Subgroups — C₂×#1418×C₁₃₇ #46863007

Order 35072.

302 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#367×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1418×C₁₃₇	C₂	1	4	yes	maximal, direct product factor
5	C₂×#135×C₁₃₇	—	2	1	no	—
6	#508×C₁₃₇	—	2	1	no	—
7	#508×C₁₃₇	—	2	1	no	—
8	#508×C₁₃₇	C₄	1	2	yes	complemented
9	#508×C₁₃₇	C₂²	1	2	yes	—
10	C₂×Q₃₂×C₁₃₇	C₄	1	1	yes	characteristic, complemented
11	C₂×SD₃₂×C₁₃₇	—	2	1	no	—
12	C₂×D₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#107×C₁₃₇	—	2	1	no	—
15	M₆₄×C₁₃₇	—	2	1	no	—
16	M₆₄×C₁₃₇	C₂²	1	2	yes	complemented
17	C₂×C₃₂×C₁₃₇	—	2	1	no	—
18	#367×C₁₃₇	—	2	1	no	—
19	#367×C₁₃₇	C₂²	1	2	yes	complemented
20	C₂×#41×C₁₃₇	—	4	1	no	—
21	#135×C₁₃₇	—	2	2	no	—
22	#135×C₁₃₇	—	2	2	no	—
23	#135×C₁₃₇	—	4	1	no	—
24	#135×C₁₃₇	—	4	1	no	—
25	C₂×Q₁₆×C₁₃₇	—	2	1	no	—
26	C₂×SD₁₆×C₁₃₇	—	4	1	no	—
27	C₂×D₈×C₁₃₇	—	2	1	no	—
28	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₃₇	—	4	1	no	—
30	Q₃₂×C₁₃₇	—	2	1	no	—
31	Q₃₂×C₁₃₇	C₂×C₄	1	2	yes	complemented
32	SD₃₂×C₁₃₇	—	2	2	no	—
33	SD₃₂×C₁₃₇	—	2	2	no	—
34	D₁₆×C₁₃₇	—	2	1	no	—
35	D₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
36	C₂×C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
42	#107×C₁₃₇	—	2	2	no	—
43	#107×C₁₃₇	—	4	1	no	—
44	C₃₂×C₁₃₇	—	2	2	no	—
45	#41×C₁₃₇	—	4	2	no	—
46	#41×C₁₃₇	—	4	2	no	—
47	#41×C₁₃₇	—	8	1	no	—
48	#41×C₁₃₇	—	8	1	no	—
49	C₂×Q₈×C₁₃₇	—	4	1	no	—
50	C₂×D₄×C₁₃₇	—	4	1	no	—
51	C₂×D₄×C₁₃₇	—	8	1	no	—
52	C₂²×C₄×C₁₃₇	—	8	1	no	—
53	C₂²×C₄×C₁₃₇	—	8	1	no	—
54	C₂²×C₄×C₁₃₇	D₈	1	1	yes	characteristic
55	Q₁₆×C₁₃₇	—	2	2	no	—
56	Q₁₆×C₁₃₇	—	4	1	no	—
57	SD₁₆×C₁₃₇	—	4	2	no	—
58	SD₁₆×C₁₃₇	—	4	2	no	—
59	D₈×C₁₃₇	—	2	2	no	—
60	D₈×C₁₃₇	—	4	1	no	—
61	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃₇	D₈	1	1	yes	characteristic
64	C₂×C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
65	C₂×C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
66	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₁₃₇	—	4	2	no	—
68	C₄⋊C₄×C₁₃₇	—	8	1	no	—
69	C₁₆×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₁₆×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₁₆×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
72	C₁₆×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	C₂³×C₁₃₇	—	8	1	no	—
74	C₂³×C₁₃₇	D₁₆	1	1	yes	characteristic
75	Q₈×C₁₃₇	—	4	2	no	—
76	Q₈×C₁₃₇	—	8	1	no	—
77	D₄×C₁₃₇	—	4	2	no	—
78	D₄×C₁₃₇	—	8	2	no	—
79	D₄×C₁₃₇	—	8	2	no	—
80	D₄×C₁₃₇	—	8	1	no	—
81	C₂×C₄×C₁₃₇	—	8	1	no	—
82	C₂×C₄×C₁₃₇	—	8	2	no	—
83	C₂×C₄×C₁₃₇	—	8	2	no	—
84	C₂×C₄×C₁₃₇	—	8	2	no	—
85	C₂×C₄×C₁₃₇	—	16	2	no	—
86	C₂×C₄×C₁₃₇	#102	1	1	yes	characteristic
87	C₂×C₄×C₁₃₇	#102	1	1	yes	characteristic
88	C₂×C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	SD₃₂	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	SD₃₂	1	1	yes	characteristic
91	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
92	C₈×C₁₃₇	#102	1	1	yes	characteristic
93	C₈×C₁₃₇	#102	1	1	yes	characteristic
94	C₈×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₈×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₂²×C₁₃₇	—	2	2	no	—
97	C₂²×C₁₃₇	—	8	2	no	—
98	C₂²×C₁₃₇	—	16	2	no	—
99	C₂²×C₁₃₇	#357	1	1	yes	characteristic
100	C₂²×C₁₃₇	#357	1	1	yes	characteristic
101	C₂²×C₁₃₇	C₂×D₁₆	1	1	yes	characteristic
102	C₄×C₁₃₇	—	8	2	no	—
103	C₄×C₁₃₇	—	16	2	no	—
104	C₄×C₁₃₇	#357	1	1	yes	characteristic
105	C₄×C₁₃₇	#357	1	1	yes	characteristic
106	C₄×C₁₃₇	C₂×#102	1	1	yes	characteristic
107	C₄×C₁₃₇	C₂×SD₃₂	1	1	yes	characteristic
108	C₂×C₁₃₇	—	2	1	no	—
109	C₂×C₁₃₇	—	2	1	no	—
110	C₂×C₁₃₇	—	16	2	no	—
111	C₂×C₁₃₇	#1418	1	2	yes	direct product factor
112	C₂×C₁₃₇	C₂×#357	1	1	yes	characteristic
113	C₂×#1418	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₁₃₇	C₂×#1418	1	1	yes	characteristic, direct product factor, Sylow
115	C₂×#508	C₂×C₁₃₇	1	1	yes	characteristic
116	C₂×M₆₄	C₂×C₁₃₇	1	1	yes	characteristic, complemented
117	C₂×#367	C₂×C₁₃₇	1	1	yes	characteristic, complemented
118	#1418	C₂×C₁₃₇	1	4	yes	direct product factor
119	C₂×#135	—	2	1	no	—
120	#508	—	2	1	no	—
121	#508	—	2	1	no	—
122	#508	C₄×C₁₃₇	1	2	yes	complemented
123	#508	C₂²×C₁₃₇	1	2	yes	—
124	C₂×Q₃₂	C₄×C₁₃₇	1	1	yes	characteristic, complemented
125	C₂×SD₃₂	—	2	1	no	—
126	C₂×D₁₆	C₄×C₁₃₇	1	1	yes	characteristic, complemented
127	C₂²×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
128	C₂×#107	—	2	1	no	—
129	M₆₄	—	2	1	no	—
130	M₆₄	C₂²×C₁₃₇	1	2	yes	complemented
131	C₂×C₃₂	—	2	1	no	—
132	#367	—	2	1	no	—
133	#367	C₂²×C₁₃₇	1	2	yes	complemented
134	C₂×#41	—	4	1	no	—
135	#135	—	2	2	no	—
136	#135	—	2	2	no	—
137	#135	—	4	1	no	—
138	#135	—	4	1	no	—
139	C₂×Q₁₆	—	2	1	no	—
140	C₂×SD₁₆	—	4	1	no	—
141	C₂×D₈	—	2	1	no	—
142	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	4	1	no	—
144	Q₃₂	—	2	1	no	—
145	Q₃₂	C₂×C₄×C₁₃₇	1	2	yes	complemented
146	SD₃₂	—	2	2	no	—
147	SD₃₂	—	2	2	no	—
148	D₁₆	—	2	1	no	—
149	D₁₆	C₂×C₄×C₁₃₇	1	2	yes	complemented
150	C₂×C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
151	C₂×C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
152	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
153	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
154	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
155	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
156	#107	—	2	2	no	—
157	#107	—	4	1	no	—
158	C₃₂	—	2	2	no	—
159	#41	—	4	2	no	—
160	#41	—	4	2	no	—
161	#41	—	8	1	no	—
162	#41	—	8	1	no	—
163	C₂×Q₈	—	4	1	no	—
164	C₂×D₄	—	4	1	no	—
165	C₂×D₄	—	8	1	no	—
166	C₂²×C₄	—	8	1	no	—
167	C₂²×C₄	—	8	1	no	—
168	C₂²×C₄	D₈×C₁₃₇	1	1	yes	characteristic
169	Q₁₆	—	2	2	no	—
170	Q₁₆	—	4	1	no	—
171	SD₁₆	—	4	2	no	—
172	SD₁₆	—	4	2	no	—
173	D₈	—	2	2	no	—
174	D₈	—	4	1	no	—
175	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
176	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
177	C₂×C₈	D₈×C₁₃₇	1	1	yes	characteristic
178	C₂×C₈	SD₁₆×C₁₃₇	1	1	yes	characteristic
179	C₂×C₈	SD₁₆×C₁₃₇	1	1	yes	characteristic
180	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	8	1	no	—
183	C₁₆	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
184	C₁₆	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
185	C₁₆	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
186	C₁₆	C₂×D₄×C₁₃₇	1	1	yes	characteristic
187	C₂³	—	8	1	no	—
188	C₂³	D₁₆×C₁₃₇	1	1	yes	characteristic
189	Q₈	—	4	2	no	—
190	Q₈	—	8	1	no	—
191	D₄	—	4	2	no	—
192	D₄	—	8	2	no	—
193	D₄	—	8	2	no	—
194	D₄	—	8	1	no	—
195	C₂×C₄	—	8	1	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	16	2	no	—
200	C₂×C₄	#102×C₁₃₇	1	1	yes	characteristic
201	C₂×C₄	#102×C₁₃₇	1	1	yes	characteristic
202	C₂×C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
203	C₂×C₄	SD₃₂×C₁₃₇	1	1	yes	characteristic
204	C₂×C₄	SD₃₂×C₁₃₇	1	1	yes	characteristic
205	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
206	C₈	#102×C₁₃₇	1	1	yes	characteristic
207	C₈	#102×C₁₃₇	1	1	yes	characteristic
208	C₈	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
209	C₈	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
210	C₂²	—	2	2	no	—
211	C₂²	—	8	2	no	—
212	C₂²	—	16	2	no	—
213	C₂²	#357×C₁₃₇	1	1	yes	characteristic
214	C₂²	#357×C₁₃₇	1	1	yes	characteristic
215	C₂²	C₂×D₁₆×C₁₃₇	1	1	yes	characteristic
216	C₄	—	8	2	no	—
217	C₄	—	16	2	no	—
218	C₄	#357×C₁₃₇	1	1	yes	characteristic
219	C₄	#357×C₁₃₇	1	1	yes	characteristic
220	C₄	C₂×#102×C₁₃₇	1	1	yes	characteristic
221	C₄	C₂×SD₃₂×C₁₃₇	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	—	16	2	no	—
225	C₂	#1418×C₁₃₇	1	2	yes	direct product factor
226	C₂	C₂×#357×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#357
commutator	C₁₆	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃₇
prime radical	C₂×#508×C₁₃₇	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂²×C₈×C₁₃₇ ◁ C₂²×C₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×M₆₄×C₁₃₇	C₂	semidirect
3. C₂×#367×C₁₃₇	C₂	semidirect
4. #1418×C₁₃₇	C₂	direct (N × H)
8. #508×C₁₃₇	C₄	semidirect
10. C₂×Q₃₂×C₁₃₇	C₄	semidirect
12. C₂×D₁₆×C₁₃₇	C₄	semidirect
16. M₆₄×C₁₃₇	C₂²	semidirect
19. #367×C₁₃₇	C₂²	semidirect
31. Q₃₂×C₁₃₇	C₂×C₄	semidirect
35. D₁₆×C₁₃₇	C₂×C₄	semidirect
111. C₂×C₁₃₇	#1418	direct (N × H)
113. C₂×#1418	C₁₃₇	direct (N × H)
114. C₁₃₇	C₂×#1418	direct (N × H)
116. C₂×M₆₄	C₂×C₁₃₇	semidirect
117. C₂×#367	C₂×C₁₃₇	semidirect
118. #1418	C₂×C₁₃₇	direct (N × H)
122. #508	C₄×C₁₃₇	semidirect
124. C₂×Q₃₂	C₄×C₁₃₇	semidirect
126. C₂×D₁₆	C₄×C₁₃₇	semidirect
130. M₆₄	C₂²×C₁₃₇	semidirect
133. #367	C₂²×C₁₃₇	semidirect
145. Q₃₂	C₂×C₄×C₁₃₇	semidirect
149. D₁₆	C₂×C₄×C₁₃₇	semidirect
225. C₂	#1418×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×#508×C₁₃₇	2
C₂×M₆₄×C₁₃₇	2
C₂×#367×C₁₃₇	2
#1418×C₁₃₇	2
C₂×#1418	137

Subgroups — C2×#1418×C137 #46863007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1418×C₁₃₇ #46863007