Subgroups — C₁₃₇×#11370 #46860756

Order 35072.

794 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2285×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#339×C₁₃₇	C₂	1	4	yes	maximal
3	#1728×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#95×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
6	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
9	#384×C₁₃₇	C₄	1	1	yes	characteristic, complemented
10	#384×C₁₃₇	C₄	1	1	yes	characteristic, complemented
11	#382×C₁₃₇	C₄	1	2	yes	complemented
12	#380×C₁₃₇	—	2	2	no	—
13	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
14	C₂×#95×C₁₃₇	—	2	2	no	—
15	#339×C₁₃₇	—	2	8	no	—
16	#336×C₁₃₇	—	2	2	no	—
17	C₂³×C₄×C₁₃₇	—	2	2	no	—
18	C₂³×C₄×C₁₃₇	D₄	1	2	yes	—
19	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₃₇	—	2	2	no	—
21	C₂×M₁₆×C₁₃₇	—	2	4	no	—
22	C₂×M₁₆×C₁₃₇	—	2	4	no	—
23	C₂×M₁₆×C₁₃₇	Q₈	1	4	yes	—
24	C₂²×C₈×C₁₃₇	—	2	2	no	—
25	#117×C₁₃₇	—	2	4	no	—
26	#117×C₁₃₇	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
32	C₂×C₄²×C₁₃₇	—	2	4	no	—
33	C₂×C₄²×C₁₃₇	—	2	4	no	—
34	C₂×C₄²×C₁₃₇	—	2	4	no	—
35	C₂×C₄²×C₁₃₇	—	2	2	no	—
36	C₂×C₄²×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	4	yes	—
40	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
41	#95×C₁₃₇	—	2	8	no	—
42	#95×C₁₃₇	—	2	4	no	—
43	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₇	—	2	8	no	—
45	C₂²×C₄×C₁₃₇	—	2	4	no	—
46	C₂²×C₄×C₁₃₇	—	2	4	no	—
47	C₂²×C₄×C₁₃₇	—	2	4	no	—
48	C₂²×C₄×C₁₃₇	—	2	2	no	—
49	C₂²×C₄×C₁₃₇	—	2	2	no	—
50	C₂²×C₄×C₁₃₇	—	4	2	no	—
51	C₂²×C₄×C₁₃₇	—	4	4	no	—
52	C₂²×C₄×C₁₃₇	—	4	2	no	—
53	C₂²×C₄×C₁₃₇	—	4	2	no	—
54	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
61	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
62	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
63	M₁₆×C₁₃₇	—	2	8	no	—
64	M₁₆×C₁₃₇	—	4	4	no	—
65	C₂×C₈×C₁₃₇	—	2	4	no	—
66	C₂×C₈×C₁₃₇	—	4	2	no	—
67	C₂×C₈×C₁₃₇	—	4	2	no	—
68	C₄⋊C₄×C₁₃₇	—	2	4	no	—
69	C₄⋊C₄×C₁₃₇	—	2	4	no	—
70	C₄⋊C₄×C₁₃₇	—	4	4	no	—
71	C₄⋊C₄×C₁₃₇	—	4	4	no	—
72	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
73	C₄²×C₁₃₇	—	2	2	no	—
74	C₄²×C₁₃₇	—	2	2	no	—
75	C₄²×C₁₃₇	—	2	4	no	—
76	C₄²×C₁₃₇	—	2	8	no	—
77	C₄²×C₁₃₇	—	4	2	no	—
78	C₄²×C₁₃₇	—	4	4	no	—
79	C₄²×C₁₃₇	—	4	2	no	—
80	C₂³×C₁₃₇	—	2	2	no	—
81	C₂³×C₁₃₇	—	2	2	no	—
82	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂³×C₁₃₇	#121	1	2	yes	—
84	C₂³×C₁₃₇	#123	1	2	yes	—
85	C₂³×C₁₃₇	#124	1	1	yes	characteristic
86	C₂³×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃₇	—	2	2	no	—
88	C₂×C₄×C₁₃₇	—	2	2	no	—
89	C₂×C₄×C₁₃₇	—	2	8	no	—
90	C₂×C₄×C₁₃₇	—	2	2	no	—
91	C₂×C₄×C₁₃₇	—	2	2	no	—
92	C₂×C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₄×C₁₃₇	—	4	8	no	—
94	C₂×C₄×C₁₃₇	—	4	8	no	—
95	C₂×C₄×C₁₃₇	—	4	4	no	—
96	C₂×C₄×C₁₃₇	—	4	8	no	—
97	C₂×C₄×C₁₃₇	—	4	4	no	—
98	C₂×C₄×C₁₃₇	—	4	2	no	—
99	C₂×C₄×C₁₃₇	—	4	2	no	—
100	C₂×C₄×C₁₃₇	—	4	4	no	—
101	C₂×C₄×C₁₃₇	—	4	4	no	—
102	C₂×C₄×C₁₃₇	#104	1	4	yes	—
103	C₂×C₄×C₁₃₇	#117	1	2	yes	—
104	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₁₃₇	#123	1	4	yes	—
109	C₂×C₄×C₁₃₇	#123	1	2	yes	—
110	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	#125	1	4	yes	—
112	C₂×C₄×C₁₃₇	#126	1	4	yes	—
113	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
114	C₈×C₁₃₇	—	4	4	no	—
115	C₂²×C₁₃₇	—	2	4	no	—
116	C₂²×C₁₃₇	—	2	2	no	—
117	C₂²×C₁₃₇	—	2	2	no	—
118	C₂²×C₁₃₇	—	2	2	no	—
119	C₂²×C₁₃₇	—	2	2	no	—
120	C₂²×C₁₃₇	#381	1	1	yes	characteristic
121	C₂²×C₁₃₇	#390	1	1	yes	characteristic
122	C₂²×C₁₃₇	#391	1	1	yes	characteristic
123	C₂²×C₁₃₇	#397	1	2	yes	—
124	C₂²×C₁₃₇	#398	1	2	yes	—
125	C₂²×C₁₃₇	C₂×#104	1	2	yes	—
126	C₂²×C₁₃₇	#421	1	2	yes	—
127	C₄×C₁₃₇	—	2	4	no	—
128	C₄×C₁₃₇	—	4	8	no	—
129	C₄×C₁₃₇	—	4	2	no	—
130	C₄×C₁₃₇	—	8	2	no	—
131	C₄×C₁₃₇	—	8	2	no	—
132	C₄×C₁₃₇	#387	1	4	yes	—
133	C₄×C₁₃₇	#399	1	2	yes	—
134	C₄×C₁₃₇	#400	1	2	yes	—
135	C₂×C₁₃₇	—	2	2	no	—
136	C₂×C₁₃₇	—	2	2	no	—
137	C₂×C₁₃₇	#1445	1	1	yes	characteristic
138	C₂×C₁₃₇	#1826	1	2	yes	—
139	C₂×C₁₃₇	#1962	1	1	yes	characteristic
140	C₂×C₁₃₇	#1963	1	1	yes	characteristic
141	C₂×C₁₃₇	#1986	1	2	yes	—
142	#11370	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₁₃₇	#11370	1	1	yes	characteristic, direct product factor, Sylow
144	#2285	C₂×C₁₃₇	1	1	yes	characteristic
145	C₂×#339	C₂×C₁₃₇	1	4	yes	—
146	#1728	C₂×C₁₃₇	1	1	yes	characteristic
147	C₄×#95	C₂×C₁₃₇	1	1	yes	characteristic
148	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
149	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
150	C₂²×C₄²	C₂²×C₁₃₇	1	2	yes	—
151	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
152	#384	C₄×C₁₃₇	1	1	yes	characteristic, complemented
153	#384	C₄×C₁₃₇	1	1	yes	characteristic, complemented
154	#382	C₄×C₁₃₇	1	2	yes	complemented
155	#380	—	2	2	no	—
156	C₄×C₄⋊C₄	—	2	2	no	—
157	C₂×#95	—	2	2	no	—
158	#339	—	2	8	no	—
159	#336	—	2	2	no	—
160	C₂³×C₄	—	2	2	no	—
161	C₂³×C₄	D₄×C₁₃₇	1	2	yes	—
162	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	Q₈×C₁₃₇	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	#117	—	2	4	no	—
169	#117	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
174	C₂×C₂²⋊C₄	—	2	2	no	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	—	2	4	no	—
177	C₂×C₄²	—	2	4	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₁₃₇	1	4	yes	—
183	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
184	#95	—	2	8	no	—
185	#95	—	2	4	no	—
186	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
198	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
203	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
204	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
205	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
206	M₁₆	—	2	8	no	—
207	M₁₆	—	4	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	8	no	—
220	C₄²	—	4	2	no	—
221	C₄²	—	4	4	no	—
222	C₄²	—	4	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
226	C₂³	#121×C₁₃₇	1	2	yes	—
227	C₂³	#123×C₁₃₇	1	2	yes	—
228	C₂³	#124×C₁₃₇	1	1	yes	characteristic
229	C₂³	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	#104×C₁₃₇	1	4	yes	—
246	C₂×C₄	#117×C₁₃₇	1	2	yes	—
247	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
248	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
249	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
250	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
251	C₂×C₄	#123×C₁₃₇	1	4	yes	—
252	C₂×C₄	#123×C₁₃₇	1	2	yes	—
253	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
254	C₂×C₄	#125×C₁₃₇	1	4	yes	—
255	C₂×C₄	#126×C₁₃₇	1	4	yes	—
256	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
257	C₈	—	4	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	#381×C₁₃₇	1	1	yes	characteristic
264	C₂²	#390×C₁₃₇	1	1	yes	characteristic
265	C₂²	#391×C₁₃₇	1	1	yes	characteristic
266	C₂²	#397×C₁₃₇	1	2	yes	—
267	C₂²	#398×C₁₃₇	1	2	yes	—
268	C₂²	C₂×#104×C₁₃₇	1	2	yes	—
269	C₂²	#421×C₁₃₇	1	2	yes	—
270	C₄	—	2	4	no	—
271	C₄	—	4	8	no	—
272	C₄	—	4	2	no	—
273	C₄	—	8	2	no	—
274	C₄	—	8	2	no	—
275	C₄	#387×C₁₃₇	1	4	yes	—
276	C₄	#399×C₁₃₇	1	2	yes	—
277	C₄	#400×C₁₃₇	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	#1445×C₁₃₇	1	1	yes	characteristic
281	C₂	#1826×C₁₃₇	1	2	yes	—
282	C₂	#1962×C₁₃₇	1	1	yes	characteristic
283	C₂	#1963×C₁₃₇	1	1	yes	characteristic
284	C₂	#1986×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
9. #384×C₁₃₇	C₄	semidirect
10. #384×C₁₃₇	C₄	semidirect
11. #382×C₁₃₇	C₄	semidirect
142. #11370	C₁₃₇	direct (N × H)
143. C₁₃₇	#11370	direct (N × H)
152. #384	C₄×C₁₃₇	semidirect
153. #384	C₄×C₁₃₇	semidirect
154. #382	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2285×C₁₃₇	2
C₂×#339×C₁₃₇	2
#1728×C₁₃₇	2
C₄×#95×C₁₃₇	2
#11370	137

Subgroups — C137×#11370 #46860756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#11370 #46860756