Subgroups — C₁₃₇×#28732 #46878118

Order 35072.

698 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×#498×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3141×C₁₃₇	C₂	1	2	yes	maximal
3	#2931×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2079×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1997×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1815×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₄×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₁₃₇	C₂²	1	2	yes	—
9	#532×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
11	#498×C₁₃₇	—	2	2	no	—
12	#498×C₁₃₇	—	2	2	no	—
13	#489×C₁₃₇	—	2	4	no	—
14	#487×C₁₃₇	—	2	2	no	—
15	#486×C₁₃₇	—	2	2	no	—
16	#436×C₁₃₇	—	2	2	no	—
17	C₈×D₄×C₁₃₇	—	2	2	no	—
18	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
19	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
20	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
21	#423×C₁₃₇	C₂²	1	1	yes	characteristic
22	#417×C₁₃₇	C₂²	1	8	yes	—
23	#408×C₁₃₇	C₂²	1	2	yes	—
24	C₄×M₁₆×C₁₃₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	#343×C₁₃₇	C₂²	1	8	yes	—
27	#341×C₁₃₇	C₂²	1	2	yes	—
28	C₂×#41×C₁₃₇	C₂³	1	2	yes	—
29	C₂²×Q₈×C₁₃₇	—	2	2	no	—
30	#131×C₁₃₇	—	2	8	no	—
31	C₂×M₁₆×C₁₃₇	—	2	2	no	—
32	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
33	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃₇	—	2	2	no	—
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
38	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
39	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
40	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
41	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
42	#127×C₁₃₇	D₄	1	1	yes	characteristic
43	#124×C₁₃₇	D₄	1	2	yes	—
44	#121×C₁₃₇	—	2	2	no	—
45	C₄×D₄×C₁₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
49	#108×C₁₃₇	—	2	4	no	—
50	#106×C₁₃₇	—	2	4	no	—
51	#106×C₁₃₇	—	2	2	no	—
52	#106×C₁₃₇	—	2	2	no	—
53	C₄⋊C₈×C₁₃₇	—	2	2	no	—
54	#103×C₁₃₇	—	2	8	no	—
55	#102×C₁₃₇	—	2	8	no	—
56	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
57	#97×C₁₃₇	—	2	2	no	—
58	C₄×C₈×C₁₃₇	—	2	2	no	—
59	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
60	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
61	#41×C₁₃₇	—	2	4	no	—
62	#41×C₁₃₇	—	2	4	no	—
63	C₂×Q₈×C₁₃₇	—	4	2	no	—
64	C₂×Q₈×C₁₃₇	—	4	2	no	—
65	C₂×Q₈×C₁₃₇	—	8	4	no	—
66	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
67	C₂×D₄×C₁₃₇	—	2	2	no	—
68	C₂×D₄×C₁₃₇	—	2	2	no	—
69	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃₇	—	2	2	no	—
71	C₂²×C₄×C₁₃₇	—	2	4	no	—
72	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
73	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
74	M₁₆×C₁₃₇	—	2	4	no	—
75	M₁₆×C₁₃₇	—	2	4	no	—
76	M₁₆×C₁₃₇	—	2	4	no	—
77	M₁₆×C₁₃₇	—	4	4	no	—
78	C₂×C₈×C₁₃₇	—	2	2	no	—
79	C₂×C₈×C₁₃₇	—	2	4	no	—
80	C₂×C₈×C₁₃₇	—	2	2	no	—
81	C₂×C₈×C₁₃₇	—	4	4	no	—
82	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃₇	#41	1	4	yes	—
86	C₄⋊C₄×C₁₃₇	—	2	2	no	—
87	C₄⋊C₄×C₁₃₇	—	2	4	no	—
88	C₄⋊C₄×C₁₃₇	—	2	4	no	—
89	C₄⋊C₄×C₁₃₇	—	4	2	no	—
90	C₄⋊C₄×C₁₃₇	—	4	2	no	—
91	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
92	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
94	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
95	C₂³×C₁₃₇	—	2	2	no	—
96	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
97	Q₈×C₁₃₇	—	2	4	no	—
98	Q₈×C₁₃₇	—	8	4	no	—
99	Q₈×C₁₃₇	—	8	4	no	—
100	D₄×C₁₃₇	—	2	4	no	—
101	D₄×C₁₃₇	—	4	4	no	—
102	D₄×C₁₃₇	—	4	4	no	—
103	C₂×C₄×C₁₃₇	—	2	4	no	—
104	C₂×C₄×C₁₃₇	—	2	2	no	—
105	C₂×C₄×C₁₃₇	—	4	2	no	—
106	C₂×C₄×C₁₃₇	—	4	2	no	—
107	C₂×C₄×C₁₃₇	—	4	4	no	—
108	C₂×C₄×C₁₃₇	—	8	4	no	—
109	C₂×C₄×C₁₃₇	#121	1	2	yes	—
110	C₂×C₄×C₁₃₇	#121	1	2	yes	—
111	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
114	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
115	C₈×C₁₃₇	—	2	2	no	—
116	C₈×C₁₃₇	—	4	2	no	—
117	C₈×C₁₃₇	—	4	4	no	—
118	C₈×C₁₃₇	#124	1	2	yes	—
119	C₈×C₁₃₇	#124	1	2	yes	—
120	C₂²×C₁₃₇	—	2	2	no	—
121	C₂²×C₁₃₇	—	4	2	no	—
122	C₂²×C₁₃₇	—	4	4	no	—
123	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
124	C₂²×C₁₃₇	#556	1	2	yes	—
125	C₄×C₁₃₇	—	2	2	no	—
126	C₄×C₁₃₇	—	4	2	no	—
127	C₄×C₁₃₇	—	8	4	no	—
128	C₄×C₁₃₇	#471	1	4	yes	—
129	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
130	C₄×C₁₃₇	#536	1	1	yes	characteristic
131	C₄×C₁₃₇	#547	1	2	yes	—
132	C₂×C₁₃₇	—	2	2	no	—
133	C₂×C₁₃₇	—	8	2	no	—
134	C₂×C₁₃₇	#2547	1	1	yes	characteristic
135	C₂×C₁₃₇	C₂×#471	1	2	yes	—
136	#28732	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₁₃₇	#28732	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×#498	C₂×C₁₃₇	1	1	yes	characteristic, complemented
139	#3141	C₂×C₁₃₇	1	2	yes	—
140	#2931	C₂×C₁₃₇	1	1	yes	characteristic
141	#2079	C₂×C₁₃₇	1	4	yes	complemented
142	#1997	C₂×C₁₃₇	1	2	yes	complemented
143	#1815	C₂×C₁₃₇	1	4	yes	complemented
144	C₄×#108	C₂×C₁₃₇	1	1	yes	characteristic, complemented
145	C₂×#131	C₂²×C₁₃₇	1	2	yes	—
146	#532	C₂²×C₁₃₇	1	1	yes	characteristic
147	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
148	#498	—	2	2	no	—
149	#498	—	2	2	no	—
150	#489	—	2	4	no	—
151	#487	—	2	2	no	—
152	#486	—	2	2	no	—
153	#436	—	2	2	no	—
154	C₈×D₄	—	2	2	no	—
155	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
156	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
157	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
158	#423	C₂²×C₁₃₇	1	1	yes	characteristic
159	#417	C₂²×C₁₃₇	1	8	yes	—
160	#408	C₂²×C₁₃₇	1	2	yes	—
161	C₄×M₁₆	C₂²×C₁₃₇	1	2	yes	—
162	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
163	#343	C₂²×C₁₃₇	1	8	yes	—
164	#341	C₂²×C₁₃₇	1	2	yes	—
165	C₂×#41	C₂³×C₁₃₇	1	2	yes	—
166	C₂²×Q₈	—	2	2	no	—
167	#131	—	2	8	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
170	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
179	#127	D₄×C₁₃₇	1	1	yes	characteristic
180	#124	D₄×C₁₃₇	1	2	yes	—
181	#121	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
185	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
186	#108	—	2	4	no	—
187	#106	—	2	4	no	—
188	#106	—	2	2	no	—
189	#106	—	2	2	no	—
190	C₄⋊C₈	—	2	2	no	—
191	#103	—	2	8	no	—
192	#102	—	2	8	no	—
193	C₂²⋊C₈	—	2	4	no	—
194	#97	—	2	2	no	—
195	C₄×C₈	—	2	2	no	—
196	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
197	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
198	#41	—	2	4	no	—
199	#41	—	2	4	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	—	8	4	no	—
203	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
210	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	4	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	4	no	—
219	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
220	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
221	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
222	C₂×C₈	#41×C₁₃₇	1	4	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	2	4	no	—
229	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
230	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
231	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
232	C₂³	—	2	2	no	—
233	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
234	Q₈	—	2	4	no	—
235	Q₈	—	8	4	no	—
236	Q₈	—	8	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	4	no	—
239	D₄	—	4	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	#121×C₁₃₇	1	2	yes	—
247	C₂×C₄	#121×C₁₃₇	1	2	yes	—
248	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
250	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
251	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
252	C₈	—	2	2	no	—
253	C₈	—	4	2	no	—
254	C₈	—	4	4	no	—
255	C₈	#124×C₁₃₇	1	2	yes	—
256	C₈	#124×C₁₃₇	1	2	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	4	no	—
260	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
261	C₂²	#556×C₁₃₇	1	2	yes	—
262	C₄	—	2	2	no	—
263	C₄	—	4	2	no	—
264	C₄	—	8	4	no	—
265	C₄	#471×C₁₃₇	1	4	yes	—
266	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
267	C₄	#536×C₁₃₇	1	1	yes	characteristic
268	C₄	#547×C₁₃₇	1	2	yes	—
269	C₂	—	2	2	no	—
270	C₂	—	8	2	no	—
271	C₂	#2547×C₁₃₇	1	1	yes	characteristic
272	C₂	C₂×#471×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	#532×C₁₃₇	C₂²
LCS	G ▷ C₂²×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. C₂×#498×C₁₃₇	C₂	semidirect
4. #2079×C₁₃₇	C₂	semidirect
5. #1997×C₁₃₇	C₂	semidirect
6. #1815×C₁₃₇	C₂	semidirect
7. C₄×#108×C₁₃₇	C₂	semidirect
136. #28732	C₁₃₇	direct (N × H)
137. C₁₃₇	#28732	direct (N × H)
138. C₂×#498	C₂×C₁₃₇	semidirect
141. #2079	C₂×C₁₃₇	semidirect
142. #1997	C₂×C₁₃₇	semidirect
143. #1815	C₂×C₁₃₇	semidirect
144. C₄×#108	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×#498×C₁₃₇	2
#3141×C₁₃₇	2
#2931×C₁₃₇	2
#2079×C₁₃₇	2
#1997×C₁₃₇	2
#1815×C₁₃₇	2
C₄×#108×C₁₃₇	2
#28732	137

Subgroups — C137×#28732 #46878118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28732 #46878118