Subgroups — M₁₆²×C₁₃₇ #46866719

Order 35072.

900 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2920×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₁₃₇	C₂	1	2	yes	maximal
3	#1455×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	#1451×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	C₈×M₁₆×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₁₃₇	C₄	1	4	yes	—
7	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
8	#432×C₁₃₇	C₄	1	4	yes	—
9	#431×C₁₃₇	C₂²	1	4	yes	—
10	#423×C₁₃₇	C₄	1	2	yes	—
11	#407×C₁₃₇	C₄	1	2	yes	—
12	C₄×M₁₆×C₁₃₇	C₄	1	4	yes	—
13	C₄×M₁₆×C₁₃₇	C₄	1	4	yes	—
14	C₄×M₁₆×C₁₃₇	C₂²	1	2	yes	—
15	C₄×M₁₆×C₁₃₇	C₂²	1	2	yes	—
16	C₄×M₁₆×C₁₃₇	C₂²	1	4	yes	—
17	C₄×M₁₆×C₁₃₇	C₂²	1	2	yes	—
18	C₂×#97×C₁₃₇	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	4	yes	—
20	#322×C₁₃₇	C₂²	1	8	yes	complemented
21	#322×C₁₃₇	C₂²	1	4	yes	complemented
22	C₈²×C₁₃₇	C₂²	1	4	yes	complemented
23	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×M₁₆×C₁₃₇	—	2	16	no	—
26	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₁₃₇	C₂×C₄	1	8	yes	—
28	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
37	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
39	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
40	#97×C₁₃₇	C₂×C₄	1	4	yes	—
41	#97×C₁₃₇	C₂×C₄	1	8	yes	—
42	#97×C₁₃₇	C₂×C₄	1	4	yes	—
43	#97×C₁₃₇	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₁₃₇	C₂³	1	4	yes	—
46	C₄×C₈×C₁₃₇	C₂³	1	4	yes	—
47	C₄×C₈×C₁₃₇	C₂³	1	4	yes	—
48	C₂⁴×C₁₃₇	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₇	—	2	8	no	—
50	C₂²×C₄×C₁₃₇	—	4	2	no	—
51	C₂²×C₄×C₁₃₇	C₄²	1	2	yes	—
52	C₂²×C₄×C₁₃₇	C₄²	1	2	yes	—
53	C₂²×C₄×C₁₃₇	C₄²	1	2	yes	—
54	C₂²×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₇	C₄²	1	2	yes	—
56	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
61	M₁₆×C₁₃₇	—	2	8	no	—
62	M₁₆×C₁₃₇	—	2	16	no	—
63	M₁₆×C₁₃₇	—	4	4	no	—
64	M₁₆×C₁₃₇	M₁₆	1	16	yes	direct product factor
65	C₂×C₈×C₁₃₇	—	2	16	no	—
66	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
67	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
68	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
69	C₄²×C₁₃₇	C₄²	1	1	yes	characteristic
70	C₄²×C₁₃₇	C₄²	1	2	yes	—
71	C₄²×C₁₃₇	C₄²	1	2	yes	—
72	C₄²×C₁₃₇	C₄²	1	1	yes	characteristic
73	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
79	C₂³×C₁₃₇	—	2	4	no	—
80	C₂³×C₁₃₇	—	4	1	no	—
81	C₂³×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
82	C₂³×C₁₃₇	C₂×C₄²	1	2	yes	—
83	C₂×C₄×C₁₃₇	—	2	4	no	—
84	C₂×C₄×C₁₃₇	—	2	4	no	—
85	C₂×C₄×C₁₃₇	—	2	4	no	—
86	C₂×C₄×C₁₃₇	—	2	2	no	—
87	C₂×C₄×C₁₃₇	—	2	4	no	—
88	C₂×C₄×C₁₃₇	—	2	4	no	—
89	C₂×C₄×C₁₃₇	—	2	4	no	—
90	C₂×C₄×C₁₃₇	—	4	2	no	—
91	C₂×C₄×C₁₃₇	—	4	1	no	—
92	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
94	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
97	C₂×C₄×C₁₃₇	C₂×M₁₆	1	8	yes	—
98	C₂×C₄×C₁₃₇	C₂³×C₄	1	2	yes	—
99	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
100	C₈×C₁₃₇	—	2	8	no	—
101	C₈×C₁₃₇	—	2	8	no	—
102	C₈×C₁₃₇	C₂×M₁₆	1	16	yes	complemented
103	C₂²×C₁₃₇	—	2	2	no	—
104	C₂²×C₁₃₇	—	2	2	no	—
105	C₂²×C₁₃₇	—	2	2	no	—
106	C₂²×C₁₃₇	—	2	1	no	—
107	C₂²×C₁₃₇	—	4	4	no	—
108	C₂²×C₁₃₇	C₄×M₁₆	1	4	yes	—
109	C₂²×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
110	C₄×C₁₃₇	—	2	2	no	—
111	C₄×C₁₃₇	—	2	2	no	—
112	C₄×C₁₃₇	—	2	1	no	—
113	C₄×C₁₃₇	—	2	2	no	—
114	C₄×C₁₃₇	C₄×M₁₆	1	4	yes	—
115	C₄×C₁₃₇	C₂²×M₁₆	1	4	yes	—
116	C₄×C₁₃₇	#568	1	2	yes	—
117	C₂×C₁₃₇	—	2	4	no	—
118	C₂×C₁₃₇	—	4	1	no	—
119	C₂×C₁₃₇	C₂×C₄×M₁₆	1	2	yes	—
120	C₂×C₁₃₇	#2873	1	1	yes	characteristic
121	M₁₆²	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₁₃₇	M₁₆²	1	1	yes	characteristic, direct product factor, Sylow
123	#2920	C₂×C₁₃₇	1	1	yes	characteristic
124	C₂×C₄×M₁₆	C₂×C₁₃₇	1	2	yes	—
125	#1455	C₂×C₁₃₇	1	4	yes	complemented
126	#1451	C₂×C₁₃₇	1	4	yes	complemented
127	C₈×M₁₆	C₂×C₁₃₇	1	4	yes	complemented
128	C₂²×M₁₆	C₄×C₁₃₇	1	4	yes	—
129	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
130	#432	C₄×C₁₃₇	1	4	yes	—
131	#431	C₂²×C₁₃₇	1	4	yes	—
132	#423	C₄×C₁₃₇	1	2	yes	—
133	#407	C₄×C₁₃₇	1	2	yes	—
134	C₄×M₁₆	C₄×C₁₃₇	1	4	yes	—
135	C₄×M₁₆	C₄×C₁₃₇	1	4	yes	—
136	C₄×M₁₆	C₂²×C₁₃₇	1	2	yes	—
137	C₄×M₁₆	C₂²×C₁₃₇	1	2	yes	—
138	C₄×M₁₆	C₂²×C₁₃₇	1	4	yes	—
139	C₄×M₁₆	C₂²×C₁₃₇	1	2	yes	—
140	C₂×#97	C₄×C₁₃₇	1	4	yes	—
141	C₂×C₄×C₈	C₂²×C₁₃₇	1	4	yes	—
142	#322	C₂²×C₁₃₇	1	8	yes	complemented
143	#322	C₂²×C₁₃₇	1	4	yes	complemented
144	C₈²	C₂²×C₁₃₇	1	4	yes	complemented
145	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
146	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
147	C₂×M₁₆	—	2	16	no	—
148	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
149	C₂×M₁₆	C₂×C₄×C₁₃₇	1	8	yes	—
150	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
151	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
152	C₂²×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
153	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
154	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
156	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
157	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
158	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
159	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
160	C₄⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
161	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
162	#97	C₂×C₄×C₁₃₇	1	4	yes	—
163	#97	C₂×C₄×C₁₃₇	1	8	yes	—
164	#97	C₂×C₄×C₁₃₇	1	4	yes	—
165	#97	C₂×C₄×C₁₃₇	1	4	yes	—
166	C₄×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
167	C₄×C₈	C₂³×C₁₃₇	1	4	yes	—
168	C₄×C₈	C₂³×C₁₃₇	1	4	yes	—
169	C₄×C₈	C₂³×C₁₃₇	1	4	yes	—
170	C₂⁴	C₄²×C₁₃₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	8	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₄²×C₁₃₇	1	2	yes	—
174	C₂²×C₄	C₄²×C₁₃₇	1	2	yes	—
175	C₂²×C₄	C₄²×C₁₃₇	1	2	yes	—
176	C₂²×C₄	C₄²×C₁₃₇	1	1	yes	characteristic
177	C₂²×C₄	C₄²×C₁₃₇	1	2	yes	—
178	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
180	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
181	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
182	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
183	M₁₆	—	2	8	no	—
184	M₁₆	—	2	16	no	—
185	M₁₆	—	4	4	no	—
186	M₁₆	M₁₆×C₁₃₇	1	16	yes	direct product factor
187	C₂×C₈	—	2	16	no	—
188	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
189	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
190	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
191	C₄²	C₄²×C₁₃₇	1	1	yes	characteristic
192	C₄²	C₄²×C₁₃₇	1	2	yes	—
193	C₄²	C₄²×C₁₃₇	1	2	yes	—
194	C₄²	C₄²×C₁₃₇	1	1	yes	characteristic
195	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
196	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
197	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
198	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
199	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
200	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
201	C₂³	—	2	4	no	—
202	C₂³	—	4	1	no	—
203	C₂³	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
204	C₂³	C₂×C₄²×C₁₃₇	1	2	yes	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
215	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
216	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
217	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
219	C₂×C₄	C₂×M₁₆×C₁₃₇	1	8	yes	—
220	C₂×C₄	C₂³×C₄×C₁₃₇	1	2	yes	—
221	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
222	C₈	—	2	8	no	—
223	C₈	—	2	8	no	—
224	C₈	C₂×M₁₆×C₁₃₇	1	16	yes	complemented
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	4	4	no	—
230	C₂²	C₄×M₁₆×C₁₃₇	1	4	yes	—
231	C₂²	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	2	no	—
236	C₄	C₄×M₁₆×C₁₃₇	1	4	yes	—
237	C₄	C₂²×M₁₆×C₁₃₇	1	4	yes	—
238	C₄	#568×C₁₃₇	1	2	yes	—
239	C₂	—	2	4	no	—
240	C₂	—	4	1	no	—
241	C₂	C₂×C₄×M₁₆×C₁₃₇	1	2	yes	—
242	C₂	#2873×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₁₃₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	C₂²×C₄²
prime radical	C₂⁴×C₁₃₇	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1455×C₁₃₇	C₂	semidirect
4. #1451×C₁₃₇	C₂	semidirect
5. C₈×M₁₆×C₁₃₇	C₂	semidirect
20. #322×C₁₃₇	C₂²	semidirect
21. #322×C₁₃₇	C₂²	semidirect
22. C₈²×C₁₃₇	C₂²	semidirect
64. M₁₆×C₁₃₇	M₁₆	direct (N × H)
102. C₈×C₁₃₇	C₂×M₁₆	semidirect
121. M₁₆²	C₁₃₇	direct (N × H)
122. C₁₃₇	M₁₆²	direct (N × H)
125. #1455	C₂×C₁₃₇	semidirect
126. #1451	C₂×C₁₃₇	semidirect
127. C₈×M₁₆	C₂×C₁₃₇	semidirect
142. #322	C₂²×C₁₃₇	semidirect
143. #322	C₂²×C₁₃₇	semidirect
144. C₈²	C₂²×C₁₃₇	semidirect
186. M₁₆	M₁₆×C₁₃₇	direct (N × H)
224. C₈	C₂×M₁₆×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2920×C₁₃₇	2
C₂×C₄×M₁₆×C₁₃₇	2
#1455×C₁₃₇	2
#1451×C₁₃₇	2
C₈×M₁₆×C₁₃₇	2
M₁₆²	137

Subgroups — M162×C137 #46866719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆²×C₁₃₇ #46866719