Subgroups — C₁₃₇×#28705 #46878091

Order 35072.

838 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	C₂×#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2563×C₁₃₇	C₂	1	2	yes	maximal
4	#2076×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1987×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1808×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₄×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₈×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	C₈×D₄×C₁₃₇	—	2	4	no	—
13	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#107×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	C₂²×C₈×C₁₃₇	—	2	4	no	—
30	C₂²×C₈×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	#107×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	2	no	—
59	C₂×D₄×C₁₃₇	Q₁₆	1	4	yes	—
60	C₂²×C₄×C₁₃₇	—	2	2	no	—
61	C₂²×C₄×C₁₃₇	—	2	2	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	—	2	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	C₂⁴	1	1	yes	characteristic
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	—	4	8	no	—
74	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃₇	#41	1	4	yes	—
77	C₄⋊C₄×C₁₃₇	—	2	2	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	—	2	4	no	—
82	C₄⋊C₄×C₁₃₇	—	4	4	no	—
83	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
84	C₂²⋊C₄×C₁₃₇	—	8	8	no	—
85	C₄²×C₁₃₇	—	4	4	no	—
86	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
87	C₄²×C₁₃₇	#41	1	2	yes	—
88	C₂³×C₁₃₇	—	2	2	no	—
89	C₂³×C₁₃₇	—	2	4	no	—
90	C₂³×C₁₃₇	—	2	2	no	—
91	C₂³×C₁₃₇	—	2	4	no	—
92	C₂³×C₁₃₇	C₂×Q₁₆	1	4	yes	—
93	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₁₃₇	—	8	8	no	—
95	D₄×C₁₃₇	—	2	8	no	—
96	C₂×C₄×C₁₃₇	—	2	2	no	—
97	C₂×C₄×C₁₃₇	—	2	2	no	—
98	C₂×C₄×C₁₃₇	—	2	4	no	—
99	C₂×C₄×C₁₃₇	—	4	2	no	—
100	C₂×C₄×C₁₃₇	—	4	2	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	4	no	—
105	C₂×C₄×C₁₃₇	—	4	4	no	—
106	C₂×C₄×C₁₃₇	—	4	8	no	—
107	C₂×C₄×C₁₃₇	#121	1	2	yes	—
108	C₂×C₄×C₁₃₇	#121	1	2	yes	—
109	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₁₃₇	#135	1	2	yes	—
111	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
113	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
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	#124	1	4	yes	—
118	C₂²×C₁₃₇	—	2	8	no	—
119	C₂²×C₁₃₇	—	2	4	no	—
120	C₂²×C₁₃₇	—	4	4	no	—
121	C₂²×C₁₃₇	—	4	2	no	—
122	C₂²×C₁₃₇	—	4	2	no	—
123	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
124	C₂²×C₁₃₇	#547	1	2	yes	—
125	C₂²×C₁₃₇	C₂²×Q₁₆	1	1	yes	characteristic
126	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
127	C₂²×C₁₃₇	#576	1	2	yes	—
128	C₄×C₁₃₇	—	2	2	no	—
129	C₄×C₁₃₇	—	4	2	no	—
130	C₄×C₁₃₇	—	8	4	no	—
131	C₄×C₁₃₇	—	8	4	no	—
132	C₄×C₁₃₇	#467	1	4	yes	—
133	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
134	C₄×C₁₃₇	#536	1	1	yes	characteristic
135	C₄×C₁₃₇	#556	1	2	yes	—
136	C₂×C₁₃₇	—	4	4	no	—
137	C₂×C₁₃₇	#2547	1	1	yes	characteristic
138	C₂×C₁₃₇	C₂×#467	1	1	yes	characteristic
139	C₂×C₁₃₇	#3058	1	1	yes	characteristic
140	C₂×C₁₃₇	#3338	1	2	yes	—
141	C₂×C₁₃₇	#3341	1	2	yes	—
142	#28705	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₁₃₇	#28705	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic, complemented
145	C₂×C₈×D₄	C₂×C₁₃₇	1	1	yes	characteristic
146	#2563	C₂×C₁₃₇	1	2	yes	—
147	#2076	C₂×C₁₃₇	1	4	yes	complemented
148	#1987	C₂×C₁₃₇	1	2	yes	complemented
149	#1808	C₂×C₁₃₇	1	4	yes	complemented
150	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic, complemented
151	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
152	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
153	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
154	#500	—	2	4	no	—
155	C₈×D₄	—	2	4	no	—
156	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
157	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
158	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
159	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
160	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
162	#393	—	2	4	no	—
163	#390	—	2	2	no	—
164	#382	—	2	2	no	—
165	#380	C₂²×C₁₃₇	1	8	yes	—
166	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
167	#340	C₂²×C₁₃₇	1	8	yes	—
168	#340	C₂²×C₁₃₇	1	2	yes	—
169	C₂²×Q₈	—	2	2	no	—
170	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
171	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
172	C₂²×C₈	—	2	4	no	—
173	C₂²×C₈	—	2	8	no	—
174	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
175	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
177	C₄⋊Q₈	—	2	4	no	—
178	C₄⋊Q₈	—	2	4	no	—
179	C₄×D₄	—	2	2	no	—
180	C₄×D₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
184	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
185	C₂×C₂²⋊C₄	—	2	8	no	—
186	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
189	#107	—	2	4	no	—
190	#107	—	2	4	no	—
191	#107	—	2	4	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₂²⋊C₈	—	2	4	no	—
194	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
195	#95	—	2	2	no	—
196	#95	—	2	8	no	—
197	C₂⁴	C₂×D₄×C₁₃₇	1	2	yes	—
198	C₂×Q₈	—	4	4	no	—
199	C₂×Q₈	—	8	4	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	Q₁₆×C₁₃₇	1	4	yes	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
212	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	8	no	—
217	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
219	C₂×C₈	#41×C₁₃₇	1	4	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	8	8	no	—
228	C₄²	—	4	4	no	—
229	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
230	C₄²	#41×C₁₃₇	1	2	yes	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	4	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	4	no	—
235	C₂³	C₂×Q₁₆×C₁₃₇	1	4	yes	—
236	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
237	Q₈	—	8	8	no	—
238	D₄	—	2	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	#121×C₁₃₇	1	2	yes	—
251	C₂×C₄	#121×C₁₃₇	1	2	yes	—
252	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
253	C₂×C₄	#135×C₁₃₇	1	2	yes	—
254	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
255	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
256	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
257	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
258	C₈	—	2	2	no	—
259	C₈	—	4	2	no	—
260	C₈	#124×C₁₃₇	1	4	yes	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	8	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	4	no	—
266	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
267	C₂²	#547×C₁₃₇	1	2	yes	—
268	C₂²	C₂²×Q₁₆×C₁₃₇	1	1	yes	characteristic
269	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
270	C₂²	#576×C₁₃₇	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	8	4	no	—
274	C₄	—	8	4	no	—
275	C₄	#467×C₁₃₇	1	4	yes	—
276	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
277	C₄	#536×C₁₃₇	1	1	yes	characteristic
278	C₄	#556×C₁₃₇	1	2	yes	—
279	C₂	—	4	4	no	—
280	C₂	#2547×C₁₃₇	1	1	yes	characteristic
281	C₂	C₂×#467×C₁₃₇	1	1	yes	characteristic
282	C₂	#3058×C₁₃₇	1	1	yes	characteristic
283	C₂	#3338×C₁₃₇	1	2	yes	—
284	C₂	#3341×C₁₃₇	1	2	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. #2076×C₁₃₇	C₂	semidirect
5. #1987×C₁₃₇	C₂	semidirect
6. #1808×C₁₃₇	C₂	semidirect
7. C₄×#107×C₁₃₇	C₂	semidirect
142. #28705	C₁₃₇	direct (N × H)
143. C₁₃₇	#28705	direct (N × H)
144. C₂×#500	C₂×C₁₃₇	semidirect
147. #2076	C₂×C₁₃₇	semidirect
148. #1987	C₂×C₁₃₇	semidirect
149. #1808	C₂×C₁₃₇	semidirect
150. C₄×#107	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×C₈×D₄×C₁₃₇	2
#2563×C₁₃₇	2
#2076×C₁₃₇	2
#1987×C₁₃₇	2
#1808×C₁₃₇	2
C₄×#107×C₁₃₇	2
#28705	137

Subgroups — C137×#28705 #46878091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28705 #46878091