Subgroups — C₁₃₇×#8194 #46857580

Order 35072.

516 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2965×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₄×#104×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1476×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1451×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#1278×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1276×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₁₃₇	C₂²	1	1	yes	characteristic
9	#446×C₁₃₇	C₄	1	2	yes	complemented
10	#436×C₁₃₇	C₄	1	2	yes	complemented
11	#435×C₁₃₇	—	2	2	no	—
12	#431×C₁₃₇	—	2	1	no	—
13	#431×C₁₃₇	C₂²	1	1	yes	characteristic
14	#431×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#104×C₁₃₇	C₄	1	2	yes	—
16	C₄×M₁₆×C₁₃₇	—	2	1	no	—
17	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₇	—	2	2	no	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
22	#339×C₁₃₇	C₄	1	2	yes	—
23	Q₈⋊C₈×C₁₃₇	C₄	1	2	yes	complemented
24	D₄⋊C₈×C₁₃₇	C₄	1	2	yes	complemented
25	#322×C₁₃₇	—	2	1	no	—
26	#322×C₁₃₇	—	2	1	no	—
27	C₂×#41×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₁₃₇	—	2	2	no	—
29	C₂×M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃₇	—	2	1	no	—
31	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
32	C₄×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₄×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₄×D₄×C₁₃₇	—	2	1	no	—
35	C₄×D₄×C₁₃₇	—	2	1	no	—
36	C₄×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	#117×C₁₃₇	—	2	1	no	—
39	#117×C₁₃₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₇	—	2	1	no	—
41	C₂×C₄²×C₁₃₇	—	2	1	no	—
42	C₂×C₄²×C₁₃₇	—	2	1	no	—
43	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₁₃₇	—	2	2	no	—
46	C₄⋊C₈×C₁₃₇	C₂×C₄	1	2	yes	—
47	C₄⋊C₈×C₁₃₇	C₂×C₄	1	2	yes	—
48	#104×C₁₃₇	—	2	4	no	—
49	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
50	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
51	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
52	#97×C₁₃₇	—	2	2	no	—
53	#97×C₁₃₇	—	2	1	no	—
54	C₄×C₈×C₁₃₇	—	2	8	no	—
55	C₄×C₈×C₁₃₇	—	2	1	no	—
56	C₄×C₈×C₁₃₇	—	2	1	no	—
57	C₄×C₈×C₁₃₇	—	2	1	no	—
58	C₄×C₈×C₁₃₇	—	2	1	no	—
59	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
63	#41×C₁₃₇	—	2	2	no	—
64	#41×C₁₃₇	—	2	2	no	—
65	C₂×Q₈×C₁₃₇	C₄²	1	1	yes	characteristic
66	C₂×D₄×C₁₃₇	—	2	1	no	—
67	C₂×D₄×C₁₃₇	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₇	—	2	2	no	—
69	C₂²×C₄×C₁₃₇	—	2	1	no	—
70	C₂²×C₄×C₁₃₇	—	2	1	no	—
71	C₂²×C₄×C₁₃₇	—	2	1	no	—
72	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
75	M₁₆×C₁₃₇	—	2	2	no	—
76	M₁₆×C₁₃₇	—	2	2	no	—
77	M₁₆×C₁₃₇	—	4	4	no	—
78	C₂×C₈×C₁₃₇	—	2	4	no	—
79	C₂×C₈×C₁₃₇	—	2	2	no	—
80	C₂×C₈×C₁₃₇	—	2	4	no	—
81	C₂×C₈×C₁₃₇	—	2	2	no	—
82	C₂×C₈×C₁₃₇	—	2	2	no	—
83	C₂×C₈×C₁₃₇	—	2	2	no	—
84	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
86	C₄⋊C₄×C₁₃₇	—	2	1	no	—
87	C₄⋊C₄×C₁₃₇	—	2	1	no	—
88	C₄⋊C₄×C₁₃₇	C₄²	1	1	yes	characteristic
89	C₄⋊C₄×C₁₃₇	C₄²	1	1	yes	characteristic
90	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
91	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
92	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
93	C₄²×C₁₃₇	—	2	4	no	—
94	C₄²×C₁₃₇	—	2	4	no	—
95	C₄²×C₁₃₇	—	2	1	no	—
96	C₄²×C₁₃₇	—	2	1	no	—
97	C₄²×C₁₃₇	—	2	1	no	—
98	C₄²×C₁₃₇	—	2	2	no	—
99	C₄²×C₁₃₇	M₁₆	1	4	yes	complemented
100	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃₇	#41	1	1	yes	characteristic
104	C₂³×C₁₃₇	—	2	1	no	—
105	C₂³×C₁₃₇	C₄×D₄	1	1	yes	characteristic
106	Q₈×C₁₃₇	—	2	1	no	—
107	Q₈×C₁₃₇	—	2	1	no	—
108	D₄×C₁₃₇	—	2	1	no	—
109	D₄×C₁₃₇	—	2	1	no	—
110	D₄×C₁₃₇	—	4	2	no	—
111	C₂×C₄×C₁₃₇	—	2	4	no	—
112	C₂×C₄×C₁₃₇	—	2	4	no	—
113	C₂×C₄×C₁₃₇	—	2	4	no	—
114	C₂×C₄×C₁₃₇	—	2	2	no	—
115	C₂×C₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	2	1	no	—
117	C₂×C₄×C₁₃₇	—	2	1	no	—
118	C₂×C₄×C₁₃₇	—	2	1	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	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₁₃₇	#131	1	2	yes	—
130	C₈×C₁₃₇	—	2	4	no	—
131	C₈×C₁₃₇	—	2	2	no	—
132	C₈×C₁₃₇	—	4	2	no	—
133	C₈×C₁₃₇	—	4	2	no	—
134	C₈×C₁₃₇	—	4	2	no	—
135	C₈×C₁₃₇	#104	1	4	yes	complemented
136	C₂²×C₁₃₇	—	2	1	no	—
137	C₂²×C₁₃₇	—	2	1	no	—
138	C₂²×C₁₃₇	—	4	1	no	—
139	C₂²×C₁₃₇	—	4	1	no	—
140	C₂²×C₁₃₇	—	4	1	no	—
141	C₂²×C₁₃₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₁₃₇	#435	1	2	yes	—
143	C₄×C₁₃₇	—	2	8	no	—
144	C₄×C₁₃₇	—	2	1	no	—
145	C₄×C₁₃₇	—	2	1	no	—
146	C₄×C₁₃₇	—	4	1	no	—
147	C₄×C₁₃₇	—	4	1	no	—
148	C₄×C₁₃₇	—	4	1	no	—
149	C₄×C₁₃₇	C₄×M₁₆	1	1	yes	characteristic
150	C₄×C₁₃₇	#405	1	1	yes	characteristic
151	C₄×C₁₃₇	C₂×#104	1	2	yes	—
152	C₄×C₁₃₇	#436	1	2	yes	—
153	C₄×C₁₃₇	#443	1	2	yes	—
154	C₂×C₁₃₇	—	2	2	no	—
155	C₂×C₁₃₇	—	8	1	no	—
156	C₂×C₁₃₇	#1753	1	1	yes	characteristic
157	C₂×C₁₃₇	C₄×#104	1	1	yes	characteristic
158	C₂×C₁₃₇	#1764	1	1	yes	characteristic
159	#8194	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₁₃₇	#8194	1	1	yes	characteristic, direct product factor, Sylow
161	#2965	C₂×C₁₃₇	1	1	yes	characteristic
162	C₄×#104	C₂×C₁₃₇	1	1	yes	characteristic
163	C₄²×C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
164	#1476	C₂×C₁₃₇	1	1	yes	characteristic
165	#1451	C₂×C₁₃₇	1	1	yes	characteristic, complemented
166	#1278	C₂×C₁₃₇	1	1	yes	characteristic, complemented
167	#1276	C₂×C₁₃₇	1	1	yes	characteristic, complemented
168	C₄×#41	C₂²×C₁₃₇	1	1	yes	characteristic
169	#446	C₄×C₁₃₇	1	2	yes	complemented
170	#436	C₄×C₁₃₇	1	2	yes	complemented
171	#435	—	2	2	no	—
172	#431	—	2	1	no	—
173	#431	C₂²×C₁₃₇	1	1	yes	characteristic
174	#431	C₂²×C₁₃₇	1	1	yes	characteristic
175	C₂×#104	C₄×C₁₃₇	1	2	yes	—
176	C₄×M₁₆	—	2	1	no	—
177	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
178	C₂×C₄×C₈	—	2	2	no	—
179	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
181	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
182	#339	C₄×C₁₃₇	1	2	yes	—
183	Q₈⋊C₈	C₄×C₁₃₇	1	2	yes	complemented
184	D₄⋊C₈	C₄×C₁₃₇	1	2	yes	complemented
185	#322	—	2	1	no	—
186	#322	—	2	1	no	—
187	C₂×#41	C₂×C₄×C₁₃₇	1	1	yes	characteristic
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
192	C₄×Q₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
193	C₄×Q₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	—	2	1	no	—
196	C₄×D₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
197	C₄×D₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
198	#117	—	2	1	no	—
199	#117	C₂×C₄×C₁₃₇	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
204	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	C₂×C₄×C₁₃₇	1	2	yes	—
207	C₄⋊C₈	C₂×C₄×C₁₃₇	1	2	yes	—
208	#104	—	2	4	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	2	2	no	—
212	#97	—	2	2	no	—
213	#97	—	2	1	no	—
214	C₄×C₈	—	2	8	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
220	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
221	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
222	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
223	#41	—	2	2	no	—
224	#41	—	2	2	no	—
225	C₂×Q₈	C₄²×C₁₃₇	1	1	yes	characteristic
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	C₄²×C₁₃₇	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
234	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	4	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
245	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	C₄²×C₁₃₇	1	1	yes	characteristic
249	C₄⋊C₄	C₄²×C₁₃₇	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	M₁₆×C₁₃₇	1	4	yes	complemented
260	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
261	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
263	C₄²	#41×C₁₃₇	1	1	yes	characteristic
264	C₂³	—	2	1	no	—
265	C₂³	C₄×D₄×C₁₃₇	1	1	yes	characteristic
266	Q₈	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	D₄	—	2	1	no	—
269	D₄	—	2	1	no	—
270	D₄	—	4	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
284	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
285	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
286	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
287	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
288	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
289	C₂×C₄	#131×C₁₃₇	1	2	yes	—
290	C₈	—	2	4	no	—
291	C₈	—	2	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₈	#104×C₁₃₇	1	4	yes	complemented
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₄×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
302	C₂²	#435×C₁₃₇	1	2	yes	—
303	C₄	—	2	8	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	C₄×M₁₆×C₁₃₇	1	1	yes	characteristic
310	C₄	#405×C₁₃₇	1	1	yes	characteristic
311	C₄	C₂×#104×C₁₃₇	1	2	yes	—
312	C₄	#436×C₁₃₇	1	2	yes	—
313	C₄	#443×C₁₃₇	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	8	1	no	—
316	C₂	#1753×C₁₃₇	1	1	yes	characteristic
317	C₂	C₄×#104×C₁₃₇	1	1	yes	characteristic
318	C₂	#1764×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₁₃₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄²×C₈×C₁₃₇	C₂	semidirect
5. #1451×C₁₃₇	C₂	semidirect
6. #1278×C₁₃₇	C₂	semidirect
7. #1276×C₁₃₇	C₂	semidirect
9. #446×C₁₃₇	C₄	semidirect
10. #436×C₁₃₇	C₄	semidirect
23. Q₈⋊C₈×C₁₃₇	C₄	semidirect
24. D₄⋊C₈×C₁₃₇	C₄	semidirect
99. C₄²×C₁₃₇	M₁₆	semidirect
135. C₈×C₁₃₇	#104	semidirect
159. #8194	C₁₃₇	direct (N × H)
160. C₁₃₇	#8194	direct (N × H)
163. C₄²×C₈	C₂×C₁₃₇	semidirect
165. #1451	C₂×C₁₃₇	semidirect
166. #1278	C₂×C₁₃₇	semidirect
167. #1276	C₂×C₁₃₇	semidirect
169. #446	C₄×C₁₃₇	semidirect
170. #436	C₄×C₁₃₇	semidirect
183. Q₈⋊C₈	C₄×C₁₃₇	semidirect
184. D₄⋊C₈	C₄×C₁₃₇	semidirect
259. C₄²	M₁₆×C₁₃₇	semidirect
295. C₈	#104×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2965×C₁₃₇	2
C₄×#104×C₁₃₇	2
C₄²×C₈×C₁₃₇	2
#1476×C₁₃₇	2
#1451×C₁₃₇	2
#1278×C₁₃₇	2
#1276×C₁₃₇	2
#8194	137

Subgroups — C137×#8194 #46857580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8194 #46857580