Subgroups — C₁₃₇×#28751 #46878137

Order 35072.

882 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×#488×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	#2858×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2043×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2041×C₁₃₇	C₂	1	2	yes	maximal
6	#1983×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	#1958×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×#124×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#124×C₁₃₇	C₂²	1	2	yes	—
14	#488×C₁₃₇	—	2	8	no	—
15	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
17	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
18	C₂×#102×C₁₃₇	C₂²	1	4	yes	—
19	C₂×#102×C₁₃₇	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
21	#390×C₁₃₇	—	2	2	no	—
22	#390×C₁₃₇	—	2	2	no	—
23	#389×C₁₃₇	C₂²	1	2	yes	—
24	#387×C₁₃₇	—	2	2	no	—
25	#382×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
27	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
28	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
29	#336×C₁₃₇	C₂²	1	2	yes	—
30	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
31	C₂²×D₄×C₁₃₇	—	2	2	no	—
32	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
34	C₄⋊Q₈×C₁₃₇	—	4	2	no	—
35	#127×C₁₃₇	—	4	2	no	—
36	#125×C₁₃₇	—	2	2	no	—
37	#125×C₁₃₇	D₄	1	4	yes	complemented
38	#124×C₁₃₇	—	2	4	no	—
39	#124×C₁₃₇	—	2	4	no	—
40	#124×C₁₃₇	—	2	8	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
46	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
47	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
49	C₂×C₄²×C₁₃₇	—	2	2	no	—
50	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
52	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₁₃₇	—	2	4	no	—
56	#103×C₁₃₇	—	2	8	no	—
57	#103×C₁₃₇	—	4	4	no	—
58	#102×C₁₃₇	—	2	8	no	—
59	#102×C₁₃₇	—	4	4	no	—
60	C₄×C₈×C₁₃₇	—	2	4	no	—
61	#95×C₁₃₇	—	2	2	no	—
62	#95×C₁₃₇	—	2	2	no	—
63	#95×C₁₃₇	—	2	2	no	—
64	C₂⁴×C₁₃₇	—	2	2	no	—
65	C₂×Q₈×C₁₃₇	—	2	2	no	—
66	C₂×Q₈×C₁₃₇	—	2	2	no	—
67	C₂×Q₈×C₁₃₇	—	2	4	no	—
68	C₂×Q₈×C₁₃₇	—	2	4	no	—
69	C₂×D₄×C₁₃₇	—	2	2	no	—
70	C₂×D₄×C₁₃₇	—	2	4	no	—
71	C₂×D₄×C₁₃₇	—	2	2	no	—
72	C₂×D₄×C₁₃₇	—	2	4	no	—
73	C₂×D₄×C₁₃₇	—	4	4	no	—
74	C₂×D₄×C₁₃₇	—	4	4	no	—
75	C₂×D₄×C₁₃₇	—	4	4	no	—
76	C₂²×C₄×C₁₃₇	—	2	2	no	—
77	C₂²×C₄×C₁₃₇	—	2	2	no	—
78	C₂²×C₄×C₁₃₇	—	2	2	no	—
79	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₁₃₇	—	2	4	no	—
85	C₂×C₈×C₁₃₇	—	2	4	no	—
86	C₂×C₈×C₁₃₇	—	2	4	no	—
87	C₄⋊C₄×C₁₃₇	—	2	8	no	—
88	C₄⋊C₄×C₁₃₇	—	2	2	no	—
89	C₄⋊C₄×C₁₃₇	—	2	4	no	—
90	C₄⋊C₄×C₁₃₇	—	4	4	no	—
91	C₄⋊C₄×C₁₃₇	—	4	4	no	—
92	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
93	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
94	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
95	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
96	C₄²×C₁₃₇	—	2	4	no	—
97	C₄²×C₁₃₇	—	2	2	no	—
98	C₄²×C₁₃₇	—	4	2	no	—
99	C₄²×C₁₃₇	—	4	4	no	—
100	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
101	C₄²×C₁₃₇	#41	1	2	yes	—
102	C₄²×C₁₃₇	#41	1	4	yes	—
103	C₂³×C₁₃₇	—	4	2	no	—
104	C₂³×C₁₃₇	—	4	2	no	—
105	C₂³×C₁₃₇	—	4	2	no	—
106	C₂³×C₁₃₇	—	4	4	no	—
107	C₂³×C₁₃₇	—	4	4	no	—
108	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₁₃₇	—	4	4	no	—
110	Q₈×C₁₃₇	—	4	4	no	—
111	D₄×C₁₃₇	—	4	4	no	—
112	D₄×C₁₃₇	—	4	4	no	—
113	D₄×C₁₃₇	—	8	8	no	—
114	C₂×C₄×C₁₃₇	—	2	2	no	—
115	C₂×C₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	2	2	no	—
117	C₂×C₄×C₁₃₇	—	2	2	no	—
118	C₂×C₄×C₁₃₇	—	2	2	no	—
119	C₂×C₄×C₁₃₇	—	2	2	no	—
120	C₂×C₄×C₁₃₇	—	2	4	no	—
121	C₂×C₄×C₁₃₇	—	4	2	no	—
122	C₂×C₄×C₁₃₇	—	4	2	no	—
123	C₂×C₄×C₁₃₇	—	4	2	no	—
124	C₂×C₄×C₁₃₇	—	4	2	no	—
125	C₂×C₄×C₁₃₇	—	4	2	no	—
126	C₂×C₄×C₁₃₇	—	4	2	no	—
127	C₂×C₄×C₁₃₇	—	4	2	no	—
128	C₂×C₄×C₁₃₇	—	4	2	no	—
129	C₂×C₄×C₁₃₇	#121	1	4	yes	—
130	C₂×C₄×C₁₃₇	#124	1	2	yes	—
131	C₂×C₄×C₁₃₇	#124	1	2	yes	—
132	C₂×C₄×C₁₃₇	#135	1	4	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	—	4	4	no	—
138	C₂²×C₁₃₇	—	8	2	no	—
139	C₂²×C₁₃₇	—	8	2	no	—
140	C₂²×C₁₃₇	—	8	4	no	—
141	C₂²×C₁₃₇	—	8	4	no	—
142	C₂²×C₁₃₇	—	8	2	no	—
143	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
144	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
145	C₂²×C₁₃₇	#535	1	1	yes	characteristic
146	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
147	C₂²×C₁₃₇	#575	1	2	yes	—
148	C₄×C₁₃₇	—	2	2	no	—
149	C₄×C₁₃₇	—	2	2	no	—
150	C₄×C₁₃₇	—	2	4	no	—
151	C₄×C₁₃₇	—	2	4	no	—
152	C₄×C₁₃₇	—	4	4	no	—
153	C₄×C₁₃₇	—	8	2	no	—
154	C₄×C₁₃₇	—	8	2	no	—
155	C₄×C₁₃₇	#550	1	2	yes	—
156	C₄×C₁₃₇	#559	1	2	yes	—
157	C₂×C₁₃₇	—	16	2	no	—
158	C₂×C₁₃₇	#2548	1	1	yes	characteristic
159	C₂×C₁₃₇	#3035	1	2	yes	—
160	C₂×C₁₃₇	#3131	1	1	yes	characteristic
161	C₂×C₁₃₇	#3142	1	1	yes	characteristic
162	C₂×C₁₃₇	#3211	1	2	yes	—
163	#28751	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₁₃₇	#28751	1	1	yes	characteristic, direct product factor, Sylow
165	C₂×#488	C₂×C₁₃₇	1	2	yes	complemented
166	#2858	C₂×C₁₃₇	1	1	yes	characteristic
167	C₄×#125	C₂×C₁₃₇	1	1	yes	characteristic, complemented
168	#2043	C₂×C₁₃₇	1	2	yes	complemented
169	#2041	C₂×C₁₃₇	1	2	yes	—
170	#1983	C₂×C₁₃₇	1	4	yes	complemented
171	#1958	C₂×C₁₃₇	1	2	yes	complemented
172	C₄×C₄⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
173	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
174	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
175	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
176	C₂×#124	C₂²×C₁₃₇	1	2	yes	—
177	C₂×#124	C₂²×C₁₃₇	1	2	yes	—
178	#488	—	2	8	no	—
179	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
180	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
181	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
182	C₂×#102	C₂²×C₁₃₇	1	4	yes	—
183	C₂×#102	C₂²×C₁₃₇	1	4	yes	—
184	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
185	#390	—	2	2	no	—
186	#390	—	2	2	no	—
187	#389	C₂²×C₁₃₇	1	2	yes	—
188	#387	—	2	2	no	—
189	#382	C₂²×C₁₃₇	1	1	yes	characteristic
190	C₄×C₄⋊C₄	—	2	2	no	—
191	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
192	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
193	#336	C₂²×C₁₃₇	1	2	yes	—
194	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
195	C₂²×D₄	—	2	2	no	—
196	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
197	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
198	C₄⋊Q₈	—	4	2	no	—
199	#127	—	4	2	no	—
200	#125	—	2	2	no	—
201	#125	D₄×C₁₃₇	1	4	yes	complemented
202	#124	—	2	4	no	—
203	#124	—	2	4	no	—
204	#124	—	2	8	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
210	C₂×C₂²⋊C₄	—	2	4	no	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
215	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
216	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
219	C₄⋊C₈	—	2	4	no	—
220	#103	—	2	8	no	—
221	#103	—	4	4	no	—
222	#102	—	2	8	no	—
223	#102	—	4	4	no	—
224	C₄×C₈	—	2	4	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	C₂⁴	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	2	4	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂×D₄	—	4	4	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
247	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	4	no	—
251	C₄⋊C₄	—	2	8	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
257	C₂²⋊C₄	—	4	8	no	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₂²⋊C₄	—	4	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	4	4	no	—
264	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
265	C₄²	#41×C₁₃₇	1	2	yes	—
266	C₄²	#41×C₁₃₇	1	4	yes	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	4	no	—
271	C₂³	—	4	4	no	—
272	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
273	Q₈	—	4	4	no	—
274	Q₈	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	4	no	—
277	D₄	—	8	8	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	#121×C₁₃₇	1	4	yes	—
294	C₂×C₄	#124×C₁₃₇	1	2	yes	—
295	C₂×C₄	#124×C₁₃₇	1	2	yes	—
296	C₂×C₄	#135×C₁₃₇	1	4	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	—	4	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	4	no	—
306	C₂²	—	8	4	no	—
307	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
308	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
309	C₂²	#535×C₁₃₇	1	1	yes	characteristic
310	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
311	C₂²	#575×C₁₃₇	1	2	yes	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	4	no	—
315	C₄	—	2	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	#550×C₁₃₇	1	2	yes	—
320	C₄	#559×C₁₃₇	1	2	yes	—
321	C₂	—	16	2	no	—
322	C₂	#2548×C₁₃₇	1	1	yes	characteristic
323	C₂	#3035×C₁₃₇	1	2	yes	—
324	C₂	#3131×C₁₃₇	1	1	yes	characteristic
325	C₂	#3142×C₁₃₇	1	1	yes	characteristic
326	C₂	#3211×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#127×C₁₃₇	C₂²
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×#488×C₁₃₇	C₂	semidirect
3. C₄×#125×C₁₃₇	C₂	semidirect
4. #2043×C₁₃₇	C₂	semidirect
6. #1983×C₁₃₇	C₂	semidirect
7. #1958×C₁₃₇	C₂	semidirect
8. C₄×C₄⋊C₈×C₁₃₇	C₂	semidirect
37. #125×C₁₃₇	D₄	semidirect
163. #28751	C₁₃₇	direct (N × H)
164. C₁₃₇	#28751	direct (N × H)
165. C₂×#488	C₂×C₁₃₇	semidirect
167. C₄×#125	C₂×C₁₃₇	semidirect
168. #2043	C₂×C₁₃₇	semidirect
170. #1983	C₂×C₁₃₇	semidirect
171. #1958	C₂×C₁₃₇	semidirect
172. C₄×C₄⋊C₈	C₂×C₁₃₇	semidirect
201. #125	D₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#488×C₁₃₇	2
#2858×C₁₃₇	2
C₄×#125×C₁₃₇	2
#2043×C₁₃₇	2
#2041×C₁₃₇	2
#1983×C₁₃₇	2
#1958×C₁₃₇	2
C₄×C₄⋊C₈×C₁₃₇	2
#28751	137

Subgroups — C137×#28751 #46878137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28751 #46878137