Subgroups — C₄×#405×C₁₃₇ #46869716

Order 35072.

1526 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405×C₁₃₇	C₂	1	4	yes	maximal
2	C₂×C₄×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1726×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	C₄×#97×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	C₄²×C₈×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	#405×C₁₃₇	C₄	1	32	yes	direct product factor
13	C₄×M₁₆×C₁₃₇	C₄	1	8	yes	complemented
14	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#97×C₁₃₇	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₁₃₇	C₄	1	8	yes	complemented
17	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	8	yes	—
21	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₃₇	C₄	1	4	yes	—
23	C₄×C₂²⋊C₄×C₁₃₇	C₄	1	4	yes	—
24	#376×C₁₃₇	C₂²	1	2	yes	—
25	C₄³×C₁₃₇	C₂²	1	2	yes	—
26	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₃₇	C₂×C₄	1	8	yes	—
29	C₂×M₁₆×C₁₃₇	C₂×C₄	1	16	yes	—
30	C₂²×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₁₃₇	C₂×C₄	1	16	yes	—
32	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
34	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
35	#117×C₁₃₇	C₂×C₄	1	16	yes	—
36	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₃₇	C₂×C₄	1	8	yes	—
39	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
45	#97×C₁₃₇	C₂×C₄	1	8	yes	—
46	#97×C₁₃₇	C₂×C₄	1	32	yes	complemented
47	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
51	C₄×C₈×C₁₃₇	C₂×C₄	1	32	yes	complemented
52	#95×C₁₃₇	C₂×C₄	1	8	yes	—
53	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₇	—	2	8	no	—
55	C₂²×C₄×C₁₃₇	C₄²	1	8	yes	—
56	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
64	M₁₆×C₁₃₇	C₄²	1	32	yes	complemented
65	C₂×C₈×C₁₃₇	C₄²	1	32	yes	complemented
66	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
69	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
70	C₂×C₈×C₁₃₇	C₂²×C₄	1	16	yes	—
71	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
72	C₄⋊C₄×C₁₃₇	C₄²	1	16	yes	—
73	C₂²⋊C₄×C₁₃₇	C₄²	1	16	yes	—
74	C₄²×C₁₃₇	—	2	16	no	—
75	C₄²×C₁₃₇	C₄²	1	8	yes	—
76	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₁₃₇	C₂²×C₄	1	16	yes	—
80	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
81	C₂³×C₁₃₇	—	2	4	no	—
82	C₂³×C₁₃₇	C₂×C₄²	1	4	yes	—
83	C₂³×C₁₃₇	C₂×C₄²	1	2	yes	—
84	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₇	—	2	16	no	—
86	C₂×C₄×C₁₃₇	—	2	4	no	—
87	C₂×C₄×C₁₃₇	—	2	16	no	—
88	C₂×C₄×C₁₃₇	C₂×C₄²	1	4	yes	—
89	C₂×C₄×C₁₃₇	C₂×C₄²	1	16	yes	—
90	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₁₃₇	C₂×C₄²	1	4	yes	—
92	C₂×C₄×C₁₃₇	C₂×C₄²	1	8	yes	—
93	C₂×C₄×C₁₃₇	C₂×C₄²	1	8	yes	—
94	C₂×C₄×C₁₃₇	C₂×C₄²	1	4	yes	—
95	C₂×C₄×C₁₃₇	#131	1	8	yes	—
96	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₇	C₂³×C₄	1	4	yes	—
99	C₈×C₁₃₇	C₂×C₄²	1	8	yes	—
100	C₈×C₁₃₇	C₂×C₄²	1	8	yes	—
101	C₈×C₁₃₇	C₂×C₄²	1	16	yes	—
102	C₂²×C₁₃₇	—	2	4	no	—
103	C₂²×C₁₃₇	—	2	8	no	—
104	C₂²×C₁₃₇	C₄³	1	4	yes	—
105	C₂²×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
106	C₂²×C₁₃₇	C₂²×C₄²	1	2	yes	—
107	C₂²×C₁₃₇	C₂×#131	1	4	yes	—
108	C₄×C₁₃₇	—	2	4	no	—
109	C₄×C₁₃₇	—	2	16	no	—
110	C₄×C₁₃₇	C₄³	1	4	yes	—
111	C₄×C₁₃₇	#405	1	8	yes	direct product factor
112	C₄×C₁₃₇	C₂²×C₄²	1	2	yes	—
113	C₄×C₁₃₇	C₂²×C₄²	1	2	yes	—
114	C₂×C₁₃₇	—	2	4	no	—
115	C₂×C₁₃₇	C₂×C₄³	1	1	yes	characteristic
116	C₂×C₁₃₇	C₂×#405	1	2	yes	—
117	C₂×C₁₃₇	C₄×#131	1	4	yes	—
118	C₄×#405	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₁₃₇	C₄×#405	1	1	yes	characteristic, direct product factor, Sylow
120	C₂×#405	C₂×C₁₃₇	1	4	yes	—
121	C₂×C₄×M₁₆	C₂×C₁₃₇	1	1	yes	characteristic
122	C₂²×C₄×C₈	C₂×C₁₃₇	1	1	yes	characteristic
123	C₄×#117	C₂×C₁₃₇	1	1	yes	characteristic
124	#1726	C₂×C₁₃₇	1	4	yes	complemented
125	C₄×#97	C₂×C₁₃₇	1	2	yes	complemented
126	C₄²×C₈	C₂×C₁₃₇	1	2	yes	complemented
127	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
128	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
129	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
130	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
131	#405	C₄×C₁₃₇	1	32	yes	direct product factor
132	C₄×M₁₆	C₄×C₁₃₇	1	8	yes	complemented
133	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
134	C₂×#97	C₂²×C₁₃₇	1	8	yes	—
135	C₂×C₄×C₈	C₄×C₁₃₇	1	8	yes	complemented
136	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₁₃₇	1	8	yes	—
141	C₄×C₄⋊C₄	C₄×C₁₃₇	1	4	yes	—
142	C₄×C₂²⋊C₄	C₄×C₁₃₇	1	4	yes	—
143	#376	C₂²×C₁₃₇	1	2	yes	—
144	C₄³	C₂²×C₁₃₇	1	2	yes	—
145	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
146	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
147	C₂×M₁₆	C₂×C₄×C₁₃₇	1	8	yes	—
148	C₂×M₁₆	C₂×C₄×C₁₃₇	1	16	yes	—
149	C₂²×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
150	C₂²×C₈	C₂×C₄×C₁₃₇	1	16	yes	—
151	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
152	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
153	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
154	#117	C₂×C₄×C₁₃₇	1	16	yes	—
155	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
156	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
157	C₂×C₄²	C₂×C₄×C₁₃₇	1	8	yes	—
158	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
159	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
160	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
161	C₂×C₄²	C₂³×C₁₃₇	1	4	yes	—
162	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
164	#97	C₂×C₄×C₁₃₇	1	8	yes	—
165	#97	C₂×C₄×C₁₃₇	1	32	yes	complemented
166	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
167	C₄×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
168	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
169	C₄×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
170	C₄×C₈	C₂×C₄×C₁₃₇	1	32	yes	complemented
171	#95	C₂×C₄×C₁₃₇	1	8	yes	—
172	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
173	C₂²×C₄	—	2	8	no	—
174	C₂²×C₄	C₄²×C₁₃₇	1	8	yes	—
175	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
176	C₂²×C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
177	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
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	8	yes	—
181	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
182	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
183	M₁₆	C₄²×C₁₃₇	1	32	yes	complemented
184	C₂×C₈	C₄²×C₁₃₇	1	32	yes	complemented
185	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
186	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
187	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
188	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
189	C₂×C₈	C₂²×C₄×C₁₃₇	1	16	yes	—
190	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
191	C₄⋊C₄	C₄²×C₁₃₇	1	16	yes	—
192	C₂²⋊C₄	C₄²×C₁₃₇	1	16	yes	—
193	C₄²	—	2	16	no	—
194	C₄²	C₄²×C₁₃₇	1	8	yes	—
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	16	yes	—
199	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
200	C₂³	—	2	4	no	—
201	C₂³	C₂×C₄²×C₁₃₇	1	4	yes	—
202	C₂³	C₂×C₄²×C₁₃₇	1	2	yes	—
203	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	16	no	—
207	C₂×C₄	C₂×C₄²×C₁₃₇	1	4	yes	—
208	C₂×C₄	C₂×C₄²×C₁₃₇	1	16	yes	—
209	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
210	C₂×C₄	C₂×C₄²×C₁₃₇	1	4	yes	—
211	C₂×C₄	C₂×C₄²×C₁₃₇	1	8	yes	—
212	C₂×C₄	C₂×C₄²×C₁₃₇	1	8	yes	—
213	C₂×C₄	C₂×C₄²×C₁₃₇	1	4	yes	—
214	C₂×C₄	#131×C₁₃₇	1	8	yes	—
215	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
216	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
217	C₂×C₄	C₂³×C₄×C₁₃₇	1	4	yes	—
218	C₈	C₂×C₄²×C₁₃₇	1	8	yes	—
219	C₈	C₂×C₄²×C₁₃₇	1	8	yes	—
220	C₈	C₂×C₄²×C₁₃₇	1	16	yes	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	8	no	—
223	C₂²	C₄³×C₁₃₇	1	4	yes	—
224	C₂²	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
225	C₂²	C₂²×C₄²×C₁₃₇	1	2	yes	—
226	C₂²	C₂×#131×C₁₃₇	1	4	yes	—
227	C₄	—	2	4	no	—
228	C₄	—	2	16	no	—
229	C₄	C₄³×C₁₃₇	1	4	yes	—
230	C₄	#405×C₁₃₇	1	8	yes	direct product factor
231	C₄	C₂²×C₄²×C₁₃₇	1	2	yes	—
232	C₄	C₂²×C₄²×C₁₃₇	1	2	yes	—
233	C₂	—	2	4	no	—
234	C₂	C₂×C₄³×C₁₃₇	1	1	yes	characteristic
235	C₂	C₂×#405×C₁₃₇	1	2	yes	—
236	C₂	C₄×#131×C₁₃₇	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈×C₁₃₇	C₂²
commutator	C₂	C₂×C₄³×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂³×C₄
prime radical	C₂⁴×C₁₃₇	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. #1726×C₁₃₇	C₂	semidirect
6. C₄×#97×C₁₃₇	C₂	semidirect
7. C₄²×C₈×C₁₃₇	C₂	semidirect
12. #405×C₁₃₇	C₄	direct (N × H)
13. C₄×M₁₆×C₁₃₇	C₄	semidirect
16. C₂×C₄×C₈×C₁₃₇	C₄	semidirect
46. #97×C₁₃₇	C₂×C₄	semidirect
51. C₄×C₈×C₁₃₇	C₂×C₄	semidirect
64. M₁₆×C₁₃₇	C₄²	semidirect
65. C₂×C₈×C₁₃₇	C₄²	semidirect
111. C₄×C₁₃₇	#405	direct (N × H)
118. C₄×#405	C₁₃₇	direct (N × H)
119. C₁₃₇	C₄×#405	direct (N × H)
124. #1726	C₂×C₁₃₇	semidirect
125. C₄×#97	C₂×C₁₃₇	semidirect
126. C₄²×C₈	C₂×C₁₃₇	semidirect
131. #405	C₄×C₁₃₇	direct (N × H)
132. C₄×M₁₆	C₄×C₁₃₇	semidirect
135. C₂×C₄×C₈	C₄×C₁₃₇	semidirect
165. #97	C₂×C₄×C₁₃₇	semidirect
170. C₄×C₈	C₂×C₄×C₁₃₇	semidirect
183. M₁₆	C₄²×C₁₃₇	semidirect
184. C₂×C₈	C₄²×C₁₃₇	semidirect
230. C₄	#405×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
C₂×#405×C₁₃₇	2
C₂×C₄×M₁₆×C₁₃₇	2
C₂²×C₄×C₈×C₁₃₇	2
C₄×#117×C₁₃₇	2
#1726×C₁₃₇	2
C₄×#97×C₁₃₇	2
C₄²×C₈×C₁₃₇	2
C₄×#405	137

Subgroups — C4×#405×C137 #46869716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#405×C₁₃₇ #46869716