Subgroups — C₁₃₇×#22815 #46872201

Order 35072.

754 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2356×C₁₃₇	C₂	1	2	yes	maximal
3	#1971×C₁₃₇	C₂	1	2	yes	maximal
4	#1928×C₁₃₇	C₂	1	4	yes	maximal
5	#1846×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1805×C₁₃₇	C₂	1	2	yes	maximal
7	#1776×C₁₃₇	C₂	1	2	yes	maximal
8	#1770×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₁₃₇	C₄	1	8	yes	complemented
12	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
17	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
18	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	#391×C₁₃₇	C₂²	1	4	yes	—
21	#387×C₁₃₇	—	2	4	no	—
22	#383×C₁₃₇	—	2	2	no	—
23	#382×C₁₃₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
25	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
26	#376×C₁₃₇	—	2	2	no	—
27	#340×C₁₃₇	C₂²	1	4	yes	—
28	#336×C₁₃₇	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
30	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
31	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
32	C₂×Q₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₁₃₇	—	2	4	no	—
37	C₄×Q₈×C₁₃₇	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃₇	—	2	2	no	—
43	C₂×C₄²×C₁₃₇	—	2	4	no	—
44	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
46	#107×C₁₃₇	—	4	2	no	—
47	#107×C₁₃₇	C₂×C₄	1	4	yes	—
48	#106×C₁₃₇	—	2	4	no	—
49	C₄⋊C₈×C₁₃₇	—	2	2	no	—
50	#103×C₁₃₇	—	4	4	no	—
51	#103×C₁₃₇	C₂×C₄	1	8	yes	—
52	#97×C₁₃₇	—	2	4	no	—
53	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
54	#95×C₁₃₇	—	2	2	no	—
55	#95×C₁₃₇	—	2	4	no	—
56	#95×C₁₃₇	—	2	4	no	—
57	#95×C₁₃₇	—	2	2	no	—
58	#95×C₁₃₇	—	2	4	no	—
59	#95×C₁₃₇	D₄	1	2	yes	—
60	C₂×Q₈×C₁₃₇	—	2	2	no	—
61	C₂×Q₈×C₁₃₇	—	2	2	no	—
62	C₂×Q₈×C₁₃₇	—	2	2	no	—
63	C₂×Q₈×C₁₃₇	—	2	2	no	—
64	C₂×Q₈×C₁₃₇	—	2	2	no	—
65	C₂×Q₈×C₁₃₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃₇	—	2	2	no	—
67	C₂²×C₄×C₁₃₇	—	2	2	no	—
68	C₂²×C₄×C₁₃₇	—	2	4	no	—
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
73	Q₁₆×C₁₃₇	—	2	8	no	—
74	C₂×C₈×C₁₃₇	—	2	1	no	—
75	C₂×C₈×C₁₃₇	—	2	1	no	—
76	C₂×C₈×C₁₃₇	—	2	2	no	—
77	C₂×C₈×C₁₃₇	—	2	2	no	—
78	C₂×C₈×C₁₃₇	—	2	2	no	—
79	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃₇	#41	1	2	yes	—
82	C₂×C₈×C₁₃₇	#41	1	2	yes	—
83	C₄⋊C₄×C₁₃₇	—	2	8	no	—
84	C₄⋊C₄×C₁₃₇	—	2	2	no	—
85	C₄⋊C₄×C₁₃₇	—	2	4	no	—
86	C₄⋊C₄×C₁₃₇	—	2	4	no	—
87	C₄⋊C₄×C₁₃₇	—	4	2	no	—
88	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₁₃₇	—	2	8	no	—
90	C₄²×C₁₃₇	—	2	2	no	—
91	C₄²×C₁₃₇	—	2	2	no	—
92	C₄²×C₁₃₇	—	4	8	no	—
93	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₃₇	#41	1	2	yes	—
95	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₁₃₇	—	2	4	no	—
97	Q₈×C₁₃₇	—	2	4	no	—
98	Q₈×C₁₃₇	—	4	2	no	—
99	Q₈×C₁₃₇	—	4	2	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	—	2	4	no	—
103	C₂×C₄×C₁₃₇	—	2	4	no	—
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	4	no	—
106	C₂×C₄×C₁₃₇	—	2	4	no	—
107	C₂×C₄×C₁₃₇	—	4	2	no	—
108	C₂×C₄×C₁₃₇	—	4	2	no	—
109	C₂×C₄×C₁₃₇	—	4	4	no	—
110	C₂×C₄×C₁₃₇	—	4	4	no	—
111	C₂×C₄×C₁₃₇	—	4	4	no	—
112	C₂×C₄×C₁₃₇	—	4	4	no	—
113	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
121	C₈×C₁₃₇	—	2	2	no	—
122	C₈×C₁₃₇	—	4	2	no	—
123	C₈×C₁₃₇	#117	1	4	yes	—
124	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₁₃₇	#547	1	1	yes	characteristic
126	C₂²×C₁₃₇	#550	1	1	yes	characteristic
127	C₂²×C₁₃₇	#575	1	2	yes	—
128	C₂²×C₁₃₇	#578	1	2	yes	—
129	C₄×C₁₃₇	—	2	2	no	—
130	C₄×C₁₃₇	—	2	2	no	—
131	C₄×C₁₃₇	—	4	4	no	—
132	C₄×C₁₃₇	—	4	4	no	—
133	C₄×C₁₃₇	—	4	4	no	—
134	C₄×C₁₃₇	—	8	4	no	—
135	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
136	C₄×C₁₃₇	#520	1	1	yes	characteristic
137	C₄×C₁₃₇	#555	1	1	yes	characteristic
138	C₄×C₁₃₇	#556	1	1	yes	characteristic
139	C₂×C₁₃₇	#2354	1	1	yes	characteristic
140	C₂×C₁₃₇	#2943	1	1	yes	characteristic
141	C₂×C₁₃₇	#2950	1	1	yes	characteristic
142	C₂×C₁₃₇	#3352	1	2	yes	—
143	C₂×C₁₃₇	#3402	1	2	yes	—
144	#22815	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₁₃₇	#22815	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×C₄×Q₁₆	C₂×C₁₃₇	1	1	yes	characteristic
147	#2356	C₂×C₁₃₇	1	2	yes	—
148	#1971	C₂×C₁₃₇	1	2	yes	—
149	#1928	C₂×C₁₃₇	1	4	yes	—
150	#1846	C₂×C₁₃₇	1	1	yes	characteristic
151	#1805	C₂×C₁₃₇	1	2	yes	—
152	#1776	C₂×C₁₃₇	1	2	yes	—
153	#1770	C₂×C₁₃₇	1	1	yes	characteristic
154	C₂²×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
155	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
156	C₄×Q₁₆	C₄×C₁₃₇	1	8	yes	complemented
157	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
158	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
159	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
160	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
161	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
162	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
163	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
165	#391	C₂²×C₁₃₇	1	4	yes	—
166	#387	—	2	4	no	—
167	#383	—	2	2	no	—
168	#382	C₂²×C₁₃₇	1	2	yes	—
169	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
170	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
171	#376	—	2	2	no	—
172	#340	C₂²×C₁₃₇	1	4	yes	—
173	#336	C₂²×C₁₃₇	1	2	yes	—
174	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
175	C₂×Q₁₆	—	2	2	no	—
176	C₂×Q₁₆	—	2	2	no	—
177	C₂×Q₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
178	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
179	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
181	C₄×Q₈	—	2	4	no	—
182	C₄×Q₈	C₂×C₄×C₁₃₇	1	8	yes	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
185	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
186	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	4	no	—
189	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
190	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
191	#107	—	4	2	no	—
192	#107	C₂×C₄×C₁₃₇	1	4	yes	—
193	#106	—	2	4	no	—
194	C₄⋊C₈	—	2	2	no	—
195	#103	—	4	4	no	—
196	#103	C₂×C₄×C₁₃₇	1	8	yes	—
197	#97	—	2	4	no	—
198	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
199	#95	—	2	2	no	—
200	#95	—	2	4	no	—
201	#95	—	2	4	no	—
202	#95	—	2	2	no	—
203	#95	—	2	4	no	—
204	#95	D₄×C₁₃₇	1	2	yes	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	C₂²×C₄×C₁₃₇	1	4	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
215	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
217	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
218	Q₁₆	—	2	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
225	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
226	C₂×C₈	#41×C₁₃₇	1	2	yes	—
227	C₂×C₈	#41×C₁₃₇	1	2	yes	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
234	C₄²	—	2	8	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	8	no	—
238	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
239	C₄²	#41×C₁₃₇	1	2	yes	—
240	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
259	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
260	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
261	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
262	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
263	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
264	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
265	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	4	2	no	—
268	C₈	#117×C₁₃₇	1	4	yes	—
269	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
270	C₂²	#547×C₁₃₇	1	1	yes	characteristic
271	C₂²	#550×C₁₃₇	1	1	yes	characteristic
272	C₂²	#575×C₁₃₇	1	2	yes	—
273	C₂²	#578×C₁₃₇	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	4	no	—
278	C₄	—	4	4	no	—
279	C₄	—	8	4	no	—
280	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
281	C₄	#520×C₁₃₇	1	1	yes	characteristic
282	C₄	#555×C₁₃₇	1	1	yes	characteristic
283	C₄	#556×C₁₃₇	1	1	yes	characteristic
284	C₂	#2354×C₁₃₇	1	1	yes	characteristic
285	C₂	#2943×C₁₃₇	1	1	yes	characteristic
286	C₂	#2950×C₁₃₇	1	1	yes	characteristic
287	C₂	#3352×C₁₃₇	1	2	yes	—
288	C₂	#3402×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₄×Q₁₆×C₁₃₇	C₄	semidirect
144. #22815	C₁₃₇	direct (N × H)
145. C₁₃₇	#22815	direct (N × H)
156. C₄×Q₁₆	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₁₃₇	2
#2356×C₁₃₇	2
#1971×C₁₃₇	2
#1928×C₁₃₇	2
#1846×C₁₃₇	2
#1805×C₁₃₇	2
#1776×C₁₃₇	2
#1770×C₁₃₇	2
#22815	137

Subgroups — C137×#22815 #46872201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#22815 #46872201