Subgroups — C₁₃₇×#23639 #46873025

Order 35072.

1014 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#408×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2311×C₁₃₇	C₂	1	2	yes	maximal
4	#2076×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#2075×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1736×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1735×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#107×C₁₃₇	—	2	4	no	—
14	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
16	#408×C₁₃₇	—	2	4	no	—
17	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
19	#391×C₁₃₇	—	2	2	no	—
20	#390×C₁₃₇	—	2	2	no	—
21	#386×C₁₃₇	—	2	4	no	—
22	#380×C₁₃₇	C₂²	1	8	yes	—
23	#340×C₁₃₇	C₂²	1	2	yes	—
24	#340×C₁₃₇	C₂²	1	8	yes	—
25	#340×C₁₃₇	C₂²	1	2	yes	—
26	C₂×#41×C₁₃₇	—	2	2	no	—
27	C₂×#41×C₁₃₇	—	2	2	no	—
28	C₂×#41×C₁₃₇	—	2	4	no	—
29	C₂×#41×C₁₃₇	—	2	4	no	—
30	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₁₃₇	D₄	1	2	yes	—
32	C₂²×D₄×C₁₃₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₃₇	—	2	2	no	—
34	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
35	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₁₃₇	—	2	2	no	—
37	C₂×M₁₆×C₁₃₇	—	2	4	no	—
38	C₂²×C₈×C₁₃₇	—	2	4	no	—
39	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
40	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
43	#117×C₁₃₇	—	2	4	no	—
44	#117×C₁₃₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
50	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
52	C₂×C₄²×C₁₃₇	—	2	2	no	—
53	#107×C₁₃₇	—	2	4	no	—
54	#107×C₁₃₇	—	2	4	no	—
55	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
56	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
57	#95×C₁₃₇	—	2	8	no	—
58	C₂⁴×C₁₃₇	C₂×D₄	1	2	yes	—
59	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
60	#41×C₁₃₇	—	4	4	no	—
61	#41×C₁₃₇	—	4	4	no	—
62	#41×C₁₃₇	—	4	8	no	—
63	C₂×Q₈×C₁₃₇	—	2	2	no	—
64	C₂×Q₈×C₁₃₇	—	2	4	no	—
65	C₂×D₄×C₁₃₇	—	2	4	no	—
66	C₂×D₄×C₁₃₇	—	2	8	no	—
67	C₂×D₄×C₁₃₇	—	2	2	no	—
68	C₂×D₄×C₁₃₇	—	2	4	no	—
69	C₂²×C₄×C₁₃₇	—	2	2	no	—
70	C₂²×C₄×C₁₃₇	—	2	2	no	—
71	C₂²×C₄×C₁₃₇	—	2	4	no	—
72	C₂²×C₄×C₁₃₇	—	2	4	no	—
73	C₂²×C₄×C₁₃₇	—	2	4	no	—
74	C₂²×C₄×C₁₃₇	—	2	4	no	—
75	C₂²×C₄×C₁₃₇	—	4	4	no	—
76	C₂²×C₄×C₁₃₇	—	8	4	no	—
77	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
81	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
82	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
83	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
84	M₁₆×C₁₃₇	—	4	4	no	—
85	C₂×C₈×C₁₃₇	—	2	8	no	—
86	C₂×C₈×C₁₃₇	—	2	2	no	—
87	C₂×C₈×C₁₃₇	—	2	2	no	—
88	C₂×C₈×C₁₃₇	—	2	2	no	—
89	C₂×C₈×C₁₃₇	—	2	4	no	—
90	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₁₃₇	—	2	4	no	—
93	C₄⋊C₄×C₁₃₇	—	2	4	no	—
94	C₄⋊C₄×C₁₃₇	—	2	4	no	—
95	C₄⋊C₄×C₁₃₇	—	2	4	no	—
96	C₄⋊C₄×C₁₃₇	—	4	4	no	—
97	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
98	C₂²⋊C₄×C₁₃₇	—	8	8	no	—
99	C₄²×C₁₃₇	—	4	4	no	—
100	C₂³×C₁₃₇	—	2	2	no	—
101	C₂³×C₁₃₇	—	2	2	no	—
102	C₂³×C₁₃₇	—	2	2	no	—
103	C₂³×C₁₃₇	—	2	4	no	—
104	C₂³×C₁₃₇	—	2	4	no	—
105	C₂³×C₁₃₇	—	2	4	no	—
106	C₂³×C₁₃₇	#121	1	2	yes	—
107	C₂³×C₁₃₇	#123	1	4	yes	—
108	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₁₃₇	—	4	4	no	—
110	D₄×C₁₃₇	—	4	8	no	—
111	D₄×C₁₃₇	—	4	4	no	—
112	C₂×C₄×C₁₃₇	—	2	2	no	—
113	C₂×C₄×C₁₃₇	—	2	4	no	—
114	C₂×C₄×C₁₃₇	—	4	2	no	—
115	C₂×C₄×C₁₃₇	—	4	2	no	—
116	C₂×C₄×C₁₃₇	—	4	4	no	—
117	C₂×C₄×C₁₃₇	—	4	4	no	—
118	C₂×C₄×C₁₃₇	—	4	4	no	—
119	C₂×C₄×C₁₃₇	—	4	4	no	—
120	C₂×C₄×C₁₃₇	—	4	8	no	—
121	C₂×C₄×C₁₃₇	—	4	4	no	—
122	C₂×C₄×C₁₃₇	—	8	8	no	—
123	C₂×C₄×C₁₃₇	#121	1	2	yes	—
124	C₂×C₄×C₁₃₇	#123	1	4	yes	—
125	C₂×C₄×C₁₃₇	#123	1	4	yes	—
126	C₂×C₄×C₁₃₇	#124	1	4	yes	—
127	C₂×C₄×C₁₃₇	#126	1	8	yes	—
128	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
130	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
131	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
132	C₈×C₁₃₇	—	2	2	no	—
133	C₈×C₁₃₇	—	2	2	no	—
134	C₈×C₁₃₇	—	4	2	no	—
135	C₂²×C₁₃₇	—	2	8	no	—
136	C₂²×C₁₃₇	—	2	2	no	—
137	C₂²×C₁₃₇	—	2	2	no	—
138	C₂²×C₁₃₇	—	4	8	no	—
139	C₂²×C₁₃₇	—	4	2	no	—
140	C₂²×C₁₃₇	—	4	2	no	—
141	C₂²×C₁₃₇	—	4	2	no	—
142	C₂²×C₁₃₇	#397	1	4	yes	—
143	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
144	C₂²×C₁₃₇	C₂×#123	1	2	yes	—
145	C₂²×C₁₃₇	#576	1	2	yes	—
146	C₂²×C₁₃₇	#578	1	2	yes	—
147	C₄×C₁₃₇	—	4	2	no	—
148	C₄×C₁₃₇	—	8	4	no	—
149	C₄×C₁₃₇	—	8	4	no	—
150	C₄×C₁₃₇	#397	1	4	yes	—
151	C₄×C₁₃₇	C₂×#123	1	1	yes	characteristic
152	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
153	C₄×C₁₃₇	C₂×#126	1	2	yes	—
154	C₂×C₁₃₇	—	2	4	no	—
155	C₂×C₁₃₇	—	8	2	no	—
156	C₂×C₁₃₇	C₂×#397	1	1	yes	characteristic
157	C₂×C₁₃₇	#3058	1	1	yes	characteristic
158	C₂×C₁₃₇	#3059	1	1	yes	characteristic
159	C₂×C₁₃₇	#3098	1	4	yes	—
160	#23639	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₁₃₇	#23639	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×#107	C₂×C₁₃₇	1	1	yes	characteristic, complemented
163	C₂×#408	C₂×C₁₃₇	1	1	yes	characteristic
164	#2311	C₂×C₁₃₇	1	2	yes	—
165	#2076	C₂×C₁₃₇	1	4	yes	complemented
166	#2075	C₂×C₁₃₇	1	4	yes	complemented
167	#1736	C₂×C₁₃₇	1	2	yes	complemented
168	#1735	C₂×C₁₃₇	1	1	yes	characteristic, complemented
169	C₂²×#41	C₂²×C₁₃₇	1	1	yes	characteristic
170	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
171	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
172	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
173	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
174	C₂×#107	—	2	4	no	—
175	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
176	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
177	#408	—	2	4	no	—
178	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
179	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
180	#391	—	2	2	no	—
181	#390	—	2	2	no	—
182	#386	—	2	4	no	—
183	#380	C₂²×C₁₃₇	1	8	yes	—
184	#340	C₂²×C₁₃₇	1	2	yes	—
185	#340	C₂²×C₁₃₇	1	8	yes	—
186	#340	C₂²×C₁₃₇	1	2	yes	—
187	C₂×#41	—	2	2	no	—
188	C₂×#41	—	2	2	no	—
189	C₂×#41	—	2	4	no	—
190	C₂×#41	—	2	4	no	—
191	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
192	C₂²×D₄	D₄×C₁₃₇	1	2	yes	—
193	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
194	C₂³×C₄	—	2	2	no	—
195	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
196	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
201	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
202	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
204	#117	—	2	4	no	—
205	#117	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
210	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₂²⋊C₄	—	2	8	no	—
213	C₂×C₄²	—	2	2	no	—
214	#107	—	2	4	no	—
215	#107	—	2	4	no	—
216	C₂²⋊C₈	—	2	4	no	—
217	C₂²⋊C₈	—	2	4	no	—
218	#95	—	2	8	no	—
219	C₂⁴	C₂×D₄×C₁₃₇	1	2	yes	—
220	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
221	#41	—	4	4	no	—
222	#41	—	4	4	no	—
223	#41	—	4	8	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	8	4	no	—
238	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
241	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
242	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
243	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
244	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
245	M₁₆	—	4	4	no	—
246	C₂×C₈	—	2	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
252	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₂²⋊C₄	—	8	8	no	—
260	C₄²	—	4	4	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	2	4	no	—
266	C₂³	—	2	4	no	—
267	C₂³	#121×C₁₃₇	1	2	yes	—
268	C₂³	#123×C₁₃₇	1	4	yes	—
269	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
270	Q₈	—	4	4	no	—
271	D₄	—	4	8	no	—
272	D₄	—	4	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	8	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	8	8	no	—
284	C₂×C₄	#121×C₁₃₇	1	2	yes	—
285	C₂×C₄	#123×C₁₃₇	1	4	yes	—
286	C₂×C₄	#123×C₁₃₇	1	4	yes	—
287	C₂×C₄	#124×C₁₃₇	1	4	yes	—
288	C₂×C₄	#126×C₁₃₇	1	8	yes	—
289	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
290	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
291	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
292	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	2	2	no	—
295	C₈	—	4	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	8	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	8	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#397×C₁₃₇	1	4	yes	—
304	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
305	C₂²	C₂×#123×C₁₃₇	1	2	yes	—
306	C₂²	#576×C₁₃₇	1	2	yes	—
307	C₂²	#578×C₁₃₇	1	2	yes	—
308	C₄	—	4	2	no	—
309	C₄	—	8	4	no	—
310	C₄	—	8	4	no	—
311	C₄	#397×C₁₃₇	1	4	yes	—
312	C₄	C₂×#123×C₁₃₇	1	1	yes	characteristic
313	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
314	C₄	C₂×#126×C₁₃₇	1	2	yes	—
315	C₂	—	2	4	no	—
316	C₂	—	8	2	no	—
317	C₂	C₂×#397×C₁₃₇	1	1	yes	characteristic
318	C₂	#3058×C₁₃₇	1	1	yes	characteristic
319	C₂	#3059×C₁₃₇	1	1	yes	characteristic
320	C₂	#3098×C₁₃₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×#41×C₁₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂²×#41×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#107×C₁₃₇	C₂	semidirect
4. #2076×C₁₃₇	C₂	semidirect
5. #2075×C₁₃₇	C₂	semidirect
6. #1736×C₁₃₇	C₂	semidirect
7. #1735×C₁₃₇	C₂	semidirect
160. #23639	C₁₃₇	direct (N × H)
161. C₁₃₇	#23639	direct (N × H)
162. C₂²×#107	C₂×C₁₃₇	semidirect
165. #2076	C₂×C₁₃₇	semidirect
166. #2075	C₂×C₁₃₇	semidirect
167. #1736	C₂×C₁₃₇	semidirect
168. #1735	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#107×C₁₃₇	2
C₂×#408×C₁₃₇	2
#2311×C₁₃₇	2
#2076×C₁₃₇	2
#2075×C₁₃₇	2
#1736×C₁₃₇	2
#1735×C₁₃₇	2
#23639	137

Subgroups — C137×#23639 #46873025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#23639 #46873025