Subgroups — C₁₃₇×#28711 #46878097

Order 35072.

806 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#436×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2563×C₁₃₇	C₂	1	2	yes	maximal
4	#2075×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1987×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1809×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
11	#500×C₁₃₇	—	2	4	no	—
12	#436×C₁₃₇	—	2	4	no	—
13	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	#393×C₁₃₇	—	2	4	no	—
20	#390×C₁₃₇	—	2	2	no	—
21	#382×C₁₃₇	—	2	2	no	—
22	#380×C₁₃₇	C₂²	1	8	yes	—
23	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	#340×C₁₃₇	C₂²	1	2	yes	—
25	#340×C₁₃₇	C₂²	1	8	yes	—
26	C₂²×Q₈×C₁₃₇	—	2	2	no	—
27	C₂²×D₄×C₁₃₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
29	C₂×M₁₆×C₁₃₇	—	2	4	no	—
30	C₂×M₁₆×C₁₃₇	—	2	8	no	—
31	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
35	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
36	C₄×D₄×C₁₃₇	—	2	2	no	—
37	C₄×D₄×C₁₃₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
42	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
43	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
44	C₂×C₄²×C₁₃₇	—	2	2	no	—
45	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
46	#107×C₁₃₇	—	2	4	no	—
47	#107×C₁₃₇	—	2	4	no	—
48	#106×C₁₃₇	—	2	4	no	—
49	C₄⋊C₈×C₁₃₇	—	2	2	no	—
50	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
51	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
52	#95×C₁₃₇	—	2	2	no	—
53	#95×C₁₃₇	—	2	8	no	—
54	C₂⁴×C₁₃₇	C₂×D₄	1	2	yes	—
55	C₂×Q₈×C₁₃₇	—	4	4	no	—
56	C₂×Q₈×C₁₃₇	—	8	4	no	—
57	C₂×D₄×C₁₃₇	—	2	2	no	—
58	C₂×D₄×C₁₃₇	—	2	4	no	—
59	C₂²×C₄×C₁₃₇	—	2	2	no	—
60	C₂²×C₄×C₁₃₇	—	2	2	no	—
61	C₂²×C₄×C₁₃₇	—	2	4	no	—
62	C₂²×C₄×C₁₃₇	—	2	4	no	—
63	C₂²×C₄×C₁₃₇	—	2	4	no	—
64	C₂²×C₄×C₁₃₇	—	2	4	no	—
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₁₃₇	—	4	8	no	—
70	C₂×C₈×C₁₃₇	—	2	2	no	—
71	C₂×C₈×C₁₃₇	—	2	2	no	—
72	C₂×C₈×C₁₃₇	—	2	4	no	—
73	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃₇	#41	1	4	yes	—
76	C₄⋊C₄×C₁₃₇	—	2	2	no	—
77	C₄⋊C₄×C₁₃₇	—	2	4	no	—
78	C₄⋊C₄×C₁₃₇	—	2	4	no	—
79	C₄⋊C₄×C₁₃₇	—	2	4	no	—
80	C₄⋊C₄×C₁₃₇	—	2	4	no	—
81	C₄⋊C₄×C₁₃₇	—	4	4	no	—
82	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
83	C₂²⋊C₄×C₁₃₇	—	8	8	no	—
84	C₄²×C₁₃₇	—	4	4	no	—
85	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
86	C₄²×C₁₃₇	#41	1	2	yes	—
87	C₂³×C₁₃₇	—	2	2	no	—
88	C₂³×C₁₃₇	—	2	4	no	—
89	C₂³×C₁₃₇	—	2	4	no	—
90	C₂³×C₁₃₇	—	2	4	no	—
91	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
92	Q₈×C₁₃₇	—	8	8	no	—
93	D₄×C₁₃₇	—	4	4	no	—
94	C₂×C₄×C₁₃₇	—	2	2	no	—
95	C₂×C₄×C₁₃₇	—	2	2	no	—
96	C₂×C₄×C₁₃₇	—	2	4	no	—
97	C₂×C₄×C₁₃₇	—	4	2	no	—
98	C₂×C₄×C₁₃₇	—	4	2	no	—
99	C₂×C₄×C₁₃₇	—	4	4	no	—
100	C₂×C₄×C₁₃₇	—	4	4	no	—
101	C₂×C₄×C₁₃₇	—	4	4	no	—
102	C₂×C₄×C₁₃₇	—	4	4	no	—
103	C₂×C₄×C₁₃₇	—	4	4	no	—
104	C₂×C₄×C₁₃₇	—	4	8	no	—
105	C₂×C₄×C₁₃₇	#121	1	2	yes	—
106	C₂×C₄×C₁₃₇	#121	1	2	yes	—
107	C₂×C₄×C₁₃₇	#137	1	4	yes	—
108	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
110	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
112	C₈×C₁₃₇	—	2	2	no	—
113	C₈×C₁₃₇	—	4	2	no	—
114	C₈×C₁₃₇	#124	1	4	yes	—
115	C₂²×C₁₃₇	—	4	8	no	—
116	C₂²×C₁₃₇	—	4	2	no	—
117	C₂²×C₁₃₇	—	4	2	no	—
118	C₂²×C₁₃₇	—	4	2	no	—
119	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
120	C₂²×C₁₃₇	#547	1	2	yes	—
121	C₂²×C₁₃₇	C₂×#137	1	2	yes	—
122	C₂²×C₁₃₇	#576	1	2	yes	—
123	C₄×C₁₃₇	—	2	2	no	—
124	C₄×C₁₃₇	—	4	2	no	—
125	C₄×C₁₃₇	—	8	4	no	—
126	C₄×C₁₃₇	—	8	4	no	—
127	C₄×C₁₃₇	#470	1	4	yes	—
128	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
129	C₄×C₁₃₇	#536	1	1	yes	characteristic
130	C₄×C₁₃₇	#556	1	2	yes	—
131	C₂×C₁₃₇	—	8	2	no	—
132	C₂×C₁₃₇	#2547	1	1	yes	characteristic
133	C₂×C₁₃₇	C₂×#470	1	1	yes	characteristic
134	C₂×C₁₃₇	#3058	1	1	yes	characteristic
135	C₂×C₁₃₇	#3357	1	4	yes	—
136	#28711	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₁₃₇	#28711	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic, complemented
139	C₂×#436	C₂×C₁₃₇	1	1	yes	characteristic
140	#2563	C₂×C₁₃₇	1	2	yes	—
141	#2075	C₂×C₁₃₇	1	4	yes	complemented
142	#1987	C₂×C₁₃₇	1	2	yes	complemented
143	#1809	C₂×C₁₃₇	1	4	yes	complemented
144	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic, complemented
145	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
146	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
147	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
148	#500	—	2	4	no	—
149	#436	—	2	4	no	—
150	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
151	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
152	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
154	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
156	#393	—	2	4	no	—
157	#390	—	2	2	no	—
158	#382	—	2	2	no	—
159	#380	C₂²×C₁₃₇	1	8	yes	—
160	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
161	#340	C₂²×C₁₃₇	1	8	yes	—
162	#340	C₂²×C₁₃₇	1	2	yes	—
163	C₂²×Q₈	—	2	2	no	—
164	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
165	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	—	2	8	no	—
168	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
169	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
171	C₄⋊Q₈	—	2	4	no	—
172	C₄⋊Q₈	—	2	4	no	—
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
179	C₂×C₂²⋊C₄	—	2	8	no	—
180	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
183	#107	—	2	4	no	—
184	#107	—	2	4	no	—
185	#106	—	2	4	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	—	2	4	no	—
188	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
189	#95	—	2	2	no	—
190	#95	—	2	8	no	—
191	C₂⁴	C₂×D₄×C₁₃₇	1	2	yes	—
192	C₂×Q₈	—	4	4	no	—
193	C₂×Q₈	—	8	4	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
205	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
206	M₁₆	—	4	8	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
211	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
212	C₂×C₈	#41×C₁₃₇	1	4	yes	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	8	8	no	—
221	C₄²	—	4	4	no	—
222	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
223	C₄²	#41×C₁₃₇	1	2	yes	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	4	no	—
228	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
229	Q₈	—	8	8	no	—
230	D₄	—	4	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	#121×C₁₃₇	1	2	yes	—
243	C₂×C₄	#121×C₁₃₇	1	2	yes	—
244	C₂×C₄	#137×C₁₃₇	1	4	yes	—
245	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
246	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
247	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
248	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
249	C₈	—	2	2	no	—
250	C₈	—	4	2	no	—
251	C₈	#124×C₁₃₇	1	4	yes	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	8	no	—
256	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
257	C₂²	#547×C₁₃₇	1	2	yes	—
258	C₂²	C₂×#137×C₁₃₇	1	2	yes	—
259	C₂²	#576×C₁₃₇	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	8	4	no	—
263	C₄	—	8	4	no	—
264	C₄	#470×C₁₃₇	1	4	yes	—
265	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
266	C₄	#536×C₁₃₇	1	1	yes	characteristic
267	C₄	#556×C₁₃₇	1	2	yes	—
268	C₂	—	8	2	no	—
269	C₂	#2547×C₁₃₇	1	1	yes	characteristic
270	C₂	C₂×#470×C₁₃₇	1	1	yes	characteristic
271	C₂	#3058×C₁₃₇	1	1	yes	characteristic
272	C₂	#3357×C₁₃₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₃₇	C₂	semidirect
4. #2075×C₁₃₇	C₂	semidirect
5. #1987×C₁₃₇	C₂	semidirect
6. #1809×C₁₃₇	C₂	semidirect
7. C₄×#106×C₁₃₇	C₂	semidirect
136. #28711	C₁₃₇	direct (N × H)
137. C₁₃₇	#28711	direct (N × H)
138. C₂×#500	C₂×C₁₃₇	semidirect
141. #2075	C₂×C₁₃₇	semidirect
142. #1987	C₂×C₁₃₇	semidirect
143. #1809	C₂×C₁₃₇	semidirect
144. C₄×#106	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₇	2
C₂×#436×C₁₃₇	2
#2563×C₁₃₇	2
#2075×C₁₃₇	2
#1987×C₁₃₇	2
#1809×C₁₃₇	2
C₄×#106×C₁₃₇	2
#28711	137

Subgroups — C137×#28711 #46878097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28711 #46878097