Subgroups — C₁₃₇×#11358 #46860744

Order 35072.

798 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#382×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#340×C₁₃₇	C₂	1	4	yes	maximal
3	C₂×#336×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#95×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
9	#382×C₁₃₇	—	2	2	no	—
10	#382×C₁₃₇	C₄	1	4	yes	complemented
11	C₂×#95×C₁₃₇	—	2	2	no	—
12	C₂×#95×C₁₃₇	—	2	2	no	—
13	#340×C₁₃₇	—	2	8	no	—
14	#336×C₁₃₇	—	2	2	no	—
15	C₂³×C₄×C₁₃₇	—	2	2	no	—
16	C₂³×C₄×C₁₃₇	—	2	2	no	—
17	C₂³×C₄×C₁₃₇	D₄	1	2	yes	—
18	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₁₃₇	—	2	2	no	—
20	C₂²×C₈×C₁₃₇	—	2	4	no	—
21	C₂²×C₈×C₁₃₇	—	2	4	no	—
22	C₂²×C₈×C₁₃₇	—	2	2	no	—
23	C₂²×C₈×C₁₃₇	Q₈	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
26	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
28	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₁₃₇	D₄	1	4	yes	—
32	C₂×C₄²×C₁₃₇	—	2	4	no	—
33	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
35	#95×C₁₃₇	—	2	8	no	—
36	#95×C₁₃₇	—	4	4	no	—
37	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃₇	—	2	4	no	—
39	C₂²×C₄×C₁₃₇	—	2	4	no	—
40	C₂²×C₄×C₁₃₇	—	2	2	no	—
41	C₂²×C₄×C₁₃₇	—	2	2	no	—
42	C₂²×C₄×C₁₃₇	—	2	4	no	—
43	C₂²×C₄×C₁₃₇	—	2	4	no	—
44	C₂²×C₄×C₁₃₇	—	4	4	no	—
45	C₂²×C₄×C₁₃₇	—	4	2	no	—
46	C₂²×C₄×C₁₃₇	—	4	2	no	—
47	C₂²×C₄×C₁₃₇	—	4	4	no	—
48	C₂²×C₄×C₁₃₇	—	4	4	no	—
49	C₂²×C₄×C₁₃₇	—	4	4	no	—
50	C₂²×C₄×C₁₃₇	—	4	2	no	—
51	C₂²×C₄×C₁₃₇	—	4	2	no	—
52	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
53	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
58	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
59	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
60	C₂×C₈×C₁₃₇	—	2	8	no	—
61	C₂×C₈×C₁₃₇	—	2	4	no	—
62	C₂×C₈×C₁₃₇	—	4	4	no	—
63	C₂×C₈×C₁₃₇	—	4	2	no	—
64	C₂×C₈×C₁₃₇	—	4	2	no	—
65	C₄⋊C₄×C₁₃₇	—	2	8	no	—
66	C₄⋊C₄×C₁₃₇	—	2	8	no	—
67	C₄⋊C₄×C₁₃₇	—	4	2	no	—
68	C₄⋊C₄×C₁₃₇	—	4	4	no	—
69	C₄⋊C₄×C₁₃₇	—	4	2	no	—
70	C₄²×C₁₃₇	—	2	4	no	—
71	C₄²×C₁₃₇	—	2	4	no	—
72	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂³×C₁₃₇	#121	1	2	yes	—
74	C₂³×C₁₃₇	#123	1	2	yes	—
75	C₂³×C₁₃₇	#124	1	1	yes	characteristic
76	C₂³×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
77	C₂³×C₁₃₇	#135	1	4	yes	—
78	C₂³×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₂³×C₁₃₇	#137	1	2	yes	—
80	C₂×C₄×C₁₃₇	—	2	4	no	—
81	C₂×C₄×C₁₃₇	—	2	4	no	—
82	C₂×C₄×C₁₃₇	—	2	4	no	—
83	C₂×C₄×C₁₃₇	—	4	4	no	—
84	C₂×C₄×C₁₃₇	—	4	4	no	—
85	C₂×C₄×C₁₃₇	—	4	4	no	—
86	C₂×C₄×C₁₃₇	—	4	4	no	—
87	C₂×C₄×C₁₃₇	—	4	4	no	—
88	C₂×C₄×C₁₃₇	—	4	4	no	—
89	C₂×C₄×C₁₃₇	—	4	8	no	—
90	C₂×C₄×C₁₃₇	—	8	4	no	—
91	C₂×C₄×C₁₃₇	—	8	4	no	—
92	C₂×C₄×C₁₃₇	—	8	4	no	—
93	C₂×C₄×C₁₃₇	—	8	2	no	—
94	C₂×C₄×C₁₃₇	—	8	2	no	—
95	C₂×C₄×C₁₃₇	#117	1	2	yes	—
96	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
98	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
99	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₁₃₇	#123	1	4	yes	—
101	C₂×C₄×C₁₃₇	#123	1	2	yes	—
102	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	#125	1	4	yes	—
104	C₂×C₄×C₁₃₇	#126	1	4	yes	—
105	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
106	C₈×C₁₃₇	—	4	4	no	—
107	C₂²×C₁₃₇	#381	1	1	yes	characteristic
108	C₂²×C₁₃₇	#390	1	1	yes	characteristic
109	C₂²×C₁₃₇	#391	1	1	yes	characteristic
110	C₂²×C₁₃₇	#397	1	2	yes	—
111	C₂²×C₁₃₇	#398	1	2	yes	—
112	C₂²×C₁₃₇	#416	1	2	yes	—
113	C₂²×C₁₃₇	#417	1	2	yes	—
114	C₂²×C₁₃₇	#463	1	4	yes	—
115	C₂²×C₁₃₇	#464	1	4	yes	—
116	C₂²×C₁₃₇	#482	1	4	yes	—
117	C₂²×C₁₃₇	#485	1	4	yes	—
118	C₂²×C₁₃₇	#488	1	2	yes	—
119	C₂²×C₁₃₇	#490	1	1	yes	characteristic
120	C₂²×C₁₃₇	#491	1	1	yes	characteristic
121	C₂²×C₁₃₇	#499	1	2	yes	—
122	C₂²×C₁₃₇	#501	1	2	yes	—
123	C₄×C₁₃₇	—	4	4	no	—
124	C₄×C₁₃₇	—	8	4	no	—
125	C₄×C₁₃₇	—	8	4	no	—
126	C₄×C₁₃₇	#387	1	4	yes	—
127	C₄×C₁₃₇	#399	1	2	yes	—
128	C₄×C₁₃₇	#400	1	2	yes	—
129	C₂×C₁₃₇	#1445	1	1	yes	characteristic
130	C₂×C₁₃₇	#1825	1	2	yes	—
131	C₂×C₁₃₇	#1983	1	2	yes	—
132	C₂×C₁₃₇	#1991	1	1	yes	characteristic
133	C₂×C₁₃₇	#1992	1	1	yes	characteristic
134	C₂×C₁₃₇	#2073	1	2	yes	—
135	C₂×C₁₃₇	#2074	1	2	yes	—
136	C₂×C₁₃₇	#2085	1	4	yes	—
137	#11358	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₁₃₇	#11358	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×#382	C₂×C₁₃₇	1	1	yes	characteristic
140	C₂×#340	C₂×C₁₃₇	1	4	yes	—
141	C₂×#336	C₂×C₁₃₇	1	1	yes	characteristic
142	C₄⋊#95	C₂×C₁₃₇	1	1	yes	characteristic
143	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
144	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
145	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
146	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
147	#382	—	2	2	no	—
148	#382	C₄×C₁₃₇	1	4	yes	complemented
149	C₂×#95	—	2	2	no	—
150	C₂×#95	—	2	2	no	—
151	#340	—	2	8	no	—
152	#336	—	2	2	no	—
153	C₂³×C₄	—	2	2	no	—
154	C₂³×C₄	—	2	2	no	—
155	C₂³×C₄	D₄×C₁₃₇	1	2	yes	—
156	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	4	no	—
159	C₂²×C₈	—	2	4	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	Q₈×C₁₃₇	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
169	C₂×C₄⋊C₄	D₄×C₁₃₇	1	4	yes	—
170	C₂×C₄²	—	2	4	no	—
171	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
172	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
173	#95	—	2	8	no	—
174	#95	—	4	4	no	—
175	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	4	4	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	4	yes	—
191	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
193	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
195	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
196	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
197	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄²	—	2	4	no	—
209	C₄²	—	2	4	no	—
210	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
211	C₂³	#121×C₁₃₇	1	2	yes	—
212	C₂³	#123×C₁₃₇	1	2	yes	—
213	C₂³	#124×C₁₃₇	1	1	yes	characteristic
214	C₂³	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
215	C₂³	#135×C₁₃₇	1	4	yes	—
216	C₂³	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
217	C₂³	#137×C₁₃₇	1	2	yes	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	#117×C₁₃₇	1	2	yes	—
234	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
235	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
236	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
237	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
238	C₂×C₄	#123×C₁₃₇	1	4	yes	—
239	C₂×C₄	#123×C₁₃₇	1	2	yes	—
240	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
241	C₂×C₄	#125×C₁₃₇	1	4	yes	—
242	C₂×C₄	#126×C₁₃₇	1	4	yes	—
243	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
244	C₈	—	4	4	no	—
245	C₂²	#381×C₁₃₇	1	1	yes	characteristic
246	C₂²	#390×C₁₃₇	1	1	yes	characteristic
247	C₂²	#391×C₁₃₇	1	1	yes	characteristic
248	C₂²	#397×C₁₃₇	1	2	yes	—
249	C₂²	#398×C₁₃₇	1	2	yes	—
250	C₂²	#416×C₁₃₇	1	2	yes	—
251	C₂²	#417×C₁₃₇	1	2	yes	—
252	C₂²	#463×C₁₃₇	1	4	yes	—
253	C₂²	#464×C₁₃₇	1	4	yes	—
254	C₂²	#482×C₁₃₇	1	4	yes	—
255	C₂²	#485×C₁₃₇	1	4	yes	—
256	C₂²	#488×C₁₃₇	1	2	yes	—
257	C₂²	#490×C₁₃₇	1	1	yes	characteristic
258	C₂²	#491×C₁₃₇	1	1	yes	characteristic
259	C₂²	#499×C₁₃₇	1	2	yes	—
260	C₂²	#501×C₁₃₇	1	2	yes	—
261	C₄	—	4	4	no	—
262	C₄	—	8	4	no	—
263	C₄	—	8	4	no	—
264	C₄	#387×C₁₃₇	1	4	yes	—
265	C₄	#399×C₁₃₇	1	2	yes	—
266	C₄	#400×C₁₃₇	1	2	yes	—
267	C₂	#1445×C₁₃₇	1	1	yes	characteristic
268	C₂	#1825×C₁₃₇	1	2	yes	—
269	C₂	#1983×C₁₃₇	1	2	yes	—
270	C₂	#1991×C₁₃₇	1	1	yes	characteristic
271	C₂	#1992×C₁₃₇	1	1	yes	characteristic
272	C₂	#2073×C₁₃₇	1	2	yes	—
273	C₂	#2074×C₁₃₇	1	2	yes	—
274	C₂	#2085×C₁₃₇	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃₇ ◁ C₂²×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
10. #382×C₁₃₇	C₄	semidirect
137. #11358	C₁₃₇	direct (N × H)
138. C₁₃₇	#11358	direct (N × H)
148. #382	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#382×C₁₃₇	2
C₂×#340×C₁₃₇	2
C₂×#336×C₁₃₇	2
C₄⋊#95×C₁₃₇	2
#11358	137

Subgroups — C137×#11358 #46860744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#11358 #46860744