Subgroups — C₁₃₇×#28764 #46878150

Order 35072.

778 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#3139×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#3138×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2960×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2046×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1977×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	#1966×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1965×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₄×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
11	C₂²×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
12	#533×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
15	#492×C₁₃₇	—	2	2	no	—
16	#492×C₁₃₇	—	2	2	no	—
17	#491×C₁₃₇	—	2	2	no	—
18	#490×C₁₃₇	—	2	2	no	—
19	#487×C₁₃₇	—	2	2	no	—
20	#486×C₁₃₇	—	2	2	no	—
21	#446×C₁₃₇	—	2	2	no	—
22	C₈×Q₈×C₁₃₇	—	2	2	no	—
23	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
24	#424×C₁₃₇	C₂²	1	2	yes	—
25	#423×C₁₃₇	C₂²	1	1	yes	characteristic
26	#419×C₁₃₇	C₂²	1	4	yes	—
27	#418×C₁₃₇	C₂²	1	4	yes	—
28	C₂×#101×C₁₃₇	C₂²	1	4	yes	—
29	C₂×#100×C₁₃₇	C₂²	1	4	yes	—
30	#405×C₁₃₇	C₂²	1	2	yes	—
31	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
32	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
34	#341×C₁₃₇	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
36	C₂²×D₄×C₁₃₇	—	2	2	no	—
37	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
38	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
39	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
40	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
41	C₂×SD₁₆×C₁₃₇	—	2	8	no	—
42	C₂×SD₁₆×C₁₃₇	—	4	4	no	—
43	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
44	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
45	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
49	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
50	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
51	C₄⋊Q₈×C₁₃₇	D₄	1	2	yes	—
52	#127×C₁₃₇	—	2	1	no	—
53	#127×C₁₃₇	—	2	1	no	—
54	#127×C₁₃₇	—	2	2	no	—
55	#125×C₁₃₇	D₄	1	2	yes	—
56	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
57	C₄×Q₈×C₁₃₇	—	2	2	no	—
58	#117×C₁₃₇	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
60	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
61	#108×C₁₃₇	—	2	4	no	—
62	C₄⋊C₈×C₁₃₇	—	2	4	no	—
63	C₄⋊C₈×C₁₃₇	—	2	2	no	—
64	#103×C₁₃₇	—	2	8	no	—
65	#102×C₁₃₇	—	2	8	no	—
66	#101×C₁₃₇	—	4	4	no	—
67	#100×C₁₃₇	—	4	4	no	—
68	#97×C₁₃₇	—	2	2	no	—
69	#97×C₁₃₇	—	2	1	no	—
70	#97×C₁₃₇	—	2	1	no	—
71	C₄×C₈×C₁₃₇	—	2	2	no	—
72	C₄×C₈×C₁₃₇	—	2	1	no	—
73	C₄×C₈×C₁₃₇	—	2	1	no	—
74	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
75	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
76	C₂⁴×C₁₃₇	—	2	2	no	—
77	C₂×Q₈×C₁₃₇	—	2	2	no	—
78	C₂×Q₈×C₁₃₇	—	2	2	no	—
79	C₂×Q₈×C₁₃₇	—	2	2	no	—
80	C₂×Q₈×C₁₃₇	—	2	4	no	—
81	C₂×Q₈×C₁₃₇	—	2	4	no	—
82	C₂×D₄×C₁₃₇	—	2	2	no	—
83	C₂×D₄×C₁₃₇	—	2	2	no	—
84	C₂×D₄×C₁₃₇	—	2	4	no	—
85	C₂×D₄×C₁₃₇	—	2	4	no	—
86	C₂×D₄×C₁₃₇	—	4	2	no	—
87	C₂×D₄×C₁₃₇	—	4	2	no	—
88	C₂×D₄×C₁₃₇	—	8	4	no	—
89	C₂²×C₄×C₁₃₇	—	2	2	no	—
90	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
91	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
92	SD₁₆×C₁₃₇	—	4	8	no	—
93	SD₁₆×C₁₃₇	—	4	4	no	—
94	SD₁₆×C₁₃₇	—	4	4	no	—
95	M₁₆×C₁₃₇	—	2	2	no	—
96	M₁₆×C₁₃₇	—	2	2	no	—
97	M₁₆×C₁₃₇	—	2	2	no	—
98	M₁₆×C₁₃₇	—	2	4	no	—
99	M₁₆×C₁₃₇	—	2	2	no	—
100	C₂×C₈×C₁₃₇	—	2	2	no	—
101	C₂×C₈×C₁₃₇	—	2	2	no	—
102	C₂×C₈×C₁₃₇	—	2	2	no	—
103	C₂×C₈×C₁₃₇	—	2	2	no	—
104	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃₇	#41	1	4	yes	—
108	C₄⋊C₄×C₁₃₇	—	2	4	no	—
109	C₄⋊C₄×C₁₃₇	—	2	2	no	—
110	C₄⋊C₄×C₁₃₇	—	4	2	no	—
111	C₄⋊C₄×C₁₃₇	—	4	2	no	—
112	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
113	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
114	C₄²×C₁₃₇	—	2	2	no	—
115	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
117	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
118	C₂³×C₁₃₇	—	4	2	no	—
119	C₂³×C₁₃₇	—	4	4	no	—
120	C₂³×C₁₃₇	—	8	4	no	—
121	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
122	Q₈×C₁₃₇	—	4	4	no	—
123	Q₈×C₁₃₇	—	4	2	no	—
124	Q₈×C₁₃₇	—	4	2	no	—
125	Q₈×C₁₃₇	—	4	4	no	—
126	D₄×C₁₃₇	—	4	2	no	—
127	D₄×C₁₃₇	—	4	2	no	—
128	D₄×C₁₃₇	—	4	4	no	—
129	D₄×C₁₃₇	—	8	4	no	—
130	D₄×C₁₃₇	—	8	4	no	—
131	C₂×C₄×C₁₃₇	—	2	4	no	—
132	C₂×C₄×C₁₃₇	—	2	4	no	—
133	C₂×C₄×C₁₃₇	—	4	2	no	—
134	C₂×C₄×C₁₃₇	—	4	4	no	—
135	C₂×C₄×C₁₃₇	#121	1	2	yes	—
136	C₂×C₄×C₁₃₇	#121	1	2	yes	—
137	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
141	C₈×C₁₃₇	—	2	2	no	—
142	C₈×C₁₃₇	—	4	2	no	—
143	C₈×C₁₃₇	—	4	2	no	—
144	C₈×C₁₃₇	—	4	2	no	—
145	C₈×C₁₃₇	#124	1	2	yes	—
146	C₈×C₁₃₇	#124	1	2	yes	—
147	C₂²×C₁₃₇	—	2	2	no	—
148	C₂²×C₁₃₇	—	8	2	no	—
149	C₂²×C₁₃₇	—	8	4	no	—
150	C₂²×C₁₃₇	—	16	4	no	—
151	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
152	C₂²×C₁₃₇	#559	1	2	yes	—
153	C₄×C₁₃₇	—	2	2	no	—
154	C₄×C₁₃₇	—	4	4	no	—
155	C₄×C₁₃₇	—	8	2	no	—
156	C₄×C₁₃₇	#471	1	4	yes	—
157	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
158	C₄×C₁₃₇	#535	1	1	yes	characteristic
159	C₄×C₁₃₇	#550	1	2	yes	—
160	C₂×C₁₃₇	—	2	2	no	—
161	C₂×C₁₃₇	—	16	2	no	—
162	C₂×C₁₃₇	#2548	1	1	yes	characteristic
163	C₂×C₁₃₇	C₂×#471	1	2	yes	—
164	#28764	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₁₃₇	#28764	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×#492	C₂×C₁₃₇	1	1	yes	characteristic
167	#3139	C₂×C₁₃₇	1	1	yes	characteristic, complemented
168	#3138	C₂×C₁₃₇	1	1	yes	characteristic
169	#2960	C₂×C₁₃₇	1	1	yes	characteristic, complemented
170	#2046	C₂×C₁₃₇	1	4	yes	complemented
171	#1977	C₂×C₁₃₇	1	4	yes	complemented
172	#1966	C₂×C₁₃₇	1	1	yes	characteristic, complemented
173	#1965	C₂×C₁₃₇	1	1	yes	characteristic
174	C₄×#108	C₂×C₁₃₇	1	1	yes	characteristic, complemented
175	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
176	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
177	#533	C₂²×C₁₃₇	1	1	yes	characteristic
178	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
179	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
180	#492	—	2	2	no	—
181	#492	—	2	2	no	—
182	#491	—	2	2	no	—
183	#490	—	2	2	no	—
184	#487	—	2	2	no	—
185	#486	—	2	2	no	—
186	#446	—	2	2	no	—
187	C₈×Q₈	—	2	2	no	—
188	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
189	#424	C₂²×C₁₃₇	1	2	yes	—
190	#423	C₂²×C₁₃₇	1	1	yes	characteristic
191	#419	C₂²×C₁₃₇	1	4	yes	—
192	#418	C₂²×C₁₃₇	1	4	yes	—
193	C₂×#101	C₂²×C₁₃₇	1	4	yes	—
194	C₂×#100	C₂²×C₁₃₇	1	4	yes	—
195	#405	C₂²×C₁₃₇	1	2	yes	—
196	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
197	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
198	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
199	#341	C₂²×C₁₃₇	1	2	yes	—
200	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
201	C₂²×D₄	—	2	2	no	—
202	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
203	C₂×SD₁₆	—	2	4	no	—
204	C₂×SD₁₆	—	2	2	no	—
205	C₂×SD₁₆	—	2	2	no	—
206	C₂×SD₁₆	—	2	8	no	—
207	C₂×SD₁₆	—	4	4	no	—
208	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
209	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
210	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
211	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	D₄×C₁₃₇	1	2	yes	—
217	#127	—	2	1	no	—
218	#127	—	2	1	no	—
219	#127	—	2	2	no	—
220	#125	D₄×C₁₃₇	1	2	yes	—
221	C₂²⋊Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	#117	C₂³×C₁₃₇	1	2	yes	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
226	#108	—	2	4	no	—
227	C₄⋊C₈	—	2	4	no	—
228	C₄⋊C₈	—	2	2	no	—
229	#103	—	2	8	no	—
230	#102	—	2	8	no	—
231	#101	—	4	4	no	—
232	#100	—	4	4	no	—
233	#97	—	2	2	no	—
234	#97	—	2	1	no	—
235	#97	—	2	1	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
240	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
241	C₂⁴	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	4	no	—
250	C₂×D₄	—	2	4	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	8	4	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
256	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
257	SD₁₆	—	4	8	no	—
258	SD₁₆	—	4	4	no	—
259	SD₁₆	—	4	4	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	2	4	no	—
264	M₁₆	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
270	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
272	C₂×C₈	#41×C₁₃₇	1	4	yes	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
282	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
283	C₂³	—	4	2	no	—
284	C₂³	—	4	4	no	—
285	C₂³	—	8	4	no	—
286	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
287	Q₈	—	4	4	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	8	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	#121×C₁₃₇	1	2	yes	—
301	C₂×C₄	#121×C₁₃₇	1	2	yes	—
302	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
304	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
305	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
306	C₈	—	2	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	2	no	—
310	C₈	#124×C₁₃₇	1	2	yes	—
311	C₈	#124×C₁₃₇	1	2	yes	—
312	C₂²	—	2	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	4	no	—
315	C₂²	—	16	4	no	—
316	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
317	C₂²	#559×C₁₃₇	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	8	2	no	—
321	C₄	#471×C₁₃₇	1	4	yes	—
322	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
323	C₄	#535×C₁₃₇	1	1	yes	characteristic
324	C₄	#550×C₁₃₇	1	2	yes	—
325	C₂	—	2	2	no	—
326	C₂	—	16	2	no	—
327	C₂	#2548×C₁₃₇	1	1	yes	characteristic
328	C₂	C₂×#471×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#127×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. #3139×C₁₃₇	C₂	semidirect
4. #2960×C₁₃₇	C₂	semidirect
5. #2046×C₁₃₇	C₂	semidirect
6. #1977×C₁₃₇	C₂	semidirect
7. #1966×C₁₃₇	C₂	semidirect
9. C₄×#108×C₁₃₇	C₂	semidirect
164. #28764	C₁₃₇	direct (N × H)
165. C₁₃₇	#28764	direct (N × H)
167. #3139	C₂×C₁₃₇	semidirect
169. #2960	C₂×C₁₃₇	semidirect
170. #2046	C₂×C₁₃₇	semidirect
171. #1977	C₂×C₁₃₇	semidirect
172. #1966	C₂×C₁₃₇	semidirect
174. 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×#492×C₁₃₇	2
#3139×C₁₃₇	2
#3138×C₁₃₇	2
#2960×C₁₃₇	2
#2046×C₁₃₇	2
#1977×C₁₃₇	2
#1966×C₁₃₇	2
#1965×C₁₃₇	2
C₄×#108×C₁₃₇	2
#28764	137

Subgroups — C137×#28764 #46878150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28764 #46878150