteristic
84	C₈×C₁₂₇	#106	1	1	yes	characteristic
85	C₈×C₁₂₇	#106	1	1	yes	characteristic
86	C₈×C₁₂₇	C₄×D₄	1	1	yes	characteristic
87	C₈×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
88	C₈×C₁₂₇	#125	1	1	yes	characteristic
89	C₈×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
90	C₂²×C₁₂₇	—	4	1	no	—
91	C₂²×C₁₂₇	#428	1	1	yes	characteristic
92	C₂²×C₁₂₇	#461	1	1	yes	characteristic
93	C₂²×C₁₂₇	#475	1	1	yes	characteristic
94	C₄×C₁₂₇	—	2	2	no	—
95	C₄×C₁₂₇	—	4	1	no	—
96	C₄×C₁₂₇	—	8	2	no	—
97	C₄×C₁₂₇	#389	1	1	yes	characteristic
98	C₄×C₁₂₇	C₂×#106	1	1	yes	characteristic
99	C₄×C₁₂₇	#460	1	1	yes	characteristic
100	C₄×C₁₂₇	#476	1	1	yes	characteristic
101	#13188	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂×C₁₂₇	—	2	1	no	—
103	C₂×C₁₂₇	—	2	1	no	—
104	C₂×C₁₂₇	—	2	1	no	—
105	C₂×C₁₂₇	#1948	1	1	yes	characteristic
106	#2155	C₂×C₁₂₇	1	2	yes	—
107	C₂×#364	C₂×C₁₂₇	1	1	yes	characteristic
108	#1848	C₂×C₁₂₇	1	1	yes	characteristic
109	#1772	C₂×C₁₂₇	1	1	yes	characteristic
110	#1388	C₂×C₁₂₇	1	2	yes	—
111	C₁₂₇	#13188	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
113	C₂×#108	—	2	1	no	—
114	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
115	C₂×#108	C₂²×C₁₂₇	1	1	yes	characteristic
116	#427	C₂²×C₁₂₇	1	1	yes	characteristic
117	#423	—	2	1	no	—
118	#405	—	2	1	no	—
119	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
120	#365	—	2	4	no	—
121	#364	C₄×C₁₂₇	1	2	yes	complemented
122	#364	C₄×C₁₂₇	1	2	yes	complemented
123	#339	—	2	1	no	—
124	C₂×M₁₆	—	2	1	no	—
125	C₂×M₁₆	—	2	2	no	—
126	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
127	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
130	M₃₂	—	2	2	no	—
131	M₃₂	C₂×C₄×C₁₂₇	1	4	yes	—
132	C₂×C₁₆	—	2	2	no	—
133	#108	—	2	2	no	—
134	#108	—	2	1	no	—
135	#108	—	2	2	no	—
136	#108	—	4	1	no	—
137	#108	—	4	1	no	—
138	#108	Q₈×C₁₂₇	1	2	yes	—
139	#107	—	2	1	no	—
140	#106	D₄×C₁₂₇	1	2	yes	—
141	C₄⋊C₈	—	4	2	no	—
142	#97	—	4	1	no	—
143	C₄×C₈	—	4	1	no	—
144	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
145	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
146	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
147	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	4	1	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
157	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
158	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
159	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
160	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
161	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
162	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
163	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
164	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
165	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
166	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
167	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
168	C₄⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
173	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₂³	#137×C₁₂₇	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
181	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
182	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
183	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
185	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	1	yes	characteristic
186	C₈	—	4	2	no	—
187	C₈	—	8	2	no	—
188	C₈	#106×C₁₂₇	1	1	yes	characteristic
189	C₈	#106×C₁₂₇	1	1	yes	characteristic
190	C₈	#106×C₁₂₇	1	1	yes	characteristic
191	C₈	#106×C₁₂₇	1	1	yes	characteristic
192	C₈	C₄×D₄×C₁₂₇	1	1	yes	characteristic
193	C₈	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
194	C₈	#125×C₁₂₇	1	1	yes	characteristic
195	C₈	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	#428×C₁₂₇	1	1	yes	characteristic
198	C₂²	#461×C₁₂₇	1	1	yes	characteristic
199	C₂²	#475×C₁₂₇	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	8	2	no	—
203	C₄	#389×C₁₂₇	1	1	yes	characteristic
204	C₄	C₂×#106×C₁₂₇	1	1	yes	characteristic
205	C₄	#460×C₁₂₇	1	1	yes	characteristic
206	C₄	#476×C₁₂₇	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	C₁₂₇×#1948	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₂₇	#476
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#1948
prime radical	C₂³×C₁₂₇	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. #364×C₁₂₇	C₄	semidirect
16. #364×C₁₂₇	C₄	semidirect
101. #13188	C₁₂₇	direct (N × H)
111. C₁₂₇	#13188	direct (N × H)
121. #364	C₄×C₁₂₇	semidirect
122. #364	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×#2155	2
C₂×#364×C₁₂₇	2
C₁₂₇×#1848	2
C₁₂₇×#1772	2
C₁₂₇×#1388	2
#13188	127

Subgroups — C127×#13188 #44576861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13188 #44576861