Subgroups — C₁₃₇×#20398 #46869784

Order 35072.

1658 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#518×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#100×C₁₃₇	C₂	1	2	yes	maximal
3	#2873×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	C₄×#100×C₁₃₇	C₂	1	8	yes	maximal, complemented
5	#1742×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂³×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₃₇	C₂²	1	4	yes	—
8	#518×C₁₃₇	—	2	8	no	—
9	C₂×C₄×D₄×C₁₃₇	C₄	1	4	yes	—
10	C₂×C₄×D₄×C₁₃₇	C₂²	1	4	yes	—
11	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊C₄×C₁₃₇	C₄	1	4	yes	—
14	C₂×#100×C₁₃₇	C₄	1	16	yes	complemented
15	C₂×#100×C₁₃₇	C₂²	1	8	yes	—
16	#405×C₁₃₇	—	2	8	no	—
17	C₄×M₁₆×C₁₃₇	C₂²	1	8	yes	complemented
18	#343×C₁₃₇	C₂²	1	8	yes	complemented
19	C₂⁵×C₁₃₇	C₂×C₄	1	2	yes	—
20	C₂²×D₄×C₁₃₇	—	2	4	no	—
21	C₂²×D₄×C₁₃₇	—	2	4	no	—
22	C₂²×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₂²×D₄×C₁₃₇	C₂×C₄	1	8	yes	—
24	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
25	C₂³×C₄×C₁₃₇	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₁₃₇	—	2	4	no	—
28	C₂×M₁₆×C₁₃₇	—	2	8	no	—
29	C₂×M₁₆×C₁₃₇	C₂×C₄	1	16	yes	complemented
30	C₂×M₁₆×C₁₃₇	C₂³	1	8	yes	—
31	C₂²×C₈×C₁₃₇	—	2	4	no	—
32	C₄×D₄×C₁₃₇	—	2	16	no	—
33	C₄×D₄×C₁₃₇	—	2	16	no	—
34	#117×C₁₃₇	—	2	4	no	—
35	#117×C₁₃₇	D₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
38	C₂×C₂²⋊C₄×C₁₃₇	—	2	16	no	—
39	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
41	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
44	#100×C₁₃₇	C₂×C₄	1	32	yes	complemented
45	#97×C₁₃₇	—	2	8	no	—
46	C₄×C₈×C₁₃₇	—	2	8	no	—
47	C₂⁴×C₁₃₇	—	2	4	no	—
48	C₂⁴×C₁₃₇	—	2	4	no	—
49	C₂⁴×C₁₃₇	—	8	4	no	—
50	C₂⁴×C₁₃₇	C₄²	1	8	yes	—
51	C₂⁴×C₁₃₇	C₂²×C₄	1	4	yes	—
52	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×D₄×C₁₃₇	—	2	4	no	—
54	C₂×D₄×C₁₃₇	—	2	4	no	—
55	C₂×D₄×C₁₃₇	—	2	16	no	—
56	C₂×D₄×C₁₃₇	—	2	16	no	—
57	C₂×D₄×C₁₃₇	—	2	4	no	—
58	C₂×D₄×C₁₃₇	—	2	4	no	—
59	C₂×D₄×C₁₃₇	C₄²	1	8	yes	—
60	C₂×D₄×C₁₃₇	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₁₃₇	—	2	16	no	—
62	C₂²×C₄×C₁₃₇	—	8	8	no	—
63	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
69	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
70	M₁₆×C₁₃₇	—	2	16	no	—
71	M₁₆×C₁₃₇	—	2	8	no	—
72	M₁₆×C₁₃₇	C₂²×C₄	1	16	yes	—
73	C₂×C₈×C₁₃₇	—	2	16	no	—
74	C₂×C₈×C₁₃₇	—	2	8	no	—
75	C₄⋊C₄×C₁₃₇	—	2	4	no	—
76	C₄⋊C₄×C₁₃₇	C₂²⋊C₄	1	8	yes	—
77	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
78	C₂²⋊C₄×C₁₃₇	—	2	16	no	—
79	C₂²⋊C₄×C₁₃₇	C₂²⋊C₄	1	8	yes	—
80	C₄²×C₁₃₇	—	2	4	no	—
81	C₄²×C₁₃₇	C₂×D₄	1	8	yes	—
82	C₂³×C₁₃₇	—	2	4	no	—
83	C₂³×C₁₃₇	—	2	4	no	—
84	C₂³×C₁₃₇	—	2	16	no	—
85	C₂³×C₁₃₇	—	4	2	no	—
86	C₂³×C₁₃₇	—	4	8	no	—
87	C₂³×C₁₃₇	—	8	8	no	—
88	C₂³×C₁₃₇	—	8	8	no	—
89	C₂³×C₁₃₇	—	8	8	no	—
90	C₂³×C₁₃₇	C₂×C₄²	1	8	yes	—
91	C₂³×C₁₃₇	C₂×C₄²	1	2	yes	—
92	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂³×C₁₃₇	#117	1	2	yes	—
94	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
95	D₄×C₁₃₇	—	2	16	no	—
96	D₄×C₁₃₇	—	2	16	no	—
97	C₂×C₄×C₁₃₇	—	2	4	no	—
98	C₂×C₄×C₁₃₇	—	4	4	no	—
99	C₂×C₄×C₁₃₇	—	4	4	no	—
100	C₂×C₄×C₁₃₇	—	8	16	no	—
101	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	8	yes	—
103	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂×C₄×C₁₃₇	#117	1	2	yes	—
105	C₂×C₄×C₁₃₇	C₄×D₄	1	8	yes	—
106	C₂×C₄×C₁₃₇	C₄×D₄	1	8	yes	—
107	C₂×C₄×C₁₃₇	C₂³×C₄	1	2	yes	—
108	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
109	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
110	C₈×C₁₃₇	—	2	16	no	—
111	C₂²×C₁₃₇	—	2	2	no	—
112	C₂²×C₁₃₇	—	2	4	no	—
113	C₂²×C₁₃₇	—	2	8	no	—
114	C₂²×C₁₃₇	—	4	2	no	—
115	C₂²×C₁₃₇	—	4	2	no	—
116	C₂²×C₁₃₇	—	4	8	no	—
117	C₂²×C₁₃₇	—	4	8	no	—
118	C₂²×C₁₃₇	—	4	8	no	—
119	C₂²×C₁₃₇	—	8	16	no	—
120	C₂²×C₁₃₇	C₄×C₂²⋊C₄	1	4	yes	—
121	C₂²×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
122	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₁₃₇	C₂×#117	1	1	yes	characteristic
124	C₄×C₁₃₇	—	2	8	no	—
125	C₄×C₁₃₇	—	4	4	no	—
126	C₄×C₁₃₇	C₄×C₂²⋊C₄	1	4	yes	—
127	C₄×C₁₃₇	C₂×C₄×D₄	1	4	yes	—
128	C₂×C₁₃₇	—	2	1	no	—
129	C₂×C₁₃₇	—	2	1	no	—
130	C₂×C₁₃₇	—	2	4	no	—
131	C₂×C₁₃₇	—	4	8	no	—
132	C₂×C₁₃₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₁₃₇	#2888	1	2	yes	—
134	#20398	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₁₃₇	#20398	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×#518	C₂×C₁₃₇	1	1	yes	characteristic
137	C₂²×#100	C₂×C₁₃₇	1	2	yes	—
138	#2873	C₂×C₁₃₇	1	2	yes	complemented
139	C₄×#100	C₂×C₁₃₇	1	8	yes	complemented
140	#1742	C₂×C₁₃₇	1	2	yes	complemented
141	C₂³×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
142	C₂²×M₁₆	C₂²×C₁₃₇	1	4	yes	—
143	#518	—	2	8	no	—
144	C₂×C₄×D₄	C₄×C₁₃₇	1	4	yes	—
145	C₂×C₄×D₄	C₂²×C₁₃₇	1	4	yes	—
146	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
147	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
148	C₂²×C₂²⋊C₄	C₄×C₁₃₇	1	4	yes	—
149	C₂×#100	C₄×C₁₃₇	1	16	yes	complemented
150	C₂×#100	C₂²×C₁₃₇	1	8	yes	—
151	#405	—	2	8	no	—
152	C₄×M₁₆	C₂²×C₁₃₇	1	8	yes	complemented
153	#343	C₂²×C₁₃₇	1	8	yes	complemented
154	C₂⁵	C₂×C₄×C₁₃₇	1	2	yes	—
155	C₂²×D₄	—	2	4	no	—
156	C₂²×D₄	—	2	4	no	—
157	C₂²×D₄	C₂×C₄×C₁₃₇	1	2	yes	—
158	C₂²×D₄	C₂×C₄×C₁₃₇	1	8	yes	—
159	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
160	C₂³×C₄	C₂×C₄×C₁₃₇	1	8	yes	—
161	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	—	2	8	no	—
164	C₂×M₁₆	C₂×C₄×C₁₃₇	1	16	yes	complemented
165	C₂×M₁₆	C₂³×C₁₃₇	1	8	yes	—
166	C₂²×C₈	—	2	4	no	—
167	C₄×D₄	—	2	16	no	—
168	C₄×D₄	—	2	16	no	—
169	#117	—	2	4	no	—
170	#117	D₄×C₁₃₇	1	8	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
172	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
173	C₂×C₂²⋊C₄	—	2	16	no	—
174	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	—
176	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
178	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
179	#100	C₂×C₄×C₁₃₇	1	32	yes	complemented
180	#97	—	2	8	no	—
181	C₄×C₈	—	2	8	no	—
182	C₂⁴	—	2	4	no	—
183	C₂⁴	—	2	4	no	—
184	C₂⁴	—	8	4	no	—
185	C₂⁴	C₄²×C₁₃₇	1	8	yes	—
186	C₂⁴	C₂²×C₄×C₁₃₇	1	4	yes	—
187	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	16	no	—
191	C₂×D₄	—	2	16	no	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	C₄²×C₁₃₇	1	8	yes	—
195	C₂×D₄	C₂²×C₄×C₁₃₇	1	8	yes	—
196	C₂²×C₄	—	2	16	no	—
197	C₂²×C₄	—	8	8	no	—
198	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
200	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
202	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
203	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
204	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
205	M₁₆	—	2	16	no	—
206	M₁₆	—	2	8	no	—
207	M₁₆	C₂²×C₄×C₁₃₇	1	16	yes	—
208	C₂×C₈	—	2	16	no	—
209	C₂×C₈	—	2	8	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	C₂²⋊C₄×C₁₃₇	1	8	yes	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	2	16	no	—
214	C₂²⋊C₄	C₂²⋊C₄×C₁₃₇	1	8	yes	—
215	C₄²	—	2	4	no	—
216	C₄²	C₂×D₄×C₁₃₇	1	8	yes	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	16	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	8	no	—
222	C₂³	—	8	8	no	—
223	C₂³	—	8	8	no	—
224	C₂³	—	8	8	no	—
225	C₂³	C₂×C₄²×C₁₃₇	1	8	yes	—
226	C₂³	C₂×C₄²×C₁₃₇	1	2	yes	—
227	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
228	C₂³	#117×C₁₃₇	1	2	yes	—
229	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
230	D₄	—	2	16	no	—
231	D₄	—	2	16	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	16	no	—
236	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	8	yes	—
238	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
239	C₂×C₄	#117×C₁₃₇	1	2	yes	—
240	C₂×C₄	C₄×D₄×C₁₃₇	1	8	yes	—
241	C₂×C₄	C₄×D₄×C₁₃₇	1	8	yes	—
242	C₂×C₄	C₂³×C₄×C₁₃₇	1	2	yes	—
243	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
244	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
245	C₈	—	2	16	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	8	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	8	no	—
252	C₂²	—	4	8	no	—
253	C₂²	—	4	8	no	—
254	C₂²	—	8	16	no	—
255	C₂²	C₄×C₂²⋊C₄×C₁₃₇	1	4	yes	—
256	C₂²	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
257	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
258	C₂²	C₂×#117×C₁₃₇	1	1	yes	characteristic
259	C₄	—	2	8	no	—
260	C₄	—	4	4	no	—
261	C₄	C₄×C₂²⋊C₄×C₁₃₇	1	4	yes	—
262	C₄	C₂×C₄×D₄×C₁₃₇	1	4	yes	—
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	4	no	—
266	C₂	—	4	8	no	—
267	C₂	C₂×C₄×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
268	C₂	#2888×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂³×D₄×C₁₃₇	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×#117×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #2873×C₁₃₇	C₂	semidirect
4. C₄×#100×C₁₃₇	C₂	semidirect
5. #1742×C₁₃₇	C₂	semidirect
14. C₂×#100×C₁₃₇	C₄	semidirect
17. C₄×M₁₆×C₁₃₇	C₂²	semidirect
18. #343×C₁₃₇	C₂²	semidirect
29. C₂×M₁₆×C₁₃₇	C₂×C₄	semidirect
44. #100×C₁₃₇	C₂×C₄	semidirect
134. #20398	C₁₃₇	direct (N × H)
135. C₁₃₇	#20398	direct (N × H)
138. #2873	C₂×C₁₃₇	semidirect
139. C₄×#100	C₂×C₁₃₇	semidirect
140. #1742	C₂×C₁₃₇	semidirect
149. C₂×#100	C₄×C₁₃₇	semidirect
152. C₄×M₁₆	C₂²×C₁₃₇	semidirect
153. #343	C₂²×C₁₃₇	semidirect
164. C₂×M₁₆	C₂×C₄×C₁₃₇	semidirect
179. #100	C₂×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×#518×C₁₃₇	2
C₂²×#100×C₁₃₇	2
#2873×C₁₃₇	2
C₄×#100×C₁₃₇	2
#1742×C₁₃₇	2
#20398	137

Subgroups — C137×#20398 #46869784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20398 #46869784