Subgroups — C₁₃₇×#12927 #46862313

Order 35072.

378 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	#2916×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2161×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#368×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#350×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1814×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1389×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1382×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#568×C₁₃₇	—	2	1	no	—
9	C₂×#131×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
12	#430×C₁₃₇	—	2	1	no	—
13	#430×C₁₃₇	—	2	2	no	—
14	C₂×#108×C₁₃₇	—	2	2	no	—
15	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
18	#427×C₁₃₇	C₂²	1	1	yes	characteristic
19	#421×C₁₃₇	—	2	1	no	—
20	#368×C₁₃₇	—	2	2	no	—
21	#368×C₁₃₇	—	2	2	no	—
22	#350×C₁₃₇	C₄	1	4	yes	complemented
23	#344×C₁₃₇	—	2	1	no	—
24	C₂×#41×C₁₃₇	D₄	1	1	yes	characteristic
25	#131×C₁₃₇	—	2	1	no	—
26	#131×C₁₃₇	—	2	1	no	—
27	#131×C₁₃₇	—	4	2	no	—
28	#131×C₁₃₇	—	4	2	no	—
29	#131×C₁₃₇	C₂×C₄	1	2	yes	—
30	#131×C₁₃₇	Q₈	1	2	yes	—
31	C₂×M₁₆×C₁₃₇	—	2	1	no	—
32	C₂×M₁₆×C₁₃₇	—	2	4	no	—
33	C₂×M₁₆×C₁₃₇	—	2	1	no	—
34	C₂×M₁₆×C₁₃₇	—	2	1	no	—
35	C₂×M₁₆×C₁₃₇	—	2	1	no	—
36	C₂×M₁₆×C₁₃₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₇	—	2	1	no	—
38	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
39	#117×C₁₃₇	—	2	1	no	—
40	M₃₂×C₁₃₇	—	2	1	no	—
41	M₃₂×C₁₃₇	C₂×C₄	1	2	yes	—
42	C₂×C₁₆×C₁₃₇	—	2	1	no	—
43	C₂×C₁₆×C₁₃₇	—	2	1	no	—
44	C₂×C₁₆×C₁₃₇	—	2	1	no	—
45	C₂×C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
46	#108×C₁₃₇	—	2	4	no	—
47	#108×C₁₃₇	—	2	1	no	—
48	#108×C₁₃₇	—	2	1	no	—
49	#108×C₁₃₇	—	2	1	no	—
50	#108×C₁₃₇	—	2	1	no	—
51	#108×C₁₃₇	D₄	1	2	yes	—
52	#108×C₁₃₇	D₄	1	2	yes	—
53	#107×C₁₃₇	—	2	1	no	—
54	#106×C₁₃₇	—	2	1	no	—
55	#104×C₁₃₇	—	4	2	no	—
56	#41×C₁₃₇	—	2	1	no	—
57	#41×C₁₃₇	—	2	1	no	—
58	#41×C₁₃₇	C₄⋊C₄	1	2	yes	—
59	#41×C₁₃₇	D₈	1	2	yes	—
60	C₂×Q₈×C₁₃₇	Q₁₆	1	1	yes	characteristic
61	C₂×D₄×C₁₃₇	—	2	1	no	—
62	C₂×D₄×C₁₃₇	Q₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₇	—	2	1	no	—
64	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
65	M₁₆×C₁₃₇	—	2	1	no	—
66	M₁₆×C₁₃₇	—	2	2	no	—
67	M₁₆×C₁₃₇	—	2	2	no	—
68	M₁₆×C₁₃₇	—	2	2	no	—
69	M₁₆×C₁₃₇	—	2	2	no	—
70	M₁₆×C₁₃₇	—	4	4	no	—
71	M₁₆×C₁₃₇	—	4	1	no	—
72	M₁₆×C₁₃₇	—	4	1	no	—
73	M₁₆×C₁₃₇	—	4	1	no	—
74	M₁₆×C₁₃₇	—	4	1	no	—
75	M₁₆×C₁₃₇	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₃₇	—	2	1	no	—
77	C₂×C₈×C₁₃₇	—	2	2	no	—
78	C₂×C₈×C₁₃₇	—	2	2	no	—
79	C₂×C₈×C₁₃₇	—	4	4	no	—
80	C₂×C₈×C₁₃₇	—	4	1	no	—
81	C₂×C₈×C₁₃₇	—	4	1	no	—
82	C₂×C₈×C₁₃₇	—	4	1	no	—
83	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
88	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
89	C₄⋊C₄×C₁₃₇	—	4	1	no	—
90	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
91	C₄²×C₁₃₇	—	4	1	no	—
92	C₁₆×C₁₃₇	—	2	2	no	—
93	C₁₆×C₁₃₇	—	2	2	no	—
94	C₂³×C₁₃₇	—	2	1	no	—
95	C₂³×C₁₃₇	#137	1	1	yes	characteristic
96	Q₈×C₁₃₇	—	2	1	no	—
97	Q₈×C₁₃₇	#107	1	2	yes	—
98	D₄×C₁₃₇	—	2	2	no	—
99	D₄×C₁₃₇	—	2	2	no	—
100	D₄×C₁₃₇	—	2	1	no	—
101	D₄×C₁₃₇	#107	1	2	yes	—
102	C₂×C₄×C₁₃₇	—	2	1	no	—
103	C₂×C₄×C₁₃₇	—	2	2	no	—
104	C₂×C₄×C₁₃₇	—	2	2	no	—
105	C₂×C₄×C₁₃₇	—	8	1	no	—
106	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	#123	1	1	yes	characteristic
109	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₈×C₁₃₇	—	2	2	no	—
113	C₈×C₁₃₇	—	4	2	no	—
114	C₈×C₁₃₇	—	4	2	no	—
115	C₈×C₁₃₇	—	8	1	no	—
116	C₈×C₁₃₇	C₄×D₄	1	1	yes	characteristic
117	C₈×C₁₃₇	C₄×D₄	1	1	yes	characteristic
118	C₈×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂²×C₁₃₇	—	2	2	no	—
121	C₂²×C₁₃₇	—	4	1	no	—
122	C₂²×C₁₃₇	—	8	1	no	—
123	C₂²×C₁₃₇	#428	1	1	yes	characteristic
124	C₂²×C₁₃₇	#451	1	1	yes	characteristic
125	C₂²×C₁₃₇	#480	1	1	yes	characteristic
126	C₄×C₁₃₇	—	2	2	no	—
127	C₄×C₁₃₇	—	16	1	no	—
128	C₄×C₁₃₇	#385	1	1	yes	characteristic
129	C₄×C₁₃₇	C₂×#107	1	1	yes	characteristic
130	C₄×C₁₃₇	#447	1	1	yes	characteristic
131	C₄×C₁₃₇	#484	1	1	yes	characteristic
132	C₂×C₁₃₇	—	2	1	no	—
133	C₂×C₁₃₇	—	2	1	no	—
134	C₂×C₁₃₇	—	2	1	no	—
135	C₂×C₁₃₇	—	4	2	no	—
136	C₂×C₁₃₇	#1852	1	1	yes	characteristic
137	#12927	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₁₃₇	#12927	1	1	yes	characteristic, direct product factor, Sylow
139	#2916	C₂×C₁₃₇	1	1	yes	characteristic
140	#2161	C₂×C₁₃₇	1	1	yes	characteristic, complemented
141	C₂×#368	C₂×C₁₃₇	1	1	yes	characteristic, complemented
142	C₂×#350	C₂×C₁₃₇	1	1	yes	characteristic
143	#1814	C₂×C₁₃₇	1	1	yes	characteristic
144	#1389	C₂×C₁₃₇	1	1	yes	characteristic, complemented
145	#1382	C₂×C₁₃₇	1	1	yes	characteristic, complemented
146	#568	—	2	1	no	—
147	C₂×#131	C₂²×C₁₃₇	1	1	yes	characteristic
148	C₂×M₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
149	C₂²×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
150	#430	—	2	1	no	—
151	#430	—	2	2	no	—
152	C₂×#108	—	2	2	no	—
153	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
154	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
155	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
156	#427	C₂²×C₁₃₇	1	1	yes	characteristic
157	#421	—	2	1	no	—
158	#368	—	2	2	no	—
159	#368	—	2	2	no	—
160	#350	C₄×C₁₃₇	1	4	yes	complemented
161	#344	—	2	1	no	—
162	C₂×#41	D₄×C₁₃₇	1	1	yes	characteristic
163	#131	—	2	1	no	—
164	#131	—	2	1	no	—
165	#131	—	4	2	no	—
166	#131	—	4	2	no	—
167	#131	C₂×C₄×C₁₃₇	1	2	yes	—
168	#131	Q₈×C₁₃₇	1	2	yes	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	D₄×C₁₃₇	1	1	yes	characteristic
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
177	#117	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	C₂×C₄×C₁₃₇	1	2	yes	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
184	#108	—	2	4	no	—
185	#108	—	2	1	no	—
186	#108	—	2	1	no	—
187	#108	—	2	1	no	—
188	#108	—	2	1	no	—
189	#108	D₄×C₁₃₇	1	2	yes	—
190	#108	D₄×C₁₃₇	1	2	yes	—
191	#107	—	2	1	no	—
192	#106	—	2	1	no	—
193	#104	—	4	2	no	—
194	#41	—	2	1	no	—
195	#41	—	2	1	no	—
196	#41	C₄⋊C₄×C₁₃₇	1	2	yes	—
197	#41	D₈×C₁₃₇	1	2	yes	—
198	C₂×Q₈	Q₁₆×C₁₃₇	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	Q₁₆×C₁₃₇	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	4	4	no	—
209	M₁₆	—	4	1	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	C₄⋊C₄×C₁₃₇	1	2	yes	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	4	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
222	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
224	C₂×C₈	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
225	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
226	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
227	C₄⋊C₄	—	4	1	no	—
228	C₂²⋊C₄	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₁₆	—	2	2	no	—
231	C₁₆	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	#137×C₁₃₇	1	1	yes	characteristic
234	Q₈	—	2	1	no	—
235	Q₈	#107×C₁₃₇	1	2	yes	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	#107×C₁₃₇	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
245	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
246	C₂×C₄	#123×C₁₃₇	1	1	yes	characteristic
247	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
248	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
249	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	4	2	no	—
252	C₈	—	4	2	no	—
253	C₈	—	8	1	no	—
254	C₈	C₄×D₄×C₁₃₇	1	1	yes	characteristic
255	C₈	C₄×D₄×C₁₃₇	1	1	yes	characteristic
256	C₈	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
257	C₈	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
258	C₂²	—	2	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	8	1	no	—
261	C₂²	#428×C₁₃₇	1	1	yes	characteristic
262	C₂²	#451×C₁₃₇	1	1	yes	characteristic
263	C₂²	#480×C₁₃₇	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	16	1	no	—
266	C₄	#385×C₁₃₇	1	1	yes	characteristic
267	C₄	C₂×#107×C₁₃₇	1	1	yes	characteristic
268	C₄	#447×C₁₃₇	1	1	yes	characteristic
269	C₄	#484×C₁₃₇	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	#1852×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₇	#447
commutator	C₂×C₈	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
socle	C₂×C₁₃₇	#1852
prime radical	C₂×#41×C₁₃₇	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂×#131×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2161×C₁₃₇	C₂	semidirect
3. C₂×#368×C₁₃₇	C₂	semidirect
6. #1389×C₁₃₇	C₂	semidirect
7. #1382×C₁₃₇	C₂	semidirect
22. #350×C₁₃₇	C₄	semidirect
137. #12927	C₁₃₇	direct (N × H)
138. C₁₃₇	#12927	direct (N × H)
140. #2161	C₂×C₁₃₇	semidirect
141. C₂×#368	C₂×C₁₃₇	semidirect
144. #1389	C₂×C₁₃₇	semidirect
145. #1382	C₂×C₁₃₇	semidirect
160. #350	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2916×C₁₃₇	2
#2161×C₁₃₇	2
C₂×#368×C₁₃₇	2
C₂×#350×C₁₃₇	2
#1814×C₁₃₇	2
#1389×C₁₃₇	2
#1382×C₁₃₇	2
#12927	137

Subgroups — C137×#12927 #46862313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#12927 #46862313