Subgroups — C₁₃₇×#28743 #46878129

Order 35072.

1002 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#446×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2548×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#2054×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1982×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1972×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂²×D₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#127×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#124×C₁₃₇	—	2	2	no	—
12	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	#493×C₁₃₇	—	2	4	no	—
14	#446×C₁₃₇	—	2	4	no	—
15	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#102×C₁₃₇	C₂²	1	8	yes	—
19	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	#394×C₁₃₇	—	2	4	no	—
22	#390×C₁₃₇	—	2	2	no	—
23	#390×C₁₃₇	C₂²	1	8	yes	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
25	#340×C₁₃₇	C₂²	1	2	yes	—
26	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₁₃₇	—	2	2	no	—
28	C₂²×D₄×C₁₃₇	—	2	2	no	—
29	C₂²×D₄×C₁₃₇	C₂³	1	4	yes	—
30	C₂×D₈×C₁₃₇	—	2	4	no	—
31	C₂×D₈×C₁₃₇	—	2	4	no	—
32	C₂×D₈×C₁₃₇	—	2	8	no	—
33	C₂×D₈×C₁₃₇	—	4	4	no	—
34	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
37	#127×C₁₃₇	—	2	4	no	—
38	#127×C₁₃₇	—	2	4	no	—
39	#124×C₁₃₇	—	4	8	no	—
40	C₄×Q₈×C₁₃₇	—	2	2	no	—
41	C₄×Q₈×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
45	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
46	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
47	C₂×C₄²×C₁₃₇	—	2	2	no	—
48	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
50	#106×C₁₃₇	—	2	4	no	—
51	C₄⋊C₈×C₁₃₇	—	2	4	no	—
52	C₄⋊C₈×C₁₃₇	—	2	2	no	—
53	#102×C₁₃₇	—	4	8	no	—
54	#97×C₁₃₇	—	2	4	no	—
55	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
56	#95×C₁₃₇	—	2	2	no	—
57	C₂⁴×C₁₃₇	—	2	4	no	—
58	C₂×Q₈×C₁₃₇	—	2	2	no	—
59	C₂×Q₈×C₁₃₇	—	2	4	no	—
60	C₂×D₄×C₁₃₇	—	2	4	no	—
61	C₂×D₄×C₁₃₇	—	2	4	no	—
62	C₂×D₄×C₁₃₇	—	2	8	no	—
63	C₂×D₄×C₁₃₇	—	2	8	no	—
64	C₂×D₄×C₁₃₇	—	4	4	no	—
65	C₂×D₄×C₁₃₇	—	4	4	no	—
66	C₂×D₄×C₁₃₇	—	4	4	no	—
67	C₂×D₄×C₁₃₇	—	4	4	no	—
68	C₂×D₄×C₁₃₇	—	8	4	no	—
69	C₂²×C₄×C₁₃₇	—	2	4	no	—
70	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
75	D₈×C₁₃₇	—	4	8	no	—
76	D₈×C₁₃₇	—	4	8	no	—
77	C₂×C₈×C₁₃₇	—	2	2	no	—
78	C₂×C₈×C₁₃₇	—	2	2	no	—
79	C₂×C₈×C₁₃₇	—	2	4	no	—
80	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃₇	#41	1	4	yes	—
83	C₄⋊C₄×C₁₃₇	—	2	2	no	—
84	C₄⋊C₄×C₁₃₇	—	2	4	no	—
85	C₄⋊C₄×C₁₃₇	—	2	4	no	—
86	C₄⋊C₄×C₁₃₇	—	2	4	no	—
87	C₂²⋊C₄×C₁₃₇	—	4	16	no	—
88	C₂²⋊C₄×C₁₃₇	—	8	8	no	—
89	C₄²×C₁₃₇	—	2	2	no	—
90	C₄²×C₁₃₇	—	2	2	no	—
91	C₄²×C₁₃₇	—	4	4	no	—
92	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
93	C₄²×C₁₃₇	#41	1	2	yes	—
94	C₂³×C₁₃₇	—	4	4	no	—
95	C₂³×C₁₃₇	—	4	8	no	—
96	C₂³×C₁₃₇	—	4	4	no	—
97	C₂³×C₁₃₇	—	4	4	no	—
98	C₂³×C₁₃₇	—	4	8	no	—
99	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
100	Q₈×C₁₃₇	—	4	4	no	—
101	D₄×C₁₃₇	—	4	4	no	—
102	D₄×C₁₃₇	—	4	4	no	—
103	D₄×C₁₃₇	—	4	8	no	—
104	D₄×C₁₃₇	—	8	8	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	4	no	—
110	C₂×C₄×C₁₃₇	—	2	4	no	—
111	C₂×C₄×C₁₃₇	—	4	4	no	—
112	C₂×C₄×C₁₃₇	—	4	4	no	—
113	C₂×C₄×C₁₃₇	—	4	4	no	—
114	C₂×C₄×C₁₃₇	#121	1	2	yes	—
115	C₂×C₄×C₁₃₇	#121	1	2	yes	—
116	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
117	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
119	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
121	C₈×C₁₃₇	—	2	2	no	—
122	C₈×C₁₃₇	—	4	2	no	—
123	C₈×C₁₃₇	#124	1	4	yes	complemented
124	C₂²×C₁₃₇	—	8	8	no	—
125	C₂²×C₁₃₇	—	8	8	no	—
126	C₂²×C₁₃₇	—	8	4	no	—
127	C₂²×C₁₃₇	—	8	4	no	—
128	C₂²×C₁₃₇	—	8	4	no	—
129	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
130	C₂²×C₁₃₇	#550	1	2	yes	—
131	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
132	C₂²×C₁₃₇	#576	1	2	yes	—
133	C₄×C₁₃₇	—	2	2	no	—
134	C₄×C₁₃₇	—	4	2	no	—
135	C₄×C₁₃₇	—	4	2	no	—
136	C₄×C₁₃₇	—	8	4	no	—
137	C₄×C₁₃₇	#469	1	4	yes	—
138	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
139	C₄×C₁₃₇	#535	1	1	yes	characteristic
140	C₄×C₁₃₇	#559	1	2	yes	—
141	C₂×C₁₃₇	—	16	4	no	—
142	C₂×C₁₃₇	#2548	1	1	yes	characteristic
143	C₂×C₁₃₇	C₂×#469	1	1	yes	characteristic
144	C₂×C₁₃₇	#3058	1	1	yes	characteristic
145	C₂×C₁₃₇	#3399	1	4	yes	—
146	#28743	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₁₃₇	#28743	1	1	yes	characteristic, direct product factor, Sylow
148	C₂×#493	C₂×C₁₃₇	1	1	yes	characteristic
149	C₂×#446	C₂×C₁₃₇	1	1	yes	characteristic, complemented
150	#2548	C₂×C₁₃₇	1	2	yes	complemented
151	#2054	C₂×C₁₃₇	1	4	yes	complemented
152	#1982	C₂×C₁₃₇	1	2	yes	complemented
153	#1972	C₂×C₁₃₇	1	4	yes	complemented
154	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic, complemented
155	C₂²×D₈	C₂²×C₁₃₇	1	2	yes	—
156	C₂²×D₈	C₂²×C₁₃₇	1	2	yes	—
157	C₂×#127	C₂²×C₁₃₇	1	2	yes	—
158	C₂×#124	—	2	2	no	—
159	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
160	#493	—	2	4	no	—
161	#446	—	2	4	no	—
162	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
164	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
165	C₂×#102	C₂²×C₁₃₇	1	8	yes	—
166	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
168	#394	—	2	4	no	—
169	#390	—	2	2	no	—
170	#390	C₂²×C₁₃₇	1	8	yes	—
171	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
172	#340	C₂²×C₁₃₇	1	2	yes	—
173	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
174	C₂²×D₄	—	2	2	no	—
175	C₂²×D₄	—	2	2	no	—
176	C₂²×D₄	C₂³×C₁₃₇	1	4	yes	—
177	C₂×D₈	—	2	4	no	—
178	C₂×D₈	—	2	4	no	—
179	C₂×D₈	—	2	8	no	—
180	C₂×D₈	—	4	4	no	—
181	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
182	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
184	#127	—	2	4	no	—
185	#127	—	2	4	no	—
186	#124	—	4	8	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×Q₈	—	2	2	no	—
189	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
190	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
191	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
192	C₂×C₂²⋊C₄	—	2	8	no	—
193	C₂×C₂²⋊C₄	—	2	8	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
196	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
197	#106	—	2	4	no	—
198	C₄⋊C₈	—	2	4	no	—
199	C₄⋊C₈	—	2	2	no	—
200	#102	—	4	8	no	—
201	#97	—	2	4	no	—
202	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
203	#95	—	2	2	no	—
204	C₂⁴	—	2	4	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	4	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	8	no	—
210	C₂×D₄	—	2	8	no	—
211	C₂×D₄	—	4	4	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	—	4	4	no	—
214	C₂×D₄	—	4	4	no	—
215	C₂×D₄	—	8	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
221	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
222	D₈	—	4	8	no	—
223	D₈	—	4	8	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
228	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
229	C₂×C₈	#41×C₁₃₇	1	4	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	4	16	no	—
235	C₂²⋊C₄	—	8	8	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	4	4	no	—
239	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
240	C₄²	#41×C₁₃₇	1	2	yes	—
241	C₂³	—	4	4	no	—
242	C₂³	—	4	8	no	—
243	C₂³	—	4	4	no	—
244	C₂³	—	4	4	no	—
245	C₂³	—	4	8	no	—
246	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
247	Q₈	—	4	4	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	8	no	—
251	D₄	—	8	8	no	—
252	D₄	—	8	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	#121×C₁₃₇	1	2	yes	—
262	C₂×C₄	#121×C₁₃₇	1	2	yes	—
263	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
264	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
265	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
266	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
267	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
268	C₈	—	2	2	no	—
269	C₈	—	4	2	no	—
270	C₈	#124×C₁₃₇	1	4	yes	complemented
271	C₂²	—	8	4	no	—
272	C₂²	—	8	4	no	—
273	C₂²	—	8	4	no	—
274	C₂²	—	8	8	no	—
275	C₂²	—	8	8	no	—
276	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
277	C₂²	#550×C₁₃₇	1	2	yes	—
278	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
279	C₂²	#576×C₁₃₇	1	2	yes	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	2	no	—
283	C₄	—	8	4	no	—
284	C₄	#469×C₁₃₇	1	4	yes	—
285	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
286	C₄	#535×C₁₃₇	1	1	yes	characteristic
287	C₄	#559×C₁₃₇	1	2	yes	—
288	C₂	—	16	4	no	—
289	C₂	#2548×C₁₃₇	1	1	yes	characteristic
290	C₂	C₂×#469×C₁₃₇	1	1	yes	characteristic
291	C₂	#3058×C₁₃₇	1	1	yes	characteristic
292	C₂	#3399×C₁₃₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#493×C₁₃₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄×Q₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#446×C₁₃₇	C₂	semidirect
3. #2548×C₁₃₇	C₂	semidirect
4. #2054×C₁₃₇	C₂	semidirect
5. #1982×C₁₃₇	C₂	semidirect
6. #1972×C₁₃₇	C₂	semidirect
7. C₄×#106×C₁₃₇	C₂	semidirect
123. C₈×C₁₃₇	#124	semidirect
146. #28743	C₁₃₇	direct (N × H)
147. C₁₃₇	#28743	direct (N × H)
149. C₂×#446	C₂×C₁₃₇	semidirect
150. #2548	C₂×C₁₃₇	semidirect
151. #2054	C₂×C₁₃₇	semidirect
152. #1982	C₂×C₁₃₇	semidirect
153. #1972	C₂×C₁₃₇	semidirect
154. C₄×#106	C₂×C₁₃₇	semidirect
270. 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×#493×C₁₃₇	2
C₂×#446×C₁₃₇	2
#2548×C₁₃₇	2
#2054×C₁₃₇	2
#1982×C₁₃₇	2
#1972×C₁₃₇	2
C₄×#106×C₁₃₇	2
#28743	137

Subgroups — C137×#28743 #46878129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28743 #46878129