Subgroups — C₁₃₇×#28765 #46878151

Order 35072.

746 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#495×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#3140×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#2960×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2047×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2045×C₁₃₇	C₂	1	2	yes	maximal
6	#1977×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	#1967×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
11	C₂²×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	#533×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#124×C₁₃₇	C₂²	1	2	yes	—
14	#495×C₁₃₇	—	2	2	no	—
15	#495×C₁₃₇	—	2	2	no	—
16	#489×C₁₃₇	—	2	4	no	—
17	#488×C₁₃₇	—	2	4	no	—
18	#446×C₁₃₇	—	2	2	no	—
19	C₈×Q₈×C₁₃₇	—	2	2	no	—
20	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
21	#424×C₁₃₇	C₂²	1	2	yes	—
22	#423×C₁₃₇	C₂²	1	1	yes	characteristic
23	#419×C₁₃₇	C₂²	1	4	yes	—
24	#418×C₁₃₇	C₂²	1	4	yes	—
25	C₂×#101×C₁₃₇	C₂²	1	4	yes	—
26	C₂×#100×C₁₃₇	C₂²	1	4	yes	—
27	#405×C₁₃₇	C₂²	1	2	yes	—
28	C₄×M₁₆×C₁₃₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
30	#341×C₁₃₇	C₂²	1	2	yes	—
31	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
32	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
33	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
34	C₂×Q₁₆×C₁₃₇	—	4	2	no	—
35	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
36	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
37	C₂×SD₁₆×C₁₃₇	—	2	8	no	—
38	C₂×D₈×C₁₃₇	—	2	2	no	—
39	C₂×D₈×C₁₃₇	—	4	2	no	—
40	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
41	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈×C₁₃₇	D₄	1	2	yes	—
45	#125×C₁₃₇	D₄	1	2	yes	—
46	#124×C₁₃₇	—	2	2	no	—
47	#124×C₁₃₇	—	2	2	no	—
48	#124×C₁₃₇	—	2	4	no	—
49	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
50	C₄×Q₈×C₁₃₇	—	2	2	no	—
51	#117×C₁₃₇	C₂³	1	2	yes	—
52	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
53	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
54	#108×C₁₃₇	—	2	4	no	—
55	C₄⋊C₈×C₁₃₇	—	2	4	no	—
56	C₄⋊C₈×C₁₃₇	—	2	2	no	—
57	#103×C₁₃₇	—	2	8	no	—
58	#102×C₁₃₇	—	2	8	no	—
59	#101×C₁₃₇	—	4	4	no	—
60	#100×C₁₃₇	—	4	4	no	—
61	#97×C₁₃₇	—	2	2	no	—
62	#97×C₁₃₇	—	2	2	no	—
63	C₄×C₈×C₁₃₇	—	2	2	no	—
64	C₄×C₈×C₁₃₇	—	2	2	no	—
65	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
66	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
67	C₂⁴×C₁₃₇	—	2	2	no	—
68	C₂×Q₈×C₁₃₇	—	2	2	no	—
69	C₂×Q₈×C₁₃₇	—	2	2	no	—
70	C₂×Q₈×C₁₃₇	—	2	2	no	—
71	C₂×Q₈×C₁₃₇	—	2	4	no	—
72	C₂×Q₈×C₁₃₇	—	2	4	no	—
73	C₂×D₄×C₁₃₇	—	2	2	no	—
74	C₂×D₄×C₁₃₇	—	2	2	no	—
75	C₂×D₄×C₁₃₇	—	2	4	no	—
76	C₂×D₄×C₁₃₇	—	2	4	no	—
77	C₂²×C₄×C₁₃₇	—	2	2	no	—
78	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
79	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
80	Q₁₆×C₁₃₇	—	4	4	no	—
81	SD₁₆×C₁₃₇	—	4	4	no	—
82	SD₁₆×C₁₃₇	—	4	4	no	—
83	D₈×C₁₃₇	—	4	4	no	—
84	M₁₆×C₁₃₇	—	2	4	no	—
85	M₁₆×C₁₃₇	—	2	4	no	—
86	M₁₆×C₁₃₇	—	2	4	no	—
87	C₂×C₈×C₁₃₇	—	2	2	no	—
88	C₂×C₈×C₁₃₇	—	2	4	no	—
89	C₂×C₈×C₁₃₇	—	2	2	no	—
90	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃₇	#41	1	4	yes	—
94	C₄⋊C₄×C₁₃₇	—	2	4	no	—
95	C₄⋊C₄×C₁₃₇	—	2	2	no	—
96	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
97	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
98	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
99	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
100	C₄²×C₁₃₇	—	2	2	no	—
101	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
103	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
104	C₂³×C₁₃₇	—	4	2	no	—
105	C₂³×C₁₃₇	—	4	4	no	—
106	C₂³×C₁₃₇	—	8	4	no	—
107	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
108	Q₈×C₁₃₇	—	4	4	no	—
109	Q₈×C₁₃₇	—	4	2	no	—
110	Q₈×C₁₃₇	—	4	2	no	—
111	Q₈×C₁₃₇	—	4	4	no	—
112	D₄×C₁₃₇	—	4	2	no	—
113	D₄×C₁₃₇	—	4	2	no	—
114	D₄×C₁₃₇	—	4	4	no	—
115	C₂×C₄×C₁₃₇	—	2	4	no	—
116	C₂×C₄×C₁₃₇	—	4	2	no	—
117	C₂×C₄×C₁₃₇	—	4	2	no	—
118	C₂×C₄×C₁₃₇	—	4	4	no	—
119	C₂×C₄×C₁₃₇	#121	1	2	yes	—
120	C₂×C₄×C₁₃₇	#121	1	2	yes	—
121	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
125	C₈×C₁₃₇	—	2	2	no	—
126	C₈×C₁₃₇	—	4	2	no	—
127	C₈×C₁₃₇	—	4	4	no	—
128	C₈×C₁₃₇	#124	1	2	yes	—
129	C₈×C₁₃₇	#124	1	2	yes	—
130	C₂²×C₁₃₇	—	2	2	no	—
131	C₂²×C₁₃₇	—	8	2	no	—
132	C₂²×C₁₃₇	—	8	4	no	—
133	C₂²×C₁₃₇	—	16	4	no	—
134	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
135	C₂²×C₁₃₇	#559	1	2	yes	—
136	C₄×C₁₃₇	—	2	2	no	—
137	C₄×C₁₃₇	—	2	2	no	—
138	C₄×C₁₃₇	—	4	4	no	—
139	C₄×C₁₃₇	—	8	2	no	—
140	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
141	C₄×C₁₃₇	#535	1	1	yes	characteristic
142	C₄×C₁₃₇	#550	1	2	yes	—
143	C₂×C₁₃₇	—	2	2	no	—
144	C₂×C₁₃₇	—	16	2	no	—
145	C₂×C₁₃₇	#2548	1	1	yes	characteristic
146	C₂×C₁₃₇	#3067	1	2	yes	—
147	#28765	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₁₃₇	#28765	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×#495	C₂×C₁₃₇	1	1	yes	characteristic
150	#3140	C₂×C₁₃₇	1	2	yes	complemented
151	#2960	C₂×C₁₃₇	1	1	yes	characteristic, complemented
152	#2047	C₂×C₁₃₇	1	2	yes	complemented
153	#2045	C₂×C₁₃₇	1	2	yes	—
154	#1977	C₂×C₁₃₇	1	4	yes	complemented
155	#1967	C₂×C₁₃₇	1	2	yes	complemented
156	C₄×#108	C₂×C₁₃₇	1	1	yes	characteristic, complemented
157	C₂²×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
158	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
159	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
160	#533	C₂²×C₁₃₇	1	1	yes	characteristic
161	C₂×#124	C₂²×C₁₃₇	1	2	yes	—
162	#495	—	2	2	no	—
163	#495	—	2	2	no	—
164	#489	—	2	4	no	—
165	#488	—	2	4	no	—
166	#446	—	2	2	no	—
167	C₈×Q₈	—	2	2	no	—
168	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
169	#424	C₂²×C₁₃₇	1	2	yes	—
170	#423	C₂²×C₁₃₇	1	1	yes	characteristic
171	#419	C₂²×C₁₃₇	1	4	yes	—
172	#418	C₂²×C₁₃₇	1	4	yes	—
173	C₂×#101	C₂²×C₁₃₇	1	4	yes	—
174	C₂×#100	C₂²×C₁₃₇	1	4	yes	—
175	#405	C₂²×C₁₃₇	1	2	yes	—
176	C₄×M₁₆	C₂²×C₁₃₇	1	2	yes	—
177	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
178	#341	C₂²×C₁₃₇	1	2	yes	—
179	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
180	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	4	2	no	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×SD₁₆	—	2	2	no	—
185	C₂×SD₁₆	—	2	8	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	4	2	no	—
188	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
189	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
190	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
192	C₄⋊Q₈	D₄×C₁₃₇	1	2	yes	—
193	#125	D₄×C₁₃₇	1	2	yes	—
194	#124	—	2	2	no	—
195	#124	—	2	2	no	—
196	#124	—	2	4	no	—
197	C₂²⋊Q₈	—	2	2	no	—
198	C₄×Q₈	—	2	2	no	—
199	#117	C₂³×C₁₃₇	1	2	yes	—
200	C₂×C₂²⋊C₄	—	2	4	no	—
201	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
202	#108	—	2	4	no	—
203	C₄⋊C₈	—	2	4	no	—
204	C₄⋊C₈	—	2	2	no	—
205	#103	—	2	8	no	—
206	#102	—	2	8	no	—
207	#101	—	4	4	no	—
208	#100	—	4	4	no	—
209	#97	—	2	2	no	—
210	#97	—	2	2	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
214	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
215	C₂⁴	—	2	2	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
227	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
228	Q₁₆	—	4	4	no	—
229	SD₁₆	—	4	4	no	—
230	SD₁₆	—	4	4	no	—
231	D₈	—	4	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
239	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
240	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
241	C₂×C₈	#41×C₁₃₇	1	4	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₂²⋊C₄	—	4	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
251	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
252	C₂³	—	4	2	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	8	4	no	—
255	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
256	Q₈	—	4	4	no	—
257	Q₈	—	4	2	no	—
258	Q₈	—	4	2	no	—
259	Q₈	—	4	4	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	#121×C₁₃₇	1	2	yes	—
268	C₂×C₄	#121×C₁₃₇	1	2	yes	—
269	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
270	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
272	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
273	C₈	—	2	2	no	—
274	C₈	—	4	2	no	—
275	C₈	—	4	4	no	—
276	C₈	#124×C₁₃₇	1	2	yes	—
277	C₈	#124×C₁₃₇	1	2	yes	—
278	C₂²	—	2	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	4	no	—
281	C₂²	—	16	4	no	—
282	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
283	C₂²	#559×C₁₃₇	1	2	yes	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	4	no	—
287	C₄	—	8	2	no	—
288	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
289	C₄	#535×C₁₃₇	1	1	yes	characteristic
290	C₄	#550×C₁₃₇	1	2	yes	—
291	C₂	—	2	2	no	—
292	C₂	—	16	2	no	—
293	C₂	#2548×C₁₃₇	1	1	yes	characteristic
294	C₂	#3067×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₈×C₁₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #3140×C₁₃₇	C₂	semidirect
3. #2960×C₁₃₇	C₂	semidirect
4. #2047×C₁₃₇	C₂	semidirect
6. #1977×C₁₃₇	C₂	semidirect
7. #1967×C₁₃₇	C₂	semidirect
8. C₄×#108×C₁₃₇	C₂	semidirect
147. #28765	C₁₃₇	direct (N × H)
148. C₁₃₇	#28765	direct (N × H)
150. #3140	C₂×C₁₃₇	semidirect
151. #2960	C₂×C₁₃₇	semidirect
152. #2047	C₂×C₁₃₇	semidirect
154. #1977	C₂×C₁₃₇	semidirect
155. #1967	C₂×C₁₃₇	semidirect
156. C₄×#108	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#495×C₁₃₇	2
#3140×C₁₃₇	2
#2960×C₁₃₇	2
#2047×C₁₃₇	2
#2045×C₁₃₇	2
#1977×C₁₃₇	2
#1967×C₁₃₇	2
C₄×#108×C₁₃₇	2
#28765	137

Subgroups — C137×#28765 #46878151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28765 #46878151