Subgroups — C₂×C₁₂₇×#1370 #44575676

Order 32512.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₁₆×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×C₈²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1370	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₂₇	—	2	2	no	—
5	C₂×C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
9	C₄⋊C₁₆×C₁₂₇	C₂²	1	4	yes	—
10	C₈²×C₁₂₇	C₄	1	2	yes	—
11	C₈²×C₁₂₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₂₇	—	2	1	no	—
13	C₂²×C₈×C₁₂₇	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₁₂₇	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₁₂₇	—	2	4	no	—
19	C₂×C₁₆×C₁₂₇	—	4	4	no	—
20	C₄×C₈×C₁₂₇	—	2	1	no	—
21	C₄×C₈×C₁₂₇	C₈	1	2	yes	—
22	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₁₂₇	—	2	4	no	—
32	C₂×C₈×C₁₂₇	—	2	2	no	—
33	C₂×C₈×C₁₂₇	—	2	2	no	—
34	C₂×C₈×C₁₂₇	—	2	2	no	—
35	C₂×C₈×C₁₂₇	C₁₆	1	4	yes	complemented
36	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
41	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
42	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
47	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	C₁₆×C₁₂₇	—	4	4	no	—
49	C₂³×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	—	2	2	no	—
51	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	#107	1	2	yes	—
55	C₂×C₄×C₁₂₇	#108	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₁₂₇	M₃₂	1	2	yes	—
58	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₁₂₇	—	2	4	no	—
62	C₈×C₁₂₇	—	2	1	no	—
63	C₈×C₁₂₇	—	2	1	no	—
64	C₈×C₁₂₇	—	2	2	no	—
65	C₈×C₁₂₇	#107	1	2	yes	—
66	C₈×C₁₂₇	C₂×C₁₆	1	4	yes	complemented
67	C₈×C₁₂₇	C₂×D₈	1	1	yes	characteristic
68	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₂²×C₁₂₇	#335	1	2	yes	—
70	C₂²×C₁₂₇	C₄⋊C₁₆	1	2	yes	—
71	C₂²×C₁₂₇	#364	1	2	yes	—
72	C₂²×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₁₂₇	—	2	1	no	—
74	C₄×C₁₂₇	—	2	1	no	—
75	C₄×C₁₂₇	#335	1	2	yes	—
76	C₄×C₁₂₇	C₄⋊C₁₆	1	2	yes	—
77	C₄×C₁₂₇	C₂×#107	1	1	yes	characteristic
78	C₄×C₁₂₇	C₂×#108	1	1	yes	characteristic
79	C₄×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
80	C₄×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
81	C₂×#1370	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₁₂₇	#1370	1	4	yes	direct product factor
83	C₂×C₁₂₇	C₂×#335	1	1	yes	characteristic
84	C₂×C₁₂₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
85	C₂×C₁₂₇	C₂×#364	1	1	yes	characteristic
86	C₂×C₄⋊C₁₆	C₂×C₁₂₇	1	2	yes	—
87	C₂×C₈²	C₂×C₁₂₇	1	1	yes	characteristic
88	#1370	C₂×C₁₂₇	1	4	yes	direct product factor
89	C₁₂₇	C₂×#1370	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×C₁₆	—	2	2	no	—
91	C₂×C₄×C₈	C₄×C₁₂₇	1	1	yes	characteristic
92	C₂×C₄×C₈	C₄×C₁₂₇	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
94	C₄⋊C₁₆	—	2	2	no	—
95	C₄⋊C₁₆	C₂²×C₁₂₇	1	4	yes	—
96	C₈²	C₄×C₁₂₇	1	2	yes	—
97	C₈²	C₂²×C₁₂₇	1	2	yes	—
98	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	C₈×C₁₂₇	1	1	yes	characteristic
100	C₂²×C₈	C₈×C₁₂₇	1	1	yes	characteristic
101	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
102	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
103	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
104	C₂×C₁₆	—	2	4	no	—
105	C₂×C₁₆	—	4	4	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	C₈×C₁₂₇	1	2	yes	—
108	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
109	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
110	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
111	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
112	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
113	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
115	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
116	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	C₁₆×C₁₂₇	1	4	yes	complemented
122	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
123	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
124	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
125	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
126	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
127	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
128	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
129	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
130	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
131	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
132	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
133	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
134	C₁₆	—	4	4	no	—
135	C₂³	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
138	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
139	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
140	C₂×C₄	#107×C₁₂₇	1	2	yes	—
141	C₂×C₄	#108×C₁₂₇	1	2	yes	—
142	C₂×C₄	C₂×C₁₆×C₁₂₇	1	2	yes	—
143	C₂×C₄	M₃₂×C₁₂₇	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
145	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
146	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
147	C₈	—	2	4	no	—
148	C₈	—	2	2	no	—
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	#107×C₁₂₇	1	2	yes	—
152	C₈	C₂×C₁₆×C₁₂₇	1	4	yes	complemented
153	C₈	C₂×D₈×C₁₂₇	1	1	yes	characteristic
154	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
155	C₂²	#335×C₁₂₇	1	2	yes	—
156	C₂²	C₄⋊C₁₆×C₁₂₇	1	2	yes	—
157	C₂²	#364×C₁₂₇	1	2	yes	—
158	C₂²	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#335×C₁₂₇	1	2	yes	—
162	C₄	C₄⋊C₁₆×C₁₂₇	1	2	yes	—
163	C₄	C₂×#107×C₁₂₇	1	1	yes	characteristic
164	C₄	C₂×#108×C₁₂₇	1	1	yes	characteristic
165	C₄	C₂²×C₁₆×C₁₂₇	1	1	yes	characteristic
166	C₄	C₂×M₃₂×C₁₂₇	1	1	yes	characteristic
167	C₂	C₁₂₇×#1370	1	4	yes	direct product factor
168	C₂	C₂×#335×C₁₂₇	1	1	yes	characteristic
169	C₂	C₂×C₄⋊C₁₆×C₁₂₇	1	1	yes	characteristic
170	C₂	C₂×#364×C₁₂₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1370	C₂	direct (N × H)
35. C₂×C₈×C₁₂₇	C₁₆	semidirect
66. C₈×C₁₂₇	C₂×C₁₆	semidirect
81. C₂×#1370	C₁₂₇	direct (N × H)
82. C₂×C₁₂₇	#1370	direct (N × H)
88. #1370	C₂×C₁₂₇	direct (N × H)
89. C₁₂₇	C₂×#1370	direct (N × H)
121. C₂×C₈	C₁₆×C₁₂₇	semidirect
152. C₈	C₂×C₁₆×C₁₂₇	semidirect
167. C₂	C₁₂₇×#1370	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×C₄⋊C₁₆×C₁₂₇	2
C₂×C₈²×C₁₂₇	2
C₁₂₇×#1370	2
C₂×#1370	127

Subgroups — C2×C127×#1370 #44575676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1370 #44575676