Subgroups — C₁₃₇×#23842 #46873228

Order 35072.

662 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2331×C₁₃₇	C₂	1	2	yes	maximal
2	#2085×C₁₃₇	C₂	1	2	yes	maximal
3	#2085×C₁₃₇	C₂	1	2	yes	maximal
4	#2058×C₁₃₇	C₂	1	2	yes	maximal
5	#2057×C₁₃₇	C₂	1	2	yes	maximal
6	#1776×C₁₃₇	C₂	1	2	yes	maximal
7	C₄×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#1746×C₁₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	#391×C₁₃₇	—	2	2	no	—
18	#391×C₁₃₇	—	2	2	no	—
19	#389×C₁₃₇	C₂²	1	2	yes	—
20	#389×C₁₃₇	C₂²	1	2	yes	—
21	#387×C₁₃₇	—	2	2	no	—
22	#387×C₁₃₇	—	2	2	no	—
23	#382×C₁₃₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
26	#376×C₁₃₇	C₂²	1	1	yes	characteristic
27	#340×C₁₃₇	C₂²	1	1	yes	characteristic
28	#340×C₁₃₇	C₂²	1	1	yes	characteristic
29	#340×C₁₃₇	C₂²	1	1	yes	characteristic
30	#340×C₁₃₇	C₂²	1	1	yes	characteristic
31	#340×C₁₃₇	C₂²	1	2	yes	—
32	#340×C₁₃₇	C₂²	1	2	yes	—
33	#336×C₁₃₇	C₂²	1	2	yes	—
34	#336×C₁₃₇	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃₇	—	2	2	no	—
36	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
40	C₄⋊Q₈×C₁₃₇	—	2	4	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×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₁₃₇	—	2	2	no	—
49	C₂×C₄²×C₁₃₇	—	2	2	no	—
50	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
52	#107×C₁₃₇	—	2	4	no	—
53	#107×C₁₃₇	—	2	4	no	—
54	#106×C₁₃₇	—	2	4	no	—
55	#106×C₁₃₇	—	2	4	no	—
56	#97×C₁₃₇	Q₈	1	4	yes	complemented
57	C₄×C₈×C₁₃₇	Q₈	1	4	yes	complemented
58	#95×C₁₃₇	—	2	2	no	—
59	#95×C₁₃₇	—	2	2	no	—
60	#95×C₁₃₇	—	2	2	no	—
61	#95×C₁₃₇	—	2	2	no	—
62	#95×C₁₃₇	—	2	2	no	—
63	#95×C₁₃₇	—	2	2	no	—
64	#95×C₁₃₇	D₄	1	1	yes	characteristic
65	#95×C₁₃₇	D₄	1	2	yes	—
66	#95×C₁₃₇	D₄	1	1	yes	characteristic
67	C₂×Q₈×C₁₃₇	—	2	8	no	—
68	C₂×Q₈×C₁₃₇	—	8	2	no	—
69	C₂×Q₈×C₁₃₇	—	8	2	no	—
70	C₂²×C₄×C₁₃₇	—	2	2	no	—
71	C₂²×C₄×C₁₃₇	—	2	2	no	—
72	C₂²×C₄×C₁₃₇	—	2	2	no	—
73	C₂²×C₄×C₁₃₇	—	2	2	no	—
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₁₃₇	—	2	2	no	—
80	C₂×C₈×C₁₃₇	—	2	2	no	—
81	C₂×C₈×C₁₃₇	—	2	2	no	—
82	C₂×C₈×C₁₃₇	—	2	2	no	—
83	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
86	C₄⋊C₄×C₁₃₇	—	2	8	no	—
87	C₄⋊C₄×C₁₃₇	—	2	2	no	—
88	C₄⋊C₄×C₁₃₇	—	2	2	no	—
89	C₄⋊C₄×C₁₃₇	—	2	2	no	—
90	C₄⋊C₄×C₁₃₇	—	2	2	no	—
91	C₄⋊C₄×C₁₃₇	—	2	2	no	—
92	C₄⋊C₄×C₁₃₇	—	2	2	no	—
93	C₄⋊C₄×C₁₃₇	—	8	2	no	—
94	C₄⋊C₄×C₁₃₇	—	8	2	no	—
95	C₄⋊C₄×C₁₃₇	#41	1	4	yes	—
96	C₄⋊C₄×C₁₃₇	#41	1	4	yes	—
97	C₄²×C₁₃₇	—	2	2	no	—
98	C₄²×C₁₃₇	—	2	2	no	—
99	C₄²×C₁₃₇	—	8	2	no	—
100	C₄²×C₁₃₇	—	8	2	no	—
101	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
102	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
103	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₁₃₇	—	8	8	no	—
105	C₂×C₄×C₁₃₇	—	2	1	no	—
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	2	1	no	—
108	C₂×C₄×C₁₃₇	—	2	2	no	—
109	C₂×C₄×C₁₃₇	—	2	2	no	—
110	C₂×C₄×C₁₃₇	—	2	4	no	—
111	C₂×C₄×C₁₃₇	—	2	4	no	—
112	C₂×C₄×C₁₃₇	—	2	2	no	—
113	C₂×C₄×C₁₃₇	—	2	2	no	—
114	C₂×C₄×C₁₃₇	—	2	2	no	—
115	C₂×C₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	4	2	no	—
117	C₂×C₄×C₁₃₇	—	4	2	no	—
118	C₂×C₄×C₁₃₇	—	4	2	no	—
119	C₂×C₄×C₁₃₇	—	4	2	no	—
120	C₂×C₄×C₁₃₇	—	4	2	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	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
131	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
132	C₈×C₁₃₇	—	2	2	no	—
133	C₈×C₁₃₇	—	2	4	no	—
134	C₈×C₁₃₇	—	2	2	no	—
135	C₂²×C₁₃₇	#525	1	2	yes	—
136	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
137	C₂²×C₁₃₇	#576	1	1	yes	characteristic
138	C₂²×C₁₃₇	#577	1	2	yes	—
139	C₂²×C₁₃₇	#578	1	1	yes	characteristic
140	C₄×C₁₃₇	—	2	2	no	—
141	C₄×C₁₃₇	—	2	2	no	—
142	C₄×C₁₃₇	—	4	2	no	—
143	C₄×C₁₃₇	—	4	4	no	—
144	C₄×C₁₃₇	—	4	4	no	—
145	C₄×C₁₃₇	—	4	2	no	—
146	C₄×C₁₃₇	—	8	2	no	—
147	C₄×C₁₃₇	—	8	2	no	—
148	C₄×C₁₃₇	#533	1	1	yes	characteristic
149	C₄×C₁₃₇	#533	1	2	yes	—
150	C₄×C₁₃₇	#533	1	1	yes	characteristic
151	C₂×C₁₃₇	#2399	1	1	yes	characteristic
152	C₂×C₁₃₇	#3118	1	2	yes	—
153	C₂×C₁₃₇	#3121	1	2	yes	—
154	C₂×C₁₃₇	#3163	1	1	yes	characteristic
155	C₂×C₁₃₇	#3164	1	1	yes	characteristic
156	#23842	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₁₃₇	#23842	1	1	yes	characteristic, direct product factor, Sylow
158	#2331	C₂×C₁₃₇	1	2	yes	—
159	#2085	C₂×C₁₃₇	1	2	yes	—
160	#2085	C₂×C₁₃₇	1	2	yes	—
161	#2058	C₂×C₁₃₇	1	2	yes	—
162	#2057	C₂×C₁₃₇	1	2	yes	—
163	#1776	C₂×C₁₃₇	1	2	yes	—
164	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic
165	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic
166	#1746	C₂×C₁₃₇	1	1	yes	characteristic
167	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
168	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
169	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
170	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
171	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
172	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
174	#391	—	2	2	no	—
175	#391	—	2	2	no	—
176	#389	C₂²×C₁₃₇	1	2	yes	—
177	#389	C₂²×C₁₃₇	1	2	yes	—
178	#387	—	2	2	no	—
179	#387	—	2	2	no	—
180	#382	C₂²×C₁₃₇	1	2	yes	—
181	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
182	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
183	#376	C₂²×C₁₃₇	1	1	yes	characteristic
184	#340	C₂²×C₁₃₇	1	1	yes	characteristic
185	#340	C₂²×C₁₃₇	1	2	yes	—
186	#340	C₂²×C₁₃₇	1	1	yes	characteristic
187	#340	C₂²×C₁₃₇	1	1	yes	characteristic
188	#340	C₂²×C₁₃₇	1	1	yes	characteristic
189	#340	C₂²×C₁₃₇	1	2	yes	—
190	#336	C₂²×C₁₃₇	1	2	yes	—
191	#336	C₂²×C₁₃₇	1	2	yes	—
192	C₂²×Q₈	—	2	2	no	—
193	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
196	C₄⋊Q₈	—	2	4	no	—
197	C₄⋊Q₈	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
208	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
209	#107	—	2	4	no	—
210	#107	—	2	4	no	—
211	#106	—	2	4	no	—
212	#106	—	2	4	no	—
213	#97	Q₈×C₁₃₇	1	4	yes	complemented
214	C₄×C₈	Q₈×C₁₃₇	1	4	yes	complemented
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	D₄×C₁₃₇	1	1	yes	characteristic
222	#95	D₄×C₁₃₇	1	2	yes	—
223	#95	D₄×C₁₃₇	1	1	yes	characteristic
224	C₂×Q₈	—	2	8	no	—
225	C₂×Q₈	—	8	2	no	—
226	C₂×Q₈	—	8	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
241	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
242	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	8	2	no	—
251	C₄⋊C₄	—	8	2	no	—
252	C₄⋊C₄	#41×C₁₃₇	1	4	yes	—
253	C₄⋊C₄	#41×C₁₃₇	1	4	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	8	2	no	—
257	C₄²	—	8	2	no	—
258	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
259	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
260	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
261	Q₈	—	8	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
284	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
285	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
286	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
287	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
288	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
289	C₈	—	2	2	no	—
290	C₈	—	2	4	no	—
291	C₈	—	2	2	no	—
292	C₂²	#525×C₁₃₇	1	2	yes	—
293	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
294	C₂²	#576×C₁₃₇	1	1	yes	characteristic
295	C₂²	#577×C₁₃₇	1	2	yes	—
296	C₂²	#578×C₁₃₇	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	#533×C₁₃₇	1	1	yes	characteristic
306	C₄	#533×C₁₃₇	1	2	yes	—
307	C₄	#533×C₁₃₇	1	1	yes	characteristic
308	C₂	#2399×C₁₃₇	1	1	yes	characteristic
309	C₂	#3118×C₁₃₇	1	2	yes	—
310	C₂	#3121×C₁₃₇	1	2	yes	—
311	C₂	#3163×C₁₃₇	1	1	yes	characteristic
312	C₂	#3164×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ #376×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
56. #97×C₁₃₇	Q₈	semidirect
57. C₄×C₈×C₁₃₇	Q₈	semidirect
156. #23842	C₁₃₇	direct (N × H)
157. C₁₃₇	#23842	direct (N × H)
213. #97	Q₈×C₁₃₇	semidirect
214. C₄×C₈	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2331×C₁₃₇	2
#2085×C₁₃₇	2
#2058×C₁₃₇	2
#2057×C₁₃₇	2
#1776×C₁₃₇	2
C₄×#107×C₁₃₇	2
C₄×#106×C₁₃₇	2
#1746×C₁₃₇	2
#23842	137

Subgroups — C137×#23842 #46873228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#23842 #46873228