Subgroups — C₁₃₇×#9764 #46859150

Order 35072.

418 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3169×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1746×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#1568×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1530×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1505×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#532×C₁₃₇	C₂²	1	2	yes	—
7	#500×C₁₃₇	C₄	1	1	yes	characteristic, complemented
8	#493×C₁₃₇	C₄	1	1	yes	characteristic, complemented
9	#486×C₁₃₇	C₄	1	2	yes	complemented
10	#466×C₁₃₇	—	2	1	no	—
11	#466×C₁₃₇	—	2	1	no	—
12	C₄×D₈×C₁₃₇	—	2	1	no	—
13	C₄×D₈×C₁₃₇	—	2	1	no	—
14	#432×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	#376×C₁₃₇	C₂²	1	1	yes	characteristic
18	#336×C₁₃₇	—	2	1	no	—
19	#336×C₁₃₇	—	2	1	no	—
20	#335×C₁₃₇	—	2	1	no	—
21	#334×C₁₃₇	—	2	1	no	—
22	#329×C₁₃₇	—	2	4	no	—
23	#327×C₁₃₇	—	2	4	no	—
24	C₂×#41×C₁₃₇	—	2	2	no	—
25	C₂×D₈×C₁₃₇	—	2	1	no	—
26	C₂×D₈×C₁₃₇	—	2	1	no	—
27	C₂²×C₈×C₁₃₇	—	2	1	no	—
28	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
31	#127×C₁₃₇	C₂×C₄	1	2	yes	—
32	#124×C₁₃₇	—	2	2	no	—
33	#121×C₁₃₇	—	2	2	no	—
34	#121×C₁₃₇	—	2	2	no	—
35	C₄×D₄×C₁₃₇	—	2	2	no	—
36	C₄×D₄×C₁₃₇	—	2	2	no	—
37	C₂×C₄²×C₁₃₇	—	2	1	no	—
38	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
39	#107×C₁₃₇	—	2	1	no	—
40	#107×C₁₃₇	—	2	1	no	—
41	C₄⋊C₈×C₁₃₇	—	2	2	no	—
42	#102×C₁₃₇	—	2	2	no	—
43	#102×C₁₃₇	—	2	2	no	—
44	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
45	#97×C₁₃₇	—	2	2	no	—
46	#97×C₁₃₇	—	2	1	no	—
47	#97×C₁₃₇	D₄	1	2	yes	—
48	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
52	#95×C₁₃₇	—	2	1	no	—
53	#95×C₁₃₇	D₄	1	2	yes	—
54	#41×C₁₃₇	—	8	4	no	—
55	C₂×Q₈×C₁₃₇	—	2	2	no	—
56	C₂×D₄×C₁₃₇	—	2	2	no	—
57	C₂×D₄×C₁₃₇	—	2	2	no	—
58	C₂×D₄×C₁₃₇	—	2	2	no	—
59	C₂×D₄×C₁₃₇	—	4	2	no	—
60	C₂²×C₄×C₁₃₇	—	2	2	no	—
61	C₂²×C₄×C₁₃₇	—	4	2	no	—
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
65	D₈×C₁₃₇	—	4	2	no	—
66	D₈×C₁₃₇	—	4	2	no	—
67	C₂×C₈×C₁₃₇	—	2	2	no	—
68	C₂×C₈×C₁₃₇	—	4	1	no	—
69	C₂×C₈×C₁₃₇	—	4	1	no	—
70	C₂×C₈×C₁₃₇	—	4	1	no	—
71	C₂×C₈×C₁₃₇	—	4	1	no	—
72	C₂×C₈×C₁₃₇	—	4	2	no	—
73	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₃₇	—	2	2	no	—
78	C₄⋊C₄×C₁₃₇	—	2	2	no	—
79	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
80	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
81	C₄²×C₁₃₇	—	4	1	no	—
82	C₄²×C₁₃₇	—	4	1	no	—
83	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₃₇	#41	1	1	yes	characteristic
87	C₂³×C₁₃₇	—	4	2	no	—
88	C₂³×C₁₃₇	#121	1	1	yes	characteristic
89	Q₈×C₁₃₇	—	8	2	no	—
90	D₄×C₁₃₇	—	4	2	no	—
91	D₄×C₁₃₇	—	4	2	no	—
92	D₄×C₁₃₇	—	8	2	no	—
93	D₄×C₁₃₇	—	8	2	no	—
94	D₄×C₁₃₇	—	8	4	no	—
95	C₂×C₄×C₁₃₇	—	2	2	no	—
96	C₂×C₄×C₁₃₇	—	2	1	no	—
97	C₂×C₄×C₁₃₇	—	2	1	no	—
98	C₂×C₄×C₁₃₇	—	4	1	no	—
99	C₂×C₄×C₁₃₇	—	4	2	no	—
100	C₂×C₄×C₁₃₇	—	4	2	no	—
101	C₂×C₄×C₁₃₇	—	4	2	no	—
102	C₂×C₄×C₁₃₇	—	4	2	no	—
103	C₂×C₄×C₁₃₇	—	8	4	no	—
104	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
109	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
110	C₈×C₁₃₇	—	2	1	no	—
111	C₈×C₁₃₇	—	2	1	no	—
112	C₈×C₁₃₇	—	2	1	no	—
113	C₈×C₁₃₇	—	2	1	no	—
114	C₈×C₁₃₇	—	4	2	no	—
115	C₈×C₁₃₇	—	8	2	no	—
116	C₂²×C₁₃₇	—	2	1	no	—
117	C₂²×C₁₃₇	—	2	1	no	—
118	C₂²×C₁₃₇	—	8	2	no	—
119	C₂²×C₁₃₇	—	8	2	no	—
120	C₂²×C₁₃₇	—	8	2	no	—
121	C₂²×C₁₃₇	#390	1	1	yes	characteristic
122	C₂²×C₁₃₇	#472	1	2	yes	—
123	C₄×C₁₃₇	—	2	1	no	—
124	C₄×C₁₃₇	—	2	1	no	—
125	C₄×C₁₃₇	—	2	1	no	—
126	C₄×C₁₃₇	—	2	1	no	—
127	C₄×C₁₃₇	—	2	1	no	—
128	C₄×C₁₃₇	—	8	2	no	—
129	C₄×C₁₃₇	—	8	2	no	—
130	C₄×C₁₃₇	#421	1	1	yes	characteristic
131	C₄×C₁₃₇	#421	1	1	yes	characteristic
132	C₂×C₁₃₇	—	2	2	no	—
133	C₂×C₁₃₇	—	16	2	no	—
134	C₂×C₁₃₇	#1963	1	1	yes	characteristic
135	C₂×C₁₃₇	#1976	1	1	yes	characteristic
136	C₂×C₁₃₇	#1978	1	1	yes	characteristic
137	#9764	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₁₃₇	#9764	1	1	yes	characteristic, direct product factor, Sylow
139	#3169	C₂×C₁₃₇	1	1	yes	characteristic
140	#1746	C₂×C₁₃₇	1	1	yes	characteristic, complemented
141	#1568	C₂×C₁₃₇	1	1	yes	characteristic, complemented
142	#1530	C₂×C₁₃₇	1	2	yes	complemented
143	#1505	C₂×C₁₃₇	1	2	yes	complemented
144	#532	C₂²×C₁₃₇	1	2	yes	—
145	#500	C₄×C₁₃₇	1	1	yes	characteristic, complemented
146	#493	C₄×C₁₃₇	1	1	yes	characteristic, complemented
147	#486	C₄×C₁₃₇	1	2	yes	complemented
148	#466	—	2	1	no	—
149	#466	—	2	1	no	—
150	C₄×D₈	—	2	1	no	—
151	C₄×D₈	—	2	1	no	—
152	#432	C₂²×C₁₃₇	1	2	yes	—
153	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
155	#376	C₂²×C₁₃₇	1	1	yes	characteristic
156	#336	—	2	1	no	—
157	#336	—	2	1	no	—
158	#335	—	2	1	no	—
159	#334	—	2	1	no	—
160	#329	—	2	4	no	—
161	#327	—	2	4	no	—
162	C₂×#41	—	2	2	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
168	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	2	yes	—
169	#127	C₂×C₄×C₁₃₇	1	2	yes	—
170	#124	—	2	2	no	—
171	#121	—	2	2	no	—
172	#121	—	2	2	no	—
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
177	#107	—	2	1	no	—
178	#107	—	2	1	no	—
179	C₄⋊C₈	—	2	2	no	—
180	#102	—	2	2	no	—
181	#102	—	2	2	no	—
182	C₂²⋊C₈	—	2	2	no	—
183	#97	—	2	2	no	—
184	#97	—	2	1	no	—
185	#97	D₄×C₁₃₇	1	2	yes	—
186	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
187	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
188	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
189	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
190	#95	—	2	1	no	—
191	#95	D₄×C₁₃₇	1	2	yes	—
192	#41	—	8	4	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
202	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
203	D₈	—	4	2	no	—
204	D₈	—	4	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	4	1	no	—
221	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
222	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
224	C₄²	#41×C₁₃₇	1	1	yes	characteristic
225	C₂³	—	4	2	no	—
226	C₂³	#121×C₁₃₇	1	1	yes	characteristic
227	Q₈	—	8	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	8	2	no	—
231	D₄	—	8	2	no	—
232	D₄	—	8	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
243	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
244	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
245	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
246	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
247	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	4	2	no	—
253	C₈	—	8	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	#390×C₁₃₇	1	1	yes	characteristic
260	C₂²	#472×C₁₃₇	1	2	yes	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	8	2	no	—
267	C₄	—	8	2	no	—
268	C₄	#421×C₁₃₇	1	1	yes	characteristic
269	C₄	#421×C₁₃₇	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	16	2	no	—
272	C₂	#1963×C₁₃₇	1	1	yes	characteristic
273	C₂	#1976×C₁₃₇	1	1	yes	characteristic
274	C₂	#1978×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#390
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	#3169×C₁₃₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ 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
2. #1746×C₁₃₇	C₂	semidirect
3. #1568×C₁₃₇	C₂	semidirect
4. #1530×C₁₃₇	C₂	semidirect
5. #1505×C₁₃₇	C₂	semidirect
7. #500×C₁₃₇	C₄	semidirect
8. #493×C₁₃₇	C₄	semidirect
9. #486×C₁₃₇	C₄	semidirect
137. #9764	C₁₃₇	direct (N × H)
138. C₁₃₇	#9764	direct (N × H)
140. #1746	C₂×C₁₃₇	semidirect
141. #1568	C₂×C₁₃₇	semidirect
142. #1530	C₂×C₁₃₇	semidirect
143. #1505	C₂×C₁₃₇	semidirect
145. #500	C₄×C₁₃₇	semidirect
146. #493	C₄×C₁₃₇	semidirect
147. #486	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3169×C₁₃₇	2
#1746×C₁₃₇	2
#1568×C₁₃₇	2
#1530×C₁₃₇	2
#1505×C₁₃₇	2
#9764	137

Subgroups — C137×#9764 #46859150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9764 #46859150