Subgroups — C₁₃₇×#20713 #46870099

Order 35072.

1244 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	#3428×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#413×C₁₃₇	C₂	1	2	yes	maximal
3	#1830×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	#1829×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	C₂×#344×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	C₂²×#41×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₃₇	C₂²	1	4	yes	—
8	#520×C₁₃₇	—	2	4	no	—
9	#519×C₁₃₇	C₂²	1	2	yes	—
10	#518×C₁₃₇	C₄	1	4	yes	—
11	#518×C₁₃₇	C₂²	1	2	yes	—
12	C₄×#41×C₁₃₇	—	2	4	no	—
13	C₂×#117×C₁₃₇	C₄	1	4	yes	—
14	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
15	#413×C₁₃₇	—	2	4	no	—
16	#413×C₁₃₇	—	2	4	no	—
17	C₂×#101×C₁₃₇	C₂²	1	4	yes	—
18	C₂×#100×C₁₃₇	C₂²	1	4	yes	—
19	#344×C₁₃₇	C₂²	1	16	yes	—
20	C₂×#41×C₁₃₇	—	2	4	no	—
21	C₂×#41×C₁₃₇	—	2	4	no	—
22	C₂×#41×C₁₃₇	D₄	1	4	yes	—
23	C₂×#41×C₁₃₇	D₄	1	4	yes	—
24	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₂²×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₁₃₇	—	2	4	no	—
30	C₂×M₁₆×C₁₃₇	—	2	4	no	—
31	C₂×M₁₆×C₁₃₇	—	2	4	no	—
32	C₂×M₁₆×C₁₃₇	—	2	4	no	—
33	C₂×M₁₆×C₁₃₇	—	2	4	no	—
34	C₂×M₁₆×C₁₃₇	C₂³	1	8	yes	—
35	C₂²×C₈×C₁₃₇	—	2	4	no	—
36	C₄×Q₈×C₁₃₇	—	2	8	no	—
37	C₄×D₄×C₁₃₇	—	2	16	no	—
38	C₄×D₄×C₁₃₇	—	2	8	no	—
39	#117×C₁₃₇	—	2	4	no	—
40	#117×C₁₃₇	—	2	4	no	—
41	#117×C₁₃₇	—	2	2	no	—
42	#117×C₁₃₇	—	2	2	no	—
43	#117×C₁₃₇	C₂×C₄	1	4	yes	—
44	#117×C₁₃₇	C₂×C₄	1	8	yes	—
45	#117×C₁₃₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
49	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₁₃₇	—	2	4	no	—
51	C₂×C₄²×C₁₃₇	—	2	2	no	—
52	#101×C₁₃₇	—	2	8	no	—
53	#100×C₁₃₇	—	2	8	no	—
54	C₂⁴×C₁₃₇	C₂²×C₄	1	2	yes	—
55	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
56	#41×C₁₃₇	—	2	8	no	—
57	#41×C₁₃₇	—	2	8	no	—
58	#41×C₁₃₇	—	2	8	no	—
59	#41×C₁₃₇	—	2	8	no	—
60	C₂×Q₈×C₁₃₇	—	2	4	no	—
61	C₂×Q₈×C₁₃₇	C₂×D₄	1	4	yes	—
62	C₂×D₄×C₁₃₇	—	2	8	no	—
63	C₂×D₄×C₁₃₇	—	2	4	no	—
64	C₂×D₄×C₁₃₇	C₂²⋊C₄	1	8	yes	—
65	C₂×D₄×C₁₃₇	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₁₃₇	—	2	4	no	—
67	C₂²×C₄×C₁₃₇	—	2	4	no	—
68	C₂²×C₄×C₁₃₇	—	2	8	no	—
69	C₂²×C₄×C₁₃₇	—	2	4	no	—
70	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
71	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
72	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
77	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
78	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₁₃₇	—	2	8	no	—
80	M₁₆×C₁₃₇	—	2	8	no	—
81	M₁₆×C₁₃₇	—	4	4	no	—
82	M₁₆×C₁₃₇	—	4	4	no	—
83	C₂×C₈×C₁₃₇	—	2	8	no	—
84	C₂×C₈×C₁₃₇	—	4	4	no	—
85	C₂×C₈×C₁₃₇	—	4	4	no	—
86	C₄⋊C₄×C₁₃₇	—	2	4	no	—
87	C₄⋊C₄×C₁₃₇	—	2	8	no	—
88	C₄⋊C₄×C₁₃₇	—	2	4	no	—
89	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
90	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
91	C₂²⋊C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
92	C₄²×C₁₃₇	—	2	4	no	—
93	C₄²×C₁₃₇	—	2	8	no	—
94	C₄²×C₁₃₇	—	2	4	no	—
95	C₂³×C₁₃₇	—	2	4	no	—
96	C₂³×C₁₃₇	—	8	1	no	—
97	C₂³×C₁₃₇	—	8	2	no	—
98	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂³×C₁₃₇	#117	1	4	yes	—
101	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
102	Q₈×C₁₃₇	—	2	8	no	—
103	D₄×C₁₃₇	—	2	16	no	—
104	D₄×C₁₃₇	—	2	8	no	—
105	C₂×C₄×C₁₃₇	—	2	4	no	—
106	C₂×C₄×C₁₃₇	—	2	4	no	—
107	C₂×C₄×C₁₃₇	—	2	8	no	—
108	C₂×C₄×C₁₃₇	—	2	4	no	—
109	C₂×C₄×C₁₃₇	—	2	8	no	—
110	C₂×C₄×C₁₃₇	—	8	4	no	—
111	C₂×C₄×C₁₃₇	—	8	8	no	—
112	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	#117	1	2	yes	—
116	C₂×C₄×C₁₃₇	#117	1	2	yes	—
117	C₂×C₄×C₁₃₇	#123	1	4	yes	—
118	C₂×C₄×C₁₃₇	#123	1	4	yes	—
119	C₂×C₄×C₁₃₇	#123	1	4	yes	—
120	C₂×C₄×C₁₃₇	#123	1	4	yes	—
121	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₁₃₇	C₂×#41	1	4	yes	—
123	C₈×C₁₃₇	—	4	8	no	—
124	C₂²×C₁₃₇	—	2	4	no	—
125	C₂²×C₁₃₇	—	4	2	no	—
126	C₂²×C₁₃₇	—	4	1	no	—
127	C₂²×C₁₃₇	—	4	4	no	—
128	C₂²×C₁₃₇	—	4	4	no	—
129	C₂²×C₁₃₇	—	8	4	no	—
130	C₂²×C₁₃₇	#381	1	4	yes	—
131	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₁₃₇	C₂×#117	1	2	yes	—
133	C₄×C₁₃₇	—	2	4	no	—
134	C₄×C₁₃₇	—	4	8	no	—
135	C₄×C₁₃₇	#381	1	2	yes	—
136	C₄×C₁₃₇	#381	1	1	yes	characteristic
137	C₄×C₁₃₇	#381	1	1	yes	characteristic
138	C₄×C₁₃₇	C₂×#123	1	4	yes	—
139	C₂×C₁₃₇	—	2	1	no	—
140	C₂×C₁₃₇	—	2	2	no	—
141	C₂×C₁₃₇	—	2	4	no	—
142	C₂×C₁₃₇	—	4	4	no	—
143	C₂×C₁₃₇	C₂×#381	1	1	yes	characteristic
144	#20713	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₁₃₇	#20713	1	1	yes	characteristic, direct product factor, Sylow
146	#3428	C₂×C₁₃₇	1	1	yes	characteristic
147	C₂×#413	C₂×C₁₃₇	1	2	yes	—
148	#1830	C₂×C₁₃₇	1	4	yes	complemented
149	#1829	C₂×C₁₃₇	1	4	yes	complemented
150	C₂×#344	C₂×C₁₃₇	1	4	yes	complemented
151	C₂²×#41	C₂²×C₁₃₇	1	1	yes	characteristic
152	C₂²×M₁₆	C₂²×C₁₃₇	1	4	yes	—
153	#520	—	2	4	no	—
154	#519	C₂²×C₁₃₇	1	2	yes	—
155	#518	C₄×C₁₃₇	1	4	yes	—
156	#518	C₂²×C₁₃₇	1	2	yes	—
157	C₄×#41	—	2	4	no	—
158	C₂×#117	C₄×C₁₃₇	1	4	yes	—
159	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
160	#413	—	2	4	no	—
161	#413	—	2	4	no	—
162	C₂×#101	C₂²×C₁₃₇	1	4	yes	—
163	C₂×#100	C₂²×C₁₃₇	1	4	yes	—
164	#344	C₂²×C₁₃₇	1	16	yes	—
165	C₂×#41	—	2	4	no	—
166	C₂×#41	—	2	4	no	—
167	C₂×#41	D₄×C₁₃₇	1	4	yes	—
168	C₂×#41	D₄×C₁₃₇	1	4	yes	—
169	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
170	C₂²×D₄	C₂×C₄×C₁₃₇	1	2	yes	—
171	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
172	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
173	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	—	2	4	no	—
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	—	2	4	no	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	C₂³×C₁₃₇	1	8	yes	—
180	C₂²×C₈	—	2	4	no	—
181	C₄×Q₈	—	2	8	no	—
182	C₄×D₄	—	2	16	no	—
183	C₄×D₄	—	2	8	no	—
184	#117	—	2	4	no	—
185	#117	—	2	4	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	C₂×C₄×C₁₃₇	1	4	yes	—
189	#117	C₂×C₄×C₁₃₇	1	8	yes	—
190	#117	C₂³×C₁₃₇	1	4	yes	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	—
194	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
195	C₂×C₄²	—	2	4	no	—
196	C₂×C₄²	—	2	2	no	—
197	#101	—	2	8	no	—
198	#100	—	2	8	no	—
199	C₂⁴	C₂²×C₄×C₁₃₇	1	2	yes	—
200	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
201	#41	—	2	8	no	—
202	#41	—	2	8	no	—
203	#41	—	2	8	no	—
204	#41	—	2	8	no	—
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	C₂×D₄×C₁₃₇	1	4	yes	—
207	C₂×D₄	—	2	8	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	C₂²⋊C₄×C₁₃₇	1	8	yes	—
210	C₂×D₄	C₂×D₄×C₁₃₇	1	4	yes	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	4	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
221	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
222	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
223	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
224	M₁₆	—	2	8	no	—
225	M₁₆	—	2	8	no	—
226	M₁₆	—	4	4	no	—
227	M₁₆	—	4	4	no	—
228	C₂×C₈	—	2	8	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	4	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	8	no	—
239	C₄²	—	2	4	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	8	1	no	—
242	C₂³	—	8	2	no	—
243	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
244	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
245	C₂³	#117×C₁₃₇	1	4	yes	—
246	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
247	Q₈	—	2	8	no	—
248	D₄	—	2	16	no	—
249	D₄	—	2	8	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
259	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
260	C₂×C₄	#117×C₁₃₇	1	2	yes	—
261	C₂×C₄	#117×C₁₃₇	1	2	yes	—
262	C₂×C₄	#123×C₁₃₇	1	4	yes	—
263	C₂×C₄	#123×C₁₃₇	1	4	yes	—
264	C₂×C₄	#123×C₁₃₇	1	4	yes	—
265	C₂×C₄	#123×C₁₃₇	1	4	yes	—
266	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
267	C₂×C₄	C₂×#41×C₁₃₇	1	4	yes	—
268	C₈	—	4	8	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	4	no	—
273	C₂²	—	4	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	#381×C₁₃₇	1	4	yes	—
276	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
277	C₂²	C₂×#117×C₁₃₇	1	2	yes	—
278	C₄	—	2	4	no	—
279	C₄	—	4	8	no	—
280	C₄	#381×C₁₃₇	1	1	yes	characteristic
281	C₄	#381×C₁₃₇	1	2	yes	—
282	C₄	#381×C₁₃₇	1	1	yes	characteristic
283	C₄	C₂×#123×C₁₃₇	1	4	yes	—
284	C₂	—	2	1	no	—
285	C₂	—	2	2	no	—
286	C₂	—	2	4	no	—
287	C₂	—	4	4	no	—
288	C₂	C₂×#381×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₇	#381
commutator	C₂³	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂×C₁₃₇	C₂×#381
prime radical	C₂²×#41×C₁₃₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1830×C₁₃₇	C₂	semidirect
4. #1829×C₁₃₇	C₂	semidirect
5. C₂×#344×C₁₃₇	C₂	semidirect
144. #20713	C₁₃₇	direct (N × H)
145. C₁₃₇	#20713	direct (N × H)
148. #1830	C₂×C₁₃₇	semidirect
149. #1829	C₂×C₁₃₇	semidirect
150. C₂×#344	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3428×C₁₃₇	2
C₂×#413×C₁₃₇	2
#1830×C₁₃₇	2
#1829×C₁₃₇	2
C₂×#344×C₁₃₇	2
#20713	137

Subgroups — C137×#20713 #46870099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20713 #46870099