Subgroups — C₁₃₇×#23674 #46873060

Order 35072.

766 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2911×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#427×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#424×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2090×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#2089×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1742×C₁₃₇	C₂	1	2	yes	maximal
7	C₂×#341×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
10	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
13	#428×C₁₃₇	—	2	4	no	—
14	#428×C₁₃₇	C₂²	1	8	yes	—
15	#427×C₁₃₇	—	2	2	no	—
16	#427×C₁₃₇	—	2	2	no	—
17	#427×C₁₃₇	—	2	4	no	—
18	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
19	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
20	#424×C₁₃₇	—	2	2	no	—
21	#424×C₁₃₇	C₂²	1	2	yes	—
22	#424×C₁₃₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈×C₁₃₇	—	2	2	no	—
24	#343×C₁₃₇	C₂²	1	8	yes	—
25	#341×C₁₃₇	C₂²	1	2	yes	—
26	#341×C₁₃₇	C₂²	1	2	yes	—
27	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₃₇	—	2	2	no	—
29	C₂×M₁₆×C₁₃₇	—	2	4	no	—
30	C₂×M₁₆×C₁₃₇	—	2	1	no	—
31	C₂×M₁₆×C₁₃₇	—	2	4	no	—
32	C₂×M₁₆×C₁₃₇	Q₈	1	8	yes	—
33	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
34	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₃₇	—	2	2	no	—
36	C₂²×C₈×C₁₃₇	—	2	4	no	—
37	C₂²×C₈×C₁₃₇	—	2	1	no	—
38	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
39	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
40	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
42	#117×C₁₃₇	—	2	1	no	—
43	#117×C₁₃₇	—	2	1	no	—
44	#117×C₁₃₇	—	2	2	no	—
45	#117×C₁₃₇	—	2	2	no	—
46	#117×C₁₃₇	—	2	2	no	—
47	#117×C₁₃₇	—	2	2	no	—
48	#117×C₁₃₇	D₄	1	2	yes	—
49	#117×C₁₃₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
53	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
54	C₂×C₄²×C₁₃₇	—	2	1	no	—
55	C₂×C₄²×C₁₃₇	—	2	2	no	—
56	#107×C₁₃₇	—	2	4	no	—
57	#107×C₁₃₇	—	2	8	no	—
58	#106×C₁₃₇	—	2	4	no	—
59	#106×C₁₃₇	—	2	8	no	—
60	C₄⋊C₈×C₁₃₇	—	2	4	no	—
61	C₄⋊C₈×C₁₃₇	—	2	4	no	—
62	C₂⁴×C₁₃₇	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₇	—	2	2	no	—
64	C₂²×C₄×C₁₃₇	—	2	4	no	—
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
72	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₁₃₇	—	2	8	no	—
74	M₁₆×C₁₃₇	—	2	4	no	—
75	M₁₆×C₁₃₇	—	2	4	no	—
76	M₁₆×C₁₃₇	—	4	2	no	—
77	M₁₆×C₁₃₇	C₂×Q₈	1	8	yes	—
78	C₂×C₈×C₁₃₇	—	2	8	no	—
79	C₂×C₈×C₁₃₇	—	2	4	no	—
80	C₂×C₈×C₁₃₇	—	2	4	no	—
81	C₂×C₈×C₁₃₇	—	2	4	no	—
82	C₂×C₈×C₁₃₇	—	2	2	no	—
83	C₂×C₈×C₁₃₇	—	2	2	no	—
84	C₂×C₈×C₁₃₇	—	2	2	no	—
85	C₂×C₈×C₁₃₇	—	4	2	no	—
86	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₁₃₇	—	2	2	no	—
89	C₄⋊C₄×C₁₃₇	—	2	4	no	—
90	C₄⋊C₄×C₁₃₇	—	2	4	no	—
91	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
92	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
93	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
94	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
95	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
96	C₄²×C₁₃₇	—	2	2	no	—
97	C₄²×C₁₃₇	—	2	2	no	—
98	C₄²×C₁₃₇	—	4	2	no	—
99	C₄²×C₁₃₇	—	4	2	no	—
100	C₂³×C₁₃₇	—	4	2	no	—
101	C₂³×C₁₃₇	#121	1	1	yes	characteristic
102	C₂³×C₁₃₇	#121	1	1	yes	characteristic
103	C₂³×C₁₃₇	#125	1	4	yes	—
104	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	—	2	4	no	—
106	C₂×C₄×C₁₃₇	—	4	4	no	—
107	C₂×C₄×C₁₃₇	—	4	4	no	—
108	C₂×C₄×C₁₃₇	—	8	4	no	—
109	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
112	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
113	C₂×C₄×C₁₃₇	#123	1	2	yes	—
114	C₂×C₄×C₁₃₇	#123	1	2	yes	—
115	C₂×C₄×C₁₃₇	#125	1	4	yes	—
116	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
121	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
122	C₈×C₁₃₇	—	2	8	no	—
123	C₈×C₁₃₇	—	2	2	no	—
124	C₈×C₁₃₇	—	2	2	no	—
125	C₈×C₁₃₇	—	4	2	no	—
126	C₂²×C₁₃₇	—	2	2	no	—
127	C₂²×C₁₃₇	—	2	2	no	—
128	C₂²×C₁₃₇	—	2	2	no	—
129	C₂²×C₁₃₇	—	4	4	no	—
130	C₂²×C₁₃₇	#398	1	2	yes	—
131	C₂²×C₁₃₇	#398	1	2	yes	—
132	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
133	C₂²×C₁₃₇	C₂×#125	1	2	yes	—
134	C₄×C₁₃₇	—	4	4	no	—
135	C₄×C₁₃₇	—	8	4	no	—
136	C₄×C₁₃₇	#398	1	2	yes	—
137	C₄×C₁₃₇	#398	1	2	yes	—
138	C₄×C₁₃₇	C₂×C₂²⋊Q₈	1	2	yes	—
139	C₄×C₁₃₇	C₂×#123	1	1	yes	characteristic
140	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
141	C₂×C₁₃₇	—	2	2	no	—
142	C₂×C₁₃₇	—	2	2	no	—
143	C₂×C₁₃₇	—	2	2	no	—
144	C₂×C₁₃₇	C₂×#398	1	1	yes	characteristic
145	C₂×C₁₃₇	#3067	1	2	yes	—
146	#23674	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₁₃₇	#23674	1	1	yes	characteristic, direct product factor, Sylow
148	#2911	C₂×C₁₃₇	1	2	yes	—
149	C₂×#427	C₂×C₁₃₇	1	1	yes	characteristic
150	C₂×#424	C₂×C₁₃₇	1	1	yes	characteristic
151	#2090	C₂×C₁₃₇	1	4	yes	complemented
152	#2089	C₂×C₁₃₇	1	4	yes	complemented
153	#1742	C₂×C₁₃₇	1	2	yes	—
154	C₂×#341	C₂×C₁₃₇	1	1	yes	characteristic
155	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
156	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
157	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
158	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
159	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
160	#428	—	2	4	no	—
161	#428	C₂²×C₁₃₇	1	8	yes	—
162	#427	—	2	2	no	—
163	#427	—	2	2	no	—
164	#427	—	2	4	no	—
165	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
166	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
167	#424	—	2	2	no	—
168	#424	C₂²×C₁₃₇	1	2	yes	—
169	#424	C₂²×C₁₃₇	1	2	yes	—
170	C₂×C₄⋊C₈	—	2	2	no	—
171	#343	C₂²×C₁₃₇	1	8	yes	—
172	#341	C₂²×C₁₃₇	1	2	yes	—
173	#341	C₂²×C₁₃₇	1	2	yes	—
174	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	Q₈×C₁₃₇	1	8	yes	—
180	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
181	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
186	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
187	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
189	#117	—	2	1	no	—
190	#117	—	2	1	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	D₄×C₁₃₇	1	2	yes	—
196	#117	C₂³×C₁₃₇	1	2	yes	—
197	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
199	C₂×C₂²⋊C₄	—	2	1	no	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	2	2	no	—
203	#107	—	2	4	no	—
204	#107	—	2	8	no	—
205	#106	—	2	4	no	—
206	#106	—	2	8	no	—
207	C₄⋊C₈	—	2	4	no	—
208	C₄⋊C₈	—	2	4	no	—
209	C₂⁴	#41×C₁₃₇	1	1	yes	characteristic
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
217	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
218	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
219	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
220	M₁₆	—	2	8	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	C₂×Q₈×C₁₃₇	1	8	yes	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	#121×C₁₃₇	1	1	yes	characteristic
249	C₂³	#121×C₁₃₇	1	1	yes	characteristic
250	C₂³	#125×C₁₃₇	1	4	yes	—
251	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
257	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
258	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
259	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
260	C₂×C₄	#123×C₁₃₇	1	2	yes	—
261	C₂×C₄	#123×C₁₃₇	1	2	yes	—
262	C₂×C₄	#125×C₁₃₇	1	4	yes	—
263	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
264	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
268	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
269	C₈	—	2	8	no	—
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	—	4	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	4	no	—
277	C₂²	#398×C₁₃₇	1	2	yes	—
278	C₂²	#398×C₁₃₇	1	2	yes	—
279	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
280	C₂²	C₂×#125×C₁₃₇	1	2	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	8	4	no	—
283	C₄	#398×C₁₃₇	1	2	yes	—
284	C₄	#398×C₁₃₇	1	2	yes	—
285	C₄	C₂×C₂²⋊Q₈×C₁₃₇	1	2	yes	—
286	C₄	C₂×#123×C₁₃₇	1	1	yes	characteristic
287	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	C₂×#398×C₁₃₇	1	1	yes	characteristic
292	C₂	#3067×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	C₂×#121
prime radical	C₂⁴×C₁₃₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. #2090×C₁₃₇	C₂	semidirect
5. #2089×C₁₃₇	C₂	semidirect
146. #23674	C₁₃₇	direct (N × H)
147. C₁₃₇	#23674	direct (N × H)
151. #2090	C₂×C₁₃₇	semidirect
152. #2089	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2911×C₁₃₇	2
C₂×#427×C₁₃₇	2
C₂×#424×C₁₃₇	2
#2090×C₁₃₇	2
#2089×C₁₃₇	2
#1742×C₁₃₇	2
C₂×#341×C₁₃₇	2
#23674	137

Subgroups — C137×#23674 #46873060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#23674 #46873060