Subgroups — C₁₃₇×#9102 #46858488

Order 35072.

546 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2982×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#2326×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1478×C₁₃₇	C₂	1	2	yes	maximal
4	#1461×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1274×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂×#124×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₁₃₇	—	2	4	no	—
10	#431×C₁₃₇	—	2	2	no	—
11	#431×C₁₃₇	C₂²	1	2	yes	—
12	#431×C₁₃₇	C₂²	1	2	yes	—
13	#391×C₁₃₇	C₄	1	1	yes	characteristic
14	#390×C₁₃₇	C₄	1	1	yes	characteristic
15	#388×C₁₃₇	—	2	2	no	—
16	#385×C₁₃₇	—	2	2	no	—
17	#384×C₁₃₇	C₄	1	1	yes	characteristic
18	#376×C₁₃₇	C₂²	1	1	yes	characteristic
19	D₄⋊C₈×C₁₃₇	—	2	4	no	—
20	#324×C₁₃₇	—	2	2	no	—
21	C₂²×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃₇	—	2	1	no	—
24	#124×C₁₃₇	—	2	2	no	—
25	#124×C₁₃₇	C₈	1	4	yes	complemented
26	C₄×Q₈×C₁₃₇	—	2	2	no	—
27	C₄×Q₈×C₁₃₇	—	2	2	no	—
28	C₄×D₄×C₁₃₇	—	2	2	no	—
29	C₄×D₄×C₁₃₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
37	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
40	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₁₃₇	—	2	4	no	—
42	C₄⋊C₈×C₁₃₇	—	2	2	no	—
43	C₄⋊C₈×C₁₃₇	—	2	2	no	—
44	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
45	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
46	C₄×C₈×C₁₃₇	—	2	4	no	—
47	C₄×C₈×C₁₃₇	—	2	2	no	—
48	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
49	#95×C₁₃₇	—	2	2	no	—
50	#95×C₁₃₇	—	2	1	no	—
51	#95×C₁₃₇	—	2	1	no	—
52	C₂⁴×C₁₃₇	—	2	1	no	—
53	C₂×Q₈×C₁₃₇	—	2	2	no	—
54	C₂×Q₈×C₁₃₇	—	2	2	no	—
55	C₂×Q₈×C₁₃₇	C₂×C₈	1	2	yes	—
56	C₂×Q₈×C₁₃₇	M₁₆	1	2	yes	—
57	C₂×D₄×C₁₃₇	—	2	2	no	—
58	C₂×D₄×C₁₃₇	—	2	1	no	—
59	C₂×D₄×C₁₃₇	—	2	2	no	—
60	C₂×D₄×C₁₃₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₃₇	—	2	1	no	—
62	C₂²×C₄×C₁₃₇	—	2	1	no	—
63	C₂²×C₄×C₁₃₇	—	2	1	no	—
64	C₂²×C₄×C₁₃₇	—	4	2	no	—
65	C₂²×C₄×C₁₃₇	—	8	2	no	—
66	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃₇	—	2	4	no	—
72	C₂×C₈×C₁₃₇	—	4	2	no	—
73	C₄⋊C₄×C₁₃₇	—	2	2	no	—
74	C₄⋊C₄×C₁₃₇	—	2	4	no	—
75	C₄⋊C₄×C₁₃₇	—	2	2	no	—
76	C₄⋊C₄×C₁₃₇	—	8	2	no	—
77	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
78	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
79	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
80	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
81	C₄²×C₁₃₇	—	2	2	no	—
82	C₄²×C₁₃₇	—	2	2	no	—
83	C₄²×C₁₃₇	—	2	1	no	—
84	C₄²×C₁₃₇	—	2	2	no	—
85	C₄²×C₁₃₇	C₂×C₈	1	2	yes	—
86	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
87	C₄²×C₁₃₇	#41	1	2	yes	—
88	C₂³×C₁₃₇	—	2	2	no	—
89	C₂³×C₁₃₇	—	4	2	no	—
90	C₂³×C₁₃₇	—	4	2	no	—
91	C₂³×C₁₃₇	—	4	1	no	—
92	C₂³×C₁₃₇	—	4	1	no	—
93	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	Q₈×C₁₃₇	—	2	4	no	—
95	Q₈×C₁₃₇	—	2	4	no	—
96	D₄×C₁₃₇	—	2	4	no	—
97	D₄×C₁₃₇	—	4	2	no	—
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	—	4	1	no	—
103	C₂×C₄×C₁₃₇	—	4	1	no	—
104	C₂×C₄×C₁₃₇	—	4	1	no	—
105	C₂×C₄×C₁₃₇	—	4	1	no	—
106	C₂×C₄×C₁₃₇	—	4	2	no	—
107	C₂×C₄×C₁₃₇	—	4	1	no	—
108	C₂×C₄×C₁₃₇	—	4	2	no	—
109	C₂×C₄×C₁₃₇	—	4	1	no	—
110	C₂×C₄×C₁₃₇	—	4	1	no	—
111	C₂×C₄×C₁₃₇	—	4	1	no	—
112	C₂×C₄×C₁₃₇	—	8	4	no	—
113	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	4	yes	—
114	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
116	C₂×C₄×C₁₃₇	#121	1	2	yes	—
117	C₂×C₄×C₁₃₇	#123	1	1	yes	characteristic
118	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	#131	1	2	yes	—
121	C₈×C₁₃₇	—	4	4	no	—
122	C₈×C₁₃₇	—	8	2	no	—
123	C₂²×C₁₃₇	—	2	1	no	—
124	C₂²×C₁₃₇	—	2	1	no	—
125	C₂²×C₁₃₇	—	4	4	no	—
126	C₂²×C₁₃₇	—	4	2	no	—
127	C₂²×C₁₃₇	—	8	2	no	—
128	C₂²×C₁₃₇	—	8	1	no	—
129	C₂²×C₁₃₇	—	8	1	no	—
130	C₂²×C₁₃₇	#386	1	1	yes	characteristic
131	C₂²×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₁₃₇	#408	1	1	yes	characteristic
133	C₄×C₁₃₇	—	2	4	no	—
134	C₄×C₁₃₇	—	2	2	no	—
135	C₄×C₁₃₇	—	2	1	no	—
136	C₄×C₁₃₇	—	2	1	no	—
137	C₄×C₁₃₇	—	4	2	no	—
138	C₄×C₁₃₇	—	4	2	no	—
139	C₄×C₁₃₇	—	8	1	no	—
140	C₄×C₁₃₇	—	8	1	no	—
141	C₄×C₁₃₇	C₈×D₄	1	2	yes	—
142	C₄×C₁₃₇	#435	1	2	yes	—
143	C₂×C₁₃₇	—	2	1	no	—
144	C₂×C₁₃₇	—	2	1	no	—
145	C₂×C₁₃₇	—	8	2	no	—
146	C₂×C₁₃₇	#1869	1	1	yes	characteristic
147	C₂×C₁₃₇	#1910	1	1	yes	characteristic
148	C₂×C₁₃₇	#1912	1	1	yes	characteristic
149	#9102	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₁₃₇	#9102	1	1	yes	characteristic, direct product factor, Sylow
151	#2982	C₂×C₁₃₇	1	1	yes	characteristic, complemented
152	#2326	C₂×C₁₃₇	1	1	yes	characteristic
153	#1478	C₂×C₁₃₇	1	2	yes	—
154	#1461	C₂×C₁₃₇	1	1	yes	characteristic, complemented
155	#1274	C₂×C₁₃₇	1	2	yes	complemented
156	C₂×#124	C₄×C₁₃₇	1	1	yes	characteristic
157	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
158	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
159	C₈×Q₈	—	2	4	no	—
160	#431	—	2	2	no	—
161	#431	C₂²×C₁₃₇	1	2	yes	—
162	#431	C₂²×C₁₃₇	1	2	yes	—
163	#391	C₄×C₁₃₇	1	1	yes	characteristic
164	#390	C₄×C₁₃₇	1	1	yes	characteristic
165	#388	—	2	2	no	—
166	#385	—	2	2	no	—
167	#384	C₄×C₁₃₇	1	1	yes	characteristic
168	#376	C₂²×C₁₃₇	1	1	yes	characteristic
169	D₄⋊C₈	—	2	4	no	—
170	#324	—	2	2	no	—
171	C₂²×Q₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
172	C₂²×D₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
173	C₂³×C₄	—	2	1	no	—
174	#124	—	2	2	no	—
175	#124	C₈×C₁₃₇	1	4	yes	complemented
176	C₄×Q₈	—	2	2	no	—
177	C₄×Q₈	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
183	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
190	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
191	C₄⋊C₈	—	2	4	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₄×C₈	—	2	4	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
199	#95	—	2	2	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	C₂⁴	—	2	1	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	C₂×C₈×C₁₃₇	1	2	yes	—
206	C₂×Q₈	M₁₆×C₁₃₇	1	2	yes	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂×C₈×C₁₃₇	1	2	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	8	2	no	—
216	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	8	2	no	—
227	C₂²⋊C₄	—	2	8	no	—
228	C₂²⋊C₄	—	4	1	no	—
229	C₂²⋊C₄	—	4	1	no	—
230	C₂²⋊C₄	—	8	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	2	no	—
235	C₄²	C₂×C₈×C₁₃₇	1	2	yes	—
236	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
237	C₄²	#41×C₁₃₇	1	2	yes	—
238	C₂³	—	2	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	4	1	no	—
242	C₂³	—	4	1	no	—
243	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	D₄	—	2	4	no	—
247	D₄	—	4	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	4	yes	—
264	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
265	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
266	C₂×C₄	#121×C₁₃₇	1	2	yes	—
267	C₂×C₄	#123×C₁₃₇	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
269	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
270	C₂×C₄	#131×C₁₃₇	1	2	yes	—
271	C₈	—	4	4	no	—
272	C₈	—	8	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	1	no	—
279	C₂²	—	8	1	no	—
280	C₂²	#386×C₁₃₇	1	1	yes	characteristic
281	C₂²	C₂×C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
282	C₂²	#408×C₁₃₇	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	8	1	no	—
290	C₄	—	8	1	no	—
291	C₄	C₈×D₄×C₁₃₇	1	2	yes	—
292	C₄	#435×C₁₃₇	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	2	no	—
296	C₂	#1869×C₁₃₇	1	1	yes	characteristic
297	C₂	#1910×C₁₃₇	1	1	yes	characteristic
298	C₂	#1912×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#386
commutator	C₂×C₄	C₂²×C₈×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	C₂²×D₄×C₁₃₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #2982×C₁₃₇	C₂	semidirect
4. #1461×C₁₃₇	C₂	semidirect
5. #1274×C₁₃₇	C₂	semidirect
25. #124×C₁₃₇	C₈	semidirect
149. #9102	C₁₃₇	direct (N × H)
150. C₁₃₇	#9102	direct (N × H)
151. #2982	C₂×C₁₃₇	semidirect
154. #1461	C₂×C₁₃₇	semidirect
155. #1274	C₂×C₁₃₇	semidirect
175. #124	C₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2982×C₁₃₇	2
#2326×C₁₃₇	2
#1478×C₁₃₇	2
#1461×C₁₃₇	2
#1274×C₁₃₇	2
#9102	137

Subgroups — C137×#9102 #46858488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9102 #46858488