Subgroups — C₁₃₇×#13002 #46862388

Order 35072.

528 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3050×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#360×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#2112×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1386×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	#525×C₁₃₇	—	2	1	no	—
9	C₂×#117×C₁₃₇	—	2	1	no	—
10	C₂×M₃₂×C₁₃₇	C₂²	1	2	yes	—
11	#467×C₁₃₇	C₄	1	2	yes	complemented
12	#466×C₁₃₇	C₄	1	2	yes	complemented
13	#465×C₁₃₇	—	2	2	no	—
14	#427×C₁₃₇	—	2	4	no	—
15	#427×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#107×C₁₃₇	—	2	1	no	—
17	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
18	#416×C₁₃₇	—	2	1	no	—
19	#360×C₁₃₇	—	2	4	no	—
20	C₂²⋊C₁₆×C₁₃₇	—	2	2	no	—
21	C₂×#41×C₁₃₇	—	2	1	no	—
22	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
23	#135×C₁₃₇	—	2	2	no	—
24	#135×C₁₃₇	—	2	2	no	—
25	C₂×Q₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂×SD₁₆×C₁₃₇	—	2	1	no	—
27	C₂×D₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₇	—	2	2	no	—
29	C₂²×C₈×C₁₃₇	—	2	2	no	—
30	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
34	#123×C₁₃₇	—	4	1	no	—
35	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
36	#121×C₁₃₇	—	2	2	no	—
37	#120×C₁₃₇	—	4	1	no	—
38	C₄×D₄×C₁₃₇	—	4	2	no	—
39	#117×C₁₃₇	—	2	4	no	—
40	#117×C₁₃₇	—	2	1	no	—
41	#117×C₁₃₇	—	4	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₁₃₇	—	4	1	no	—
44	C₂×C₄²×C₁₃₇	—	4	1	no	—
45	M₃₂×C₁₃₇	—	2	2	no	—
46	M₃₂×C₁₃₇	—	2	2	no	—
47	C₂×C₁₆×C₁₃₇	—	2	2	no	—
48	#107×C₁₃₇	—	2	4	no	—
49	#107×C₁₃₇	C₂×C₄	1	2	yes	—
50	#106×C₁₃₇	—	2	2	no	—
51	#106×C₁₃₇	—	2	1	no	—
52	#106×C₁₃₇	D₄	1	4	yes	complemented
53	#103×C₁₃₇	—	2	2	no	—
54	#102×C₁₃₇	—	2	2	no	—
55	C₂⁴×C₁₃₇	D₈	1	1	yes	characteristic
56	#41×C₁₃₇	—	4	1	no	—
57	#41×C₁₃₇	—	4	1	no	—
58	#41×C₁₃₇	—	4	1	no	—
59	#41×C₁₃₇	—	4	1	no	—
60	C₂×Q₈×C₁₃₇	—	2	1	no	—
61	C₂×D₄×C₁₃₇	—	2	1	no	—
62	C₂×D₄×C₁₃₇	—	4	2	no	—
63	C₂×D₄×C₁₃₇	—	4	1	no	—
64	C₂²×C₄×C₁₃₇	—	2	1	no	—
65	C₂²×C₄×C₁₃₇	—	2	1	no	—
66	C₂²×C₄×C₁₃₇	—	2	1	no	—
67	C₂²×C₄×C₁₃₇	—	2	1	no	—
68	C₂²×C₄×C₁₃₇	—	4	2	no	—
69	C₂²×C₄×C₁₃₇	—	4	1	no	—
70	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₃₇	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₁₃₇	—	2	2	no	—
75	SD₁₆×C₁₃₇	—	4	2	no	—
76	D₈×C₁₃₇	—	2	2	no	—
77	C₂×C₈×C₁₃₇	—	2	1	no	—
78	C₂×C₈×C₁₃₇	—	2	1	no	—
79	C₂×C₈×C₁₃₇	—	2	4	no	—
80	C₂×C₈×C₁₃₇	—	2	4	no	—
81	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃₇	#41	1	2	yes	—
86	C₄⋊C₄×C₁₃₇	—	2	2	no	—
87	C₄⋊C₄×C₁₃₇	—	2	4	no	—
88	C₄⋊C₄×C₁₃₇	—	4	2	no	—
89	C₄⋊C₄×C₁₃₇	—	4	2	no	—
90	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
91	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
92	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
93	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
94	C₄²×C₁₃₇	—	4	4	no	—
95	C₄²×C₁₃₇	—	4	1	no	—
96	C₁₆×C₁₃₇	—	4	2	no	—
97	C₂³×C₁₃₇	—	2	1	no	—
98	C₂³×C₁₃₇	—	2	1	no	—
99	C₂³×C₁₃₇	—	4	2	no	—
100	C₂³×C₁₃₇	—	4	1	no	—
101	C₂³×C₁₃₇	#102	1	2	yes	—
102	C₂³×C₁₃₇	C₂×D₈	1	1	yes	characteristic
103	Q₈×C₁₃₇	—	4	1	no	—
104	Q₈×C₁₃₇	—	4	1	no	—
105	D₄×C₁₃₇	—	4	1	no	—
106	D₄×C₁₃₇	—	4	1	no	—
107	D₄×C₁₃₇	—	8	4	no	—
108	D₄×C₁₃₇	—	8	1	no	—
109	D₄×C₁₃₇	—	8	1	no	—
110	C₂×C₄×C₁₃₇	—	2	1	no	—
111	C₂×C₄×C₁₃₇	—	2	1	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	1	no	—
118	C₂×C₄×C₁₃₇	—	4	4	no	—
119	C₂×C₄×C₁₃₇	—	8	4	no	—
120	C₂×C₄×C₁₃₇	—	8	1	no	—
121	C₂×C₄×C₁₃₇	—	8	1	no	—
122	C₂×C₄×C₁₃₇	#102	1	2	yes	—
123	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
125	C₂×C₄×C₁₃₇	#123	1	1	yes	characteristic
126	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₁₃₇	#137	1	1	yes	characteristic
129	C₈×C₁₃₇	—	2	2	no	—
130	C₈×C₁₃₇	C₄×D₄	1	2	yes	—
131	C₈×C₁₃₇	#121	1	2	yes	—
132	C₂²×C₁₃₇	—	2	2	no	—
133	C₂²×C₁₃₇	—	2	2	no	—
134	C₂²×C₁₃₇	—	2	2	no	—
135	C₂²×C₁₃₇	—	2	2	no	—
136	C₂²×C₁₃₇	—	4	2	no	—
137	C₂²×C₁₃₇	—	4	2	no	—
138	C₂²×C₁₃₇	—	8	1	no	—
139	C₂²×C₁₃₇	—	8	2	no	—
140	C₂²×C₁₃₇	C₂×#102	1	1	yes	characteristic
141	C₂²×C₁₃₇	#447	1	1	yes	characteristic
142	C₂²×C₁₃₇	#480	1	1	yes	characteristic
143	C₄×C₁₃₇	—	2	2	no	—
144	C₄×C₁₃₇	—	8	4	no	—
145	C₄×C₁₃₇	—	8	1	no	—
146	C₄×C₁₃₇	—	8	2	no	—
147	C₄×C₁₃₇	#386	1	1	yes	characteristic
148	C₄×C₁₃₇	#417	1	1	yes	characteristic
149	C₄×C₁₃₇	#448	1	1	yes	characteristic
150	C₄×C₁₃₇	#481	1	1	yes	characteristic
151	C₂×C₁₃₇	—	2	1	no	—
152	C₂×C₁₃₇	—	2	1	no	—
153	C₂×C₁₃₇	—	4	2	no	—
154	C₂×C₁₃₇	—	16	1	no	—
155	C₂×C₁₃₇	#1874	1	1	yes	characteristic
156	C₂×C₁₃₇	#2145	1	2	yes	—
157	#13002	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₁₃₇	#13002	1	1	yes	characteristic, direct product factor, Sylow
159	#3050	C₂×C₁₃₇	1	1	yes	characteristic
160	C₂×#427	C₂×C₁₃₇	1	1	yes	characteristic, complemented
161	C₂×#360	C₂×C₁₃₇	1	2	yes	complemented
162	#2112	C₂×C₁₃₇	1	1	yes	characteristic, complemented
163	#1386	C₂×C₁₃₇	1	2	yes	complemented
164	C₂×#135	C₂²×C₁₃₇	1	1	yes	characteristic
165	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
166	#525	—	2	1	no	—
167	C₂×#117	—	2	1	no	—
168	C₂×M₃₂	C₂²×C₁₃₇	1	2	yes	—
169	#467	C₄×C₁₃₇	1	2	yes	complemented
170	#466	C₄×C₁₃₇	1	2	yes	complemented
171	#465	—	2	2	no	—
172	#427	—	2	4	no	—
173	#427	C₂²×C₁₃₇	1	1	yes	characteristic
174	C₂×#107	—	2	1	no	—
175	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
176	#416	—	2	1	no	—
177	#360	—	2	4	no	—
178	C₂²⋊C₁₆	—	2	2	no	—
179	C₂×#41	—	2	1	no	—
180	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
181	#135	—	2	2	no	—
182	#135	—	2	2	no	—
183	C₂×Q₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
184	C₂×SD₁₆	—	2	1	no	—
185	C₂×D₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
189	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
190	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
191	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
192	#123	—	4	1	no	—
193	C₂²⋊Q₈	—	2	2	no	—
194	#121	—	2	2	no	—
195	#120	—	4	1	no	—
196	C₄×D₄	—	4	2	no	—
197	#117	—	2	4	no	—
198	#117	—	2	1	no	—
199	#117	—	4	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	—	4	1	no	—
202	C₂×C₄²	—	4	1	no	—
203	M₃₂	—	2	2	no	—
204	M₃₂	—	2	2	no	—
205	C₂×C₁₆	—	2	2	no	—
206	#107	—	2	4	no	—
207	#107	C₂×C₄×C₁₃₇	1	2	yes	—
208	#106	—	2	2	no	—
209	#106	—	2	1	no	—
210	#106	D₄×C₁₃₇	1	4	yes	complemented
211	#103	—	2	2	no	—
212	#102	—	2	2	no	—
213	C₂⁴	D₈×C₁₃₇	1	1	yes	characteristic
214	#41	—	4	1	no	—
215	#41	—	4	1	no	—
216	#41	—	4	1	no	—
217	#41	—	4	1	no	—
218	C₂×Q₈	—	2	1	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
229	C₂²×C₄	D₈×C₁₃₇	1	1	yes	characteristic
230	C₂²×C₄	SD₁₆×C₁₃₇	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
232	Q₁₆	—	2	2	no	—
233	SD₁₆	—	4	2	no	—
234	D₈	—	2	2	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
240	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
242	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
243	C₂×C₈	#41×C₁₃₇	1	2	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	4	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	1	no	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₄²	—	4	4	no	—
253	C₄²	—	4	1	no	—
254	C₁₆	—	4	2	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	1	no	—
259	C₂³	#102×C₁₃₇	1	2	yes	—
260	C₂³	C₂×D₈×C₁₃₇	1	1	yes	characteristic
261	Q₈	—	4	1	no	—
262	Q₈	—	4	1	no	—
263	D₄	—	4	1	no	—
264	D₄	—	4	1	no	—
265	D₄	—	8	4	no	—
266	D₄	—	8	1	no	—
267	D₄	—	8	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	—	8	1	no	—
279	C₂×C₄	—	8	1	no	—
280	C₂×C₄	#102×C₁₃₇	1	2	yes	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
282	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
283	C₂×C₄	#123×C₁₃₇	1	1	yes	characteristic
284	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
285	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
286	C₂×C₄	#137×C₁₃₇	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	C₄×D₄×C₁₃₇	1	2	yes	—
289	C₈	#121×C₁₃₇	1	2	yes	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	8	1	no	—
297	C₂²	—	8	2	no	—
298	C₂²	C₂×#102×C₁₃₇	1	1	yes	characteristic
299	C₂²	#447×C₁₃₇	1	1	yes	characteristic
300	C₂²	#480×C₁₃₇	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	8	4	no	—
303	C₄	—	8	1	no	—
304	C₄	—	8	2	no	—
305	C₄	#386×C₁₃₇	1	1	yes	characteristic
306	C₄	#417×C₁₃₇	1	1	yes	characteristic
307	C₄	#448×C₁₃₇	1	1	yes	characteristic
308	C₄	#481×C₁₃₇	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	4	2	no	—
312	C₂	—	16	1	no	—
313	C₂	#1874×C₁₃₇	1	1	yes	characteristic
314	C₂	#2145×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
prime radical	#3050×C₁₃₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ C₂³×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#427×C₁₃₇	C₂	semidirect
3. C₂×#360×C₁₃₇	C₂	semidirect
4. #2112×C₁₃₇	C₂	semidirect
5. #1386×C₁₃₇	C₂	semidirect
11. #467×C₁₃₇	C₄	semidirect
12. #466×C₁₃₇	C₄	semidirect
52. #106×C₁₃₇	D₄	semidirect
157. #13002	C₁₃₇	direct (N × H)
158. C₁₃₇	#13002	direct (N × H)
160. C₂×#427	C₂×C₁₃₇	semidirect
161. C₂×#360	C₂×C₁₃₇	semidirect
162. #2112	C₂×C₁₃₇	semidirect
163. #1386	C₂×C₁₃₇	semidirect
169. #467	C₄×C₁₃₇	semidirect
170. #466	C₄×C₁₃₇	semidirect
210. #106	D₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3050×C₁₃₇	2
C₂×#427×C₁₃₇	2
C₂×#360×C₁₃₇	2
#2112×C₁₃₇	2
#1386×C₁₃₇	2
#13002	137

Subgroups — C137×#13002 #46862388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13002 #46862388