Subgroups — C₁₃₇×#12599 #46861985

Order 35072.

590 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₁₆×C₁₃₇	C₂	1	2	yes	maximal
3	C₂×#346×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#1374×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₁₃₇	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃₇	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₁₃₇	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₁₃₇	—	2	4	no	—
10	C₂×C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²⋊C₁₆×C₁₃₇	C₄	1	8	yes	complemented
15	#346×C₁₃₇	—	2	4	no	—
16	C₂³×C₄×C₁₃₇	C₈	1	2	yes	—
17	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃₇	—	2	2	no	—
19	C₂²×C₈×C₁₃₇	—	2	2	no	—
20	C₂²×C₈×C₁₃₇	—	2	2	no	—
21	C₂²×C₈×C₁₃₇	—	2	2	no	—
22	C₂²×C₈×C₁₃₇	C₈	1	4	yes	—
23	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃₇	—	2	2	no	—
29	C₂×C₄²×C₁₃₇	—	2	2	no	—
30	C₂×C₄²×C₁₃₇	C₈	1	2	yes	—
31	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₁₃₇	—	2	4	no	—
34	C₂×C₁₆×C₁₃₇	—	2	4	no	—
35	C₂×C₁₆×C₁₃₇	C₂×C₄	1	8	yes	complemented
36	C₄×C₈×C₁₃₇	—	2	2	no	—
37	C₄×C₈×C₁₃₇	—	2	2	no	—
38	C₄×C₈×C₁₃₇	—	2	2	no	—
39	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
40	C₂⁴×C₁₃₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₇	—	2	4	no	—
42	C₂²×C₄×C₁₃₇	—	2	4	no	—
43	C₂²×C₄×C₁₃₇	—	2	4	no	—
44	C₂²×C₄×C₁₃₇	—	2	1	no	—
45	C₂²×C₄×C₁₃₇	—	2	1	no	—
46	C₂²×C₄×C₁₃₇	—	2	1	no	—
47	C₂²×C₄×C₁₃₇	—	2	1	no	—
48	C₂²×C₄×C₁₃₇	C₄²	1	2	yes	—
49	C₂²×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃₇	—	2	1	no	—
55	C₂×C₈×C₁₃₇	—	2	1	no	—
56	C₂×C₈×C₁₃₇	—	2	1	no	—
57	C₂×C₈×C₁₃₇	—	2	1	no	—
58	C₂×C₈×C₁₃₇	—	2	8	no	—
59	C₂×C₈×C₁₃₇	—	2	8	no	—
60	C₂×C₈×C₁₃₇	C₄²	1	2	yes	—
61	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₃₇	C₂×C₈	1	2	yes	—
63	C₂×C₈×C₁₃₇	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₁₃₇	M₁₆	1	2	yes	—
65	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₃₇	#41	1	2	yes	—
68	C₄²×C₁₃₇	—	2	4	no	—
69	C₄²×C₁₃₇	—	2	1	no	—
70	C₄²×C₁₃₇	—	2	1	no	—
71	C₄²×C₁₃₇	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₁₃₇	M₁₆	1	2	yes	—
73	C₁₆×C₁₃₇	—	2	8	no	—
74	C₂³×C₁₃₇	—	2	1	no	—
75	C₂³×C₁₃₇	—	2	1	no	—
76	C₂³×C₁₃₇	—	2	2	no	—
77	C₂³×C₁₃₇	C₄×C₈	1	2	yes	—
78	C₂³×C₁₃₇	M₃₂	1	4	yes	—
79	C₂³×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃₇	—	2	16	no	—
81	C₂×C₄×C₁₃₇	—	2	2	no	—
82	C₂×C₄×C₁₃₇	—	2	1	no	—
83	C₂×C₄×C₁₃₇	—	2	1	no	—
84	C₂×C₄×C₁₃₇	—	2	2	no	—
85	C₂×C₄×C₁₃₇	—	2	2	no	—
86	C₂×C₄×C₁₃₇	—	2	2	no	—
87	C₂×C₄×C₁₃₇	—	2	2	no	—
88	C₂×C₄×C₁₃₇	—	2	2	no	—
89	C₂×C₄×C₁₃₇	—	2	2	no	—
90	C₂×C₄×C₁₃₇	—	2	2	no	—
91	C₂×C₄×C₁₃₇	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	4	yes	—
93	C₂×C₄×C₁₃₇	M₃₂	1	4	yes	—
94	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
97	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
100	C₈×C₁₃₇	—	2	2	no	—
101	C₈×C₁₃₇	—	2	2	no	—
102	C₈×C₁₃₇	C₂²⋊C₈	1	4	yes	—
103	C₈×C₁₃₇	C₄×D₄	1	4	yes	—
104	C₂²×C₁₃₇	—	2	2	no	—
105	C₂²×C₁₃₇	—	2	4	no	—
106	C₂²×C₁₃₇	—	2	4	no	—
107	C₂²×C₁₃₇	—	2	2	no	—
108	C₂²×C₁₃₇	#346	1	4	yes	complemented
109	C₂²×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
110	C₂²×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
111	C₂²×C₁₃₇	#431	1	1	yes	characteristic
112	C₂²×C₁₃₇	C₂×M₃₂	1	2	yes	—
113	C₂²×C₁₃₇	C₂×M₃₂	1	2	yes	—
114	C₄×C₁₃₇	—	2	4	no	—
115	C₄×C₁₃₇	—	2	4	no	—
116	C₄×C₁₃₇	—	2	2	no	—
117	C₄×C₁₃₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
118	C₄×C₁₃₇	C₄×M₁₆	1	1	yes	characteristic
119	C₄×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₁₃₇	#431	1	1	yes	characteristic
121	C₂×C₁₃₇	—	2	4	no	—
122	C₂×C₁₃₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
123	C₂×C₁₃₇	C₂×#346	1	1	yes	characteristic
124	C₂×C₁₃₇	C₄×M₃₂	1	1	yes	characteristic
125	C₂×C₁₃₇	#2112	1	2	yes	—
126	C₂×C₁₃₇	#2163	1	2	yes	—
127	#12599	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₁₃₇	#12599	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×C₄×C₈	C₂×C₁₃₇	1	1	yes	characteristic
130	C₂×C₂²⋊C₁₆	C₂×C₁₃₇	1	2	yes	—
131	C₂×#346	C₂×C₁₃₇	1	2	yes	complemented
132	#1374	C₂×C₁₃₇	1	2	yes	complemented
133	C₂³×C₈	C₄×C₁₃₇	1	1	yes	characteristic
134	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
135	C₂²×C₄²	C₄×C₁₃₇	1	1	yes	characteristic
136	C₂²×C₁₆	C₂²×C₁₃₇	1	4	yes	—
137	C₂×C₄×C₈	—	2	4	no	—
138	C₂×C₄×C₈	C₄×C₁₃₇	1	1	yes	characteristic
139	C₂×C₄×C₈	C₄×C₁₃₇	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
142	C₂²⋊C₁₆	C₄×C₁₃₇	1	8	yes	complemented
143	#346	—	2	4	no	—
144	C₂³×C₄	C₈×C₁₃₇	1	2	yes	—
145	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	C₈×C₁₃₇	1	4	yes	—
151	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
152	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
153	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
154	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	C₈×C₁₃₇	1	2	yes	—
159	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
161	C₂×C₁₆	—	2	4	no	—
162	C₂×C₁₆	—	2	4	no	—
163	C₂×C₁₆	C₂×C₄×C₁₃₇	1	8	yes	complemented
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
168	C₂⁴	C₂×C₈×C₁₃₇	1	1	yes	characteristic
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	C₄²×C₁₃₇	1	2	yes	—
177	C₂²×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₈×C₁₃₇	1	2	yes	—
179	C₂²×C₄	C₂×C₈×C₁₃₇	1	2	yes	—
180	C₂²×C₄	C₂×C₈×C₁₃₇	1	2	yes	—
181	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	8	no	—
187	C₂×C₈	—	2	8	no	—
188	C₂×C₈	C₄²×C₁₃₇	1	2	yes	—
189	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
190	C₂×C₈	C₂×C₈×C₁₃₇	1	2	yes	—
191	C₂×C₈	C₂×C₈×C₁₃₇	1	2	yes	—
192	C₂×C₈	M₁₆×C₁₃₇	1	2	yes	—
193	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
194	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
195	C₂×C₈	#41×C₁₃₇	1	2	yes	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	C₂²⋊C₄×C₁₃₇	1	2	yes	—
200	C₄²	M₁₆×C₁₃₇	1	2	yes	—
201	C₁₆	—	2	8	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	2	no	—
205	C₂³	C₄×C₈×C₁₃₇	1	2	yes	—
206	C₂³	M₃₂×C₁₃₇	1	4	yes	—
207	C₂³	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
208	C₂×C₄	—	2	16	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	C₄×C₈×C₁₃₇	1	2	yes	—
220	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	4	yes	—
221	C₂×C₄	M₃₂×C₁₃₇	1	4	yes	—
222	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
224	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
226	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
227	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	C₂²⋊C₈×C₁₃₇	1	4	yes	—
231	C₈	C₄×D₄×C₁₃₇	1	4	yes	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	4	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	#346×C₁₃₇	1	4	yes	complemented
237	C₂²	C₂×C₄×C₈×C₁₃₇	1	1	yes	characteristic
238	C₂²	C₂×C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
239	C₂²	#431×C₁₃₇	1	1	yes	characteristic
240	C₂²	C₂×M₃₂×C₁₃₇	1	2	yes	—
241	C₂²	C₂×M₃₂×C₁₃₇	1	2	yes	—
242	C₄	—	2	4	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	C₄×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
246	C₄	C₄×M₁₆×C₁₃₇	1	1	yes	characteristic
247	C₄	C₂×C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
248	C₄	#431×C₁₃₇	1	1	yes	characteristic
249	C₂	—	2	4	no	—
250	C₂	C₄×C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
251	C₂	C₂×#346×C₁₃₇	1	1	yes	characteristic
252	C₂	C₄×M₃₂×C₁₃₇	1	1	yes	characteristic
253	C₂	#2112×C₁₃₇	1	2	yes	—
254	C₂	#2163×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×#346×C₁₃₇	C₂	semidirect
4. #1374×C₁₃₇	C₂	semidirect
14. C₂²⋊C₁₆×C₁₃₇	C₄	semidirect
35. C₂×C₁₆×C₁₃₇	C₂×C₄	semidirect
108. C₂²×C₁₃₇	#346	semidirect
127. #12599	C₁₃₇	direct (N × H)
128. C₁₃₇	#12599	direct (N × H)
131. C₂×#346	C₂×C₁₃₇	semidirect
132. #1374	C₂×C₁₃₇	semidirect
142. C₂²⋊C₁₆	C₄×C₁₃₇	semidirect
163. C₂×C₁₆	C₂×C₄×C₁₃₇	semidirect
236. C₂²	#346×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₁₃₇	2
C₂×C₂²⋊C₁₆×C₁₃₇	2
C₂×#346×C₁₃₇	2
#1374×C₁₃₇	2
#12599	137

Subgroups — C137×#12599 #46861985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#12599 #46861985