Subgroups — C₁₃₇×#8659 #46858045

Order 35072.

394 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1848×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1564×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1487×C₁₃₇	C₂	1	2	yes	maximal
5	#1301×C₁₃₇	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₃₇	—	2	1	no	—
8	C₈×Q₈×C₁₃₇	—	2	1	no	—
9	C₈×Q₈×C₁₃₇	C₄	1	2	yes	—
10	C₈×Q₈×C₁₃₇	C₄	1	2	yes	—
11	#431×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#108×C₁₃₇	—	2	1	no	—
13	C₂×#108×C₁₃₇	—	2	1	no	—
14	#423×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₁₃₇	—	2	1	no	—
16	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃₇	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	#335×C₁₃₇	—	2	1	no	—
20	#334×C₁₃₇	—	2	1	no	—
21	Q₈⋊C₈×C₁₃₇	—	2	4	no	—
22	C₂²×Q₈×C₁₃₇	Q₈	1	1	yes	characteristic
23	C₂×M₁₆×C₁₃₇	—	2	2	no	—
24	C₂²×C₈×C₁₃₇	—	2	1	no	—
25	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₁₃₇	—	2	1	no	—
28	C₄×Q₈×C₁₃₇	—	2	1	no	—
29	C₄×Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₄×Q₈×C₁₃₇	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₃₇	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₁₃₇	—	2	1	no	—
34	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
35	#108×C₁₃₇	—	4	2	no	—
36	#108×C₁₃₇	—	4	2	no	—
37	C₄⋊C₈×C₁₃₇	—	2	2	no	—
38	C₄⋊C₈×C₁₃₇	—	2	2	no	—
39	C₄⋊C₈×C₁₃₇	—	2	1	no	—
40	C₄⋊C₈×C₁₃₇	—	2	2	no	—
41	C₄⋊C₈×C₁₃₇	—	2	2	no	—
42	C₄⋊C₈×C₁₃₇	—	2	2	no	—
43	C₄⋊C₈×C₁₃₇	C₂×C₄	1	2	yes	—
44	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
45	C₄×C₈×C₁₃₇	—	2	2	no	—
46	C₄×C₈×C₁₃₇	—	2	2	no	—
47	C₄×C₈×C₁₃₇	—	2	2	no	—
48	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
52	C₂×Q₈×C₁₃₇	—	2	4	no	—
53	C₂×Q₈×C₁₃₇	—	2	1	no	—
54	C₂×Q₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₃₇	—	2	1	no	—
56	C₂²×C₄×C₁₃₇	—	2	1	no	—
57	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
60	M₁₆×C₁₃₇	—	4	2	no	—
61	M₁₆×C₁₃₇	—	4	2	no	—
62	C₂×C₈×C₁₃₇	—	2	1	no	—
63	C₂×C₈×C₁₃₇	—	2	1	no	—
64	C₂×C₈×C₁₃₇	—	2	2	no	—
65	C₂×C₈×C₁₃₇	—	4	1	no	—
66	C₂×C₈×C₁₃₇	—	4	1	no	—
67	C₂×C₈×C₁₃₇	—	4	2	no	—
68	C₂×C₈×C₁₃₇	—	4	2	no	—
69	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃₇	SD₁₆	1	2	yes	—
74	C₂×C₈×C₁₃₇	Q₁₆	1	2	yes	—
75	C₄⋊C₄×C₁₃₇	—	2	2	no	—
76	C₄⋊C₄×C₁₃₇	—	2	2	no	—
77	C₄⋊C₄×C₁₃₇	—	2	1	no	—
78	C₄⋊C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
79	C₄²×C₁₃₇	—	2	2	no	—
80	C₄²×C₁₃₇	—	2	2	no	—
81	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃₇	#41	1	1	yes	characteristic
85	C₂³×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
86	Q₈×C₁₃₇	—	2	4	no	—
87	Q₈×C₁₃₇	—	2	2	no	—
88	Q₈×C₁₃₇	#108	1	4	yes	complemented
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	1	no	—
94	C₂×C₄×C₁₃₇	—	4	2	no	—
95	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₁₃₇	#137	1	1	yes	characteristic
105	C₈×C₁₃₇	—	2	1	no	—
106	C₈×C₁₃₇	—	2	1	no	—
107	C₈×C₁₃₇	—	4	2	no	—
108	C₈×C₁₃₇	—	8	2	no	—
109	C₈×C₁₃₇	—	8	2	no	—
110	C₈×C₁₃₇	#103	1	2	yes	—
111	C₈×C₁₃₇	#103	1	2	yes	—
112	C₂²×C₁₃₇	—	2	1	no	—
113	C₂²×C₁₃₇	—	2	1	no	—
114	C₂²×C₁₃₇	#380	1	1	yes	characteristic
115	C₂²×C₁₃₇	#475	1	1	yes	characteristic
116	C₂²×C₁₃₇	#477	1	1	yes	characteristic
117	C₄×C₁₃₇	—	2	4	no	—
118	C₄×C₁₃₇	—	2	1	no	—
119	C₄×C₁₃₇	—	2	1	no	—
120	C₄×C₁₃₇	—	2	1	no	—
121	C₄×C₁₃₇	—	4	2	no	—
122	C₄×C₁₃₇	C₂×#103	1	1	yes	characteristic
123	C₄×C₁₃₇	#417	1	1	yes	characteristic
124	C₄×C₁₃₇	C₂×#108	1	2	yes	—
125	C₄×C₁₃₇	#460	1	1	yes	characteristic
126	C₄×C₁₃₇	#462	1	1	yes	characteristic
127	C₂×C₁₃₇	—	2	1	no	—
128	C₂×C₁₃₇	—	2	1	no	—
129	C₂×C₁₃₇	#1803	1	1	yes	characteristic
130	C₂×C₁₃₇	#1810	1	1	yes	characteristic
131	C₂×C₁₃₇	#1814	1	1	yes	characteristic
132	#8659	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₁₃₇	#8659	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×C₈×Q₈	C₂×C₁₃₇	1	1	yes	characteristic
135	#1848	C₂×C₁₃₇	1	1	yes	characteristic
136	#1564	C₂×C₁₃₇	1	1	yes	characteristic
137	#1487	C₂×C₁₃₇	1	2	yes	—
138	#1301	C₂×C₁₃₇	1	2	yes	—
139	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
140	C₈×Q₈	—	2	1	no	—
141	C₈×Q₈	—	2	1	no	—
142	C₈×Q₈	C₄×C₁₃₇	1	2	yes	—
143	C₈×Q₈	C₄×C₁₃₇	1	2	yes	—
144	#431	C₂²×C₁₃₇	1	2	yes	—
145	C₂×#108	—	2	1	no	—
146	C₂×#108	—	2	1	no	—
147	#423	C₂²×C₁₃₇	1	2	yes	—
148	C₂×C₄⋊C₈	—	2	1	no	—
149	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
150	C₂×C₄×C₈	—	2	1	no	—
151	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
152	#335	—	2	1	no	—
153	#334	—	2	1	no	—
154	Q₈⋊C₈	—	2	4	no	—
155	C₂²×Q₈	Q₈×C₁₃₇	1	1	yes	characteristic
156	C₂×M₁₆	—	2	2	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
159	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×Q₈	C₂×C₄×C₁₃₇	1	2	yes	—
163	C₄×Q₈	D₄×C₁₃₇	1	2	yes	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	Q₈×C₁₃₇	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
168	#108	—	4	2	no	—
169	#108	—	4	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	C₂×C₄×C₁₃₇	1	2	yes	—
177	C₂²⋊C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
182	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
183	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
184	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	2	1	no	—
187	C₂×Q₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
191	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
192	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
206	C₂×C₈	SD₁₆×C₁₃₇	1	2	yes	—
207	C₂×C₈	Q₁₆×C₁₃₇	1	2	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
215	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
216	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
217	C₄²	#41×C₁₃₇	1	1	yes	characteristic
218	C₂³	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
219	Q₈	—	2	4	no	—
220	Q₈	—	2	2	no	—
221	Q₈	#108×C₁₃₇	1	4	yes	complemented
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
230	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
231	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
232	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
233	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
234	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
236	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
237	C₂×C₄	#137×C₁₃₇	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	2	no	—
241	C₈	—	8	2	no	—
242	C₈	—	8	2	no	—
243	C₈	#103×C₁₃₇	1	2	yes	—
244	C₈	#103×C₁₃₇	1	2	yes	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	#380×C₁₃₇	1	1	yes	characteristic
248	C₂²	#475×C₁₃₇	1	1	yes	characteristic
249	C₂²	#477×C₁₃₇	1	1	yes	characteristic
250	C₄	—	2	4	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	2	no	—
255	C₄	C₂×#103×C₁₃₇	1	1	yes	characteristic
256	C₄	#417×C₁₃₇	1	1	yes	characteristic
257	C₄	C₂×#108×C₁₃₇	1	2	yes	—
258	C₄	#460×C₁₃₇	1	1	yes	characteristic
259	C₄	#462×C₁₃₇	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#1803×C₁₃₇	1	1	yes	characteristic
263	C₂	#1810×C₁₃₇	1	1	yes	characteristic
264	C₂	#1814×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#121
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#380
prime radical	C₂³×C₁₃₇	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
88. Q₈×C₁₃₇	#108	semidirect
132. #8659	C₁₃₇	direct (N × H)
133. C₁₃₇	#8659	direct (N × H)
221. Q₈	#108×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₁₃₇	2
#1848×C₁₃₇	2
#1564×C₁₃₇	2
#1487×C₁₃₇	2
#1301×C₁₃₇	2
#8659	137

Subgroups — C137×#8659 #46858045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8659 #46858045