Subgroups — C₁₃₇×#33610 #46882996

Order 35072.

674 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#504×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#3061×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#2220×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#2216×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2215×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	C₂×#366×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#2139×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	C₂×#135×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#131×C₁₃₇	C₂²	1	1	yes	characteristic
11	#537×C₁₃₇	—	2	2	no	—
12	#508×C₁₃₇	—	2	4	no	—
13	C₂×Q₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×SD₃₂×C₁₃₇	C₂²	1	2	yes	—
15	C₂×D₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
16	#504×C₁₃₇	—	2	4	no	—
17	C₂×M₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×M₃₂×C₁₃₇	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₁₃₇	C₂²	1	2	yes	—
20	C₂²×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
21	#468×C₁₃₇	—	2	4	no	—
22	#467×C₁₃₇	C₂²	1	4	yes	—
23	#466×C₁₃₇	C₂²	1	4	yes	—
24	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
25	#417×C₁₃₇	—	2	2	no	—
26	#366×C₁₃₇	C₂²	1	2	yes	—
27	#366×C₁₃₇	C₂²	1	2	yes	—
28	#358×C₁₃₇	C₂²	1	4	yes	—
29	#357×C₁₃₇	C₂²	1	4	yes	—
30	C₂×#41×C₁₃₇	—	2	2	no	—
31	C₂×#41×C₁₃₇	D₄	1	1	yes	characteristic
32	#135×C₁₃₇	—	2	4	no	—
33	#135×C₁₃₇	—	2	4	no	—
34	C₂×Q₁₆×C₁₃₇	C₂³	1	2	yes	—
35	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
36	C₂×D₈×C₁₃₇	C₂³	1	2	yes	—
37	#131×C₁₃₇	—	2	2	no	—
38	#131×C₁₃₇	—	2	2	no	—
39	C₂×M₁₆×C₁₃₇	D₄	1	1	yes	characteristic
40	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
41	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂²⋊Q₈×C₁₃₇	—	2	4	no	—
44	C₂²⋊Q₈×C₁₃₇	—	4	2	no	—
45	#121×C₁₃₇	—	2	4	no	—
46	#121×C₁₃₇	—	4	2	no	—
47	#121×C₁₃₇	—	4	4	no	—
48	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
49	Q₃₂×C₁₃₇	—	2	2	no	—
50	SD₃₂×C₁₃₇	—	2	4	no	—
51	D₁₆×C₁₃₇	—	2	2	no	—
52	M₃₂×C₁₃₇	—	2	2	no	—
53	M₃₂×C₁₃₇	—	2	4	no	—
54	C₂×C₁₆×C₁₃₇	—	2	4	no	—
55	C₂×C₁₆×C₁₃₇	D₄	1	4	yes	complemented
56	C₂×C₁₆×C₁₃₇	C₂³	1	2	yes	—
57	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
59	#107×C₁₃₇	—	2	2	no	—
60	#107×C₁₃₇	C₂³	1	4	yes	—
61	#103×C₁₃₇	—	2	4	no	—
62	#102×C₁₃₇	—	2	4	no	—
63	#41×C₁₃₇	—	2	2	no	—
64	#41×C₁₃₇	—	2	2	no	—
65	#41×C₁₃₇	—	4	4	no	—
66	#41×C₁₃₇	—	4	4	no	—
67	C₂×Q₈×C₁₃₇	—	2	2	no	—
68	C₂×Q₈×C₁₃₇	D₈	1	1	yes	characteristic
69	C₂×D₄×C₁₃₇	—	2	2	no	—
70	C₂×D₄×C₁₃₇	—	4	2	no	—
71	C₂×D₄×C₁₃₇	—	4	4	no	—
72	C₂×D₄×C₁₃₇	D₈	1	2	yes	—
73	C₂×D₄×C₁₃₇	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₇	—	4	2	no	—
75	C₂²×C₄×C₁₃₇	—	4	2	no	—
76	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₁₃₇	—	2	2	no	—
79	Q₁₆×C₁₃₇	—	2	2	no	—
80	SD₁₆×C₁₃₇	—	4	4	no	—
81	D₈×C₁₃₇	—	2	2	no	—
82	D₈×C₁₃₇	—	2	2	no	—
83	M₁₆×C₁₃₇	—	2	2	no	—
84	M₁₆×C₁₃₇	—	2	4	no	—
85	C₂×C₈×C₁₃₇	—	2	2	no	—
86	C₂×C₈×C₁₃₇	—	2	2	no	—
87	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃₇	#41	1	2	yes	—
91	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₁₃₇	—	2	4	no	—
93	C₄⋊C₄×C₁₃₇	—	4	2	no	—
94	C₄⋊C₄×C₁₃₇	—	4	4	no	—
95	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
96	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
97	C₁₆×C₁₃₇	—	2	2	no	—
98	C₁₆×C₁₃₇	C₂×D₄	1	2	yes	—
99	C₁₆×C₁₃₇	C₂×D₄	1	2	yes	—
100	C₂³×C₁₃₇	—	4	2	no	—
101	C₂³×C₁₃₇	C₂×D₈	1	2	yes	—
102	C₂³×C₁₃₇	C₂×D₈	1	1	yes	characteristic
103	Q₈×C₁₃₇	—	2	2	no	—
104	Q₈×C₁₃₇	—	4	2	no	—
105	Q₈×C₁₃₇	—	4	2	no	—
106	D₄×C₁₃₇	—	2	4	no	—
107	D₄×C₁₃₇	—	2	2	no	—
108	D₄×C₁₃₇	—	4	2	no	—
109	D₄×C₁₃₇	—	4	2	no	—
110	D₄×C₁₃₇	—	8	4	no	—
111	D₄×C₁₃₇	—	16	4	no	—
112	C₂×C₄×C₁₃₇	—	2	2	no	—
113	C₂×C₄×C₁₃₇	—	2	2	no	—
114	C₂×C₄×C₁₃₇	—	4	4	no	—
115	C₂×C₄×C₁₃₇	—	4	2	no	—
116	C₂×C₄×C₁₃₇	—	8	4	no	—
117	C₂×C₄×C₁₃₇	—	16	2	no	—
118	C₂×C₄×C₁₃₇	#121	1	2	yes	—
119	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	#135	1	2	yes	—
122	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₁₃₇	—	2	2	no	—
124	C₈×C₁₃₇	#121	1	2	yes	—
125	C₈×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₁₃₇	C₂×#41	1	1	yes	characteristic
127	C₂²×C₁₃₇	—	2	1	no	—
128	C₂²×C₁₃₇	—	2	1	no	—
129	C₂²×C₁₃₇	—	2	2	no	—
130	C₂²×C₁₃₇	—	2	2	no	—
131	C₂²×C₁₃₇	—	2	2	no	—
132	C₂²×C₁₃₇	—	8	2	no	—
133	C₂²×C₁₃₇	—	8	2	no	—
134	C₂²×C₁₃₇	—	8	2	no	—
135	C₂²×C₁₃₇	#466	1	2	yes	—
136	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
137	C₄×C₁₃₇	—	2	2	no	—
138	C₄×C₁₃₇	—	8	4	no	—
139	C₄×C₁₃₇	—	8	2	no	—
140	C₄×C₁₃₇	#466	1	2	yes	—
141	C₄×C₁₃₇	C₂×#121	1	1	yes	characteristic
142	C₄×C₁₃₇	C₂×#135	1	1	yes	characteristic
143	C₂×C₁₃₇	—	2	2	no	—
144	C₂×C₁₃₇	—	4	2	no	—
145	C₂×C₁₃₇	—	16	2	no	—
146	C₂×C₁₃₇	C₂×#466	1	1	yes	characteristic
147	C₂×C₁₃₇	#3415	1	1	yes	characteristic
148	C₂×C₁₃₇	#3417	1	1	yes	characteristic
149	#33610	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₁₃₇	#33610	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×#508	C₂×C₁₃₇	1	1	yes	characteristic, complemented
152	C₂×#504	C₂×C₁₃₇	1	1	yes	characteristic, complemented
153	#3061	C₂×C₁₃₇	1	2	yes	complemented
154	#2220	C₂×C₁₃₇	1	4	yes	complemented
155	#2216	C₂×C₁₃₇	1	2	yes	complemented
156	#2215	C₂×C₁₃₇	1	2	yes	complemented
157	C₂×#366	C₂×C₁₃₇	1	1	yes	characteristic, complemented
158	#2139	C₂×C₁₃₇	1	2	yes	complemented
159	C₂×#135	C₂²×C₁₃₇	1	2	yes	—
160	C₂×#131	C₂²×C₁₃₇	1	1	yes	characteristic
161	#537	—	2	2	no	—
162	#508	—	2	4	no	—
163	C₂×Q₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
164	C₂×SD₃₂	C₂²×C₁₃₇	1	2	yes	—
165	C₂×D₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
166	#504	—	2	4	no	—
167	C₂×M₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
168	C₂×M₃₂	C₂²×C₁₃₇	1	2	yes	—
169	C₂²×C₁₆	C₂²×C₁₃₇	1	2	yes	—
170	C₂²×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
171	#468	—	2	4	no	—
172	#467	C₂²×C₁₃₇	1	4	yes	—
173	#466	C₂²×C₁₃₇	1	4	yes	—
174	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
175	#417	—	2	2	no	—
176	#366	C₂²×C₁₃₇	1	2	yes	—
177	#366	C₂²×C₁₃₇	1	2	yes	—
178	#358	C₂²×C₁₃₇	1	4	yes	—
179	#357	C₂²×C₁₃₇	1	4	yes	—
180	C₂×#41	—	2	2	no	—
181	C₂×#41	D₄×C₁₃₇	1	1	yes	characteristic
182	#135	—	2	4	no	—
183	#135	—	2	4	no	—
184	C₂×Q₁₆	C₂³×C₁₃₇	1	2	yes	—
185	C₂×SD₁₆	—	2	2	no	—
186	C₂×D₈	C₂³×C₁₃₇	1	2	yes	—
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	C₂×M₁₆	D₄×C₁₃₇	1	1	yes	characteristic
190	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
191	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
192	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
193	C₂²⋊Q₈	—	2	4	no	—
194	C₂²⋊Q₈	—	4	2	no	—
195	#121	—	2	4	no	—
196	#121	—	4	2	no	—
197	#121	—	4	4	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	Q₃₂	—	2	2	no	—
200	SD₃₂	—	2	4	no	—
201	D₁₆	—	2	2	no	—
202	M₃₂	—	2	2	no	—
203	M₃₂	—	2	4	no	—
204	C₂×C₁₆	—	2	4	no	—
205	C₂×C₁₆	D₄×C₁₃₇	1	4	yes	complemented
206	C₂×C₁₆	C₂³×C₁₃₇	1	2	yes	—
207	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
208	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
209	#107	—	2	2	no	—
210	#107	C₂³×C₁₃₇	1	4	yes	—
211	#103	—	2	4	no	—
212	#102	—	2	4	no	—
213	#41	—	2	2	no	—
214	#41	—	2	2	no	—
215	#41	—	4	4	no	—
216	#41	—	4	4	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	D₈×C₁₃₇	1	1	yes	characteristic
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	4	no	—
222	C₂×D₄	D₈×C₁₃₇	1	2	yes	—
223	C₂×D₄	D₈×C₁₃₇	1	1	yes	characteristic
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
228	Q₁₆	—	2	2	no	—
229	Q₁₆	—	2	2	no	—
230	SD₁₆	—	4	4	no	—
231	D₈	—	2	2	no	—
232	D₈	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
238	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
240	C₂×C₈	#41×C₁₃₇	1	2	yes	—
241	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	C₂×D₄×C₁₃₇	1	2	yes	—
249	C₁₆	C₂×D₄×C₁₃₇	1	2	yes	—
250	C₂³	—	4	2	no	—
251	C₂³	C₂×D₈×C₁₃₇	1	2	yes	—
252	C₂³	C₂×D₈×C₁₃₇	1	1	yes	characteristic
253	Q₈	—	2	2	no	—
254	Q₈	—	4	2	no	—
255	Q₈	—	4	2	no	—
256	D₄	—	2	4	no	—
257	D₄	—	2	2	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	8	4	no	—
261	D₄	—	16	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	—	16	2	no	—
268	C₂×C₄	#121×C₁₃₇	1	2	yes	—
269	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
270	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
271	C₂×C₄	#135×C₁₃₇	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	#121×C₁₃₇	1	2	yes	—
275	C₈	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
276	C₈	C₂×#41×C₁₃₇	1	1	yes	characteristic
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	#466×C₁₃₇	1	2	yes	—
286	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	8	2	no	—
289	C₄	—	8	4	no	—
290	C₄	#466×C₁₃₇	1	2	yes	—
291	C₄	C₂×#121×C₁₃₇	1	1	yes	characteristic
292	C₄	C₂×#135×C₁₃₇	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	—	16	2	no	—
296	C₂	C₂×#466×C₁₃₇	1	1	yes	characteristic
297	C₂	#3415×C₁₃₇	1	1	yes	characteristic
298	C₂	#3417×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×#41×C₁₃₇ ◁ C₂×#131×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#508×C₁₃₇	C₂	semidirect
2. C₂×#504×C₁₃₇	C₂	semidirect
3. #3061×C₁₃₇	C₂	semidirect
4. #2220×C₁₃₇	C₂	semidirect
5. #2216×C₁₃₇	C₂	semidirect
6. #2215×C₁₃₇	C₂	semidirect
7. C₂×#366×C₁₃₇	C₂	semidirect
8. #2139×C₁₃₇	C₂	semidirect
55. C₂×C₁₆×C₁₃₇	D₄	semidirect
149. #33610	C₁₃₇	direct (N × H)
150. C₁₃₇	#33610	direct (N × H)
151. C₂×#508	C₂×C₁₃₇	semidirect
152. C₂×#504	C₂×C₁₃₇	semidirect
153. #3061	C₂×C₁₃₇	semidirect
154. #2220	C₂×C₁₃₇	semidirect
155. #2216	C₂×C₁₃₇	semidirect
156. #2215	C₂×C₁₃₇	semidirect
157. C₂×#366	C₂×C₁₃₇	semidirect
158. #2139	C₂×C₁₃₇	semidirect
205. C₂×C₁₆	D₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₁₃₇	2
C₂×#504×C₁₃₇	2
#3061×C₁₃₇	2
#2220×C₁₃₇	2
#2216×C₁₃₇	2
#2215×C₁₃₇	2
C₂×#366×C₁₃₇	2
#2139×C₁₃₇	2
#33610	137

Subgroups — C137×#33610 #46882996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33610 #46882996