Subgroups — C₁₃₇×#28742 #46878128

Order 35072.

846 conjugacy classes of subgroups.

The subgroup lattice has 328 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	C₂×#446×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2592×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#2055×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1983×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1972×C₁₃₇	C₂	1	2	yes	maximal
7	#1971×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×#106×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	C₂×#124×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	#495×C₁₃₇	—	2	4	no	—
15	#446×C₁₃₇	—	2	4	no	—
16	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
20	C₂×#102×C₁₃₇	C₂²	1	4	yes	—
21	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
23	#397×C₁₃₇	—	2	2	no	—
24	#393×C₁₃₇	—	2	2	no	—
25	#391×C₁₃₇	C₂²	1	4	yes	—
26	#390×C₁₃₇	C₂²	1	4	yes	—
27	#388×C₁₃₇	—	2	4	no	—
28	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
29	#340×C₁₃₇	C₂²	1	2	yes	—
30	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
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	—	2	4	no	—
35	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
36	C₂×SD₁₆×C₁₃₇	—	4	4	no	—
37	C₂×D₈×C₁₃₇	—	2	2	no	—
38	C₂×D₈×C₁₃₇	—	2	4	no	—
39	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
42	#124×C₁₃₇	—	2	4	no	—
43	#124×C₁₃₇	—	2	4	no	—
44	C₄×Q₈×C₁₃₇	—	2	2	no	—
45	C₄×Q₈×C₁₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
51	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
53	C₂×C₄²×C₁₃₇	—	2	2	no	—
54	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
56	#106×C₁₃₇	—	2	4	no	—
57	C₄⋊C₈×C₁₃₇	—	2	4	no	—
58	C₄⋊C₈×C₁₃₇	—	2	2	no	—
59	#103×C₁₃₇	—	4	4	no	—
60	#102×C₁₃₇	—	4	4	no	—
61	#97×C₁₃₇	—	2	4	no	—
62	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
63	#95×C₁₃₇	—	2	2	no	—
64	#95×C₁₃₇	—	2	4	no	—
65	#95×C₁₃₇	—	2	4	no	—
66	C₂⁴×C₁₃₇	—	2	2	no	—
67	C₂×Q₈×C₁₃₇	—	2	2	no	—
68	C₂×Q₈×C₁₃₇	—	2	2	no	—
69	C₂×Q₈×C₁₃₇	—	2	2	no	—
70	C₂×Q₈×C₁₃₇	—	2	4	no	—
71	C₂×Q₈×C₁₃₇	—	2	4	no	—
72	C₂×Q₈×C₁₃₇	—	4	2	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	—	2	4	no	—
79	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
84	Q₁₆×C₁₃₇	—	4	4	no	—
85	SD₁₆×C₁₃₇	—	4	8	no	—
86	D₈×C₁₃₇	—	4	4	no	—
87	C₂×C₈×C₁₃₇	—	2	2	no	—
88	C₂×C₈×C₁₃₇	—	2	2	no	—
89	C₂×C₈×C₁₃₇	—	2	2	no	—
90	C₂×C₈×C₁₃₇	—	2	2	no	—
91	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃₇	#41	1	4	yes	—
94	C₄⋊C₄×C₁₃₇	—	2	2	no	—
95	C₄⋊C₄×C₁₃₇	—	2	4	no	—
96	C₄⋊C₄×C₁₃₇	—	2	4	no	—
97	C₄⋊C₄×C₁₃₇	—	2	4	no	—
98	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
99	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
100	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
101	C₂²⋊C₄×C₁₃₇	—	8	4	no	—
102	C₄²×C₁₃₇	—	2	2	no	—
103	C₄²×C₁₃₇	—	2	2	no	—
104	C₄²×C₁₃₇	—	8	2	no	—
105	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
106	C₄²×C₁₃₇	#41	1	2	yes	—
107	C₂³×C₁₃₇	—	4	2	no	—
108	C₂³×C₁₃₇	—	4	4	no	—
109	C₂³×C₁₃₇	—	4	4	no	—
110	C₂³×C₁₃₇	—	4	2	no	—
111	C₂³×C₁₃₇	—	4	2	no	—
112	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₁₃₇	—	4	4	no	—
114	Q₈×C₁₃₇	—	4	2	no	—
115	Q₈×C₁₃₇	—	4	2	no	—
116	Q₈×C₁₃₇	—	4	4	no	—
117	D₄×C₁₃₇	—	4	2	no	—
118	D₄×C₁₃₇	—	4	2	no	—
119	D₄×C₁₃₇	—	4	4	no	—
120	C₂×C₄×C₁₃₇	—	2	2	no	—
121	C₂×C₄×C₁₃₇	—	2	2	no	—
122	C₂×C₄×C₁₃₇	—	2	2	no	—
123	C₂×C₄×C₁₃₇	—	2	2	no	—
124	C₂×C₄×C₁₃₇	—	2	4	no	—
125	C₂×C₄×C₁₃₇	—	2	4	no	—
126	C₂×C₄×C₁₃₇	—	4	2	no	—
127	C₂×C₄×C₁₃₇	—	4	4	no	—
128	C₂×C₄×C₁₃₇	—	4	4	no	—
129	C₂×C₄×C₁₃₇	—	4	4	no	—
130	C₂×C₄×C₁₃₇	—	4	4	no	—
131	C₂×C₄×C₁₃₇	#121	1	2	yes	—
132	C₂×C₄×C₁₃₇	#121	1	2	yes	—
133	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
135	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
137	C₈×C₁₃₇	—	2	2	no	—
138	C₈×C₁₃₇	—	4	2	no	—
139	C₈×C₁₃₇	#124	1	4	yes	—
140	C₂²×C₁₃₇	—	8	4	no	—
141	C₂²×C₁₃₇	—	8	4	no	—
142	C₂²×C₁₃₇	—	8	2	no	—
143	C₂²×C₁₃₇	—	8	2	no	—
144	C₂²×C₁₃₇	—	8	2	no	—
145	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
146	C₂²×C₁₃₇	#550	1	2	yes	—
147	C₂²×C₁₃₇	#575	1	2	yes	—
148	C₂²×C₁₃₇	#577	1	2	yes	—
149	C₄×C₁₃₇	—	2	2	no	—
150	C₄×C₁₃₇	—	2	2	no	—
151	C₄×C₁₃₇	—	4	2	no	—
152	C₄×C₁₃₇	—	4	2	no	—
153	C₄×C₁₃₇	—	8	4	no	—
154	C₄×C₁₃₇	—	8	2	no	—
155	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
156	C₄×C₁₃₇	#535	1	1	yes	characteristic
157	C₄×C₁₃₇	#559	1	2	yes	—
158	C₂×C₁₃₇	—	16	2	no	—
159	C₂×C₁₃₇	#2548	1	1	yes	characteristic
160	C₂×C₁₃₇	#3054	1	1	yes	characteristic
161	C₂×C₁₃₇	#3060	1	1	yes	characteristic
162	C₂×C₁₃₇	#3401	1	4	yes	—
163	#28742	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₁₃₇	#28742	1	1	yes	characteristic, direct product factor, Sylow
165	C₂×#495	C₂×C₁₃₇	1	1	yes	characteristic
166	C₂×#446	C₂×C₁₃₇	1	1	yes	characteristic, complemented
167	#2592	C₂×C₁₃₇	1	2	yes	complemented
168	#2055	C₂×C₁₃₇	1	4	yes	complemented
169	#1983	C₂×C₁₃₇	1	2	yes	complemented
170	#1972	C₂×C₁₃₇	1	2	yes	—
171	#1971	C₂×C₁₃₇	1	2	yes	complemented
172	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic, complemented
173	C₂²×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
174	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
175	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
176	C₂×#124	C₂²×C₁₃₇	1	2	yes	—
177	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
178	#495	—	2	4	no	—
179	#446	—	2	4	no	—
180	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
181	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
182	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
183	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
184	C₂×#102	C₂²×C₁₃₇	1	4	yes	—
185	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
186	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
187	#397	—	2	2	no	—
188	#393	—	2	2	no	—
189	#391	C₂²×C₁₃₇	1	4	yes	—
190	#390	C₂²×C₁₃₇	1	4	yes	—
191	#388	—	2	4	no	—
192	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
193	#340	C₂²×C₁₃₇	1	2	yes	—
194	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
195	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
196	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×Q₁₆	—	2	4	no	—
199	C₂×SD₁₆	—	2	4	no	—
200	C₂×SD₁₆	—	4	4	no	—
201	C₂×D₈	—	2	2	no	—
202	C₂×D₈	—	2	4	no	—
203	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
204	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
206	#124	—	2	4	no	—
207	#124	—	2	4	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
211	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₂²⋊C₄	—	2	4	no	—
216	C₂×C₂²⋊C₄	—	2	4	no	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
219	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
220	#106	—	2	4	no	—
221	C₄⋊C₈	—	2	4	no	—
222	C₄⋊C₈	—	2	2	no	—
223	#103	—	4	4	no	—
224	#102	—	4	4	no	—
225	#97	—	2	4	no	—
226	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
227	#95	—	2	2	no	—
228	#95	—	2	4	no	—
229	#95	—	2	4	no	—
230	C₂⁴	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
247	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
248	Q₁₆	—	4	4	no	—
249	SD₁₆	—	4	8	no	—
250	D₈	—	4	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
256	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
257	C₂×C₈	#41×C₁₃₇	1	4	yes	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	4	8	no	—
263	C₂²⋊C₄	—	4	4	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	8	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	8	2	no	—
269	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
270	C₄²	#41×C₁₃₇	1	2	yes	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	4	no	—
273	C₂³	—	4	4	no	—
274	C₂³	—	4	2	no	—
275	C₂³	—	4	2	no	—
276	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
277	Q₈	—	4	4	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	4	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	#121×C₁₃₇	1	2	yes	—
296	C₂×C₄	#121×C₁₃₇	1	2	yes	—
297	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
298	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
299	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
300	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
301	C₈	—	2	2	no	—
302	C₈	—	4	2	no	—
303	C₈	#124×C₁₃₇	1	4	yes	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	4	no	—
308	C₂²	—	8	4	no	—
309	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
310	C₂²	#550×C₁₃₇	1	2	yes	—
311	C₂²	#575×C₁₃₇	1	2	yes	—
312	C₂²	#577×C₁₃₇	1	2	yes	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	4	no	—
319	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
320	C₄	#535×C₁₃₇	1	1	yes	characteristic
321	C₄	#559×C₁₃₇	1	2	yes	—
322	C₂	—	16	2	no	—
323	C₂	#2548×C₁₃₇	1	1	yes	characteristic
324	C₂	#3054×C₁₃₇	1	1	yes	characteristic
325	C₂	#3060×C₁₃₇	1	1	yes	characteristic
326	C₂	#3401×C₁₃₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₈×C₁₃₇	C₂²
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. #2592×C₁₃₇	C₂	semidirect
4. #2055×C₁₃₇	C₂	semidirect
5. #1983×C₁₃₇	C₂	semidirect
7. #1971×C₁₃₇	C₂	semidirect
8. C₄×#106×C₁₃₇	C₂	semidirect
163. #28742	C₁₃₇	direct (N × H)
164. C₁₃₇	#28742	direct (N × H)
166. C₂×#446	C₂×C₁₃₇	semidirect
167. #2592	C₂×C₁₃₇	semidirect
168. #2055	C₂×C₁₃₇	semidirect
169. #1983	C₂×C₁₃₇	semidirect
171. #1971	C₂×C₁₃₇	semidirect
172. C₄×#106	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
C₂×#446×C₁₃₇	2
#2592×C₁₃₇	2
#2055×C₁₃₇	2
#1983×C₁₃₇	2
#1972×C₁₃₇	2
#1971×C₁₃₇	2
C₄×#106×C₁₃₇	2
#28742	137

Subgroups — C137×#28742 #46878128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28742 #46878128