Subgroups — C₂×#1330×C₁₃₇ #46860971

Order 35072.

422 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#346×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1330×C₁₃₇	C₂	1	4	yes	maximal, direct product factor
5	C₂²×M₁₆×C₁₃₇	—	2	1	no	—
6	C₂×C₄×D₄×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₃₇	—	2	1	no	—
8	C₂²×C₁₆×C₁₃₇	—	2	1	no	—
9	#436×C₁₃₇	—	2	1	no	—
10	#436×C₁₃₇	—	2	1	no	—
11	#436×C₁₃₇	C₄	1	2	yes	—
12	#436×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₁₃₇	C₄	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₁₃₇	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₁₃₇	—	2	1	no	—
17	C₄⋊C₁₆×C₁₃₇	C₂²	1	2	yes	complemented
18	#346×C₁₃₇	—	2	1	no	—
19	#346×C₁₃₇	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₁₃₇	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₃₇	—	2	1	no	—
22	C₂×M₁₆×C₁₃₇	—	2	2	no	—
23	C₂×M₁₆×C₁₃₇	—	2	2	no	—
24	C₂×M₁₆×C₁₃₇	—	4	2	no	—
25	C₂×M₁₆×C₁₃₇	—	4	2	no	—
26	C₂²×C₈×C₁₃₇	—	2	1	no	—
27	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
29	C₄×D₄×C₁₃₇	—	2	1	no	—
30	C₄×D₄×C₁₃₇	—	2	1	no	—
31	C₄×D₄×C₁₃₇	C₈	1	2	yes	—
32	C₄×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₃₇	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
35	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₁₃₇	—	2	4	no	—
37	C₂×C₁₆×C₁₃₇	—	2	2	no	—
38	C₂×C₁₆×C₁₃₇	—	2	2	no	—
39	C₂×C₁₆×C₁₃₇	—	4	2	no	—
40	C₄⋊C₈×C₁₃₇	—	2	1	no	—
41	C₄⋊C₈×C₁₃₇	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
43	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
44	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂⁴×C₁₃₇	—	2	1	no	—
49	C₂×D₄×C₁₃₇	—	2	1	no	—
50	C₂×D₄×C₁₃₇	—	4	2	no	—
51	C₂×D₄×C₁₃₇	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₃₇	—	2	1	no	—
53	C₂²×C₄×C₁₃₇	—	2	2	no	—
54	C₂²×C₄×C₁₃₇	—	2	2	no	—
55	C₂²×C₄×C₁₃₇	—	4	2	no	—
56	C₂²×C₄×C₁₃₇	—	4	2	no	—
57	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
60	M₁₆×C₁₃₇	—	4	4	no	—
61	M₁₆×C₁₃₇	—	4	2	no	—
62	M₁₆×C₁₃₇	—	4	2	no	—
63	C₂×C₈×C₁₃₇	—	2	2	no	—
64	C₂×C₈×C₁₃₇	—	2	2	no	—
65	C₂×C₈×C₁₃₇	—	4	2	no	—
66	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃₇	D₈	1	2	yes	—
69	C₂×C₈×C₁₃₇	SD₁₆	1	2	yes	—
70	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₃₇	—	2	1	no	—
73	C₄⋊C₄×C₁₃₇	C₂×C₈	1	2	yes	—
74	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
75	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
76	C₄²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₁₃₇	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₁₃₇	M₁₆	1	1	yes	characteristic
79	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
80	C₁₆×C₁₃₇	—	2	4	no	—
81	C₁₆×C₁₃₇	—	4	2	no	—
82	C₂³×C₁₃₇	—	2	2	no	—
83	C₂³×C₁₃₇	—	4	2	no	—
84	C₂³×C₁₃₇	—	4	2	no	—
85	C₂³×C₁₃₇	—	4	2	no	—
86	C₂³×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
87	D₄×C₁₃₇	—	4	2	no	—
88	D₄×C₁₃₇	—	4	2	no	—
89	C₂×C₄×C₁₃₇	—	2	2	no	—
90	C₂×C₄×C₁₃₇	—	2	2	no	—
91	C₂×C₄×C₁₃₇	—	2	2	no	—
92	C₂×C₄×C₁₃₇	—	4	2	no	—
93	C₂×C₄×C₁₃₇	—	4	4	no	—
94	C₂×C₄×C₁₃₇	—	4	2	no	—
95	C₂×C₄×C₁₃₇	—	4	2	no	—
96	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	#102	1	2	yes	—
100	C₂×C₄×C₁₃₇	#104	1	2	yes	—
101	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
104	C₈×C₁₃₇	—	2	1	no	—
105	C₈×C₁₃₇	—	2	1	no	—
106	C₈×C₁₃₇	—	4	2	no	—
107	C₈×C₁₃₇	#102	1	2	yes	—
108	C₈×C₁₃₇	C₂×D₈	1	1	yes	characteristic
109	C₈×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
110	C₂²×C₁₃₇	—	4	2	no	—
111	C₂²×C₁₃₇	—	4	2	no	—
112	C₂²×C₁₃₇	—	4	2	no	—
113	C₂²×C₁₃₇	—	8	4	no	—
114	C₂²×C₁₃₇	D₄⋊C₈	1	2	yes	—
115	C₂²×C₁₃₇	#349	1	2	yes	—
116	C₂²×C₁₃₇	#350	1	2	yes	—
117	C₂²×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₄×C₁₃₇	—	2	1	no	—
119	C₄×C₁₃₇	—	2	1	no	—
120	C₄×C₁₃₇	—	2	1	no	—
121	C₄×C₁₃₇	—	2	1	no	—
122	C₄×C₁₃₇	—	4	2	no	—
123	C₄×C₁₃₇	D₄⋊C₈	1	2	yes	—
124	C₄×C₁₃₇	C₂×#102	1	1	yes	characteristic
125	C₄×C₁₃₇	C₂×#104	1	1	yes	characteristic
126	C₂×C₁₃₇	—	8	2	no	—
127	C₂×C₁₃₇	#1330	1	4	yes	direct product factor
128	C₂×C₁₃₇	C₂×D₄⋊C₈	1	1	yes	characteristic
129	C₂×C₁₃₇	C₂×#349	1	1	yes	characteristic
130	C₂×C₁₃₇	C₂×#350	1	1	yes	characteristic
131	C₂×#1330	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₁₃₇	C₂×#1330	1	1	yes	characteristic, direct product factor, Sylow
133	C₂×#436	C₂×C₁₃₇	1	1	yes	characteristic
134	C₂×C₄⋊C₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
135	C₂×#346	C₂×C₁₃₇	1	1	yes	characteristic, complemented
136	#1330	C₂×C₁₃₇	1	4	yes	direct product factor
137	C₂²×M₁₆	—	2	1	no	—
138	C₂×C₄×D₄	C₄×C₁₃₇	1	1	yes	characteristic
139	C₂²×C₁₆	—	2	1	no	—
140	C₂²×C₁₆	—	2	1	no	—
141	#436	—	2	1	no	—
142	#436	—	2	1	no	—
143	#436	C₄×C₁₃₇	1	2	yes	—
144	#436	C₂²×C₁₃₇	1	2	yes	—
145	C₂×C₄⋊C₈	C₄×C₁₃₇	1	1	yes	characteristic
146	C₂×C₂²⋊C₈	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
148	C₄⋊C₁₆	—	2	1	no	—
149	C₄⋊C₁₆	C₂²×C₁₃₇	1	2	yes	complemented
150	#346	—	2	1	no	—
151	#346	C₂²×C₁₃₇	1	2	yes	complemented
152	C₂²×D₄	C₈×C₁₃₇	1	1	yes	characteristic
153	C₂³×C₄	—	2	1	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	4	2	no	—
157	C₂×M₁₆	—	4	2	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
160	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	C₈×C₁₃₇	1	2	yes	—
164	C₄×D₄	C₂×C₄×C₁₃₇	1	2	yes	—
165	C₂×C₄⋊C₄	C₈×C₁₃₇	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	—	2	1	no	—
167	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
168	C₂×C₁₆	—	2	4	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	4	2	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	C₂×C₄×C₁₃₇	1	2	yes	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
177	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
178	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
179	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
180	C₂⁴	—	2	1	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	C₂×C₈×C₁₃₇	1	2	yes	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
191	C₂²×C₄	M₁₆×C₁₃₇	1	1	yes	characteristic
192	M₁₆	—	4	4	no	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
200	C₂×C₈	D₈×C₁₃₇	1	2	yes	—
201	C₂×C₈	SD₁₆×C₁₃₇	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
203	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	C₂×C₈×C₁₃₇	1	2	yes	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₄²	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
209	C₄²	C₂×C₈×C₁₃₇	1	1	yes	characteristic
210	C₄²	M₁₆×C₁₃₇	1	1	yes	characteristic
211	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
212	C₁₆	—	2	4	no	—
213	C₁₆	—	4	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	2	no	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
219	D₄	—	4	2	no	—
220	D₄	—	4	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
229	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
230	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
231	C₂×C₄	#102×C₁₃₇	1	2	yes	—
232	C₂×C₄	#104×C₁₃₇	1	2	yes	—
233	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
235	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	4	2	no	—
239	C₈	#102×C₁₃₇	1	2	yes	—
240	C₈	C₂×D₈×C₁₃₇	1	1	yes	characteristic
241	C₈	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	8	4	no	—
246	C₂²	D₄⋊C₈×C₁₃₇	1	2	yes	—
247	C₂²	#349×C₁₃₇	1	2	yes	—
248	C₂²	#350×C₁₃₇	1	2	yes	—
249	C₂²	C₂×C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	2	no	—
255	C₄	D₄⋊C₈×C₁₃₇	1	2	yes	—
256	C₄	C₂×#102×C₁₃₇	1	1	yes	characteristic
257	C₄	C₂×#104×C₁₃₇	1	1	yes	characteristic
258	C₂	—	8	2	no	—
259	C₂	#1330×C₁₃₇	1	4	yes	direct product factor
260	C₂	C₂×D₄⋊C₈×C₁₃₇	1	1	yes	characteristic
261	C₂	C₂×#349×C₁₃₇	1	1	yes	characteristic
262	C₂	C₂×#350×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₃₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₃₇
socle	C₂³×C₁₃₇	C₂²⋊C₈
prime radical	C₂²×D₄×C₁₃₇	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊C₁₆×C₁₃₇	C₂	semidirect
3. C₂×#346×C₁₃₇	C₂	semidirect
4. #1330×C₁₃₇	C₂	direct (N × H)
17. C₄⋊C₁₆×C₁₃₇	C₂²	semidirect
19. #346×C₁₃₇	C₂²	semidirect
127. C₂×C₁₃₇	#1330	direct (N × H)
131. C₂×#1330	C₁₃₇	direct (N × H)
132. C₁₃₇	C₂×#1330	direct (N × H)
134. C₂×C₄⋊C₁₆	C₂×C₁₃₇	semidirect
135. C₂×#346	C₂×C₁₃₇	semidirect
136. #1330	C₂×C₁₃₇	direct (N × H)
149. C₄⋊C₁₆	C₂²×C₁₃₇	semidirect
151. #346	C₂²×C₁₃₇	semidirect
259. C₂	#1330×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#436×C₁₃₇	2
C₂×C₄⋊C₁₆×C₁₃₇	2
C₂×#346×C₁₃₇	2
#1330×C₁₃₇	2
C₂×#1330	137

Subgroups — C2×#1330×C137 #46860971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1330×C₁₃₇ #46860971