Subgroups — C₁₃₇×#20534 #46869920

Order 35072.

1010 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1771×C₁₃₇	C₂	1	8	yes	maximal, complemented
4	#1749×C₁₃₇	C₂	1	2	yes	maximal
5	C₄×C₂²⋊C₈×C₁₃₇	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₁₃₇	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
10	#432×C₁₃₇	C₄	1	16	yes	complemented
11	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	4	yes	—
13	C₂×#97×C₁₃₇	C₂²	1	4	yes	—
14	C₂×#97×C₁₃₇	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	4	yes	—
16	C₄×C₄⋊C₄×C₁₃₇	C₄	1	4	yes	—
17	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
18	C₂×#95×C₁₃₇	C₄	1	2	yes	—
19	#336×C₁₃₇	C₂²	1	4	yes	—
20	#336×C₁₃₇	C₂²	1	4	yes	—
21	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₇	C₂³	1	8	yes	—
27	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
28	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
29	C₂×C₄²×C₁₃₇	—	4	2	no	—
30	C₂×C₄²×C₁₃₇	—	4	2	no	—
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	8	yes	—
35	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
36	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
37	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
38	C₄⋊C₈×C₁₃₇	C₂×C₄	1	16	yes	complemented
39	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	16	yes	—
40	#97×C₁₃₇	—	4	4	no	—
41	#97×C₁₃₇	C₂×C₄	1	16	yes	complemented
42	C₄×C₈×C₁₃₇	—	4	4	no	—
43	#95×C₁₃₇	C₂×C₄	1	8	yes	—
44	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₇	—	2	4	no	—
46	C₂²×C₄×C₁₃₇	—	2	2	no	—
47	C₂²×C₄×C₁₃₇	—	2	2	no	—
48	C₂²×C₄×C₁₃₇	—	2	4	no	—
49	C₂²×C₄×C₁₃₇	—	4	2	no	—
50	C₂²×C₄×C₁₃₇	—	4	2	no	—
51	C₂²×C₄×C₁₃₇	—	4	2	no	—
52	C₂²×C₄×C₁₃₇	—	4	2	no	—
53	C₂²×C₄×C₁₃₇	—	4	2	no	—
54	C₂²×C₄×C₁₃₇	C₄²	1	4	yes	—
55	C₂²×C₄×C₁₃₇	C₄²	1	4	yes	—
56	C₂²×C₄×C₁₃₇	M₁₆	1	8	yes	—
57	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	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	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₁₃₇	—	2	8	no	—
65	C₂×C₈×C₁₃₇	—	2	8	no	—
66	C₂×C₈×C₁₃₇	C₂²×C₄	1	16	yes	—
67	C₄⋊C₄×C₁₃₇	—	2	16	no	—
68	C₄²×C₁₃₇	—	2	4	no	—
69	C₄²×C₁₃₇	—	2	4	no	—
70	C₄²×C₁₃₇	—	4	2	no	—
71	C₄²×C₁₃₇	—	4	2	no	—
72	C₄²×C₁₃₇	C₄²	1	8	yes	—
73	C₄²×C₁₃₇	C₂²×C₄	1	4	yes	—
74	C₄²×C₁₃₇	C₂²×C₄	1	4	yes	—
75	C₂³×C₁₃₇	—	2	1	no	—
76	C₂³×C₁₃₇	—	2	1	no	—
77	C₂³×C₁₃₇	—	2	2	no	—
78	C₂³×C₁₃₇	C₂×C₄²	1	2	yes	—
79	C₂³×C₁₃₇	C₂×M₁₆	1	4	yes	—
80	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃₇	—	2	4	no	—
82	C₂×C₄×C₁₃₇	—	2	1	no	—
83	C₂×C₄×C₁₃₇	—	2	1	no	—
84	C₂×C₄×C₁₃₇	—	2	4	no	—
85	C₂×C₄×C₁₃₇	—	2	4	no	—
86	C₂×C₄×C₁₃₇	—	2	4	no	—
87	C₂×C₄×C₁₃₇	—	2	8	no	—
88	C₂×C₄×C₁₃₇	—	2	8	no	—
89	C₂×C₄×C₁₃₇	—	2	4	no	—
90	C₂×C₄×C₁₃₇	—	2	4	no	—
91	C₂×C₄×C₁₃₇	—	2	4	no	—
92	C₂×C₄×C₁₃₇	—	4	4	no	—
93	C₂×C₄×C₁₃₇	—	4	8	no	—
94	C₂×C₄×C₁₃₇	—	4	8	no	—
95	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
96	C₂×C₄×C₁₃₇	C₂×C₄²	1	4	yes	—
97	C₂×C₄×C₁₃₇	C₂×C₄²	1	4	yes	—
98	C₂×C₄×C₁₃₇	C₂×M₁₆	1	8	yes	—
99	C₂×C₄×C₁₃₇	C₂³×C₄	1	2	yes	—
100	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
101	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
102	C₈×C₁₃₇	—	2	16	no	—
103	C₂²×C₁₃₇	—	2	2	no	—
104	C₂²×C₁₃₇	—	2	4	no	—
105	C₂²×C₁₃₇	—	2	4	no	—
106	C₂²×C₁₃₇	—	2	2	no	—
107	C₂²×C₁₃₇	C₄×M₁₆	1	4	yes	—
108	C₂²×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
109	C₂²×C₁₃₇	#518	1	1	yes	characteristic
110	C₂²×C₁₃₇	#519	1	1	yes	characteristic
111	C₂²×C₁₃₇	C₂²×M₁₆	1	2	yes	—
112	C₂²×C₁₃₇	#568	1	2	yes	—
113	C₄×C₁₃₇	—	2	4	no	—
114	C₄×C₁₃₇	—	2	4	no	—
115	C₄×C₁₃₇	—	2	8	no	—
116	C₄×C₁₃₇	—	2	2	no	—
117	C₄×C₁₃₇	—	4	4	no	—
118	C₄×C₁₃₇	#520	1	4	yes	—
119	C₂×C₁₃₇	—	2	4	no	—
120	C₂×C₁₃₇	#2274	1	1	yes	characteristic
121	C₂×C₁₃₇	C₂×C₄×M₁₆	1	1	yes	characteristic
122	C₂×C₁₃₇	#2873	1	1	yes	characteristic
123	C₂×C₁₃₇	#2973	1	2	yes	—
124	C₂×C₁₃₇	#2983	1	2	yes	—
125	#20534	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₁₃₇	#20534	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×#432	C₂×C₁₃₇	1	2	yes	—
128	C₂×C₄×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
129	#1771	C₂×C₁₃₇	1	8	yes	complemented
130	#1749	C₂×C₁₃₇	1	2	yes	—
131	C₄×C₂²⋊C₈	C₂×C₁₃₇	1	2	yes	—
132	C₂²×C₄⋊C₄	C₄×C₁₃₇	1	2	yes	—
133	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
134	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
135	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
136	#432	C₄×C₁₃₇	1	16	yes	complemented
137	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
138	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	4	yes	—
139	C₂×#97	C₂²×C₁₃₇	1	4	yes	—
140	C₂×#97	C₂²×C₁₃₇	1	4	yes	—
141	C₂×C₄×C₈	C₂²×C₁₃₇	1	4	yes	—
142	C₄×C₄⋊C₄	C₄×C₁₃₇	1	4	yes	—
143	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
144	C₂×#95	C₄×C₁₃₇	1	2	yes	—
145	#336	C₂²×C₁₃₇	1	4	yes	—
146	#336	C₂²×C₁₃₇	1	4	yes	—
147	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
148	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
149	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
150	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
151	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₁₃₇	1	8	yes	—
153	C₂×C₄⋊C₄	—	2	8	no	—
154	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	—
155	C₂×C₄²	—	4	2	no	—
156	C₂×C₄²	—	4	2	no	—
157	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
158	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
159	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₁₃₇	1	8	yes	—
161	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
162	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
163	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
164	C₄⋊C₈	C₂×C₄×C₁₃₇	1	16	yes	complemented
165	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	16	yes	—
166	#97	—	4	4	no	—
167	#97	C₂×C₄×C₁₃₇	1	16	yes	complemented
168	C₄×C₈	—	4	4	no	—
169	#95	C₂×C₄×C₁₃₇	1	8	yes	—
170	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₄²×C₁₃₇	1	4	yes	—
181	C₂²×C₄	C₄²×C₁₃₇	1	4	yes	—
182	C₂²×C₄	M₁₆×C₁₃₇	1	8	yes	—
183	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
184	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
186	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
187	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
189	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
190	C₂×C₈	—	2	8	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	C₂²×C₄×C₁₃₇	1	16	yes	—
193	C₄⋊C₄	—	2	16	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	4	no	—
196	C₄²	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	C₄²×C₁₃₇	1	8	yes	—
199	C₄²	C₂²×C₄×C₁₃₇	1	4	yes	—
200	C₄²	C₂²×C₄×C₁₃₇	1	4	yes	—
201	C₂³	—	2	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	2	no	—
204	C₂³	C₂×C₄²×C₁₃₇	1	2	yes	—
205	C₂³	C₂×M₁₆×C₁₃₇	1	4	yes	—
206	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
222	C₂×C₄	C₂×C₄²×C₁₃₇	1	4	yes	—
223	C₂×C₄	C₂×C₄²×C₁₃₇	1	4	yes	—
224	C₂×C₄	C₂×M₁₆×C₁₃₇	1	8	yes	—
225	C₂×C₄	C₂³×C₄×C₁₃₇	1	2	yes	—
226	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
227	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
228	C₈	—	2	16	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	4	no	—
233	C₂²	C₄×M₁₆×C₁₃₇	1	4	yes	—
234	C₂²	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
235	C₂²	#518×C₁₃₇	1	1	yes	characteristic
236	C₂²	#519×C₁₃₇	1	1	yes	characteristic
237	C₂²	C₂²×M₁₆×C₁₃₇	1	2	yes	—
238	C₂²	#568×C₁₃₇	1	2	yes	—
239	C₄	—	2	2	no	—
240	C₄	—	2	4	no	—
241	C₄	—	2	4	no	—
242	C₄	—	2	8	no	—
243	C₄	—	4	4	no	—
244	C₄	#520×C₁₃₇	1	4	yes	—
245	C₂	—	2	4	no	—
246	C₂	#2274×C₁₃₇	1	1	yes	characteristic
247	C₂	C₂×C₄×M₁₆×C₁₃₇	1	1	yes	characteristic
248	C₂	#2873×C₁₃₇	1	1	yes	characteristic
249	C₂	#2973×C₁₃₇	1	2	yes	—
250	C₂	#2983×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂³×C₄
prime radical	C₂⁴×C₁₃₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1771×C₁₃₇	C₂	semidirect
10. #432×C₁₃₇	C₄	semidirect
38. C₄⋊C₈×C₁₃₇	C₂×C₄	semidirect
41. #97×C₁₃₇	C₂×C₄	semidirect
125. #20534	C₁₃₇	direct (N × H)
126. C₁₃₇	#20534	direct (N × H)
129. #1771	C₂×C₁₃₇	semidirect
136. #432	C₄×C₁₃₇	semidirect
164. C₄⋊C₈	C₂×C₄×C₁₃₇	semidirect
167. #97	C₂×C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₁₃₇	2
C₂×C₄×C₄⋊C₄×C₁₃₇	2
#1771×C₁₃₇	2
#1749×C₁₃₇	2
C₄×C₂²⋊C₈×C₁₃₇	2
#20534	137

Subgroups — C137×#20534 #46869920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20534 #46869920