Subgroups — C₁₃₇×#10889 #46860275

Order 35072.

734 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2873×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2317×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1740×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1736×C₁₃₇	C₂	1	2	yes	maximal
5	C₂×H(3,ℤ/4)×C₁₃₇	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
7	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
8	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
11	#405×C₁₃₇	—	2	4	no	—
12	C₄×M₁₆×C₁₃₇	—	2	2	no	—
13	#389×C₁₃₇	—	2	2	no	—
14	#385×C₁₃₇	—	2	2	no	—
15	#384×C₁₃₇	C₄	1	2	yes	complemented
16	#382×C₁₃₇	C₄	1	2	yes	complemented
17	#342×C₁₃₇	—	2	2	no	—
18	#340×C₁₃₇	—	2	4	no	—
19	H(3,ℤ/4)×C₁₃₇	C₄	1	8	yes	complemented
20	C₂³×C₄×C₁₃₇	—	2	1	no	—
21	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₃₇	—	2	2	no	—
23	C₂×M₁₆×C₁₃₇	—	2	2	no	—
24	C₂×M₁₆×C₁₃₇	—	2	2	no	—
25	C₂×M₁₆×C₁₃₇	—	2	2	no	—
26	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₁₃₇	Q₈	1	4	yes	—
28	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
29	#117×C₁₃₇	—	2	2	no	—
30	#117×C₁₃₇	—	2	2	no	—
31	#117×C₁₃₇	—	2	2	no	—
32	#117×C₁₃₇	—	2	2	no	—
33	#117×C₁₃₇	—	2	2	no	—
34	#117×C₁₃₇	—	2	2	no	—
35	#117×C₁₃₇	C₂×C₄	1	4	yes	—
36	#117×C₁₃₇	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
49	C₂×C₄²×C₁₃₇	—	2	2	no	—
50	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
52	#97×C₁₃₇	—	2	4	no	—
53	C₄×C₈×C₁₃₇	—	2	4	no	—
54	#95×C₁₃₇	—	2	4	no	—
55	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₇	—	2	2	no	—
57	C₂²×C₄×C₁₃₇	—	2	2	no	—
58	C₂²×C₄×C₁₃₇	—	4	2	no	—
59	C₂²×C₄×C₁₃₇	—	4	2	no	—
60	C₂²×C₄×C₁₃₇	—	8	2	no	—
61	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
69	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
70	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
71	M₁₆×C₁₃₇	—	2	4	no	—
72	M₁₆×C₁₃₇	—	2	4	no	—
73	M₁₆×C₁₃₇	—	2	4	no	—
74	M₁₆×C₁₃₇	—	2	4	no	—
75	C₂×C₈×C₁₃₇	—	2	4	no	—
76	C₂×C₈×C₁₃₇	—	2	4	no	—
77	C₂×C₈×C₁₃₇	C₄⋊C₄	1	8	yes	complemented
78	C₄⋊C₄×C₁₃₇	—	2	4	no	—
79	C₄⋊C₄×C₁₃₇	—	2	2	no	—
80	C₄⋊C₄×C₁₃₇	—	2	2	no	—
81	C₄⋊C₄×C₁₃₇	—	2	4	no	—
82	C₄⋊C₄×C₁₃₇	—	2	4	no	—
83	C₄⋊C₄×C₁₃₇	—	4	1	no	—
84	C₄⋊C₄×C₁₃₇	—	4	1	no	—
85	C₄⋊C₄×C₁₃₇	—	4	2	no	—
86	C₄⋊C₄×C₁₃₇	—	8	4	no	—
87	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
88	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
89	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
90	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
91	C₄²×C₁₃₇	—	2	2	no	—
92	C₄²×C₁₃₇	—	2	2	no	—
93	C₄²×C₁₃₇	—	2	2	no	—
94	C₄²×C₁₃₇	—	2	4	no	—
95	C₄²×C₁₃₇	—	4	2	no	—
96	C₄²×C₁₃₇	C₄²	1	4	yes	—
97	C₂³×C₁₃₇	—	4	1	no	—
98	C₂³×C₁₃₇	—	4	1	no	—
99	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₃₇	C₄×D₄	1	2	yes	—
101	C₂³×C₁₃₇	#121	1	2	yes	—
102	C₂³×C₁₃₇	#124	1	1	yes	characteristic
103	C₂³×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	2	no	—
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	4	2	no	—
108	C₂×C₄×C₁₃₇	—	4	2	no	—
109	C₂×C₄×C₁₃₇	—	4	2	no	—
110	C₂×C₄×C₁₃₇	—	4	2	no	—
111	C₂×C₄×C₁₃₇	—	4	4	no	—
112	C₂×C₄×C₁₃₇	—	4	2	no	—
113	C₂×C₄×C₁₃₇	—	4	4	no	—
114	C₂×C₄×C₁₃₇	—	8	2	no	—
115	C₂×C₄×C₁₃₇	—	8	2	no	—
116	C₂×C₄×C₁₃₇	—	8	4	no	—
117	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₃₇	#117	1	2	yes	—
120	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
127	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₁₃₇	#123	1	2	yes	—
130	C₂×C₄×C₁₃₇	#123	1	1	yes	characteristic
131	C₂×C₄×C₁₃₇	#125	1	2	yes	—
132	C₂×C₄×C₁₃₇	#126	1	2	yes	—
133	C₈×C₁₃₇	—	2	8	no	—
134	C₂²×C₁₃₇	—	2	1	no	—
135	C₂²×C₁₃₇	—	2	2	no	—
136	C₂²×C₁₃₇	—	2	1	no	—
137	C₂²×C₁₃₇	—	2	2	no	—
138	C₂²×C₁₃₇	—	4	2	no	—
139	C₂²×C₁₃₇	—	4	1	no	—
140	C₂²×C₁₃₇	—	4	1	no	—
141	C₂²×C₁₃₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₁₃₇	#386	1	1	yes	characteristic
143	C₂²×C₁₃₇	#388	1	2	yes	—
144	C₂²×C₁₃₇	#389	1	2	yes	—
145	C₂²×C₁₃₇	#391	1	1	yes	characteristic
146	C₄×C₁₃₇	—	4	4	no	—
147	C₄×C₁₃₇	—	4	2	no	—
148	C₄×C₁₃₇	—	4	2	no	—
149	C₄×C₁₃₇	—	8	4	no	—
150	C₄×C₁₃₇	—	8	2	no	—
151	C₄×C₁₃₇	C₄×C₄⋊C₄	1	2	yes	—
152	C₄×C₁₃₇	#385	1	2	yes	—
153	C₄×C₁₃₇	#387	1	2	yes	—
154	C₄×C₁₃₇	#387	1	2	yes	—
155	C₂×C₁₃₇	—	2	1	no	—
156	C₂×C₁₃₇	—	2	2	no	—
157	C₂×C₁₃₇	—	2	2	no	—
158	C₂×C₁₃₇	—	2	1	no	—
159	C₂×C₁₃₇	#1436	1	1	yes	characteristic
160	C₂×C₁₃₇	#1909	1	1	yes	characteristic
161	C₂×C₁₃₇	#1910	1	1	yes	characteristic
162	#10889	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₁₃₇	#10889	1	1	yes	characteristic, direct product factor, Sylow
164	#2873	C₂×C₁₃₇	1	1	yes	characteristic
165	#2317	C₂×C₁₃₇	1	1	yes	characteristic
166	#1740	C₂×C₁₃₇	1	1	yes	characteristic
167	#1736	C₂×C₁₃₇	1	2	yes	—
168	C₂×H(3,ℤ/4)	C₂×C₁₃₇	1	2	yes	—
169	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
170	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
171	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
172	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
173	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
174	#405	—	2	4	no	—
175	C₄×M₁₆	—	2	2	no	—
176	#389	—	2	2	no	—
177	#385	—	2	2	no	—
178	#384	C₄×C₁₃₇	1	2	yes	complemented
179	#382	C₄×C₁₃₇	1	2	yes	complemented
180	#342	—	2	2	no	—
181	#340	—	2	4	no	—
182	H(3,ℤ/4)	C₄×C₁₃₇	1	8	yes	complemented
183	C₂³×C₄	—	2	1	no	—
184	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
190	C₂×M₁₆	Q₈×C₁₃₇	1	4	yes	—
191	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	C₂×C₄×C₁₃₇	1	4	yes	—
199	#117	C₂×C₄×C₁₃₇	1	4	yes	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
214	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
215	#97	—	2	4	no	—
216	C₄×C₈	—	2	4	no	—
217	#95	—	2	4	no	—
218	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	8	2	no	—
224	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
225	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
231	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
232	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
233	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	C₄⋊C₄×C₁₃₇	1	8	yes	complemented
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	8	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	4	2	no	—
259	C₄²	C₄²×C₁₃₇	1	4	yes	—
260	C₂³	—	4	1	no	—
261	C₂³	—	4	1	no	—
262	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
263	C₂³	C₄×D₄×C₁₃₇	1	2	yes	—
264	C₂³	#121×C₁₃₇	1	2	yes	—
265	C₂³	#124×C₁₃₇	1	1	yes	characteristic
266	C₂³	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
282	C₂×C₄	#117×C₁₃₇	1	2	yes	—
283	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
284	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
285	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
286	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
287	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
288	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
289	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
290	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
291	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
292	C₂×C₄	#123×C₁₃₇	1	2	yes	—
293	C₂×C₄	#123×C₁₃₇	1	1	yes	characteristic
294	C₂×C₄	#125×C₁₃₇	1	2	yes	—
295	C₂×C₄	#126×C₁₃₇	1	2	yes	—
296	C₈	—	2	8	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	C₄×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
305	C₂²	#386×C₁₃₇	1	1	yes	characteristic
306	C₂²	#388×C₁₃₇	1	2	yes	—
307	C₂²	#389×C₁₃₇	1	2	yes	—
308	C₂²	#391×C₁₃₇	1	1	yes	characteristic
309	C₄	—	4	4	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	4	no	—
313	C₄	—	8	2	no	—
314	C₄	C₄×C₄⋊C₄×C₁₃₇	1	2	yes	—
315	C₄	#385×C₁₃₇	1	2	yes	—
316	C₄	#387×C₁₃₇	1	2	yes	—
317	C₄	#387×C₁₃₇	1	2	yes	—
318	C₂	—	2	1	no	—
319	C₂	—	2	2	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	#1436×C₁₃₇	1	1	yes	characteristic
323	C₂	#1909×C₁₃₇	1	1	yes	characteristic
324	C₂	#1910×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#386
commutator	C₂×C₄	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
prime radical	C₂⁴×C₁₃₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. #384×C₁₃₇	C₄	semidirect
16. #382×C₁₃₇	C₄	semidirect
19. H(3,ℤ/4)×C₁₃₇	C₄	semidirect
77. C₂×C₈×C₁₃₇	C₄⋊C₄	semidirect
162. #10889	C₁₃₇	direct (N × H)
163. C₁₃₇	#10889	direct (N × H)
178. #384	C₄×C₁₃₇	semidirect
179. #382	C₄×C₁₃₇	semidirect
182. H(3,ℤ/4)	C₄×C₁₃₇	semidirect
240. C₂×C₈	C₄⋊C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2873×C₁₃₇	2
#2317×C₁₃₇	2
#1740×C₁₃₇	2
#1736×C₁₃₇	2
C₂×H(3,ℤ/4)×C₁₃₇	2
#10889	137

Subgroups — C137×#10889 #46860275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#10889 #46860275