Subgroups — C₁₃₇×#11347 #46860733

Order 35072.

666 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	#2296×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1742×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1740×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#339×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×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
11	#389×C₁₃₇	—	2	2	no	—
12	#385×C₁₃₇	—	2	2	no	—
13	C₄×C₄⋊C₄×C₁₃₇	C₄	1	4	yes	complemented
14	#343×C₁₃₇	—	2	2	no	—
15	#342×C₁₃₇	—	2	2	no	—
16	#339×C₁₃₇	—	2	4	no	—
17	H(3,ℤ/4)×C₁₃₇	—	2	4	no	—
18	C₂³×C₄×C₁₃₇	—	2	1	no	—
19	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₃₇	—	2	2	no	—
21	C₂×M₁₆×C₁₃₇	—	2	2	no	—
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	Q₈	1	4	yes	—
26	C₂²×C₈×C₁₃₇	—	2	2	no	—
27	#117×C₁₃₇	—	2	2	no	—
28	#117×C₁₃₇	—	2	2	no	—
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	D₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
42	C₂×C₄²×C₁₃₇	—	2	2	no	—
43	C₂×C₄²×C₁₃₇	—	2	2	no	—
44	C₂×C₄²×C₁₃₇	—	2	2	no	—
45	C₂×C₄²×C₁₃₇	—	2	1	no	—
46	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
49	#95×C₁₃₇	—	2	4	no	—
50	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₇	—	2	2	no	—
52	C₂²×C₄×C₁₃₇	—	2	2	no	—
53	C₂²×C₄×C₁₃₇	—	4	4	no	—
54	C₂²×C₄×C₁₃₇	—	4	2	no	—
55	C₂²×C₄×C₁₃₇	—	4	2	no	—
56	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
62	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
63	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
64	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
65	M₁₆×C₁₃₇	—	2	4	no	—
66	M₁₆×C₁₃₇	—	2	4	no	—
67	M₁₆×C₁₃₇	—	4	2	no	—
68	M₁₆×C₁₃₇	—	4	2	no	—
69	C₂×C₈×C₁₃₇	—	2	4	no	—
70	C₂×C₈×C₁₃₇	—	4	2	no	—
71	C₂×C₈×C₁₃₇	—	4	2	no	—
72	C₄⋊C₄×C₁₃₇	—	2	4	no	—
73	C₄⋊C₄×C₁₃₇	—	2	4	no	—
74	C₄⋊C₄×C₁₃₇	—	4	8	no	—
75	C₄⋊C₄×C₁₃₇	—	4	1	no	—
76	C₄⋊C₄×C₁₃₇	—	4	2	no	—
77	C₄⋊C₄×C₁₃₇	—	4	1	no	—
78	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
79	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
80	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
81	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
82	C₄²×C₁₃₇	—	2	2	no	—
83	C₄²×C₁₃₇	—	2	2	no	—
84	C₄²×C₁₃₇	—	2	4	no	—
85	C₄²×C₁₃₇	—	2	4	no	—
86	C₄²×C₁₃₇	—	2	4	no	—
87	C₄²×C₁₃₇	—	4	1	no	—
88	C₄²×C₁₃₇	—	4	2	no	—
89	C₄²×C₁₃₇	—	4	1	no	—
90	C₂³×C₁₃₇	—	4	2	no	—
91	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂³×C₁₃₇	#121	1	2	yes	—
93	C₂³×C₁₃₇	#123	1	2	yes	—
94	C₂³×C₁₃₇	#124	1	1	yes	characteristic
95	C₂³×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃₇	—	2	4	no	—
97	C₂×C₄×C₁₃₇	—	2	4	no	—
98	C₂×C₄×C₁₃₇	—	4	2	no	—
99	C₂×C₄×C₁₃₇	—	4	8	no	—
100	C₂×C₄×C₁₃₇	—	4	2	no	—
101	C₂×C₄×C₁₃₇	—	4	4	no	—
102	C₂×C₄×C₁₃₇	—	4	4	no	—
103	C₂×C₄×C₁₃₇	—	4	2	no	—
104	C₂×C₄×C₁₃₇	—	4	4	no	—
105	C₂×C₄×C₁₃₇	—	8	4	no	—
106	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₁₃₇	#123	1	2	yes	—
114	C₂×C₄×C₁₃₇	#123	1	2	yes	—
115	C₂×C₄×C₁₃₇	#123	1	2	yes	—
116	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	#125	1	2	yes	—
118	C₂×C₄×C₁₃₇	#125	1	2	yes	—
119	C₂×C₄×C₁₃₇	#126	1	2	yes	—
120	C₂×C₄×C₁₃₇	#126	1	2	yes	—
121	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
122	C₈×C₁₃₇	—	4	4	no	—
123	C₂²×C₁₃₇	—	2	2	no	—
124	C₂²×C₁₃₇	—	2	4	no	—
125	C₂²×C₁₃₇	—	4	2	no	—
126	C₂²×C₁₃₇	—	4	1	no	—
127	C₂²×C₁₃₇	—	4	1	no	—
128	C₂²×C₁₃₇	#381	1	1	yes	characteristic
129	C₂²×C₁₃₇	#390	1	1	yes	characteristic
130	C₂²×C₁₃₇	#391	1	1	yes	characteristic
131	C₂²×C₁₃₇	#397	1	2	yes	—
132	C₂²×C₁₃₇	#398	1	2	yes	—
133	C₄×C₁₃₇	—	4	4	no	—
134	C₄×C₁₃₇	—	4	4	no	—
135	C₄×C₁₃₇	—	8	2	no	—
136	C₄×C₁₃₇	—	8	4	no	—
137	C₄×C₁₃₇	#387	1	2	yes	—
138	C₄×C₁₃₇	#387	1	2	yes	—
139	C₄×C₁₃₇	#399	1	2	yes	—
140	C₄×C₁₃₇	#400	1	2	yes	—
141	C₂×C₁₃₇	—	2	1	no	—
142	C₂×C₁₃₇	—	2	2	no	—
143	C₂×C₁₃₇	—	2	1	no	—
144	C₂×C₁₃₇	—	2	2	no	—
145	C₂×C₁₃₇	#1445	1	1	yes	characteristic
146	C₂×C₁₃₇	#1827	1	2	yes	—
147	#11347	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₁₃₇	#11347	1	1	yes	characteristic, direct product factor, Sylow
149	#2296	C₂×C₁₃₇	1	1	yes	characteristic
150	#1742	C₂×C₁₃₇	1	1	yes	characteristic
151	#1740	C₂×C₁₃₇	1	1	yes	characteristic
152	C₂×#339	C₂×C₁₃₇	1	2	yes	—
153	C₂×H(3,ℤ/4)	C₂×C₁₃₇	1	2	yes	—
154	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
155	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
156	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
157	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
158	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
159	#389	—	2	2	no	—
160	#385	—	2	2	no	—
161	C₄×C₄⋊C₄	C₄×C₁₃₇	1	4	yes	complemented
162	#343	—	2	2	no	—
163	#342	—	2	2	no	—
164	#339	—	2	4	no	—
165	H(3,ℤ/4)	—	2	4	no	—
166	C₂³×C₄	—	2	1	no	—
167	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	Q₈×C₁₃₇	1	4	yes	—
174	C₂²×C₈	—	2	2	no	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	—	2	2	no	—
182	#117	D₄×C₁₃₇	1	4	yes	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	1	no	—
189	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
196	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
197	#95	—	2	4	no	—
198	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
209	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
210	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
211	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
212	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	4	8	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	1	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
240	C₂³	#121×C₁₃₇	1	2	yes	—
241	C₂³	#123×C₁₃₇	1	2	yes	—
242	C₂³	#124×C₁₃₇	1	1	yes	characteristic
243	C₂³	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
255	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
256	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
257	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
258	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
259	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
260	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
261	C₂×C₄	#123×C₁₃₇	1	2	yes	—
262	C₂×C₄	#123×C₁₃₇	1	2	yes	—
263	C₂×C₄	#123×C₁₃₇	1	2	yes	—
264	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
265	C₂×C₄	#125×C₁₃₇	1	2	yes	—
266	C₂×C₄	#125×C₁₃₇	1	2	yes	—
267	C₂×C₄	#126×C₁₃₇	1	2	yes	—
268	C₂×C₄	#126×C₁₃₇	1	2	yes	—
269	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
270	C₈	—	4	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	#381×C₁₃₇	1	1	yes	characteristic
277	C₂²	#390×C₁₃₇	1	1	yes	characteristic
278	C₂²	#391×C₁₃₇	1	1	yes	characteristic
279	C₂²	#397×C₁₃₇	1	2	yes	—
280	C₂²	#398×C₁₃₇	1	2	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	4	4	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	#387×C₁₃₇	1	2	yes	—
286	C₄	#387×C₁₃₇	1	2	yes	—
287	C₄	#399×C₁₃₇	1	2	yes	—
288	C₄	#400×C₁₃₇	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	#1445×C₁₃₇	1	1	yes	characteristic
294	C₂	#1827×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#117
commutator	C₂²×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#381
prime radical	C₂⁴×C₁₃₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×#117×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄×C₄⋊C₄×C₁₃₇	C₄	semidirect
147. #11347	C₁₃₇	direct (N × H)
148. C₁₃₇	#11347	direct (N × H)
161. C₄×C₄⋊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
#2296×C₁₃₇	2
#1742×C₁₃₇	2
#1740×C₁₃₇	2
C₂×#339×C₁₃₇	2
C₂×H(3,ℤ/4)×C₁₃₇	2
#11347	137

Subgroups — C137×#11347 #46860733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#11347 #46860733