Subgroups — C₁₆×SD₁₆×C₁₃₇ #46861434

Order 35072.

238 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	Q₈×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₈×SD₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1367×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₈×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₆×C₁₃₇	—	2	1	no	—
9	C₈×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₄×SD₁₆×C₁₃₇	C₄	1	1	yes	characteristic
11	C₈×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₁₃₇	—	2	1	no	—
13	C₄⋊C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂²⋊C₁₆×C₁₃₇	—	2	1	no	—
16	C₄×C₁₆×C₁₃₇	—	2	1	no	—
17	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
19	#334×C₁₃₇	C₄	1	1	yes	characteristic
20	Q₈⋊C₈×C₁₃₇	C₄	1	1	yes	characteristic
21	D₄⋊C₈×C₁₃₇	C₄	1	1	yes	characteristic
22	C₈²×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×SD₁₆×C₁₃₇	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₁₃₇	—	2	1	no	—
25	C₄×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₁₃₇	—	2	1	no	—
28	C₂×C₁₆×C₁₃₇	—	2	1	no	—
29	C₂×C₁₆×C₁₃₇	—	2	1	no	—
30	C₂×C₁₆×C₁₃₇	—	4	2	no	—
31	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
33	#106×C₁₃₇	C₈	1	1	yes	characteristic
34	C₄⋊C₈×C₁₃₇	—	2	1	no	—
35	C₄⋊C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	#103×C₁₃₇	C₈	1	1	yes	characteristic
38	#102×C₁₃₇	C₈	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
40	C₄×C₈×C₁₃₇	—	2	1	no	—
41	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
45	C₂×D₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₇	—	2	1	no	—
47	SD₁₆×C₁₃₇	C₁₆	1	4	yes	direct product factor
48	C₂×C₈×C₁₃₇	—	2	1	no	—
49	C₂×C₈×C₁₃₇	—	2	1	no	—
50	C₂×C₈×C₁₃₇	—	2	1	no	—
51	C₂×C₈×C₁₃₇	—	4	1	no	—
52	C₂×C₈×C₁₃₇	—	4	1	no	—
53	C₂×C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
54	C₂×C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
55	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₁₃₇	—	2	1	no	—
58	C₄⋊C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
59	C₄⋊C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
60	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
61	C₄²×C₁₃₇	—	2	1	no	—
62	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
63	C₁₆×C₁₃₇	—	2	2	no	—
64	C₁₆×C₁₃₇	—	2	1	no	—
65	C₁₆×C₁₃₇	—	4	1	no	—
66	C₁₆×C₁₃₇	—	4	1	no	—
67	C₁₆×C₁₃₇	SD₁₆	1	2	yes	direct product factor
68	C₂³×C₁₃₇	—	2	1	no	—
69	Q₈×C₁₃₇	—	2	1	no	—
70	Q₈×C₁₃₇	C₂×C₁₆	1	2	yes	complemented
71	D₄×C₁₃₇	—	2	1	no	—
72	D₄×C₁₃₇	C₂×C₁₆	1	2	yes	—
73	C₂×C₄×C₁₃₇	—	2	1	no	—
74	C₂×C₄×C₁₃₇	—	2	1	no	—
75	C₂×C₄×C₁₃₇	—	2	1	no	—
76	C₂×C₄×C₁₃₇	—	4	1	no	—
77	C₂×C₄×C₁₃₇	—	4	1	no	—
78	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃₇	#131	1	1	yes	characteristic
81	C₈×C₁₃₇	—	2	2	no	—
82	C₈×C₁₃₇	—	2	1	no	—
83	C₈×C₁₃₇	—	2	1	no	—
84	C₈×C₁₃₇	—	4	1	no	—
85	C₈×C₁₃₇	—	4	1	no	—
86	C₈×C₁₃₇	C₂×C₁₆	1	2	yes	complemented
87	C₈×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₈×C₁₃₇	#135	1	1	yes	characteristic
89	C₂²×C₁₃₇	—	2	2	no	—
90	C₂²×C₁₃₇	—	4	1	no	—
91	C₂²×C₁₃₇	—	4	1	no	—
92	C₂²×C₁₃₇	C₈×D₄	1	1	yes	characteristic
93	C₄×C₁₃₇	—	2	2	no	—
94	C₄×C₁₃₇	—	2	1	no	—
95	C₄×C₁₃₇	—	4	1	no	—
96	C₄×C₁₃₇	—	4	1	no	—
97	C₄×C₁₃₇	C₄×SD₁₆	1	1	yes	characteristic
98	C₄×C₁₃₇	#443	1	1	yes	characteristic
99	C₄×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
100	C₄×C₁₃₇	#504	1	1	yes	characteristic
101	C₂×C₁₃₇	—	4	2	no	—
102	C₂×C₁₃₇	C₈×SD₁₆	1	1	yes	characteristic
103	C₂×C₁₃₇	D₄×C₁₆	1	1	yes	characteristic
104	C₂×C₁₃₇	#2170	1	1	yes	characteristic
105	C₁₆×SD₁₆	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₁₃₇	C₁₆×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
107	Q₈×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
108	D₄×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic
109	C₈×SD₁₆	C₂×C₁₃₇	1	1	yes	characteristic
110	#1367	C₂×C₁₃₇	1	1	yes	characteristic, complemented
111	Q₈⋊C₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
112	D₄⋊C₁₆	C₂×C₁₃₇	1	1	yes	characteristic
113	C₈×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
114	C₂²×C₁₆	—	2	1	no	—
115	C₈×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
116	C₄×SD₁₆	C₄×C₁₃₇	1	1	yes	characteristic
117	C₈×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
118	C₄⋊C₁₆	—	2	1	no	—
119	C₄⋊C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
120	C₄⋊C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
121	C₂²⋊C₁₆	—	2	1	no	—
122	C₄×C₁₆	—	2	1	no	—
123	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
124	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
125	#334	C₄×C₁₃₇	1	1	yes	characteristic
126	Q₈⋊C₈	C₄×C₁₃₇	1	1	yes	characteristic
127	D₄⋊C₈	C₄×C₁₃₇	1	1	yes	characteristic
128	C₈²	C₂²×C₁₃₇	1	1	yes	characteristic
129	C₂×SD₁₆	C₈×C₁₃₇	1	1	yes	characteristic
130	C₂²×C₈	—	2	1	no	—
131	C₄×Q₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
132	C₄×D₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
133	C₂×C₁₆	—	2	1	no	—
134	C₂×C₁₆	—	2	1	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	4	2	no	—
137	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
138	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
139	#106	C₈×C₁₃₇	1	1	yes	characteristic
140	C₄⋊C₈	—	2	1	no	—
141	C₄⋊C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
142	C₄⋊C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
143	#103	C₈×C₁₃₇	1	1	yes	characteristic
144	#102	C₈×C₁₃₇	1	1	yes	characteristic
145	C₂²⋊C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
148	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
149	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
150	C₂×Q₈	C₂×C₈×C₁₃₇	1	1	yes	characteristic
151	C₂×D₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
152	C₂²×C₄	—	2	1	no	—
153	SD₁₆	C₁₆×C₁₃₇	1	4	yes	direct product factor
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	C₂×C₈×C₁₃₇	1	1	yes	characteristic
160	C₂×C₈	C₂×C₈×C₁₃₇	1	1	yes	characteristic
161	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
162	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
165	C₄⋊C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
166	C₂²⋊C₄	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
169	C₁₆	—	2	2	no	—
170	C₁₆	—	2	1	no	—
171	C₁₆	—	4	1	no	—
172	C₁₆	—	4	1	no	—
173	C₁₆	SD₁₆×C₁₃₇	1	2	yes	direct product factor
174	C₂³	—	2	1	no	—
175	Q₈	—	2	1	no	—
176	Q₈	C₂×C₁₆×C₁₃₇	1	2	yes	complemented
177	D₄	—	2	1	no	—
178	D₄	C₂×C₁₆×C₁₃₇	1	2	yes	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
186	C₂×C₄	#131×C₁₃₇	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	4	1	no	—
191	C₈	—	4	1	no	—
192	C₈	C₂×C₁₆×C₁₃₇	1	2	yes	complemented
193	C₈	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
194	C₈	#135×C₁₃₇	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	4	1	no	—
198	C₂²	C₈×D₄×C₁₃₇	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	1	no	—
201	C₄	—	4	1	no	—
202	C₄	—	4	1	no	—
203	C₄	C₄×SD₁₆×C₁₃₇	1	1	yes	characteristic
204	C₄	#443×C₁₃₇	1	1	yes	characteristic
205	C₄	C₂²×C₁₆×C₁₃₇	1	1	yes	characteristic
206	C₄	#504×C₁₃₇	1	1	yes	characteristic
207	C₂	—	4	2	no	—
208	C₂	C₈×SD₁₆×C₁₃₇	1	1	yes	characteristic
209	C₂	D₄×C₁₆×C₁₃₇	1	1	yes	characteristic
210	C₂	#2170×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆×C₁₃₇	D₄
commutator	C₄	C₂²×C₁₆×C₁₃₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₈×D₄
prime radical	C₂×D₄×C₁₃₇	C₂×C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₃₇ ◁ C₄×C₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×C₁₆×C₁₃₇	C₂	semidirect
4. #1367×C₁₃₇	C₂	semidirect
5. Q₈⋊C₁₆×C₁₃₇	C₂	semidirect
7. C₈×C₁₆×C₁₃₇	C₂	semidirect
47. SD₁₆×C₁₃₇	C₁₆	direct (N × H)
67. C₁₆×C₁₃₇	SD₁₆	direct (N × H)
70. Q₈×C₁₃₇	C₂×C₁₆	semidirect
86. C₈×C₁₃₇	C₂×C₁₆	semidirect
105. C₁₆×SD₁₆	C₁₃₇	direct (N × H)
106. C₁₃₇	C₁₆×SD₁₆	direct (N × H)
107. Q₈×C₁₆	C₂×C₁₃₇	semidirect
110. #1367	C₂×C₁₃₇	semidirect
111. Q₈⋊C₁₆	C₂×C₁₃₇	semidirect
113. C₈×C₁₆	C₂×C₁₃₇	semidirect
153. SD₁₆	C₁₆×C₁₃₇	direct (N × H)
173. C₁₆	SD₁₆×C₁₃₇	direct (N × H)
176. Q₈	C₂×C₁₆×C₁₃₇	semidirect
192. C₈	C₂×C₁₆×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₁₆×C₁₃₇	2
D₄×C₁₆×C₁₃₇	2
C₈×SD₁₆×C₁₃₇	2
#1367×C₁₃₇	2
Q₈⋊C₁₆×C₁₃₇	2
D₄⋊C₁₆×C₁₃₇	2
C₈×C₁₆×C₁₃₇	2
C₁₆×SD₁₆	137

Subgroups — C16×SD16×C137 #46861434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×SD₁₆×C₁₃₇ #46861434