Subgroups — C₁₃₇×#20788 #46870174

Order 35072.

608 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2875×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1841×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#1833×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#1827×C₁₃₇	C₂	1	2	yes	maximal
5	#1813×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1812×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1773×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1757×C₁₃₇	C₂	1	2	yes	maximal
9	#1747×C₁₃₇	C₂	1	2	yes	maximal, complemented
10	C₂×#131×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#131×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄×#41×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
14	#427×C₁₃₇	C₂²	1	2	yes	—
15	#413×C₁₃₇	C₂²	1	2	yes	—
16	#410×C₁₃₇	C₂²	1	2	yes	—
17	#405×C₁₃₇	C₄	1	2	yes	complemented
18	#405×C₁₃₇	C₄	1	2	yes	complemented
19	#405×C₁₃₇	C₂²	1	4	yes	—
20	#405×C₁₃₇	C₂²	1	1	yes	characteristic
21	#405×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₁₃₇	C₄	1	2	yes	complemented
23	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×#97×C₁₃₇	C₄	1	2	yes	complemented
25	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
26	#344×C₁₃₇	C₂²	1	4	yes	—
27	#339×C₁₃₇	C₂²	1	4	yes	—
28	#338×C₁₃₇	C₂²	1	2	yes	—
29	H(3,ℤ/4)×C₁₃₇	C₂²	1	2	yes	—
30	H(3,ℤ/4)×C₁₃₇	C₂²	1	4	yes	—
31	C₂×#41×C₁₃₇	C₂³	1	1	yes	characteristic
32	#131×C₁₃₇	—	2	4	no	—
33	#131×C₁₃₇	—	2	2	no	—
34	#131×C₁₃₇	—	2	2	no	—
35	C₂×M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
38	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₁₃₇	—	2	1	no	—
45	C₄×D₄×C₁₃₇	—	2	2	no	—
46	C₄×D₄×C₁₃₇	—	2	1	no	—
47	#117×C₁₃₇	C₂×C₄	1	2	yes	—
48	#117×C₁₃₇	C₂³	1	4	yes	—
49	#117×C₁₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
52	#108×C₁₃₇	—	2	2	no	—
53	#108×C₁₃₇	—	2	2	no	—
54	#107×C₁₃₇	—	2	2	no	—
55	#106×C₁₃₇	—	2	2	no	—
56	#101×C₁₃₇	—	2	2	no	—
57	#100×C₁₃₇	—	2	2	no	—
58	#99×C₁₃₇	—	2	4	no	—
59	#97×C₁₃₇	—	2	4	no	—
60	#97×C₁₃₇	C₂×C₄	1	1	yes	characteristic
61	#97×C₁₃₇	C₂×C₄	1	2	yes	—
62	#97×C₁₃₇	C₂×C₄	1	2	yes	—
63	#97×C₁₃₇	C₂×C₄	1	2	yes	—
64	#97×C₁₃₇	C₂×C₄	1	1	yes	characteristic
65	#97×C₁₃₇	C₂×C₄	1	1	yes	characteristic
66	#97×C₁₃₇	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃₇	—	2	4	no	—
68	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
71	#41×C₁₃₇	—	2	2	no	—
72	#41×C₁₃₇	—	2	2	no	—
73	C₂×Q₈×C₁₃₇	#41	1	1	yes	characteristic
74	C₂×D₄×C₁₃₇	#41	1	2	yes	—
75	C₂×D₄×C₁₃₇	#41	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₇	—	2	2	no	—
77	C₂²×C₄×C₁₃₇	—	2	1	no	—
78	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
80	M₁₆×C₁₃₇	—	2	2	no	—
81	M₁₆×C₁₃₇	—	2	2	no	—
82	M₁₆×C₁₃₇	—	2	4	no	—
83	M₁₆×C₁₃₇	—	2	4	no	—
84	M₁₆×C₁₃₇	—	2	4	no	—
85	M₁₆×C₁₃₇	—	2	4	no	—
86	C₂×C₈×C₁₃₇	—	2	2	no	—
87	C₂×C₈×C₁₃₇	—	2	2	no	—
88	C₂×C₈×C₁₃₇	—	2	4	no	—
89	C₂×C₈×C₁₃₇	—	2	4	no	—
90	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃₇	#41	1	4	yes	—
97	C₂×C₈×C₁₃₇	#41	1	2	yes	—
98	C₂×C₈×C₁₃₇	#41	1	2	yes	—
99	C₄⋊C₄×C₁₃₇	—	2	2	no	—
100	C₄⋊C₄×C₁₃₇	—	2	1	no	—
101	C₄⋊C₄×C₁₃₇	—	2	4	no	—
102	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
103	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
104	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
105	C₄²×C₁₃₇	—	2	2	no	—
106	C₄²×C₁₃₇	—	2	1	no	—
107	C₄²×C₁₃₇	—	2	4	no	—
108	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
109	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
111	C₂³×C₁₃₇	#117	1	2	yes	—
112	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
113	Q₈×C₁₃₇	—	2	2	no	—
114	D₄×C₁₃₇	—	2	4	no	—
115	D₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	2	2	no	—
117	C₂×C₄×C₁₃₇	—	2	2	no	—
118	C₂×C₄×C₁₃₇	—	2	1	no	—
119	C₂×C₄×C₁₃₇	—	2	2	no	—
120	C₂×C₄×C₁₃₇	—	2	1	no	—
121	C₂×C₄×C₁₃₇	—	4	4	no	—
122	C₂×C₄×C₁₃₇	—	8	1	no	—
123	C₂×C₄×C₁₃₇	—	8	2	no	—
124	C₂×C₄×C₁₃₇	#117	1	2	yes	—
125	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
127	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
128	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
129	C₈×C₁₃₇	—	2	2	no	—
130	C₈×C₁₃₇	—	2	2	no	—
131	C₈×C₁₃₇	—	4	4	no	—
132	C₈×C₁₃₇	#117	1	1	yes	characteristic
133	C₈×C₁₃₇	#117	1	1	yes	characteristic
134	C₈×C₁₃₇	#117	1	2	yes	—
135	C₈×C₁₃₇	#117	1	2	yes	—
136	C₈×C₁₃₇	#117	1	2	yes	—
137	C₂²×C₁₃₇	—	2	2	no	—
138	C₂²×C₁₃₇	—	4	1	no	—
139	C₂²×C₁₃₇	—	4	2	no	—
140	C₂²×C₁₃₇	C₂×#117	1	1	yes	characteristic
141	C₂²×C₁₃₇	#529	1	2	yes	—
142	C₄×C₁₃₇	—	2	2	no	—
143	C₄×C₁₃₇	—	4	2	no	—
144	C₄×C₁₃₇	—	4	1	no	—
145	C₄×C₁₃₇	—	4	1	no	—
146	C₄×C₁₃₇	—	8	4	no	—
147	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
148	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
149	C₄×C₁₃₇	#529	1	2	yes	—
150	C₂×C₁₃₇	—	2	1	no	—
151	C₂×C₁₃₇	—	2	2	no	—
152	C₂×C₁₃₇	—	4	2	no	—
153	C₂×C₁₃₇	#2283	1	1	yes	characteristic
154	#20788	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₁₃₇	#20788	1	1	yes	characteristic, direct product factor, Sylow
156	#2875	C₂×C₁₃₇	1	1	yes	characteristic
157	#1841	C₂×C₁₃₇	1	2	yes	complemented
158	#1833	C₂×C₁₃₇	1	2	yes	complemented
159	#1827	C₂×C₁₃₇	1	2	yes	—
160	#1813	C₂×C₁₃₇	1	1	yes	characteristic
161	#1812	C₂×C₁₃₇	1	1	yes	characteristic
162	#1773	C₂×C₁₃₇	1	2	yes	complemented
163	#1757	C₂×C₁₃₇	1	2	yes	—
164	#1747	C₂×C₁₃₇	1	2	yes	complemented
165	C₂×#131	C₂²×C₁₃₇	1	1	yes	characteristic
166	C₂×#131	C₂²×C₁₃₇	1	1	yes	characteristic
167	C₄×#41	C₂²×C₁₃₇	1	1	yes	characteristic
168	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
169	#427	C₂²×C₁₃₇	1	2	yes	—
170	#413	C₂²×C₁₃₇	1	2	yes	—
171	#410	C₂²×C₁₃₇	1	2	yes	—
172	#405	C₄×C₁₃₇	1	2	yes	complemented
173	#405	C₄×C₁₃₇	1	2	yes	complemented
174	#405	C₂²×C₁₃₇	1	4	yes	—
175	#405	C₂²×C₁₃₇	1	1	yes	characteristic
176	#405	C₂²×C₁₃₇	1	1	yes	characteristic
177	C₄×M₁₆	C₄×C₁₃₇	1	2	yes	complemented
178	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
179	C₂×#97	C₄×C₁₃₇	1	2	yes	complemented
180	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
181	#344	C₂²×C₁₃₇	1	4	yes	—
182	#339	C₂²×C₁₃₇	1	4	yes	—
183	#338	C₂²×C₁₃₇	1	2	yes	—
184	H(3,ℤ/4)	C₂²×C₁₃₇	1	2	yes	—
185	H(3,ℤ/4)	C₂²×C₁₃₇	1	4	yes	—
186	C₂×#41	C₂³×C₁₃₇	1	1	yes	characteristic
187	#131	—	2	4	no	—
188	#131	—	2	2	no	—
189	#131	—	2	2	no	—
190	C₂×M₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
191	C₂×M₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
192	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
193	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
194	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
195	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
196	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
197	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
199	C₄×Q₈	—	2	1	no	—
200	C₄×D₄	—	2	2	no	—
201	C₄×D₄	—	2	1	no	—
202	#117	C₂×C₄×C₁₃₇	1	2	yes	—
203	#117	C₂³×C₁₃₇	1	4	yes	—
204	#117	C₂³×C₁₃₇	1	1	yes	characteristic
205	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
206	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
207	#108	—	2	2	no	—
208	#108	—	2	2	no	—
209	#107	—	2	2	no	—
210	#106	—	2	2	no	—
211	#101	—	2	2	no	—
212	#100	—	2	2	no	—
213	#99	—	2	4	no	—
214	#97	—	2	4	no	—
215	#97	C₂×C₄×C₁₃₇	1	1	yes	characteristic
216	#97	C₂×C₄×C₁₃₇	1	2	yes	—
217	#97	C₂×C₄×C₁₃₇	1	2	yes	—
218	#97	C₂×C₄×C₁₃₇	1	2	yes	—
219	#97	C₂×C₄×C₁₃₇	1	1	yes	characteristic
220	#97	C₂×C₄×C₁₃₇	1	1	yes	characteristic
221	#97	C₂×C₄×C₁₃₇	1	1	yes	characteristic
222	C₄×C₈	—	2	4	no	—
223	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
224	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
225	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	#41×C₁₃₇	1	1	yes	characteristic
229	C₂×D₄	#41×C₁₃₇	1	2	yes	—
230	C₂×D₄	#41×C₁₃₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
246	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
247	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
248	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
249	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
250	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
251	C₂×C₈	#41×C₁₃₇	1	4	yes	—
252	C₂×C₈	#41×C₁₃₇	1	2	yes	—
253	C₂×C₈	#41×C₁₃₇	1	2	yes	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	4	no	—
263	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
264	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
265	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
266	C₂³	#117×C₁₃₇	1	2	yes	—
267	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
268	Q₈	—	2	2	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	#117×C₁₃₇	1	2	yes	—
280	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
281	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
282	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
283	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	—	4	4	no	—
287	C₈	#117×C₁₃₇	1	1	yes	characteristic
288	C₈	#117×C₁₃₇	1	1	yes	characteristic
289	C₈	#117×C₁₃₇	1	2	yes	—
290	C₈	#117×C₁₃₇	1	2	yes	—
291	C₈	#117×C₁₃₇	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₂×#117×C₁₃₇	1	1	yes	characteristic
296	C₂²	#529×C₁₃₇	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	8	4	no	—
302	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
303	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
304	C₄	#529×C₁₃₇	1	2	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	#2283×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₇	C₂×#117
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂×C₁₃₇	#2283
prime radical	C₂×#41×C₁₃₇	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₇ ◁ #405×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1841×C₁₃₇	C₂	semidirect
3. #1833×C₁₃₇	C₂	semidirect
7. #1773×C₁₃₇	C₂	semidirect
9. #1747×C₁₃₇	C₂	semidirect
17. #405×C₁₃₇	C₄	semidirect
18. #405×C₁₃₇	C₄	semidirect
22. C₄×M₁₆×C₁₃₇	C₄	semidirect
24. C₂×#97×C₁₃₇	C₄	semidirect
154. #20788	C₁₃₇	direct (N × H)
155. C₁₃₇	#20788	direct (N × H)
157. #1841	C₂×C₁₃₇	semidirect
158. #1833	C₂×C₁₃₇	semidirect
162. #1773	C₂×C₁₃₇	semidirect
164. #1747	C₂×C₁₃₇	semidirect
172. #405	C₄×C₁₃₇	semidirect
173. #405	C₄×C₁₃₇	semidirect
177. C₄×M₁₆	C₄×C₁₃₇	semidirect
179. C₂×#97	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2875×C₁₃₇	2
#1841×C₁₃₇	2
#1833×C₁₃₇	2
#1827×C₁₃₇	2
#1813×C₁₃₇	2
#1812×C₁₃₇	2
#1773×C₁₃₇	2
#1757×C₁₃₇	2
#1747×C₁₃₇	2
#20788	137

Subgroups — C137×#20788 #46870174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20788 #46870174