Subgroups — C₁₂₇×#7182 #44570855

Order 32512.

258 conjugacy classes of subgroups.

The subgroup lattice has 198 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×#1772	C₂	1	1	yes	maximal, characteristic
3	C₂×#322×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₂₇	—	2	1	no	—
5	C₂×#108×C₁₂₇	C₄	1	1	yes	characteristic
6	#427×C₁₂₇	C₄	1	1	yes	characteristic
7	#405×C₁₂₇	—	2	1	no	—
8	C₂×C₄×C₈×C₁₂₇	—	2	1	no	—
9	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	#364×C₁₂₇	—	2	1	no	—
11	#364×C₁₂₇	C₄	1	2	yes	complemented
12	#339×C₁₂₇	—	2	1	no	—
13	#322×C₁₂₇	—	2	1	no	—
14	#322×C₁₂₇	C₄	1	2	yes	complemented
15	C₂×M₁₆×C₁₂₇	—	2	1	no	—
16	C₂²×C₈×C₁₂₇	—	2	2	no	—
17	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	#117×C₁₂₇	—	2	1	no	—
20	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
21	M₃₂×C₁₂₇	—	2	2	no	—
22	M₃₂×C₁₂₇	—	4	1	no	—
23	C₂×C₁₆×C₁₂₇	—	4	1	no	—
24	#108×C₁₂₇	—	2	1	no	—
25	#108×C₁₂₇	C₈	1	2	yes	complemented
26	#107×C₁₂₇	C₈	1	2	yes	complemented
27	#106×C₁₂₇	—	2	1	no	—
28	#97×C₁₂₇	—	4	1	no	—
29	C₄×C₈×C₁₂₇	—	2	2	no	—
30	C₄×C₈×C₁₂₇	—	2	2	no	—
31	C₄×C₈×C₁₂₇	—	4	1	no	—
32	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
35	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	—	2	1	no	—
37	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
38	M₁₆×C₁₂₇	—	2	2	no	—
39	M₁₆×C₁₂₇	—	4	1	no	—
40	C₂×C₈×C₁₂₇	—	2	4	no	—
41	C₂×C₈×C₁₂₇	—	4	4	no	—
42	C₂×C₈×C₁₂₇	—	4	1	no	—
43	C₂×C₈×C₁₂₇	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	D₈	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	Q₁₆	1	1	yes	characteristic
55	C₄⋊C₄×C₁₂₇	—	2	2	no	—
56	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
57	C₄²×C₁₂₇	—	2	1	no	—
58	C₄²×C₁₂₇	—	4	1	no	—
59	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
61	C₁₆×C₁₂₇	—	4	2	no	—
62	C₂³×C₁₂₇	#104	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	—	2	4	no	—
64	C₂×C₄×C₁₂₇	—	2	1	no	—
65	C₂×C₄×C₁₂₇	—	2	1	no	—
66	C₂×C₄×C₁₂₇	—	4	2	no	—
67	C₂×C₄×C₁₂₇	#95	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	#104	1	1	yes	characteristic
73	C₈×C₁₂₇	—	4	4	no	—
74	C₈×C₁₂₇	—	4	2	no	—
75	C₈×C₁₂₇	C₄×C₈	1	1	yes	characteristic
76	C₈×C₁₂₇	#97	1	1	yes	characteristic
77	C₈×C₁₂₇	#102	1	1	yes	characteristic
78	C₈×C₁₂₇	#103	1	1	yes	characteristic
79	C₈×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
80	C₈×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
81	C₈×C₁₂₇	#106	1	1	yes	characteristic
82	C₈×C₁₂₇	#107	1	1	yes	characteristic
83	C₂²×C₁₂₇	—	2	2	no	—
84	C₂²×C₁₂₇	D₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₁₂₇	Q₈⋊C₈	1	1	yes	characteristic
86	C₂²×C₁₂₇	#339	1	1	yes	characteristic
87	C₄×C₁₂₇	—	2	2	no	—
88	C₄×C₁₂₇	—	2	2	no	—
89	C₄×C₁₂₇	—	8	2	no	—
90	C₄×C₁₂₇	D₄⋊C₈	1	1	yes	characteristic
91	C₄×C₁₂₇	Q₈⋊C₈	1	1	yes	characteristic
92	C₄×C₁₂₇	#336	1	1	yes	characteristic
93	C₄×C₁₂₇	#340	1	1	yes	characteristic
94	#7182	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₁₂₇	—	2	1	no	—
96	C₂×C₁₂₇	—	2	1	no	—
97	C₂×C₁₂₇	#1279	1	1	yes	characteristic
98	C₂×C₁₂₇	#1336	1	2	yes	—
99	C₂×#364	C₂×C₁₂₇	1	1	yes	characteristic
100	#1772	C₂×C₁₂₇	1	1	yes	characteristic
101	C₂×#322	C₂×C₁₂₇	1	1	yes	characteristic
102	C₁₂₇	#7182	1	1	yes	characteristic, direct product factor, Sylow
103	C₂×M₃₂	—	2	1	no	—
104	C₂×#108	C₄×C₁₂₇	1	1	yes	characteristic
105	#427	C₄×C₁₂₇	1	1	yes	characteristic
106	#405	—	2	1	no	—
107	C₂×C₄×C₈	—	2	1	no	—
108	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
109	#364	—	2	1	no	—
110	#364	C₄×C₁₂₇	1	2	yes	complemented
111	#339	—	2	1	no	—
112	#322	—	2	1	no	—
113	#322	C₄×C₁₂₇	1	2	yes	complemented
114	C₂×M₁₆	—	2	1	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
117	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
118	#117	—	2	1	no	—
119	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
120	M₃₂	—	2	2	no	—
121	M₃₂	—	4	1	no	—
122	C₂×C₁₆	—	4	1	no	—
123	#108	—	2	1	no	—
124	#108	C₈×C₁₂₇	1	2	yes	complemented
125	#107	C₈×C₁₂₇	1	2	yes	complemented
126	#106	—	2	1	no	—
127	#97	—	4	1	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	4	1	no	—
131	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
132	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
133	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
134	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
137	M₁₆	—	2	2	no	—
138	M₁₆	—	4	1	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	4	4	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	C₄²×C₁₂₇	1	1	yes	characteristic
143	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
144	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
145	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
146	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
147	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
148	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
149	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
150	C₂×C₈	D₈×C₁₂₇	1	1	yes	characteristic
151	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
152	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
153	C₂×C₈	Q₁₆×C₁₂₇	1	1	yes	characteristic
154	C₄⋊C₄	—	2	2	no	—
155	C₂²⋊C₄	—	4	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	4	1	no	—
158	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
159	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
160	C₁₆	—	4	2	no	—
161	C₂³	#104×C₁₂₇	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	#95×C₁₂₇	1	1	yes	characteristic
167	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
168	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
169	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
171	C₂×C₄	#104×C₁₂₇	1	1	yes	characteristic
172	C₈	—	4	4	no	—
173	C₈	—	4	2	no	—
174	C₈	C₄×C₈×C₁₂₇	1	1	yes	characteristic
175	C₈	#97×C₁₂₇	1	1	yes	characteristic
176	C₈	#102×C₁₂₇	1	1	yes	characteristic
177	C₈	#103×C₁₂₇	1	1	yes	characteristic
178	C₈	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
179	C₈	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
180	C₈	#106×C₁₂₇	1	1	yes	characteristic
181	C₈	#107×C₁₂₇	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	D₄⋊C₈×C₁₂₇	1	1	yes	characteristic
184	C₂²	Q₈⋊C₈×C₁₂₇	1	1	yes	characteristic
185	C₂²	#339×C₁₂₇	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	2	no	—
188	C₄	—	8	2	no	—
189	C₄	D₄⋊C₈×C₁₂₇	1	1	yes	characteristic
190	C₄	Q₈⋊C₈×C₁₂₇	1	1	yes	characteristic
191	C₄	#336×C₁₂₇	1	1	yes	characteristic
192	C₄	#340×C₁₂₇	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₁₂₇×#1279	1	1	yes	characteristic
196	C₂	C₁₂₇×#1336	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#102
commutator	C₈	C₄×C₈×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₂₇
socle	C₂²×C₁₂₇	D₄⋊C₈
prime radical	C₂³×C₁₂₇	#104
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
11. #364×C₁₂₇	C₄	semidirect
14. #322×C₁₂₇	C₄	semidirect
25. #108×C₁₂₇	C₈	semidirect
26. #107×C₁₂₇	C₈	semidirect
94. #7182	C₁₂₇	direct (N × H)
102. C₁₂₇	#7182	direct (N × H)
110. #364	C₄×C₁₂₇	semidirect
113. #322	C₄×C₁₂₇	semidirect
124. #108	C₈×C₁₂₇	semidirect
125. #107	C₈×C₁₂₇	semidirect

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×#1772	2
C₂×#322×C₁₂₇	2
#7182	127

Subgroups — C127×#7182 #44570855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7182 #44570855