Subgroups — C₁₃₇×#20354 #46869740

Order 35072.

2410 conjugacy classes of subgroups.

The subgroup lattice has 256 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	2	yes	maximal
2	C₂×#518×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1740×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	C₂×#342×C₁₃₇	C₂	1	8	yes	maximal, complemented
5	C₂³×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
6	#518×C₁₃₇	—	2	8	no	—
7	#518×C₁₃₇	—	2	8	no	—
8	#518×C₁₃₇	C₄	1	16	yes	complemented
9	C₂×C₄×D₄×C₁₃₇	—	2	8	no	—
10	C₂×C₄×D₄×C₁₃₇	C₄	1	8	yes	complemented
11	C₂×#117×C₁₃₇	C₂²	1	4	yes	—
12	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×C₂²⋊C₄×C₁₃₇	C₂²	1	4	yes	—
14	C₂²×C₂²⋊C₄×C₁₃₇	C₂²	1	8	yes	—
15	#342×C₁₃₇	C₂²	1	16	yes	—
16	C₂⁵×C₁₃₇	C₂³	1	2	yes	—
17	C₂²×D₄×C₁₃₇	C₂×C₄	1	4	yes	—
18	C₂²×D₄×C₁₃₇	C₂×C₄	1	8	yes	—
19	C₂²×D₄×C₁₃₇	D₄	1	8	yes	—
20	C₂²×D₄×C₁₃₇	D₄	1	4	yes	—
21	C₂²×D₄×C₁₃₇	Q₈	1	4	yes	—
22	C₂³×C₄×C₁₃₇	—	2	8	no	—
23	C₂³×C₄×C₁₃₇	C₂×C₄	1	8	yes	complemented
24	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₄×D₄×C₁₃₇	—	2	64	no	—
26	C₄×D₄×C₁₃₇	—	2	32	no	—
27	#117×C₁₃₇	—	2	8	no	—
28	#117×C₁₃₇	—	2	8	no	—
29	#117×C₁₃₇	C₂×C₄	1	16	yes	complemented
30	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
31	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	complemented
32	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
33	C₂×C₂²⋊C₄×C₁₃₇	—	2	16	no	—
34	C₂×C₂²⋊C₄×C₁₃₇	—	2	16	no	—
35	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	complemented
36	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	32	yes	complemented
37	C₂×C₂²⋊C₄×C₁₃₇	C₂³	1	4	yes	—
38	C₂×C₂²⋊C₄×C₁₃₇	C₂³	1	8	yes	—
39	C₂×C₄²×C₁₃₇	—	2	4	no	—
40	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	complemented
41	C₂⁴×C₁₃₇	—	8	4	no	—
42	C₂⁴×C₁₃₇	C₂²×C₄	1	4	yes	—
43	C₂⁴×C₁₃₇	C₂²×C₄	1	8	yes	—
44	C₂⁴×C₁₃₇	C₂×D₄	1	8	yes	—
45	C₂⁴×C₁₃₇	C₂×D₄	1	4	yes	—
46	C₂⁴×C₁₃₇	C₂×Q₈	1	4	yes	—
47	C₂⁴×C₁₃₇	C₂⁴	1	1	yes	characteristic
48	C₂×D₄×C₁₃₇	C₄²	1	16	yes	—
49	C₂×D₄×C₁₃₇	C₂²⋊C₄	1	32	yes	—
50	C₂×D₄×C₁₃₇	C₂²⋊C₄	1	16	yes	—
51	C₂×D₄×C₁₃₇	C₄⋊C₄	1	16	yes	—
52	C₂×D₄×C₁₃₇	C₄⋊C₄	1	32	yes	—
53	C₂²×C₄×C₁₃₇	—	2	8	no	—
54	C₂²×C₄×C₁₃₇	—	2	16	no	—
55	C₂²×C₄×C₁₃₇	—	2	32	no	—
56	C₂²×C₄×C₁₃₇	—	8	16	no	—
57	C₂²×C₄×C₁₃₇	—	8	8	no	—
58	C₂²×C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
64	C₄⋊C₄×C₁₃₇	—	2	16	no	—
65	C₄⋊C₄×C₁₃₇	—	2	8	no	—
66	C₂²⋊C₄×C₁₃₇	—	2	32	no	—
67	C₂²⋊C₄×C₁₃₇	—	2	16	no	—
68	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
69	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
70	C₂²⋊C₄×C₁₃₇	C₂²×C₄	1	16	yes	—
71	C₄²×C₁₃₇	—	2	16	no	—
72	C₄²×C₁₃₇	—	2	8	no	—
73	C₂³×C₁₃₇	—	4	8	no	—
74	C₂³×C₁₃₇	—	4	16	no	—
75	C₂³×C₁₃₇	—	4	2	no	—
76	C₂³×C₁₃₇	—	8	8	no	—
77	C₂³×C₁₃₇	—	8	4	no	—
78	C₂³×C₁₃₇	—	8	4	no	—
79	C₂³×C₁₃₇	C₂×C₄²	1	8	yes	—
80	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	8	yes	—
81	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	16	yes	—
82	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	8	yes	—
83	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	16	yes	—
84	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
85	C₂³×C₁₃₇	C₂³×C₄	1	2	yes	—
86	C₂³×C₁₃₇	C₂²×D₄	1	2	yes	—
87	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
88	C₂³×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
89	D₄×C₁₃₇	#95	1	64	yes	—
90	C₂×C₄×C₁₃₇	—	2	16	no	—
91	C₂×C₄×C₁₃₇	—	2	8	no	—
92	C₂×C₄×C₁₃₇	—	4	16	no	—
93	C₂×C₄×C₁₃₇	—	4	8	no	—
94	C₂×C₄×C₁₃₇	—	8	16	no	—
95	C₂×C₄×C₁₃₇	—	8	32	no	—
96	C₂×C₄×C₁₃₇	C₂×C₄²	1	4	yes	—
97	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	8	yes	—
98	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	8	yes	—
101	C₂²×C₁₃₇	—	2	2	no	—
102	C₂²×C₁₃₇	—	2	4	no	—
103	C₂²×C₁₃₇	—	2	16	no	—
104	C₂²×C₁₃₇	—	4	16	no	—
105	C₂²×C₁₃₇	—	4	2	no	—
106	C₂²×C₁₃₇	—	4	1	no	—
107	C₂²×C₁₃₇	—	4	1	no	—
108	C₂²×C₁₃₇	—	4	8	no	—
109	C₂²×C₁₃₇	—	4	8	no	—
110	C₂²×C₁₃₇	—	4	16	no	—
111	C₂²×C₁₃₇	C₂×#95	1	16	yes	—
112	C₂²×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
113	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	2	yes	—
114	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	2	yes	—
117	C₄×C₁₃₇	—	4	16	no	—
118	C₄×C₁₃₇	—	4	8	no	—
119	C₄×C₁₃₇	C₂×#95	1	8	yes	—
120	C₂×C₁₃₇	—	2	2	no	—
121	C₂×C₁₃₇	—	2	1	no	—
122	C₂×C₁₃₇	—	2	2	no	—
123	C₂×C₁₃₇	—	2	1	no	—
124	C₂×C₁₃₇	—	2	16	no	—
125	C₂×C₁₃₇	C₂²×#95	1	1	yes	characteristic
126	C₂×C₁₃₇	#2883	1	2	yes	—
127	#20354	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₁₃₇	#20354	1	1	yes	characteristic, direct product factor, Sylow
129	C₂×#518	C₂×C₁₃₇	1	2	yes	—
130	C₂×#518	C₂×C₁₃₇	1	1	yes	characteristic
131	#1740	C₂×C₁₃₇	1	4	yes	complemented
132	C₂×#342	C₂×C₁₃₇	1	8	yes	complemented
133	C₂³×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
134	#518	—	2	8	no	—
135	#518	—	2	8	no	—
136	#518	C₄×C₁₃₇	1	16	yes	complemented
137	C₂×C₄×D₄	—	2	8	no	—
138	C₂×C₄×D₄	C₄×C₁₃₇	1	8	yes	complemented
139	C₂×#117	C₂²×C₁₃₇	1	4	yes	—
140	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
141	C₂²×C₂²⋊C₄	C₂²×C₁₃₇	1	4	yes	—
142	C₂²×C₂²⋊C₄	C₂²×C₁₃₇	1	8	yes	—
143	#342	C₂²×C₁₃₇	1	16	yes	—
144	C₂⁵	C₂³×C₁₃₇	1	2	yes	—
145	C₂²×D₄	C₂×C₄×C₁₃₇	1	4	yes	—
146	C₂²×D₄	C₂×C₄×C₁₃₇	1	8	yes	—
147	C₂²×D₄	D₄×C₁₃₇	1	8	yes	—
148	C₂²×D₄	D₄×C₁₃₇	1	4	yes	—
149	C₂²×D₄	Q₈×C₁₃₇	1	4	yes	—
150	C₂³×C₄	—	2	8	no	—
151	C₂³×C₄	C₂×C₄×C₁₃₇	1	8	yes	complemented
152	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
153	C₄×D₄	—	2	64	no	—
154	C₄×D₄	—	2	32	no	—
155	#117	—	2	8	no	—
156	#117	—	2	8	no	—
157	#117	C₂×C₄×C₁₃₇	1	16	yes	complemented
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	complemented
160	C₂×C₂²⋊C₄	—	2	8	no	—
161	C₂×C₂²⋊C₄	—	2	16	no	—
162	C₂×C₂²⋊C₄	—	2	16	no	—
163	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	complemented
164	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	32	yes	complemented
165	C₂×C₂²⋊C₄	C₂³×C₁₃₇	1	4	yes	—
166	C₂×C₂²⋊C₄	C₂³×C₁₃₇	1	8	yes	—
167	C₂×C₄²	—	2	4	no	—
168	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	complemented
169	C₂⁴	—	8	4	no	—
170	C₂⁴	C₂²×C₄×C₁₃₇	1	4	yes	—
171	C₂⁴	C₂²×C₄×C₁₃₇	1	8	yes	—
172	C₂⁴	C₂×D₄×C₁₃₇	1	8	yes	—
173	C₂⁴	C₂×D₄×C₁₃₇	1	4	yes	—
174	C₂⁴	C₂×Q₈×C₁₃₇	1	4	yes	—
175	C₂⁴	C₂⁴×C₁₃₇	1	1	yes	characteristic
176	C₂×D₄	C₄²×C₁₃₇	1	16	yes	—
177	C₂×D₄	C₂²⋊C₄×C₁₃₇	1	16	yes	—
178	C₂×D₄	C₂²⋊C₄×C₁₃₇	1	32	yes	—
179	C₂×D₄	C₄⋊C₄×C₁₃₇	1	16	yes	—
180	C₂×D₄	C₄⋊C₄×C₁₃₇	1	32	yes	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	16	no	—
183	C₂²×C₄	—	2	32	no	—
184	C₂²×C₄	—	8	16	no	—
185	C₂²×C₄	—	8	8	no	—
186	C₂²×C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
187	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
189	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
190	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
191	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
192	C₄⋊C₄	—	2	16	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₂²⋊C₄	—	2	16	no	—
195	C₂²⋊C₄	—	2	32	no	—
196	C₂²⋊C₄	—	2	8	no	—
197	C₂²⋊C₄	—	2	8	no	—
198	C₂²⋊C₄	C₂²×C₄×C₁₃₇	1	16	yes	—
199	C₄²	—	2	16	no	—
200	C₄²	—	2	8	no	—
201	C₂³	—	4	8	no	—
202	C₂³	—	4	16	no	—
203	C₂³	—	4	2	no	—
204	C₂³	—	8	8	no	—
205	C₂³	—	8	4	no	—
206	C₂³	—	8	4	no	—
207	C₂³	C₂×C₄²×C₁₃₇	1	8	yes	—
208	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	8	yes	—
209	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	16	yes	—
210	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	8	yes	—
211	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	16	yes	—
212	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
213	C₂³	C₂³×C₄×C₁₃₇	1	2	yes	—
214	C₂³	C₂²×D₄×C₁₃₇	1	2	yes	—
215	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
216	C₂³	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
217	D₄	#95×C₁₃₇	1	64	yes	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	16	no	—
220	C₂×C₄	—	4	16	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	—	8	16	no	—
223	C₂×C₄	—	8	32	no	—
224	C₂×C₄	C₂×C₄²×C₁₃₇	1	4	yes	—
225	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	8	yes	—
226	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
227	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
228	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	8	yes	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	16	no	—
232	C₂²	—	4	16	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	1	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	8	no	—
237	C₂²	—	4	8	no	—
238	C₂²	—	4	16	no	—
239	C₂²	C₂×#95×C₁₃₇	1	16	yes	—
240	C₂²	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
241	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	2	yes	—
242	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
243	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
244	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	2	yes	—
245	C₄	—	4	16	no	—
246	C₄	—	4	8	no	—
247	C₄	C₂×#95×C₁₃₇	1	8	yes	—
248	C₂	—	2	2	no	—
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	—	2	2	no	—
252	C₂	—	2	16	no	—
253	C₂	C₂²×#95×C₁₃₇	1	1	yes	characteristic
254	C₂	#2883×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #1740×C₁₃₇	C₂	semidirect
4. C₂×#342×C₁₃₇	C₂	semidirect
8. #518×C₁₃₇	C₄	semidirect
10. C₂×C₄×D₄×C₁₃₇	C₄	semidirect
23. C₂³×C₄×C₁₃₇	C₂×C₄	semidirect
29. #117×C₁₃₇	C₂×C₄	semidirect
31. C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	semidirect
35. C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	semidirect
36. C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	semidirect
40. C₂×C₄²×C₁₃₇	C₂×C₄	semidirect
127. #20354	C₁₃₇	direct (N × H)
128. C₁₃₇	#20354	direct (N × H)
131. #1740	C₂×C₁₃₇	semidirect
132. C₂×#342	C₂×C₁₃₇	semidirect
136. #518	C₄×C₁₃₇	semidirect
138. C₂×C₄×D₄	C₄×C₁₃₇	semidirect
151. C₂³×C₄	C₂×C₄×C₁₃₇	semidirect
157. #117	C₂×C₄×C₁₃₇	semidirect
159. C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	semidirect
163. C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	semidirect
164. C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	semidirect
168. C₂×C₄²	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
#1740×C₁₃₇	2
C₂×#342×C₁₃₇	2
#20354	137

Subgroups — C137×#20354 #46869740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20354 #46869740