Subgroups — C₂²×C₂²⋊C₁₆×C₁₃₇ #46882716

Order 35072.

1470 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₁₆×C₁₃₇	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₄×C₁₃₇	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₃₇	—	2	8	no	—
6	C₂³×C₈×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₁₃₇	C₄	1	6	yes	—
8	C₂³×C₈×C₁₃₇	C₂²	1	6	yes	—
9	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₁₃₇	—	2	8	no	—
11	C₂²×C₁₆×C₁₃₇	C₂²	1	12	yes	complemented
12	C₂²⋊C₁₆×C₁₃₇	C₂²	1	16	yes	direct product factor
13	C₂⁵×C₁₃₇	C₈	1	1	yes	characteristic
14	C₂³×C₄×C₁₃₇	—	2	4	no	—
15	C₂³×C₄×C₁₃₇	—	2	4	no	—
16	C₂³×C₄×C₁₃₇	C₈	1	1	yes	characteristic
17	C₂³×C₄×C₁₃₇	C₈	1	6	yes	—
18	C₂³×C₄×C₁₃₇	C₂×C₄	1	6	yes	—
19	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃₇	—	2	4	no	—
21	C₂²×C₈×C₁₃₇	—	2	24	no	—
22	C₂²×C₈×C₁₃₇	—	2	24	no	—
23	C₂²×C₈×C₁₃₇	C₂×C₄	1	6	yes	—
24	C₂²×C₈×C₁₃₇	C₂×C₄	1	12	yes	—
25	C₂²×C₈×C₁₃₇	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₁₃₇	D₄	1	8	yes	—
27	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
28	C₂²×C₈×C₁₃₇	C₂³	1	3	yes	—
29	C₂×C₁₆×C₁₃₇	—	2	24	no	—
30	C₂×C₁₆×C₁₃₇	C₂³	1	8	yes	complemented
31	C₂⁴×C₁₃₇	—	2	4	no	—
32	C₂⁴×C₁₃₇	—	2	4	no	—
33	C₂⁴×C₁₃₇	C₁₆	1	8	yes	complemented
34	C₂⁴×C₁₃₇	C₂×C₈	1	6	yes	—
35	C₂⁴×C₁₃₇	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₇	—	2	4	no	—
37	C₂²×C₄×C₁₃₇	—	2	24	no	—
38	C₂²×C₄×C₁₃₇	—	2	12	no	—
39	C₂²×C₄×C₁₃₇	—	2	12	no	—
40	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
41	C₂²×C₄×C₁₃₇	C₂×C₈	1	6	yes	—
42	C₂²×C₄×C₁₃₇	C₂×C₈	1	12	yes	—
43	C₂²×C₄×C₁₃₇	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₃₇	M₁₆	1	4	yes	—
45	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₁₃₇	—	2	12	no	—
48	C₂×C₈×C₁₃₇	—	2	24	no	—
49	C₂×C₈×C₁₃₇	—	2	8	no	—
50	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	12	yes	—
51	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
52	C₂×C₈×C₁₃₇	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₁₃₇	C₂×D₄	1	12	yes	—
54	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
55	C₁₆×C₁₃₇	—	2	8	no	—
56	C₂³×C₁₃₇	—	2	32	no	—
57	C₂³×C₁₃₇	—	2	12	no	—
58	C₂³×C₁₃₇	—	2	12	no	—
59	C₂³×C₁₃₇	C₂²⋊C₈	1	4	yes	—
60	C₂³×C₁₃₇	C₂×C₁₆	1	4	yes	—
61	C₂³×C₁₃₇	C₂×C₁₆	1	24	yes	complemented
62	C₂³×C₁₃₇	M₃₂	1	4	yes	—
63	C₂³×C₁₃₇	C₂²×C₈	1	4	yes	—
64	C₂³×C₁₃₇	C₂²×C₈	1	3	yes	—
65	C₂×C₄×C₁₃₇	—	2	12	no	—
66	C₂×C₄×C₁₃₇	—	2	4	no	—
67	C₂×C₄×C₁₃₇	—	2	4	no	—
68	C₂×C₄×C₁₃₇	—	2	24	no	—
69	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	12	yes	—
70	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	6	yes	—
71	C₂×C₄×C₁₃₇	C₂²×C₈	1	4	yes	—
72	C₂×C₄×C₁₃₇	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₁₃₇	C₂×M₁₆	1	6	yes	—
74	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
75	C₈×C₁₃₇	—	2	4	no	—
76	C₈×C₁₃₇	C₂×C₂²⋊C₄	1	6	yes	—
77	C₈×C₁₃₇	C₂²×D₄	1	2	yes	—
78	C₂²×C₁₃₇	—	2	4	no	—
79	C₂²×C₁₃₇	—	2	48	no	—
80	C₂²×C₁₃₇	—	2	4	no	—
81	C₂²×C₁₃₇	C₂²⋊C₁₆	1	16	yes	direct product factor
82	C₂²×C₁₃₇	C₂×C₂²⋊C₈	1	6	yes	—
83	C₂²×C₁₃₇	C₂²×C₁₆	1	6	yes	—
84	C₂²×C₁₃₇	C₂²×C₁₆	1	8	yes	complemented
85	C₂²×C₁₃₇	C₂×M₃₂	1	6	yes	—
86	C₂²×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
87	C₄×C₁₃₇	—	2	4	no	—
88	C₄×C₁₃₇	C₂×C₂²⋊C₈	1	6	yes	—
89	C₄×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
90	C₄×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
91	C₂×C₁₃₇	—	2	8	no	—
92	C₂×C₁₃₇	C₂×C₂²⋊C₁₆	1	12	yes	direct product factor
93	C₂×C₁₃₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₁₃₇	C₂³×C₁₆	1	1	yes	characteristic
95	C₂×C₁₃₇	C₂²×M₃₂	1	1	yes	characteristic
96	C₂²×C₂²⋊C₁₆	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₁₃₇	C₂²×C₂²⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
98	C₂⁴×C₈	C₂×C₁₃₇	1	1	yes	characteristic
99	C₂³×C₁₆	C₂×C₁₃₇	1	2	yes	complemented
100	C₂×C₂²⋊C₁₆	C₂×C₁₃₇	1	12	yes	direct product factor
101	C₂⁴×C₄	C₄×C₁₃₇	1	1	yes	characteristic
102	C₂³×C₈	—	2	8	no	—
103	C₂³×C₈	C₄×C₁₃₇	1	1	yes	characteristic
104	C₂³×C₈	C₄×C₁₃₇	1	6	yes	—
105	C₂³×C₈	C₂²×C₁₃₇	1	6	yes	—
106	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
107	C₂²×C₁₆	—	2	8	no	—
108	C₂²×C₁₆	C₂²×C₁₃₇	1	12	yes	complemented
109	C₂²⋊C₁₆	C₂²×C₁₃₇	1	16	yes	direct product factor
110	C₂⁵	C₈×C₁₃₇	1	1	yes	characteristic
111	C₂³×C₄	—	2	4	no	—
112	C₂³×C₄	—	2	4	no	—
113	C₂³×C₄	C₈×C₁₃₇	1	1	yes	characteristic
114	C₂³×C₄	C₈×C₁₃₇	1	6	yes	—
115	C₂³×C₄	C₂×C₄×C₁₃₇	1	6	yes	—
116	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	—	2	24	no	—
119	C₂²×C₈	—	2	24	no	—
120	C₂²×C₈	C₂×C₄×C₁₃₇	1	6	yes	—
121	C₂²×C₈	C₂×C₄×C₁₃₇	1	12	yes	—
122	C₂²×C₈	C₂×C₄×C₁₃₇	1	3	yes	—
123	C₂²×C₈	D₄×C₁₃₇	1	8	yes	—
124	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
125	C₂²×C₈	C₂³×C₁₃₇	1	3	yes	—
126	C₂×C₁₆	—	2	24	no	—
127	C₂×C₁₆	C₂³×C₁₃₇	1	8	yes	complemented
128	C₂⁴	—	2	4	no	—
129	C₂⁴	—	2	4	no	—
130	C₂⁴	C₁₆×C₁₃₇	1	8	yes	complemented
131	C₂⁴	C₂×C₈×C₁₃₇	1	6	yes	—
132	C₂⁴	C₂×C₈×C₁₃₇	1	1	yes	characteristic
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	2	24	no	—
135	C₂²×C₄	—	2	12	no	—
136	C₂²×C₄	—	2	12	no	—
137	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	4	yes	—
138	C₂²×C₄	C₂×C₈×C₁₃₇	1	6	yes	—
139	C₂²×C₄	C₂×C₈×C₁₃₇	1	12	yes	—
140	C₂²×C₄	C₂×C₈×C₁₃₇	1	3	yes	—
141	C₂²×C₄	M₁₆×C₁₃₇	1	4	yes	—
142	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
143	C₂²×C₄	C₂²×C₄×C₁₃₇	1	3	yes	—
144	C₂×C₈	—	2	12	no	—
145	C₂×C₈	—	2	24	no	—
146	C₂×C₈	—	2	8	no	—
147	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	12	yes	—
148	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
149	C₂×C₈	C₂²×C₄×C₁₃₇	1	3	yes	—
150	C₂×C₈	C₂×D₄×C₁₃₇	1	12	yes	—
151	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
152	C₁₆	—	2	8	no	—
153	C₂³	—	2	12	no	—
154	C₂³	—	2	12	no	—
155	C₂³	—	2	32	no	—
156	C₂³	C₂²⋊C₈×C₁₃₇	1	4	yes	—
157	C₂³	C₂×C₁₆×C₁₃₇	1	4	yes	—
158	C₂³	C₂×C₁₆×C₁₃₇	1	24	yes	complemented
159	C₂³	M₃₂×C₁₃₇	1	4	yes	—
160	C₂³	C₂²×C₈×C₁₃₇	1	4	yes	—
161	C₂³	C₂²×C₈×C₁₃₇	1	3	yes	—
162	C₂×C₄	—	2	12	no	—
163	C₂×C₄	—	2	24	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	12	yes	—
167	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	6	yes	—
168	C₂×C₄	C₂²×C₈×C₁₃₇	1	4	yes	—
169	C₂×C₄	C₂²×C₈×C₁₃₇	1	3	yes	—
170	C₂×C₄	C₂×M₁₆×C₁₃₇	1	6	yes	—
171	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
172	C₈	—	2	4	no	—
173	C₈	C₂×C₂²⋊C₄×C₁₃₇	1	6	yes	—
174	C₈	C₂²×D₄×C₁₃₇	1	2	yes	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	48	no	—
177	C₂²	—	2	4	no	—
178	C₂²	C₂²⋊C₁₆×C₁₃₇	1	16	yes	direct product factor
179	C₂²	C₂×C₂²⋊C₈×C₁₃₇	1	6	yes	—
180	C₂²	C₂²×C₁₆×C₁₃₇	1	6	yes	—
181	C₂²	C₂²×C₁₆×C₁₃₇	1	8	yes	complemented
182	C₂²	C₂×M₃₂×C₁₃₇	1	6	yes	—
183	C₂²	C₂³×C₈×C₁₃₇	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	C₂×C₂²⋊C₈×C₁₃₇	1	6	yes	—
186	C₄	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
187	C₄	C₂²×M₁₆×C₁₃₇	1	1	yes	characteristic
188	C₂	—	2	8	no	—
189	C₂	C₂×C₂²⋊C₁₆×C₁₃₇	1	12	yes	direct product factor
190	C₂	C₂²×C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
191	C₂	C₂³×C₁₆×C₁₃₇	1	1	yes	characteristic
192	C₂	C₂²×M₃₂×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈×C₁₃₇	C₂²
commutator	C₂	C₂³×C₁₆×C₁₃₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
socle	C₂⁴×C₁₃₇	C₂×C₈
prime radical	C₂⁵×C₁₃₇	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₁₆×C₁₃₇	C₂	semidirect
3. C₂×C₂²⋊C₁₆×C₁₃₇	C₂	direct (N × H)
11. C₂²×C₁₆×C₁₃₇	C₂²	semidirect
12. C₂²⋊C₁₆×C₁₃₇	C₂²	direct (N × H)
30. C₂×C₁₆×C₁₃₇	C₂³	semidirect
33. C₂⁴×C₁₃₇	C₁₆	semidirect
61. C₂³×C₁₃₇	C₂×C₁₆	semidirect
81. C₂²×C₁₃₇	C₂²⋊C₁₆	direct (N × H)
84. C₂²×C₁₃₇	C₂²×C₁₆	semidirect
92. C₂×C₁₃₇	C₂×C₂²⋊C₁₆	direct (N × H)
96. C₂²×C₂²⋊C₁₆	C₁₃₇	direct (N × H)
97. C₁₃₇	C₂²×C₂²⋊C₁₆	direct (N × H)
99. C₂³×C₁₆	C₂×C₁₃₇	semidirect
100. C₂×C₂²⋊C₁₆	C₂×C₁₃₇	direct (N × H)
108. C₂²×C₁₆	C₂²×C₁₃₇	semidirect
109. C₂²⋊C₁₆	C₂²×C₁₃₇	direct (N × H)
127. C₂×C₁₆	C₂³×C₁₃₇	semidirect
130. C₂⁴	C₁₆×C₁₃₇	semidirect
158. C₂³	C₂×C₁₆×C₁₃₇	semidirect
178. C₂²	C₂²⋊C₁₆×C₁₃₇	direct (N × H)
181. C₂²	C₂²×C₁₆×C₁₃₇	semidirect
189. C₂	C₂×C₂²⋊C₁₆×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₁₃₇	2
C₂³×C₁₆×C₁₃₇	2
C₂×C₂²⋊C₁₆×C₁₃₇	2
C₂²×C₂²⋊C₁₆	137

Subgroups — C22×C22⋊C16×C137 #46882716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊C₁₆×C₁₃₇ #46882716