Subgroups — C₁₃₇×#25912 #46875298

Order 35072.

1078 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#491×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#2857×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2026×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1992×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1872×C₁₃₇	C₂	1	4	yes	maximal
7	#1749×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×#125×C₁₃₇	—	2	8	no	—
10	C₂×#125×C₁₃₇	—	2	2	no	—
11	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₁₃₇	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
17	#491×C₁₃₇	—	2	8	no	—
18	C₂×#103×C₁₃₇	C₂²	1	8	yes	—
19	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
20	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
21	#391×C₁₃₇	C₂²	1	2	yes	—
22	#389×C₁₃₇	—	2	2	no	—
23	#384×C₁₃₇	C₂²	1	1	yes	characteristic
24	#380×C₁₃₇	—	2	2	no	—
25	#340×C₁₃₇	C₂²	1	8	yes	—
26	#336×C₁₃₇	C₂²	1	2	yes	—
27	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
28	C₂³×C₄×C₁₃₇	—	2	2	no	—
29	C₂³×C₄×C₁₃₇	D₄	1	2	yes	—
30	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
32	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
33	#125×C₁₃₇	—	2	16	no	—
34	#125×C₁₃₇	—	2	8	no	—
35	#125×C₁₃₇	—	2	4	no	—
36	#125×C₁₃₇	D₄	1	8	yes	—
37	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	16	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
49	C₂×C₄²×C₁₃₇	—	2	4	no	—
50	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
53	#103×C₁₃₇	—	2	16	no	—
54	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
55	#97×C₁₃₇	—	2	4	no	—
56	#95×C₁₃₇	—	2	4	no	—
57	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₁₃₇	—	2	4	no	—
59	C₂×Q₈×C₁₃₇	—	2	4	no	—
60	C₂×Q₈×C₁₃₇	—	2	4	no	—
61	C₂²×C₄×C₁₃₇	—	2	2	no	—
62	C₂²×C₄×C₁₃₇	—	2	2	no	—
63	C₂²×C₄×C₁₃₇	—	2	8	no	—
64	C₂²×C₄×C₁₃₇	—	2	4	no	—
65	C₂²×C₄×C₁₃₇	—	2	4	no	—
66	C₂²×C₄×C₁₃₇	—	2	4	no	—
67	C₂²×C₄×C₁₃₇	—	2	2	no	—
68	C₂²×C₄×C₁₃₇	—	2	2	no	—
69	C₂²×C₄×C₁₃₇	—	4	2	no	—
70	C₂²×C₄×C₁₃₇	—	4	4	no	—
71	C₂²×C₄×C₁₃₇	—	4	4	no	—
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
76	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₃₇	—	2	4	no	—
78	C₂×C₈×C₁₃₇	—	2	4	no	—
79	C₂×C₈×C₁₃₇	—	2	4	no	—
80	C₄⋊C₄×C₁₃₇	—	2	8	no	—
81	C₄⋊C₄×C₁₃₇	—	2	16	no	—
82	C₄⋊C₄×C₁₃₇	—	2	16	no	—
83	C₄⋊C₄×C₁₃₇	—	2	4	no	—
84	C₄⋊C₄×C₁₃₇	—	2	4	no	—
85	C₄⋊C₄×C₁₃₇	—	4	4	no	—
86	C₄⋊C₄×C₁₃₇	—	4	4	no	—
87	C₄⋊C₄×C₁₃₇	—	4	4	no	—
88	C₄⋊C₄×C₁₃₇	C₂×D₄	1	8	yes	—
89	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
90	C₄²×C₁₃₇	—	2	4	no	—
91	C₄²×C₁₃₇	—	2	2	no	—
92	C₄²×C₁₃₇	C₂×D₄	1	4	yes	—
93	C₂³×C₁₃₇	—	2	1	no	—
94	C₂³×C₁₃₇	—	2	1	no	—
95	C₂³×C₁₃₇	—	2	2	no	—
96	C₂³×C₁₃₇	#124	1	2	yes	—
97	C₂³×C₁₃₇	#137	1	4	yes	—
98	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
99	Q₈×C₁₃₇	—	4	8	no	—
100	C₂×C₄×C₁₃₇	—	2	1	no	—
101	C₂×C₄×C₁₃₇	—	2	1	no	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	2	2	no	—
104	C₂×C₄×C₁₃₇	—	2	8	no	—
105	C₂×C₄×C₁₃₇	—	2	2	no	—
106	C₂×C₄×C₁₃₇	—	2	8	no	—
107	C₂×C₄×C₁₃₇	—	2	2	no	—
108	C₂×C₄×C₁₃₇	—	2	2	no	—
109	C₂×C₄×C₁₃₇	—	4	8	no	—
110	C₂×C₄×C₁₃₇	—	4	16	no	—
111	C₂×C₄×C₁₃₇	—	4	4	no	—
112	C₂×C₄×C₁₃₇	—	4	4	no	—
113	C₂×C₄×C₁₃₇	—	4	2	no	—
114	C₂×C₄×C₁₃₇	—	4	2	no	—
115	C₂×C₄×C₁₃₇	—	4	2	no	—
116	C₂×C₄×C₁₃₇	—	4	2	no	—
117	C₂×C₄×C₁₃₇	—	4	4	no	—
118	C₂×C₄×C₁₃₇	#120	1	4	yes	—
119	C₂×C₄×C₁₃₇	#124	1	2	yes	—
120	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
121	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
123	C₈×C₁₃₇	—	4	4	no	—
124	C₂²×C₁₃₇	—	2	2	no	—
125	C₂²×C₁₃₇	—	2	2	no	—
126	C₂²×C₁₃₇	—	2	4	no	—
127	C₂²×C₁₃₇	—	2	4	no	—
128	C₂²×C₁₃₇	#491	1	4	yes	complemented
129	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
130	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
131	C₂²×C₁₃₇	#536	1	1	yes	characteristic
132	C₂²×C₁₃₇	C₂×#137	1	2	yes	—
133	C₂²×C₁₃₇	#575	1	2	yes	—
134	C₄×C₁₃₇	—	2	2	no	—
135	C₄×C₁₃₇	—	2	4	no	—
136	C₄×C₁₃₇	—	4	2	no	—
137	C₄×C₁₃₇	—	4	8	no	—
138	C₄×C₁₃₇	—	8	2	no	—
139	C₄×C₁₃₇	—	8	2	no	—
140	C₄×C₁₃₇	#548	1	4	yes	—
141	C₂×C₁₃₇	—	2	4	no	—
142	C₂×C₁₃₇	#2436	1	1	yes	characteristic
143	C₂×C₁₃₇	#3007	1	2	yes	—
144	C₂×C₁₃₇	C₂×#491	1	1	yes	characteristic
145	C₂×C₁₃₇	#3142	1	1	yes	characteristic
146	C₂×C₁₃₇	#3236	1	2	yes	—
147	#25912	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₁₃₇	#25912	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×#125	C₂×C₁₃₇	1	1	yes	characteristic
150	C₂×#491	C₂×C₁₃₇	1	2	yes	complemented
151	#2857	C₂×C₁₃₇	1	1	yes	characteristic
152	#2026	C₂×C₁₃₇	1	4	yes	complemented
153	#1992	C₂×C₁₃₇	1	2	yes	complemented
154	#1872	C₂×C₁₃₇	1	4	yes	—
155	#1749	C₂×C₁₃₇	1	1	yes	characteristic
156	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
157	C₂×#125	—	2	8	no	—
158	C₂×#125	—	2	2	no	—
159	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
160	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
161	C₂×C₂²⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
162	C₂²×C₄⋊C₄	—	2	2	no	—
163	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
164	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
165	#491	—	2	8	no	—
166	C₂×#103	C₂²×C₁₃₇	1	8	yes	—
167	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
168	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
169	#391	C₂²×C₁₃₇	1	2	yes	—
170	#389	—	2	2	no	—
171	#384	C₂²×C₁₃₇	1	1	yes	characteristic
172	#380	—	2	2	no	—
173	#340	C₂²×C₁₃₇	1	8	yes	—
174	#336	C₂²×C₁₃₇	1	2	yes	—
175	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
176	C₂³×C₄	—	2	2	no	—
177	C₂³×C₄	D₄×C₁₃₇	1	2	yes	—
178	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
180	C₄⋊Q₈	—	2	4	no	—
181	#125	—	2	16	no	—
182	#125	—	2	8	no	—
183	#125	—	2	4	no	—
184	#125	D₄×C₁₃₇	1	8	yes	—
185	C₂²⋊Q₈	—	2	8	no	—
186	C₂×C₄⋊C₄	—	2	16	no	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	8	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₄²	—	2	4	no	—
198	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
199	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
201	#103	—	2	16	no	—
202	C₂²⋊C₈	—	2	4	no	—
203	#97	—	2	4	no	—
204	#95	—	2	4	no	—
205	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
206	C₂×Q₈	—	2	4	no	—
207	C₂×Q₈	—	2	4	no	—
208	C₂×Q₈	—	2	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
223	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
224	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	16	no	—
230	C₄⋊C₄	—	2	16	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	C₂×D₄×C₁₃₇	1	8	yes	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	2	no	—
240	C₄²	C₂×D₄×C₁₃₇	1	4	yes	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	2	no	—
244	C₂³	#124×C₁₃₇	1	2	yes	—
245	C₂³	#137×C₁₃₇	1	4	yes	—
246	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
247	Q₈	—	4	8	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	16	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	#120×C₁₃₇	1	4	yes	—
267	C₂×C₄	#124×C₁₃₇	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
269	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
270	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
271	C₈	—	4	4	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	#491×C₁₃₇	1	4	yes	complemented
277	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
278	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
279	C₂²	#536×C₁₃₇	1	1	yes	characteristic
280	C₂²	C₂×#137×C₁₃₇	1	2	yes	—
281	C₂²	#575×C₁₃₇	1	2	yes	—
282	C₄	—	2	2	no	—
283	C₄	—	2	4	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	8	no	—
286	C₄	—	8	2	no	—
287	C₄	—	8	2	no	—
288	C₄	#548×C₁₃₇	1	4	yes	—
289	C₂	—	2	4	no	—
290	C₂	#2436×C₁₃₇	1	1	yes	characteristic
291	C₂	#3007×C₁₃₇	1	2	yes	—
292	C₂	C₂×#491×C₁₃₇	1	1	yes	characteristic
293	C₂	#3142×C₁₃₇	1	1	yes	characteristic
294	C₂	#3236×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×#491×C₁₃₇	C₂	semidirect
4. #2026×C₁₃₇	C₂	semidirect
5. #1992×C₁₃₇	C₂	semidirect
128. C₂²×C₁₃₇	#491	semidirect
147. #25912	C₁₃₇	direct (N × H)
148. C₁₃₇	#25912	direct (N × H)
150. C₂×#491	C₂×C₁₃₇	semidirect
152. #2026	C₂×C₁₃₇	semidirect
153. #1992	C₂×C₁₃₇	semidirect
276. C₂²	#491×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#125×C₁₃₇	2
C₂×#491×C₁₃₇	2
#2857×C₁₃₇	2
#2026×C₁₃₇	2
#1992×C₁₃₇	2
#1872×C₁₃₇	2
#1749×C₁₃₇	2
#25912	137

Subgroups — C137×#25912 #46875298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25912 #46875298