Subgroups — C₂×M₁₆×#10587739 #31797343

Order 17536.

656 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#10587739	C₂	1	1	yes	maximal, characteristic
3	M₁₆×#10587739	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17984142	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#10587739	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×#10587739	C₄	1	2	yes	—
8	C₂×#14555520	C₄	1	2	yes	complemented
9	D₁₃₇⋊M₁₆	C₄	1	4	yes	complemented
10	D₁₃₇⋊M₁₆	C₂²	1	4	yes	—
11	C₂×#14555519	C₄	1	2	yes	complemented
12	C₂×#14555518	C₂²	1	2	yes	—
13	C₂×C₄×#10587739	C₄	1	4	yes	—
14	C₂×C₄×#10587739	C₄	1	2	yes	—
15	C₂×C₄×#10587739	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10587739	C₂²	1	1	yes	characteristic
17	C₂×C₄×#10587739	C₂²	1	4	yes	—
18	#17984142	C₂²	1	8	yes	complemented
19	C₈×#10587739	C₂²	1	8	yes	complemented
20	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×#14555494	C₄	1	1	yes	characteristic, complemented
22	C₂×#14555493	C₄	1	2	yes	complemented
23	M₁₆×D₁₃₇	C₄	1	4	yes	complemented
24	M₁₆×D₁₃₇	C₂²	1	4	yes	—
25	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
26	C₂×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
27	C₂²×#10587739	—	2	8	no	—
28	C₂²×#10587739	C₂×C₄	1	8	yes	—
29	C₂²×#10587739	C₂×C₄	1	2	yes	—
30	C₂²×#10587739	C₂×C₄	1	2	yes	—
31	#14555520	C₂×C₄	1	8	yes	complemented
32	#14555519	C₂×C₄	1	8	yes	complemented
33	#14555518	C₂×C₄	1	4	yes	complemented
34	#14555518	C₂³	1	4	yes	—
35	C₂×#11764047	C₂×C₄	1	4	yes	complemented
36	C₄×#10587739	C₂×C₄	1	2	yes	—
37	C₄×#10587739	C₂×C₄	1	8	yes	—
38	C₄×#10587739	C₂×C₄	1	2	yes	—
39	C₄×#10587739	C₂³	1	2	yes	—
40	C₄×#10587739	C₂³	1	2	yes	—
41	C₂³×D₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
51	#14555494	C₂×C₄	1	4	yes	complemented
52	#14555493	C₂×C₄	1	8	yes	complemented
53	C₂×#11764040	C₂×C₄	1	2	yes	complemented
54	C₈×D₁₃₇	C₂×C₄	1	4	yes	complemented
55	C₈×D₁₃₇	C₂³	1	4	yes	—
56	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
57	M₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
58	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	complemented
59	C₂×#10587739	—	2	4	no	—
60	C₂×#10587739	—	2	16	no	—
61	C₂×#10587739	M₁₆	1	8	yes	direct product factor
62	C₂×#10587739	C₂²×C₄	1	4	yes	—
63	C₂×#10587739	C₂²×C₄	1	4	yes	—
64	#11764047	C₂²×C₄	1	8	yes	complemented
65	C₂²×D₁₃₇	—	2	4	no	—
66	C₂²×D₁₃₇	C₄²	1	1	yes	characteristic
67	C₂²×D₁₃₇	C₄²	1	2	yes	—
68	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
69	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×Dic₁₃₇	C₄²	1	2	yes	—
71	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
72	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
74	C₄×D₁₃₇	C₄²	1	2	yes	—
75	C₄×D₁₃₇	C₄²	1	2	yes	—
76	C₄×D₁₃₇	C₄²	1	2	yes	—
77	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₄×D₁₃₇	C₂²×C₄	1	2	yes	—
79	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
80	C₄×D₁₃₇	C₂²×C₄	1	2	yes	—
81	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₁₃₇	C₂²×C₄	1	2	yes	—
83	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
84	#11764040	C₂²×C₄	1	4	yes	complemented
85	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
86	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
87	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
88	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
90	C₈×C₁₃₇	C₂²×C₄	1	4	yes	complemented
91	C₂²×C₁₃₇	—	2	2	no	—
92	C₂²×C₁₃₇	C₂×C₄²	1	2	yes	—
93	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
94	C₂×D₁₃₇	—	2	2	no	—
95	C₂×D₁₃₇	—	2	4	no	—
96	C₂×D₁₃₇	—	2	4	no	—
97	C₂×D₁₃₇	C₂×C₄²	1	2	yes	—
98	C₂×D₁₃₇	C₂×C₄²	1	1	yes	characteristic
99	C₂×D₁₃₇	C₂×M₁₆	1	4	yes	—
100	C₂×D₁₃₇	C₂³×C₄	1	1	yes	characteristic
101	#10587739	—	2	4	no	—
102	#10587739	C₂×M₁₆	1	8	yes	direct product factor
103	C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
104	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
105	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
106	Dic₁₃₇	C₂×C₄²	1	1	yes	characteristic
107	Dic₁₃₇	C₂×C₄²	1	2	yes	—
108	Dic₁₃₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₁₃₇	—	2	2	no	—
110	C₂×C₁₃₇	C₄×M₁₆	1	2	yes	complemented
111	C₂×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
112	D₁₃₇	—	2	2	no	—
113	D₁₃₇	C₄×M₁₆	1	2	yes	complemented
114	D₁₃₇	C₂²×M₁₆	1	2	yes	—
115	C₁₃₇	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₄×M₁₆	—	137	1	no	maximal, Sylow
117	C₂²×M₁₆	—	137	1	no	—
118	C₂²×M₁₆	—	137	1	no	—
119	C₂²×C₄²	—	137	1	no	—
120	C₄×M₁₆	—	137	8	no	—
121	C₂×#97	—	137	2	no	—
122	C₂×C₄×C₈	—	137	2	no	—
123	C₂³×C₄	—	137	2	no	—
124	C₂³×C₄	—	137	1	no	—
125	C₂×M₁₆	—	137	4	no	—
126	C₂×M₁₆	—	137	2	no	—
127	C₂×M₁₆	—	137	4	no	—
128	C₂×M₁₆	—	137	2	no	—
129	C₂×M₁₆	—	137	1	no	—
130	C₂×M₁₆	—	137	4	no	—
131	C₂×M₁₆	—	137	4	no	—
132	C₂×M₁₆	—	137	2	no	—
133	C₂×M₁₆	#10587739	1	1	yes	characteristic, direct product factor
134	C₂²×C₈	—	137	2	no	—
135	C₂²×C₈	—	137	2	no	—
136	C₂×C₄²	—	137	1	no	—
137	C₂×C₄²	—	137	1	no	—
138	C₂×C₄²	—	137	4	no	—
139	C₂×C₄²	—	137	4	no	—
140	C₂×C₄²	—	137	2	no	—
141	#97	—	137	8	no	—
142	C₄×C₈	—	137	8	no	—
143	C₂⁴	—	137	1	no	—
144	C₂²×C₄	—	274	8	no	—
145	C₂²×C₄	—	137	2	no	—
146	C₂²×C₄	—	137	8	no	—
147	C₂²×C₄	—	137	2	no	—
148	C₂²×C₄	—	137	1	no	—
149	C₂²×C₄	—	137	2	no	—
150	C₂²×C₄	—	137	1	no	—
151	C₂²×C₄	—	137	1	no	—
152	C₂²×C₄	—	137	2	no	—
153	C₂²×C₄	—	137	2	no	—
154	C₂²×C₄	—	137	2	no	—
155	C₂²×C₄	—	137	1	no	—
156	C₂²×C₄	—	137	1	no	—
157	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
158	M₁₆	—	137	8	no	—
159	M₁₆	—	137	8	no	—
160	M₁₆	—	137	4	no	—
161	M₁₆	—	137	8	no	—
162	M₁₆	C₂×#10587739	1	4	yes	direct product factor
163	C₂×C₈	—	137	4	no	—
164	C₂×C₈	—	137	4	no	—
165	C₂×C₈	—	137	4	no	—
166	C₂×C₈	—	137	2	no	—
167	C₂×C₈	—	137	4	no	—
168	C₂×C₈	—	137	4	no	—
169	C₂×C₈	C₂×#10587739	1	2	yes	complemented
170	C₄²	—	137	2	no	—
171	C₄²	—	137	2	no	—
172	C₄²	—	137	2	no	—
173	C₄²	—	137	8	no	—
174	C₄²	—	137	2	no	—
175	C₂³	—	274	4	no	—
176	C₂³	—	137	2	no	—
177	C₂³	—	137	2	no	—
178	C₂³	—	137	1	no	—
179	C₂³	—	137	1	no	—
180	C₂³	C₄×#10587739	1	1	yes	characteristic
181	C₂×C₄	—	274	4	no	—
182	C₂×C₄	—	274	16	no	—
183	C₂×C₄	—	137	4	no	—
184	C₂×C₄	—	137	4	no	—
185	C₂×C₄	—	137	8	no	—
186	C₂×C₄	—	137	1	no	—
187	C₂×C₄	—	137	2	no	—
188	C₂×C₄	—	137	2	no	—
189	C₂×C₄	—	137	1	no	—
190	C₂×C₄	—	137	2	no	—
191	C₂×C₄	—	137	1	no	—
192	C₂×C₄	—	137	1	no	—
193	C₂×C₄	—	137	2	no	—
194	C₂×C₄	—	137	2	no	—
195	C₂×C₄	—	137	2	no	—
196	C₂×C₄	—	137	2	no	—
197	C₂×C₄	—	137	1	no	—
198	C₂×C₄	—	137	1	no	—
199	C₂×C₄	—	137	2	no	—
200	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
201	C₂×C₄	C₄×#10587739	1	2	yes	—
202	C₂×C₄	C₂²×#10587739	1	2	yes	—
203	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
204	C₈	—	137	8	no	—
205	C₈	—	137	4	no	—
206	C₈	C₂²×#10587739	1	4	yes	complemented
207	C₂²	—	2	2	no	—
208	C₂²	—	274	4	no	—
209	C₂²	—	274	4	no	—
210	C₂²	—	274	2	no	—
211	C₂²	—	137	2	no	—
212	C₂²	—	137	1	no	—
213	C₂²	—	137	1	no	—
214	C₂²	—	137	4	no	—
215	C₂²	C₂×C₄×#10587739	1	2	yes	—
216	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
217	C₄	—	274	4	no	—
218	C₄	—	137	8	no	—
219	C₄	—	137	1	no	—
220	C₄	—	137	1	no	—
221	C₄	—	137	2	no	—
222	C₄	C₂×C₄×#10587739	1	2	yes	—
223	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
224	C₄	C₂³×#10587739	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	—	274	2	no	—
227	C₂	—	137	2	no	—
228	C₂	—	137	2	no	—
229	C₂	M₁₆×#10587739	1	2	yes	direct product factor
230	C₂	C₂²×C₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#10587739
commutator	C₂×C₁₃₇	C₂²×C₄²
Frattini	C₄	C₂³×#10587739
Fitting	C₂×M₁₆×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄²
prime radical	C₂³×D₁₃₇	C₂×C₄
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	C₂×M₁₆	#10587739
hypocenter	C₁₃₇	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. M₁₆×#10587739	C₂	direct (N × H)
4. C₂×#17984142	C₂	semidirect
5. C₂×C₈×#10587739	C₂	semidirect
8. C₂×#14555520	C₄	semidirect
9. D₁₃₇⋊M₁₆	C₄	semidirect
11. C₂×#14555519	C₄	semidirect
18. #17984142	C₂²	semidirect
19. C₈×#10587739	C₂²	semidirect
21. C₂×#14555494	C₄	semidirect
22. C₂×#14555493	C₄	semidirect
23. M₁₆×D₁₃₇	C₄	semidirect
26. C₂×M₁₆×C₁₃₇	C₄	semidirect
31. #14555520	C₂×C₄	semidirect
32. #14555519	C₂×C₄	semidirect
33. #14555518	C₂×C₄	semidirect
35. C₂×#11764047	C₂×C₄	semidirect
51. #14555494	C₂×C₄	semidirect
52. #14555493	C₂×C₄	semidirect
53. C₂×#11764040	C₂×C₄	semidirect
54. C₈×D₁₃₇	C₂×C₄	semidirect
57. M₁₆×C₁₃₇	C₂×C₄	semidirect
58. C₂×C₈×C₁₃₇	C₂×C₄	semidirect
61. C₂×#10587739	M₁₆	direct (N × H)
64. #11764047	C₂²×C₄	semidirect
84. #11764040	C₂²×C₄	semidirect
90. C₈×C₁₃₇	C₂²×C₄	semidirect
102. #10587739	C₂×M₁₆	direct (N × H)
110. C₂×C₁₃₇	C₄×M₁₆	semidirect
113. D₁₃₇	C₄×M₁₆	semidirect
115. C₁₃₇	C₂×C₄×M₁₆	semidirect
133. C₂×M₁₆	#10587739	direct (N × H)
162. M₁₆	C₂×#10587739	direct (N × H)
169. C₂×C₈	C₂×#10587739	semidirect
206. C₈	C₂²×#10587739	semidirect
229. C₂	M₁₆×#10587739	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×D₁₃₇⋊M₁₆	2
C₂²×C₄×#10587739	2
M₁₆×#10587739	2
C₂×#17984142	2
C₂×C₈×#10587739	2
C₂×M₁₆×D₁₃₇	2
C₂×C₄×M₁₆	137

Subgroups — C2×M16×#10587739 #31797343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₁₆×#10587739 #31797343