Subgroups — C₁₃₇×#28690 #46878076

Order 35072.

710 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₇	C₂	1	2	yes	maximal
2	#2859×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2084×C₁₃₇	C₂	1	2	yes	maximal
5	#2083×C₁₃₇	C₂	1	2	yes	maximal
6	#1991×C₁₃₇	C₂	1	2	yes	maximal
7	#1949×C₁₃₇	C₂	1	4	yes	maximal
8	C₄×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
11	#499×C₁₃₇	—	2	8	no	—
12	C₂×#107×C₁₃₇	C₂²	1	4	yes	—
13	C₂×#106×C₁₃₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
16	#389×C₁₃₇	—	2	2	no	—
17	#389×C₁₃₇	—	2	2	no	—
18	#389×C₁₃₇	C₂²	1	2	yes	—
19	#387×C₁₃₇	—	2	2	no	—
20	#384×C₁₃₇	C₂²	1	2	yes	—
21	#382×C₁₃₇	C₂²	1	1	yes	characteristic
22	#382×C₁₃₇	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
25	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
26	#340×C₁₃₇	C₂²	1	4	yes	—
27	#340×C₁₃₇	C₂²	1	4	yes	—
28	#336×C₁₃₇	C₂²	1	1	yes	characteristic
29	#336×C₁₃₇	C₂²	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
32	#125×C₁₃₇	—	2	4	no	—
33	#125×C₁₃₇	—	2	2	no	—
34	#125×C₁₃₇	—	2	4	no	—
35	#125×C₁₃₇	D₄	1	4	yes	—
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	2	no	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃₇	—	2	2	no	—
45	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
47	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
50	#107×C₁₃₇	—	2	8	no	—
51	#106×C₁₃₇	—	2	8	no	—
52	C₄⋊C₈×C₁₃₇	Q₈	1	8	yes	—
53	C₄×C₈×C₁₃₇	—	2	4	no	—
54	#95×C₁₃₇	—	2	2	no	—
55	#95×C₁₃₇	—	2	2	no	—
56	#95×C₁₃₇	—	2	2	no	—
57	#95×C₁₃₇	—	2	2	no	—
58	#95×C₁₃₇	—	2	2	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	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₁₃₇	—	2	2	no	—
68	C₂×C₈×C₁₃₇	—	2	2	no	—
69	C₂×C₈×C₁₃₇	—	2	2	no	—
70	C₂×C₈×C₁₃₇	—	2	2	no	—
71	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
72	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
73	C₄⋊C₄×C₁₃₇	—	2	8	no	—
74	C₄⋊C₄×C₁₃₇	—	2	2	no	—
75	C₄⋊C₄×C₁₃₇	—	2	4	no	—
76	C₄⋊C₄×C₁₃₇	—	2	4	no	—
77	C₄⋊C₄×C₁₃₇	—	2	4	no	—
78	C₄⋊C₄×C₁₃₇	—	4	4	no	—
79	C₄⋊C₄×C₁₃₇	—	4	8	no	—
80	C₄⋊C₄×C₁₃₇	—	4	4	no	—
81	C₄⋊C₄×C₁₃₇	—	4	4	no	—
82	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
83	C₄²×C₁₃₇	—	2	4	no	—
84	C₄²×C₁₃₇	—	2	2	no	—
85	C₄²×C₁₃₇	—	4	2	no	—
86	C₄²×C₁₃₇	—	4	4	no	—
87	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃₇	C₂×Q₈	1	4	yes	—
89	C₄²×C₁₃₇	#41	1	2	yes	—
90	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₇	—	2	2	no	—
92	C₂×C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₄×C₁₃₇	—	2	2	no	—
94	C₂×C₄×C₁₃₇	—	2	2	no	—
95	C₂×C₄×C₁₃₇	—	2	2	no	—
96	C₂×C₄×C₁₃₇	—	2	2	no	—
97	C₂×C₄×C₁₃₇	—	2	4	no	—
98	C₂×C₄×C₁₃₇	—	4	2	no	—
99	C₂×C₄×C₁₃₇	—	4	2	no	—
100	C₂×C₄×C₁₃₇	—	4	4	no	—
101	C₂×C₄×C₁₃₇	—	4	2	no	—
102	C₂×C₄×C₁₃₇	—	4	2	no	—
103	C₂×C₄×C₁₃₇	—	4	4	no	—
104	C₂×C₄×C₁₃₇	—	4	2	no	—
105	C₂×C₄×C₁₃₇	—	4	4	no	—
106	C₂×C₄×C₁₃₇	#121	1	4	yes	—
107	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₁₃₇	#135	1	4	yes	—
110	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
112	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
114	C₈×C₁₃₇	—	2	4	no	—
115	C₈×C₁₃₇	—	2	4	no	—
116	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
117	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
118	C₂²×C₁₃₇	#536	1	1	yes	characteristic
119	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
120	C₂²×C₁₃₇	#575	1	2	yes	—
121	C₄×C₁₃₇	—	2	2	no	—
122	C₄×C₁₃₇	—	2	2	no	—
123	C₄×C₁₃₇	—	2	4	no	—
124	C₄×C₁₃₇	—	2	4	no	—
125	C₄×C₁₃₇	—	4	4	no	—
126	C₄×C₁₃₇	—	8	2	no	—
127	C₄×C₁₃₇	—	8	4	no	—
128	C₄×C₁₃₇	D₄×Q₈	1	2	yes	—
129	C₄×C₁₃₇	#557	1	2	yes	—
130	C₂×C₁₃₇	#2546	1	1	yes	characteristic
131	C₂×C₁₃₇	#3035	1	2	yes	—
132	C₂×C₁₃₇	#3160	1	1	yes	characteristic
133	C₂×C₁₃₇	#3166	1	1	yes	characteristic
134	C₂×C₁₃₇	#3232	1	2	yes	—
135	#28690	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₁₃₇	#28690	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×#499	C₂×C₁₃₇	1	2	yes	—
138	#2859	C₂×C₁₃₇	1	1	yes	characteristic
139	C₄×#125	C₂×C₁₃₇	1	1	yes	characteristic
140	#2084	C₂×C₁₃₇	1	2	yes	—
141	#2083	C₂×C₁₃₇	1	2	yes	—
142	#1991	C₂×C₁₃₇	1	2	yes	—
143	#1949	C₂×C₁₃₇	1	4	yes	—
144	C₄×C₄⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic
145	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
146	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
147	#499	—	2	8	no	—
148	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
149	C₂×#106	C₂²×C₁₃₇	1	4	yes	—
150	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
152	#389	—	2	2	no	—
153	#389	—	2	2	no	—
154	#389	C₂²×C₁₃₇	1	2	yes	—
155	#387	—	2	2	no	—
156	#384	C₂²×C₁₃₇	1	2	yes	—
157	#382	C₂²×C₁₃₇	1	1	yes	characteristic
158	#382	C₂²×C₁₃₇	1	2	yes	—
159	C₄×C₄⋊C₄	—	2	2	no	—
160	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
161	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
162	#340	C₂²×C₁₃₇	1	4	yes	—
163	#340	C₂²×C₁₃₇	1	4	yes	—
164	#336	C₂²×C₁₃₇	1	1	yes	characteristic
165	#336	C₂²×C₁₃₇	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
167	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
168	#125	—	2	4	no	—
169	#125	—	2	2	no	—
170	#125	—	2	4	no	—
171	#125	D₄×C₁₃₇	1	4	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
179	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
182	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
183	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
186	#107	—	2	8	no	—
187	#106	—	2	8	no	—
188	C₄⋊C₈	Q₈×C₁₃₇	1	8	yes	—
189	C₄×C₈	—	2	4	no	—
190	#95	—	2	2	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	2	no	—
194	#95	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
199	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
202	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
208	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	8	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	2	no	—
222	C₄²	—	4	4	no	—
223	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
224	C₄²	C₂×Q₈×C₁₃₇	1	4	yes	—
225	C₄²	#41×C₁₃₇	1	2	yes	—
226	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	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	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	#121×C₁₃₇	1	4	yes	—
243	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
244	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
245	C₂×C₄	#135×C₁₃₇	1	4	yes	—
246	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
247	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
248	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
249	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
250	C₈	—	2	4	no	—
251	C₈	—	2	4	no	—
252	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
253	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
254	C₂²	#536×C₁₃₇	1	1	yes	characteristic
255	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
256	C₂²	#575×C₁₃₇	1	2	yes	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	2	no	—
263	C₄	—	8	4	no	—
264	C₄	D₄×Q₈×C₁₃₇	1	2	yes	—
265	C₄	#557×C₁₃₇	1	2	yes	—
266	C₂	#2546×C₁₃₇	1	1	yes	characteristic
267	C₂	#3035×C₁₃₇	1	2	yes	—
268	C₂	#3160×C₁₃₇	1	1	yes	characteristic
269	C₂	#3166×C₁₃₇	1	1	yes	characteristic
270	C₂	#3232×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
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
135. #28690	C₁₃₇	direct (N × H)
136. C₁₃₇	#28690	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#499×C₁₃₇	2
#2859×C₁₃₇	2
C₄×#125×C₁₃₇	2
#2084×C₁₃₇	2
#2083×C₁₃₇	2
#1991×C₁₃₇	2
#1949×C₁₃₇	2
C₄×C₄⋊C₈×C₁₃₇	2
#28690	137

Subgroups — C137×#28690 #46878076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28690 #46878076