Subgroups — C₁₃₇×#28712 #46878098

Order 35072.

846 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#436×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2547×C₁₃₇	C₂	1	2	yes	maximal
4	#2076×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1988×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1809×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#124×C₁₃₇	—	2	2	no	—
11	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
12	#500×C₁₃₇	—	2	4	no	—
13	#436×C₁₃₇	—	2	4	no	—
14	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	#397×C₁₃₇	—	2	4	no	—
21	#391×C₁₃₇	—	2	2	no	—
22	#380×C₁₃₇	C₂²	1	8	yes	—
23	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	#340×C₁₃₇	C₂²	1	2	yes	—
25	#340×C₁₃₇	C₂²	1	8	yes	—
26	C₂²×Q₈×C₁₃₇	—	2	2	no	—
27	C₂²×D₄×C₁₃₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
29	C₂×M₁₆×C₁₃₇	—	2	4	no	—
30	C₂×M₁₆×C₁₃₇	—	2	8	no	—
31	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
35	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
36	#124×C₁₃₇	—	4	8	no	—
37	C₄×D₄×C₁₃₇	—	2	2	no	—
38	C₄×D₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
44	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃₇	—	2	2	no	—
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	#107×C₁₃₇	—	2	4	no	—
48	#107×C₁₃₇	—	2	4	no	—
49	#106×C₁₃₇	—	2	4	no	—
50	C₄⋊C₈×C₁₃₇	—	2	2	no	—
51	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
52	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
53	#95×C₁₃₇	—	2	2	no	—
54	#95×C₁₃₇	—	2	8	no	—
55	C₂⁴×C₁₃₇	C₂×D₄	1	2	yes	—
56	C₂×Q₈×C₁₃₇	—	4	4	no	—
57	C₂×Q₈×C₁₃₇	—	4	4	no	—
58	C₂×Q₈×C₁₃₇	—	4	4	no	—
59	C₂×D₄×C₁₃₇	—	2	2	no	—
60	C₂×D₄×C₁₃₇	—	2	4	no	—
61	C₂²×C₄×C₁₃₇	—	2	2	no	—
62	C₂²×C₄×C₁₃₇	—	2	2	no	—
63	C₂²×C₄×C₁₃₇	—	2	4	no	—
64	C₂²×C₄×C₁₃₇	—	2	4	no	—
65	C₂²×C₄×C₁₃₇	—	2	4	no	—
66	C₂²×C₄×C₁₃₇	—	2	4	no	—
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
71	M₁₆×C₁₃₇	—	4	8	no	—
72	C₂×C₈×C₁₃₇	—	2	2	no	—
73	C₂×C₈×C₁₃₇	—	2	2	no	—
74	C₂×C₈×C₁₃₇	—	2	4	no	—
75	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃₇	#41	1	4	yes	—
78	C₄⋊C₄×C₁₃₇	—	2	2	no	—
79	C₄⋊C₄×C₁₃₇	—	2	4	no	—
80	C₄⋊C₄×C₁₃₇	—	2	4	no	—
81	C₄⋊C₄×C₁₃₇	—	2	4	no	—
82	C₄⋊C₄×C₁₃₇	—	2	4	no	—
83	C₄⋊C₄×C₁₃₇	—	4	4	no	—
84	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
85	C₂²⋊C₄×C₁₃₇	—	4	16	no	—
86	C₄²×C₁₃₇	—	4	4	no	—
87	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃₇	#41	1	2	yes	—
89	C₂³×C₁₃₇	—	2	2	no	—
90	C₂³×C₁₃₇	—	2	4	no	—
91	C₂³×C₁₃₇	—	2	4	no	—
92	C₂³×C₁₃₇	—	2	4	no	—
93	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₁₃₇	—	8	8	no	—
95	D₄×C₁₃₇	—	4	4	no	—
96	C₂×C₄×C₁₃₇	—	2	2	no	—
97	C₂×C₄×C₁₃₇	—	2	2	no	—
98	C₂×C₄×C₁₃₇	—	2	4	no	—
99	C₂×C₄×C₁₃₇	—	4	2	no	—
100	C₂×C₄×C₁₃₇	—	4	2	no	—
101	C₂×C₄×C₁₃₇	—	4	4	no	—
102	C₂×C₄×C₁₃₇	—	4	4	no	—
103	C₂×C₄×C₁₃₇	—	4	4	no	—
104	C₂×C₄×C₁₃₇	—	4	4	no	—
105	C₂×C₄×C₁₃₇	—	4	4	no	—
106	C₂×C₄×C₁₃₇	—	4	4	no	—
107	C₂×C₄×C₁₃₇	—	4	4	no	—
108	C₂×C₄×C₁₃₇	#121	1	2	yes	—
109	C₂×C₄×C₁₃₇	#121	1	2	yes	—
110	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
111	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
113	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
115	C₈×C₁₃₇	—	2	2	no	—
116	C₈×C₁₃₇	—	4	2	no	—
117	C₈×C₁₃₇	#124	1	4	yes	complemented
118	C₂²×C₁₃₇	—	4	8	no	—
119	C₂²×C₁₃₇	—	4	2	no	—
120	C₂²×C₁₃₇	—	4	2	no	—
121	C₂²×C₁₃₇	—	4	2	no	—
122	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
123	C₂²×C₁₃₇	#547	1	2	yes	—
124	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
125	C₂²×C₁₃₇	#578	1	2	yes	—
126	C₄×C₁₃₇	—	2	2	no	—
127	C₄×C₁₃₇	—	4	2	no	—
128	C₄×C₁₃₇	—	8	4	no	—
129	C₄×C₁₃₇	—	8	4	no	—
130	C₄×C₁₃₇	#469	1	4	yes	—
131	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
132	C₄×C₁₃₇	#536	1	1	yes	characteristic
133	C₄×C₁₃₇	#556	1	2	yes	—
134	C₂×C₁₃₇	—	8	2	no	—
135	C₂×C₁₃₇	#2547	1	1	yes	characteristic
136	C₂×C₁₃₇	C₂×#469	1	1	yes	characteristic
137	C₂×C₁₃₇	#3059	1	1	yes	characteristic
138	C₂×C₁₃₇	#3353	1	4	yes	—
139	#28712	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₁₃₇	#28712	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic, complemented
142	C₂×#436	C₂×C₁₃₇	1	1	yes	characteristic
143	#2547	C₂×C₁₃₇	1	2	yes	—
144	#2076	C₂×C₁₃₇	1	4	yes	complemented
145	#1988	C₂×C₁₃₇	1	2	yes	complemented
146	#1809	C₂×C₁₃₇	1	4	yes	complemented
147	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic, complemented
148	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
149	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
150	C₂×#124	—	2	2	no	—
151	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
152	#500	—	2	4	no	—
153	#436	—	2	4	no	—
154	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
155	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
156	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
157	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
158	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
160	#397	—	2	4	no	—
161	#391	—	2	2	no	—
162	#380	C₂²×C₁₃₇	1	8	yes	—
163	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
164	#340	C₂²×C₁₃₇	1	8	yes	—
165	#340	C₂²×C₁₃₇	1	2	yes	—
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
168	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	—	2	8	no	—
171	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
172	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
174	C₄⋊Q₈	—	2	4	no	—
175	C₄⋊Q₈	—	2	4	no	—
176	#124	—	4	8	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
182	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
183	C₂×C₂²⋊C₄	—	2	8	no	—
184	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
187	#107	—	2	4	no	—
188	#107	—	2	4	no	—
189	#106	—	2	4	no	—
190	C₄⋊C₈	—	2	2	no	—
191	C₂²⋊C₈	—	2	4	no	—
192	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
193	#95	—	2	2	no	—
194	#95	—	2	8	no	—
195	C₂⁴	C₂×D₄×C₁₃₇	1	2	yes	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	—	4	4	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
210	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
211	M₁₆	—	4	8	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
216	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
217	C₂×C₈	#41×C₁₃₇	1	4	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	—	4	16	no	—
226	C₄²	—	4	4	no	—
227	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
228	C₄²	#41×C₁₃₇	1	2	yes	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	4	no	—
231	C₂³	—	2	4	no	—
232	C₂³	—	2	4	no	—
233	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
234	Q₈	—	8	8	no	—
235	D₄	—	4	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	#121×C₁₃₇	1	2	yes	—
249	C₂×C₄	#121×C₁₃₇	1	2	yes	—
250	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
251	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
252	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
253	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
254	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	4	2	no	—
257	C₈	#124×C₁₃₇	1	4	yes	complemented
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	8	no	—
262	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
263	C₂²	#547×C₁₃₇	1	2	yes	—
264	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
265	C₂²	#578×C₁₃₇	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	4	no	—
269	C₄	—	8	4	no	—
270	C₄	#469×C₁₃₇	1	4	yes	—
271	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
272	C₄	#536×C₁₃₇	1	1	yes	characteristic
273	C₄	#556×C₁₃₇	1	2	yes	—
274	C₂	—	8	2	no	—
275	C₂	#2547×C₁₃₇	1	1	yes	characteristic
276	C₂	C₂×#469×C₁₃₇	1	1	yes	characteristic
277	C₂	#3059×C₁₃₇	1	1	yes	characteristic
278	C₂	#3353×C₁₃₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₃₇	C₂	semidirect
4. #2076×C₁₃₇	C₂	semidirect
5. #1988×C₁₃₇	C₂	semidirect
6. #1809×C₁₃₇	C₂	semidirect
7. C₄×#106×C₁₃₇	C₂	semidirect
117. C₈×C₁₃₇	#124	semidirect
139. #28712	C₁₃₇	direct (N × H)
140. C₁₃₇	#28712	direct (N × H)
141. C₂×#500	C₂×C₁₃₇	semidirect
144. #2076	C₂×C₁₃₇	semidirect
145. #1988	C₂×C₁₃₇	semidirect
146. #1809	C₂×C₁₃₇	semidirect
147. C₄×#106	C₂×C₁₃₇	semidirect
257. C₈	#124×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₇	2
C₂×#436×C₁₃₇	2
#2547×C₁₃₇	2
#2076×C₁₃₇	2
#1988×C₁₃₇	2
#1809×C₁₃₇	2
C₄×#106×C₁₃₇	2
#28712	137

Subgroups — C137×#28712 #46878098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28712 #46878098