Subgroups — C₁₃₇×#28725 #46878111

Order 35072.

750 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#491×C₁₃₇	C₂	1	2	yes	maximal
2	#2859×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2074×C₁₃₇	C₂	1	4	yes	maximal
5	#1992×C₁₃₇	C₂	1	2	yes	maximal
6	#1803×C₁₃₇	C₂	1	4	yes	maximal
7	#1767×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
11	#491×C₁₃₇	—	2	8	no	—
12	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#103×C₁₃₇	C₂²	1	8	yes	—
14	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
15	#391×C₁₃₇	C₂²	1	2	yes	—
16	#389×C₁₃₇	—	2	2	no	—
17	#389×C₁₃₇	—	2	2	no	—
18	#389×C₁₃₇	—	2	2	no	—
19	#384×C₁₃₇	C₂²	1	1	yes	characteristic
20	#384×C₁₃₇	C₂²	1	2	yes	—
21	#382×C₁₃₇	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
23	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
24	#340×C₁₃₇	C₂²	1	8	yes	—
25	#336×C₁₃₇	C₂²	1	2	yes	—
26	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
27	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
28	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
29	C₄⋊Q₈×C₁₃₇	D₄	1	4	yes	—
30	#125×C₁₃₇	—	2	4	no	—
31	#125×C₁₃₇	—	2	4	no	—
32	C₄×Q₈×C₁₃₇	—	2	8	no	—
33	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₃₇	—	2	2	no	—
42	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
43	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
44	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₁₃₇	—	2	4	no	—
48	#103×C₁₃₇	—	2	16	no	—
49	#97×C₁₃₇	—	2	4	no	—
50	#95×C₁₃₇	—	2	4	no	—
51	#95×C₁₃₇	—	2	4	no	—
52	C₂×Q₈×C₁₃₇	—	2	2	no	—
53	C₂×Q₈×C₁₃₇	—	2	4	no	—
54	C₂×Q₈×C₁₃₇	—	2	4	no	—
55	C₂×Q₈×C₁₃₇	C₂×D₄	1	4	yes	—
56	C₂²×C₄×C₁₃₇	—	2	2	no	—
57	C₂²×C₄×C₁₃₇	—	2	2	no	—
58	C₂²×C₄×C₁₃₇	—	2	4	no	—
59	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₁₃₇	—	2	4	no	—
65	C₂×C₈×C₁₃₇	—	2	4	no	—
66	C₂×C₈×C₁₃₇	—	2	4	no	—
67	C₄⋊C₄×C₁₃₇	—	2	8	no	—
68	C₄⋊C₄×C₁₃₇	—	2	4	no	—
69	C₄⋊C₄×C₁₃₇	—	2	4	no	—
70	C₄⋊C₄×C₁₃₇	—	2	4	no	—
71	C₄⋊C₄×C₁₃₇	—	4	4	no	—
72	C₄⋊C₄×C₁₃₇	—	4	4	no	—
73	C₄⋊C₄×C₁₃₇	—	4	8	no	—
74	C₄⋊C₄×C₁₃₇	—	4	4	no	—
75	C₄⋊C₄×C₁₃₇	—	4	4	no	—
76	C₄²×C₁₃₇	—	2	4	no	—
77	C₄²×C₁₃₇	—	2	2	no	—
78	C₄²×C₁₃₇	—	4	2	no	—
79	C₄²×C₁₃₇	—	4	4	no	—
80	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
81	C₄²×C₁₃₇	#41	1	2	yes	—
82	C₄²×C₁₃₇	#41	1	4	yes	—
83	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
84	Q₈×C₁₃₇	—	2	8	no	—
85	Q₈×C₁₃₇	—	4	4	no	—
86	C₂×C₄×C₁₃₇	—	2	2	no	—
87	C₂×C₄×C₁₃₇	—	2	2	no	—
88	C₂×C₄×C₁₃₇	—	2	2	no	—
89	C₂×C₄×C₁₃₇	—	2	4	no	—
90	C₂×C₄×C₁₃₇	—	2	2	no	—
91	C₂×C₄×C₁₃₇	—	2	2	no	—
92	C₂×C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₄×C₁₃₇	—	4	2	no	—
94	C₂×C₄×C₁₃₇	—	4	2	no	—
95	C₂×C₄×C₁₃₇	—	4	4	no	—
96	C₂×C₄×C₁₃₇	—	4	2	no	—
97	C₂×C₄×C₁₃₇	—	4	4	no	—
98	C₂×C₄×C₁₃₇	—	4	4	no	—
99	C₂×C₄×C₁₃₇	—	4	4	no	—
100	C₂×C₄×C₁₃₇	#121	1	4	yes	—
101	C₂×C₄×C₁₃₇	#124	1	2	yes	—
102	C₂×C₄×C₁₃₇	#124	1	2	yes	—
103	C₂×C₄×C₁₃₇	#137	1	4	yes	—
104	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
106	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
108	C₈×C₁₃₇	—	4	4	no	—
109	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
110	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
111	C₂²×C₁₃₇	#536	1	1	yes	characteristic
112	C₂²×C₁₃₇	C₂×#137	1	2	yes	—
113	C₂²×C₁₃₇	#575	1	2	yes	—
114	C₄×C₁₃₇	—	2	2	no	—
115	C₄×C₁₃₇	—	2	4	no	—
116	C₄×C₁₃₇	—	4	2	no	—
117	C₄×C₁₃₇	—	4	4	no	—
118	C₄×C₁₃₇	—	8	2	no	—
119	C₄×C₁₃₇	—	8	4	no	—
120	C₄×C₁₃₇	#491	1	4	yes	complemented
121	C₄×C₁₃₇	#547	1	2	yes	—
122	C₄×C₁₃₇	#556	1	2	yes	—
123	C₂×C₁₃₇	#2547	1	1	yes	characteristic
124	C₂×C₁₃₇	#3037	1	2	yes	—
125	C₂×C₁₃₇	C₂×#491	1	1	yes	characteristic
126	C₂×C₁₃₇	#3142	1	1	yes	characteristic
127	C₂×C₁₃₇	#3237	1	2	yes	—
128	#28725	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₁₃₇	#28725	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×#491	C₂×C₁₃₇	1	2	yes	—
131	#2859	C₂×C₁₃₇	1	1	yes	characteristic
132	C₄×C₄⋊Q₈	C₂×C₁₃₇	1	1	yes	characteristic
133	#2074	C₂×C₁₃₇	1	4	yes	—
134	#1992	C₂×C₁₃₇	1	2	yes	—
135	#1803	C₂×C₁₃₇	1	4	yes	—
136	#1767	C₂×C₁₃₇	1	1	yes	characteristic
137	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
138	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
139	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
140	#491	—	2	8	no	—
141	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
142	C₂×#103	C₂²×C₁₃₇	1	8	yes	—
143	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
144	#391	C₂²×C₁₃₇	1	2	yes	—
145	#389	—	2	2	no	—
146	#389	—	2	2	no	—
147	#389	—	2	2	no	—
148	#384	C₂²×C₁₃₇	1	1	yes	characteristic
149	#384	C₂²×C₁₃₇	1	2	yes	—
150	#382	C₂²×C₁₃₇	1	2	yes	—
151	C₄×C₄⋊C₄	—	2	2	no	—
152	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
153	#340	C₂²×C₁₃₇	1	8	yes	—
154	#336	C₂²×C₁₃₇	1	2	yes	—
155	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
156	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	D₄×C₁₃₇	1	4	yes	—
159	#125	—	2	4	no	—
160	#125	—	2	4	no	—
161	C₄×Q₈	—	2	8	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
169	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
170	C₂×C₄²	—	2	2	no	—
171	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
172	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
173	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
176	C₄⋊C₈	—	2	4	no	—
177	#103	—	2	16	no	—
178	#97	—	2	4	no	—
179	#95	—	2	4	no	—
180	#95	—	2	4	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	C₂×D₄×C₁₃₇	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
192	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₄⋊C₄	—	2	8	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	4	8	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	4	2	no	—
208	C₄²	—	4	4	no	—
209	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
210	C₄²	#41×C₁₃₇	1	2	yes	—
211	C₄²	#41×C₁₃₇	1	4	yes	—
212	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
213	Q₈	—	2	8	no	—
214	Q₈	—	4	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	#121×C₁₃₇	1	4	yes	—
230	C₂×C₄	#124×C₁₃₇	1	2	yes	—
231	C₂×C₄	#124×C₁₃₇	1	2	yes	—
232	C₂×C₄	#137×C₁₃₇	1	4	yes	—
233	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
234	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
235	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
236	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
237	C₈	—	4	4	no	—
238	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
239	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
240	C₂²	#536×C₁₃₇	1	1	yes	characteristic
241	C₂²	C₂×#137×C₁₃₇	1	2	yes	—
242	C₂²	#575×C₁₃₇	1	2	yes	—
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	4	no	—
246	C₄	—	4	2	no	—
247	C₄	—	8	4	no	—
248	C₄	—	8	2	no	—
249	C₄	#491×C₁₃₇	1	4	yes	complemented
250	C₄	#547×C₁₃₇	1	2	yes	—
251	C₄	#556×C₁₃₇	1	2	yes	—
252	C₂	#2547×C₁₃₇	1	1	yes	characteristic
253	C₂	#3037×C₁₃₇	1	2	yes	—
254	C₂	C₂×#491×C₁₃₇	1	1	yes	characteristic
255	C₂	#3142×C₁₃₇	1	1	yes	characteristic
256	C₂	#3237×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
120. C₄×C₁₃₇	#491	semidirect
128. #28725	C₁₃₇	direct (N × H)
129. C₁₃₇	#28725	direct (N × H)
249. C₄	#491×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#491×C₁₃₇	2
#2859×C₁₃₇	2
C₄×C₄⋊Q₈×C₁₃₇	2
#2074×C₁₃₇	2
#1992×C₁₃₇	2
#1803×C₁₃₇	2
#1767×C₁₃₇	2
#28725	137

Subgroups — C137×#28725 #46878111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28725 #46878111