Subgroups — C₁₃₇×#28756 #46878142

Order 35072.

1130 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#490×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	#2867×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2041×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1982×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1961×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1767×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#127×C₁₃₇	—	2	2	no	—
9	C₂×#127×C₁₃₇	—	2	2	no	—
10	C₂×#127×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#127×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#127×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
14	#490×C₁₃₇	—	2	8	no	—
15	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#102×C₁₃₇	C₂²	1	8	yes	—
17	C₂×#102×C₁₃₇	C₂²	1	8	yes	—
18	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
19	#389×C₁₃₇	C₂²	1	2	yes	—
20	#387×C₁₃₇	—	2	2	no	—
21	#382×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
25	#336×C₁₃₇	C₂²	1	2	yes	—
26	C₂²×D₄×C₁₃₇	—	2	4	no	—
27	C₂²×D₄×C₁₃₇	—	2	2	no	—
28	C₂²×D₄×C₁₃₇	—	2	4	no	—
29	C₂²×D₄×C₁₃₇	—	2	2	no	—
30	C₂²×D₄×C₁₃₇	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
32	#127×C₁₃₇	—	2	4	no	—
33	#127×C₁₃₇	—	2	4	no	—
34	#127×C₁₃₇	—	2	8	no	—
35	#127×C₁₃₇	—	4	4	no	—
36	#127×C₁₃₇	—	4	8	no	—
37	#127×C₁₃₇	—	4	8	no	—
38	#125×C₁₃₇	—	2	2	no	—
39	#125×C₁₃₇	D₄	1	4	yes	complemented
40	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃₇	—	2	2	no	—
45	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
47	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₁₃₇	—	2	4	no	—
51	#102×C₁₃₇	—	2	16	no	—
52	#102×C₁₃₇	—	4	8	no	—
53	#97×C₁₃₇	—	2	4	no	—
54	#95×C₁₃₇	—	2	2	no	—
55	#95×C₁₃₇	—	2	2	no	—
56	#95×C₁₃₇	—	2	2	no	—
57	C₂⁴×C₁₃₇	—	2	4	no	—
58	C₂×D₄×C₁₃₇	—	2	4	no	—
59	C₂×D₄×C₁₃₇	—	2	8	no	—
60	C₂×D₄×C₁₃₇	—	2	4	no	—
61	C₂×D₄×C₁₃₇	—	2	8	no	—
62	C₂×D₄×C₁₃₇	—	4	8	no	—
63	C₂×D₄×C₁₃₇	—	4	8	no	—
64	C₂×D₄×C₁₃₇	—	4	8	no	—
65	C₂×D₄×C₁₃₇	—	4	8	no	—
66	C₂×D₄×C₁₃₇	—	4	8	no	—
67	C₂×D₄×C₁₃₇	—	4	8	no	—
68	C₂×D₄×C₁₃₇	—	4	4	no	—
69	C₂×D₄×C₁₃₇	—	4	4	no	—
70	C₂×D₄×C₁₃₇	—	4	8	no	—
71	C₂×D₄×C₁₃₇	—	4	8	no	—
72	C₂²×C₄×C₁₃₇	—	2	2	no	—
73	C₂²×C₄×C₁₃₇	—	2	2	no	—
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₁₃₇	—	2	4	no	—
80	C₂×C₈×C₁₃₇	—	2	4	no	—
81	C₂×C₈×C₁₃₇	—	2	4	no	—
82	C₄⋊C₄×C₁₃₇	—	2	8	no	—
83	C₄⋊C₄×C₁₃₇	—	2	2	no	—
84	C₄⋊C₄×C₁₃₇	—	2	4	no	—
85	C₄⋊C₄×C₁₃₇	—	4	4	no	—
86	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
87	C₄²×C₁₃₇	—	2	4	no	—
88	C₄²×C₁₃₇	—	2	4	no	—
89	C₄²×C₁₃₇	—	2	2	no	—
90	C₄²×C₁₃₇	—	4	4	no	—
91	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
92	C₄²×C₁₃₇	#41	1	2	yes	—
93	C₄²×C₁₃₇	#41	1	4	yes	—
94	C₂³×C₁₃₇	—	4	4	no	—
95	C₂³×C₁₃₇	—	4	4	no	—
96	C₂³×C₁₃₇	—	4	4	no	—
97	C₂³×C₁₃₇	—	4	8	no	—
98	C₂³×C₁₃₇	—	4	8	no	—
99	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
100	D₄×C₁₃₇	—	4	8	no	—
101	D₄×C₁₃₇	—	4	8	no	—
102	D₄×C₁₃₇	—	8	8	no	—
103	D₄×C₁₃₇	—	8	16	no	—
104	D₄×C₁₃₇	—	8	16	no	—
105	D₄×C₁₃₇	—	8	8	no	—
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	2	2	no	—
108	C₂×C₄×C₁₃₇	—	2	2	no	—
109	C₂×C₄×C₁₃₇	—	2	2	no	—
110	C₂×C₄×C₁₃₇	—	2	2	no	—
111	C₂×C₄×C₁₃₇	—	2	4	no	—
112	C₂×C₄×C₁₃₇	—	4	2	no	—
113	C₂×C₄×C₁₃₇	—	4	2	no	—
114	C₂×C₄×C₁₃₇	—	4	2	no	—
115	C₂×C₄×C₁₃₇	—	4	2	no	—
116	C₂×C₄×C₁₃₇	—	4	2	no	—
117	C₂×C₄×C₁₃₇	#121	1	4	yes	—
118	C₂×C₄×C₁₃₇	#124	1	2	yes	—
119	C₂×C₄×C₁₃₇	#124	1	2	yes	—
120	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
121	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
126	C₈×C₁₃₇	—	4	4	no	—
127	C₂²×C₁₃₇	—	8	4	no	—
128	C₂²×C₁₃₇	—	8	4	no	—
129	C₂²×C₁₃₇	—	8	8	no	—
130	C₂²×C₁₃₇	—	8	8	no	—
131	C₂²×C₁₃₇	—	8	4	no	—
132	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
133	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
134	C₂²×C₁₃₇	#535	1	1	yes	characteristic
135	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
136	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
137	C₄×C₁₃₇	—	2	2	no	—
138	C₄×C₁₃₇	—	2	4	no	—
139	C₄×C₁₃₇	—	2	4	no	—
140	C₄×C₁₃₇	—	4	4	no	—
141	C₄×C₁₃₇	—	8	2	no	—
142	C₄×C₁₃₇	#490	1	4	yes	complemented
143	C₄×C₁₃₇	#550	1	2	yes	—
144	C₄×C₁₃₇	#559	1	2	yes	—
145	C₂×C₁₃₇	—	16	4	no	—
146	C₂×C₁₃₇	#2548	1	1	yes	characteristic
147	C₂×C₁₃₇	#3038	1	2	yes	—
148	C₂×C₁₃₇	C₂×#490	1	1	yes	characteristic
149	C₂×C₁₃₇	#3138	1	1	yes	characteristic
150	C₂×C₁₃₇	#3217	1	2	yes	—
151	#28756	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₁₃₇	#28756	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×#490	C₂×C₁₃₇	1	2	yes	complemented
154	#2867	C₂×C₁₃₇	1	1	yes	characteristic
155	C₄×#125	C₂×C₁₃₇	1	1	yes	characteristic, complemented
156	#2041	C₂×C₁₃₇	1	4	yes	complemented
157	#1982	C₂×C₁₃₇	1	4	yes	complemented
158	#1961	C₂×C₁₃₇	1	2	yes	complemented
159	#1767	C₂×C₁₃₇	1	1	yes	characteristic, complemented
160	C₂×#127	—	2	2	no	—
161	C₂×#127	—	2	2	no	—
162	C₂×#127	C₂²×C₁₃₇	1	2	yes	—
163	C₂×#127	C₂²×C₁₃₇	1	2	yes	—
164	C₂×#127	C₂²×C₁₃₇	1	2	yes	—
165	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
166	#490	—	2	8	no	—
167	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
168	C₂×#102	C₂²×C₁₃₇	1	8	yes	—
169	C₂×#102	C₂²×C₁₃₇	1	8	yes	—
170	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
171	#389	C₂²×C₁₃₇	1	2	yes	—
172	#387	—	2	2	no	—
173	#382	C₂²×C₁₃₇	1	1	yes	characteristic
174	C₄×C₄⋊C₄	—	2	2	no	—
175	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
176	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
177	#336	C₂²×C₁₃₇	1	2	yes	—
178	C₂²×D₄	—	2	4	no	—
179	C₂²×D₄	—	2	2	no	—
180	C₂²×D₄	—	2	4	no	—
181	C₂²×D₄	—	2	2	no	—
182	C₂²×D₄	C₂³×C₁₃₇	1	4	yes	—
183	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
184	#127	—	2	4	no	—
185	#127	—	2	4	no	—
186	#127	—	2	8	no	—
187	#127	—	4	4	no	—
188	#127	—	4	8	no	—
189	#127	—	4	8	no	—
190	#125	—	2	2	no	—
191	#125	D₄×C₁₃₇	1	4	yes	complemented
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
198	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
199	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
202	C₄⋊C₈	—	2	4	no	—
203	#102	—	2	16	no	—
204	#102	—	4	8	no	—
205	#97	—	2	4	no	—
206	#95	—	2	2	no	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	C₂⁴	—	2	4	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	2	8	no	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	2	8	no	—
214	C₂×D₄	—	4	8	no	—
215	C₂×D₄	—	4	8	no	—
216	C₂×D₄	—	4	8	no	—
217	C₂×D₄	—	4	8	no	—
218	C₂×D₄	—	4	8	no	—
219	C₂×D₄	—	4	8	no	—
220	C₂×D₄	—	4	4	no	—
221	C₂×D₄	—	4	4	no	—
222	C₂×D₄	—	4	8	no	—
223	C₂×D₄	—	4	8	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
230	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	4	no	—
243	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
244	C₄²	#41×C₁₃₇	1	2	yes	—
245	C₄²	#41×C₁₃₇	1	4	yes	—
246	C₂³	—	4	4	no	—
247	C₂³	—	4	4	no	—
248	C₂³	—	4	4	no	—
249	C₂³	—	4	8	no	—
250	C₂³	—	4	8	no	—
251	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
252	D₄	—	4	8	no	—
253	D₄	—	4	8	no	—
254	D₄	—	8	16	no	—
255	D₄	—	8	8	no	—
256	D₄	—	8	16	no	—
257	D₄	—	8	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	#121×C₁₃₇	1	4	yes	—
270	C₂×C₄	#124×C₁₃₇	1	2	yes	—
271	C₂×C₄	#124×C₁₃₇	1	2	yes	—
272	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
273	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
274	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
275	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
276	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
277	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
278	C₈	—	4	4	no	—
279	C₂²	—	8	8	no	—
280	C₂²	—	8	4	no	—
281	C₂²	—	8	4	no	—
282	C₂²	—	8	4	no	—
283	C₂²	—	8	8	no	—
284	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
285	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
286	C₂²	#535×C₁₃₇	1	1	yes	characteristic
287	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
288	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	4	no	—
292	C₄	—	4	4	no	—
293	C₄	—	8	2	no	—
294	C₄	#490×C₁₃₇	1	4	yes	complemented
295	C₄	#550×C₁₃₇	1	2	yes	—
296	C₄	#559×C₁₃₇	1	2	yes	—
297	C₂	—	16	4	no	—
298	C₂	#2548×C₁₃₇	1	1	yes	characteristic
299	C₂	#3038×C₁₃₇	1	2	yes	—
300	C₂	C₂×#490×C₁₃₇	1	1	yes	characteristic
301	C₂	#3138×C₁₃₇	1	1	yes	characteristic
302	C₂	#3217×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	#2867×C₁₃₇	C₂
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
1. C₂×#490×C₁₃₇	C₂	semidirect
3. C₄×#125×C₁₃₇	C₂	semidirect
4. #2041×C₁₃₇	C₂	semidirect
5. #1982×C₁₃₇	C₂	semidirect
6. #1961×C₁₃₇	C₂	semidirect
7. #1767×C₁₃₇	C₂	semidirect
39. #125×C₁₃₇	D₄	semidirect
142. C₄×C₁₃₇	#490	semidirect
151. #28756	C₁₃₇	direct (N × H)
152. C₁₃₇	#28756	direct (N × H)
153. C₂×#490	C₂×C₁₃₇	semidirect
155. C₄×#125	C₂×C₁₃₇	semidirect
156. #2041	C₂×C₁₃₇	semidirect
157. #1982	C₂×C₁₃₇	semidirect
158. #1961	C₂×C₁₃₇	semidirect
159. #1767	C₂×C₁₃₇	semidirect
191. #125	D₄×C₁₃₇	semidirect
294. C₄	#490×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#490×C₁₃₇	2
#2867×C₁₃₇	2
C₄×#125×C₁₃₇	2
#2041×C₁₃₇	2
#1982×C₁₃₇	2
#1961×C₁₃₇	2
#1767×C₁₃₇	2
#28756	137

Subgroups — C137×#28756 #46878142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28756 #46878142