Subgroups — C₁₃₇×#28722 #46878108

Order 35072.

918 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#490×C₁₃₇	C₂	1	2	yes	maximal
2	#2859×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#127×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2071×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1992×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1804×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	#1767×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₃₇	—	2	2	no	—
11	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
12	#490×C₁₃₇	—	2	8	no	—
13	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#102×C₁₃₇	C₂²	1	8	yes	—
15	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
16	#390×C₁₃₇	—	2	2	no	—
17	#390×C₁₃₇	C₂²	1	2	yes	—
18	#389×C₁₃₇	—	2	2	no	—
19	#389×C₁₃₇	—	2	2	no	—
20	#384×C₁₃₇	C₂²	1	1	yes	characteristic
21	#384×C₁₃₇	C₂²	1	2	yes	—
22	#382×C₁₃₇	C₂²	1	2	yes	—
23	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
24	#340×C₁₃₇	C₂²	1	8	yes	—
25	#336×C₁₃₇	C₂²	1	2	yes	—
26	C₂²×D₄×C₁₃₇	—	2	2	no	—
27	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
28	C₂³×C₄×C₁₃₇	—	2	2	no	—
29	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
30	#127×C₁₃₇	—	2	2	no	—
31	#127×C₁₃₇	D₄	1	4	yes	—
32	#125×C₁₃₇	—	2	4	no	—
33	#125×C₁₃₇	—	2	4	no	—
34	C₄×D₄×C₁₃₇	—	2	8	no	—
35	C₄×D₄×C₁₃₇	—	4	8	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
42	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
44	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₁₃₇	—	2	4	no	—
50	#102×C₁₃₇	—	2	16	no	—
51	#97×C₁₃₇	—	2	4	no	—
52	#95×C₁₃₇	—	2	2	no	—
53	#95×C₁₃₇	—	2	2	no	—
54	C₂⁴×C₁₃₇	—	2	2	no	—
55	C₂×D₄×C₁₃₇	—	2	8	no	—
56	C₂×D₄×C₁₃₇	—	2	4	no	—
57	C₂×D₄×C₁₃₇	—	2	2	no	—
58	C₂×D₄×C₁₃₇	—	2	4	no	—
59	C₂×D₄×C₁₃₇	—	4	4	no	—
60	C₂×D₄×C₁₃₇	—	4	4	no	—
61	C₂×D₄×C₁₃₇	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₁₃₇	—	2	4	no	—
63	C₂²×C₄×C₁₃₇	—	2	2	no	—
64	C₂²×C₄×C₁₃₇	—	4	8	no	—
65	C₂²×C₄×C₁₃₇	—	4	4	no	—
66	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₁₃₇	—	2	4	no	—
72	C₂×C₈×C₁₃₇	—	2	4	no	—
73	C₂×C₈×C₁₃₇	—	2	4	no	—
74	C₄⋊C₄×C₁₃₇	—	2	4	no	—
75	C₄⋊C₄×C₁₃₇	—	2	4	no	—
76	C₄⋊C₄×C₁₃₇	—	2	4	no	—
77	C₄⋊C₄×C₁₃₇	—	4	4	no	—
78	C₄⋊C₄×C₁₃₇	—	4	4	no	—
79	C₄⋊C₄×C₁₃₇	—	4	8	no	—
80	C₄⋊C₄×C₁₃₇	—	4	4	no	—
81	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
82	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
83	C₄²×C₁₃₇	—	2	4	no	—
84	C₄²×C₁₃₇	—	2	2	no	—
85	C₄²×C₁₃₇	—	4	2	no	—
86	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
87	C₄²×C₁₃₇	#41	1	2	yes	—
88	C₄²×C₁₃₇	#41	1	4	yes	—
89	C₂³×C₁₃₇	—	2	4	no	—
90	C₂³×C₁₃₇	—	4	2	no	—
91	C₂³×C₁₃₇	—	4	2	no	—
92	C₂³×C₁₃₇	—	4	4	no	—
93	C₂³×C₁₃₇	—	4	4	no	—
94	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
95	D₄×C₁₃₇	—	2	8	no	—
96	D₄×C₁₃₇	—	4	16	no	—
97	D₄×C₁₃₇	—	4	4	no	—
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	2	2	no	—
104	C₂×C₄×C₁₃₇	—	4	4	no	—
105	C₂×C₄×C₁₃₇	—	4	2	no	—
106	C₂×C₄×C₁₃₇	—	4	4	no	—
107	C₂×C₄×C₁₃₇	—	4	4	no	—
108	C₂×C₄×C₁₃₇	—	4	4	no	—
109	C₂×C₄×C₁₃₇	—	8	8	no	—
110	C₂×C₄×C₁₃₇	#121	1	4	yes	—
111	C₂×C₄×C₁₃₇	#124	1	2	yes	—
112	C₂×C₄×C₁₃₇	#124	1	2	yes	—
113	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
114	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
116	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
118	C₈×C₁₃₇	—	4	4	no	—
119	C₂²×C₁₃₇	—	4	8	no	—
120	C₂²×C₁₃₇	—	4	4	no	—
121	C₂²×C₁₃₇	—	8	2	no	—
122	C₂²×C₁₃₇	—	8	2	no	—
123	C₂²×C₁₃₇	—	8	4	no	—
124	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
125	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
126	C₂²×C₁₃₇	#536	1	1	yes	characteristic
127	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
128	C₂²×C₁₃₇	#575	1	2	yes	—
129	C₄×C₁₃₇	—	2	2	no	—
130	C₄×C₁₃₇	—	2	2	no	—
131	C₄×C₁₃₇	—	2	4	no	—
132	C₄×C₁₃₇	—	2	4	no	—
133	C₄×C₁₃₇	—	8	2	no	—
134	C₄×C₁₃₇	—	8	4	no	—
135	C₄×C₁₃₇	#547	1	2	yes	—
136	C₄×C₁₃₇	#556	1	2	yes	—
137	C₂×C₁₃₇	—	8	4	no	—
138	C₂×C₁₃₇	#2547	1	1	yes	characteristic
139	C₂×C₁₃₇	#3036	1	2	yes	—
140	C₂×C₁₃₇	#3135	1	1	yes	characteristic
141	C₂×C₁₃₇	#3138	1	1	yes	characteristic
142	C₂×C₁₃₇	#3234	1	2	yes	—
143	#28722	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₁₃₇	#28722	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×#490	C₂×C₁₃₇	1	2	yes	—
146	#2859	C₂×C₁₃₇	1	1	yes	characteristic, complemented
147	C₄×#127	C₂×C₁₃₇	1	1	yes	characteristic
148	#2071	C₂×C₁₃₇	1	4	yes	complemented
149	#1992	C₂×C₁₃₇	1	2	yes	complemented
150	#1804	C₂×C₁₃₇	1	4	yes	complemented
151	#1767	C₂×C₁₃₇	1	1	yes	characteristic, complemented
152	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
153	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
154	C₂×C₄×D₄	—	2	2	no	—
155	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
156	#490	—	2	8	no	—
157	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
158	C₂×#102	C₂²×C₁₃₇	1	8	yes	—
159	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
160	#390	—	2	2	no	—
161	#390	C₂²×C₁₃₇	1	2	yes	—
162	#389	—	2	2	no	—
163	#389	—	2	2	no	—
164	#384	C₂²×C₁₃₇	1	1	yes	characteristic
165	#384	C₂²×C₁₃₇	1	2	yes	—
166	#382	C₂²×C₁₃₇	1	2	yes	—
167	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
168	#340	C₂²×C₁₃₇	1	8	yes	—
169	#336	C₂²×C₁₃₇	1	2	yes	—
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
172	C₂³×C₄	—	2	2	no	—
173	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
174	#127	—	2	2	no	—
175	#127	D₄×C₁₃₇	1	4	yes	—
176	#125	—	2	4	no	—
177	#125	—	2	4	no	—
178	C₄×D₄	—	2	8	no	—
179	C₄×D₄	—	4	8	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
185	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
186	C₂×C₂²⋊C₄	—	2	4	no	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
189	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
190	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
193	C₄⋊C₈	—	2	4	no	—
194	#102	—	2	16	no	—
195	#97	—	2	4	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	C₂⁴	—	2	2	no	—
199	C₂×D₄	—	2	8	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	4	4	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	C₂×D₄×C₁₃₇	1	4	yes	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	4	8	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
214	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	8	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₂²⋊C₄	—	4	4	no	—
226	C₂²⋊C₄	—	4	8	no	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	2	no	—
230	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
231	C₄²	#41×C₁₃₇	1	2	yes	—
232	C₄²	#41×C₁₃₇	1	4	yes	—
233	C₂³	—	2	4	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	—	4	4	no	—
237	C₂³	—	4	4	no	—
238	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
239	D₄	—	2	8	no	—
240	D₄	—	4	16	no	—
241	D₄	—	4	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	8	no	—
254	C₂×C₄	#121×C₁₃₇	1	4	yes	—
255	C₂×C₄	#124×C₁₃₇	1	2	yes	—
256	C₂×C₄	#124×C₁₃₇	1	2	yes	—
257	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
258	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
259	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
260	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
261	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
262	C₈	—	4	4	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	4	8	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	4	no	—
268	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
269	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
270	C₂²	#536×C₁₃₇	1	1	yes	characteristic
271	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
272	C₂²	#575×C₁₃₇	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	4	no	—
277	C₄	—	8	2	no	—
278	C₄	—	8	4	no	—
279	C₄	#547×C₁₃₇	1	2	yes	—
280	C₄	#556×C₁₃₇	1	2	yes	—
281	C₂	—	8	4	no	—
282	C₂	#2547×C₁₃₇	1	1	yes	characteristic
283	C₂	#3036×C₁₃₇	1	2	yes	—
284	C₂	#3135×C₁₃₇	1	1	yes	characteristic
285	C₂	#3138×C₁₃₇	1	1	yes	characteristic
286	C₂	#3234×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#127×C₁₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₄³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2859×C₁₃₇	C₂	semidirect
4. #2071×C₁₃₇	C₂	semidirect
5. #1992×C₁₃₇	C₂	semidirect
6. #1804×C₁₃₇	C₂	semidirect
7. #1767×C₁₃₇	C₂	semidirect
143. #28722	C₁₃₇	direct (N × H)
144. C₁₃₇	#28722	direct (N × H)
146. #2859	C₂×C₁₃₇	semidirect
148. #2071	C₂×C₁₃₇	semidirect
149. #1992	C₂×C₁₃₇	semidirect
150. #1804	C₂×C₁₃₇	semidirect
151. #1767	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×#490×C₁₃₇	2
#2859×C₁₃₇	2
C₄×#127×C₁₃₇	2
#2071×C₁₃₇	2
#1992×C₁₃₇	2
#1804×C₁₃₇	2
#1767×C₁₃₇	2
#28722	137

Subgroups — C137×#28722 #46878108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28722 #46878108