Subgroups — C₂×#1419×C₁₃₇ #46863009

Order 35072.

326 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#368×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1419×C₁₃₇	C₂	1	4	yes	maximal, direct product factor
5	C₂²×D₈×C₁₃₇	—	2	1	no	—
6	C₂×D₁₆×C₁₃₇	—	2	1	no	—
7	C₂×D₁₆×C₁₃₇	—	2	1	no	—
8	C₂×D₁₆×C₁₃₇	—	2	1	no	—
9	C₂×D₁₆×C₁₃₇	C₄	1	2	yes	—
10	C₂×D₁₆×C₁₃₇	C₄	1	2	yes	—
11	C₂×D₁₆×C₁₃₇	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₁₃₇	—	2	1	no	—
14	M₆₄×C₁₃₇	—	2	1	no	—
15	M₆₄×C₁₃₇	C₂²	1	2	yes	complemented
16	C₂×C₃₂×C₁₃₇	—	2	1	no	—
17	#368×C₁₃₇	—	2	1	no	—
18	#368×C₁₃₇	C₂²	1	2	yes	complemented
19	C₂²×D₄×C₁₃₇	—	4	1	no	—
20	C₂×D₈×C₁₃₇	—	2	2	no	—
21	C₂×D₈×C₁₃₇	—	2	2	no	—
22	C₂×D₈×C₁₃₇	—	2	2	no	—
23	C₂×D₈×C₁₃₇	—	4	1	no	—
24	C₂×D₈×C₁₃₇	—	4	1	no	—
25	C₂×D₈×C₁₃₇	—	4	1	no	—
26	C₂×M₁₆×C₁₃₇	—	4	1	no	—
27	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
28	D₁₆×C₁₃₇	—	2	2	no	—
29	D₁₆×C₁₃₇	—	2	2	no	—
30	D₁₆×C₁₃₇	—	2	2	no	—
31	D₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
32	C₂×C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
38	#108×C₁₃₇	—	2	2	no	—
39	#108×C₁₃₇	—	4	1	no	—
40	C₃₂×C₁₃₇	—	2	2	no	—
41	C₂⁴×C₁₃₇	—	8	1	no	—
42	C₂×D₄×C₁₃₇	—	4	2	no	—
43	C₂×D₄×C₁₃₇	—	4	2	no	—
44	C₂×D₄×C₁₃₇	—	4	2	no	—
45	C₂×D₄×C₁₃₇	—	8	1	no	—
46	C₂×D₄×C₁₃₇	—	8	1	no	—
47	C₂×D₄×C₁₃₇	—	8	1	no	—
48	C₂²×C₄×C₁₃₇	D₈	1	1	yes	characteristic
49	D₈×C₁₃₇	—	2	4	no	—
50	D₈×C₁₃₇	—	4	2	no	—
51	D₈×C₁₃₇	—	4	2	no	—
52	D₈×C₁₃₇	—	4	2	no	—
53	M₁₆×C₁₃₇	—	4	2	no	—
54	M₁₆×C₁₃₇	—	8	1	no	—
55	C₂×C₈×C₁₃₇	—	8	1	no	—
56	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃₇	D₈	1	1	yes	characteristic
59	C₂×C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
60	C₂×C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
61	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
62	C₁₆×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₁₆×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₁₆×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₁₆×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂³×C₁₃₇	—	8	2	no	—
67	C₂³×C₁₃₇	—	8	2	no	—
68	C₂³×C₁₃₇	—	8	2	no	—
69	C₂³×C₁₃₇	—	16	4	no	—
70	C₂³×C₁₃₇	SD₃₂	1	1	yes	characteristic
71	D₄×C₁₃₇	—	4	4	no	—
72	D₄×C₁₃₇	—	8	2	no	—
73	D₄×C₁₃₇	—	8	2	no	—
74	D₄×C₁₃₇	—	8	2	no	—
75	C₂×C₄×C₁₃₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	#102	1	1	yes	characteristic
77	C₂×C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	SD₃₂	1	1	yes	characteristic
80	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
81	C₈×C₁₃₇	—	8	2	no	—
82	C₈×C₁₃₇	#102	1	1	yes	characteristic
83	C₈×C₁₃₇	#102	1	1	yes	characteristic
84	C₈×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₈×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
86	C₂²×C₁₃₇	—	2	2	no	—
87	C₂²×C₁₃₇	—	8	4	no	—
88	C₂²×C₁₃₇	—	16	4	no	—
89	C₂²×C₁₃₇	—	16	4	no	—
90	C₂²×C₁₃₇	—	16	4	no	—
91	C₂²×C₁₃₇	#357	1	1	yes	characteristic
92	C₂²×C₁₃₇	#357	1	1	yes	characteristic
93	C₂²×C₁₃₇	C₂×SD₃₂	1	1	yes	characteristic
94	C₄×C₁₃₇	#357	1	1	yes	characteristic
95	C₄×C₁₃₇	#357	1	1	yes	characteristic
96	C₄×C₁₃₇	C₂×#102	1	1	yes	characteristic
97	C₄×C₁₃₇	C₂×D₁₆	1	1	yes	characteristic
98	C₂×C₁₃₇	—	2	1	no	—
99	C₂×C₁₃₇	—	2	1	no	—
100	C₂×C₁₃₇	—	16	4	no	—
101	C₂×C₁₃₇	#1419	1	2	yes	direct product factor
102	C₂×C₁₃₇	C₂×#357	1	1	yes	characteristic
103	C₂×#1419	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₁₃₇	C₂×#1419	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²×D₁₆	C₂×C₁₃₇	1	1	yes	characteristic
106	C₂×M₆₄	C₂×C₁₃₇	1	1	yes	characteristic, complemented
107	C₂×#368	C₂×C₁₃₇	1	1	yes	characteristic, complemented
108	#1419	C₂×C₁₃₇	1	4	yes	direct product factor
109	C₂²×D₈	—	2	1	no	—
110	C₂×D₁₆	—	2	1	no	—
111	C₂×D₁₆	—	2	1	no	—
112	C₂×D₁₆	—	2	1	no	—
113	C₂×D₁₆	C₄×C₁₃₇	1	2	yes	—
114	C₂×D₁₆	C₄×C₁₃₇	1	2	yes	—
115	C₂×D₁₆	C₂²×C₁₃₇	1	2	yes	—
116	C₂²×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
117	C₂×#108	—	2	1	no	—
118	M₆₄	—	2	1	no	—
119	M₆₄	C₂²×C₁₃₇	1	2	yes	complemented
120	C₂×C₃₂	—	2	1	no	—
121	#368	—	2	1	no	—
122	#368	C₂²×C₁₃₇	1	2	yes	complemented
123	C₂²×D₄	—	4	1	no	—
124	C₂×D₈	—	2	2	no	—
125	C₂×D₈	—	2	2	no	—
126	C₂×D₈	—	2	2	no	—
127	C₂×D₈	—	4	1	no	—
128	C₂×D₈	—	4	1	no	—
129	C₂×D₈	—	4	1	no	—
130	C₂×M₁₆	—	4	1	no	—
131	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
132	D₁₆	—	2	2	no	—
133	D₁₆	—	2	2	no	—
134	D₁₆	—	2	2	no	—
135	D₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
136	C₂×C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
137	C₂×C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
138	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
139	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
140	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
141	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
142	#108	—	2	2	no	—
143	#108	—	4	1	no	—
144	C₃₂	—	2	2	no	—
145	C₂⁴	—	8	1	no	—
146	C₂×D₄	—	4	2	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	4	2	no	—
149	C₂×D₄	—	8	1	no	—
150	C₂×D₄	—	8	1	no	—
151	C₂×D₄	—	8	1	no	—
152	C₂²×C₄	D₈×C₁₃₇	1	1	yes	characteristic
153	D₈	—	2	4	no	—
154	D₈	—	4	2	no	—
155	D₈	—	4	2	no	—
156	D₈	—	4	2	no	—
157	M₁₆	—	4	2	no	—
158	M₁₆	—	8	1	no	—
159	C₂×C₈	—	8	1	no	—
160	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
161	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
162	C₂×C₈	D₈×C₁₃₇	1	1	yes	characteristic
163	C₂×C₈	SD₁₆×C₁₃₇	1	1	yes	characteristic
164	C₂×C₈	SD₁₆×C₁₃₇	1	1	yes	characteristic
165	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
166	C₁₆	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
167	C₁₆	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
168	C₁₆	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
169	C₁₆	C₂×D₄×C₁₃₇	1	1	yes	characteristic
170	C₂³	—	8	2	no	—
171	C₂³	—	8	2	no	—
172	C₂³	—	8	2	no	—
173	C₂³	—	16	4	no	—
174	C₂³	SD₃₂×C₁₃₇	1	1	yes	characteristic
175	D₄	—	4	4	no	—
176	D₄	—	8	2	no	—
177	D₄	—	8	2	no	—
178	D₄	—	8	2	no	—
179	C₂×C₄	#102×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄	#102×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
182	C₂×C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
183	C₂×C₄	SD₃₂×C₁₃₇	1	1	yes	characteristic
184	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
185	C₈	—	8	2	no	—
186	C₈	#102×C₁₃₇	1	1	yes	characteristic
187	C₈	#102×C₁₃₇	1	1	yes	characteristic
188	C₈	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
189	C₈	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
190	C₂²	—	2	2	no	—
191	C₂²	—	8	4	no	—
192	C₂²	—	16	4	no	—
193	C₂²	—	16	4	no	—
194	C₂²	—	16	4	no	—
195	C₂²	#357×C₁₃₇	1	1	yes	characteristic
196	C₂²	#357×C₁₃₇	1	1	yes	characteristic
197	C₂²	C₂×SD₃₂×C₁₃₇	1	1	yes	characteristic
198	C₄	#357×C₁₃₇	1	1	yes	characteristic
199	C₄	#357×C₁₃₇	1	1	yes	characteristic
200	C₄	C₂×#102×C₁₃₇	1	1	yes	characteristic
201	C₄	C₂×D₁₆×C₁₃₇	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	—	16	4	no	—
205	C₂	#1419×C₁₃₇	1	2	yes	direct product factor
206	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	C₂²×D₁₆×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×#368×C₁₃₇	C₂	semidirect
4. #1419×C₁₃₇	C₂	direct (N × H)
15. M₆₄×C₁₃₇	C₂²	semidirect
18. #368×C₁₃₇	C₂²	semidirect
101. C₂×C₁₃₇	#1419	direct (N × H)
103. C₂×#1419	C₁₃₇	direct (N × H)
104. C₁₃₇	C₂×#1419	direct (N × H)
106. C₂×M₆₄	C₂×C₁₃₇	semidirect
107. C₂×#368	C₂×C₁₃₇	semidirect
108. #1419	C₂×C₁₃₇	direct (N × H)
119. M₆₄	C₂²×C₁₃₇	semidirect
122. #368	C₂²×C₁₃₇	semidirect
205. C₂	#1419×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
C₂²×D₁₆×C₁₃₇	2
C₂×M₆₄×C₁₃₇	2
C₂×#368×C₁₃₇	2
#1419×C₁₃₇	2
C₂×#1419	137

Subgroups — C2×#1419×C137 #46863009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1419×C₁₃₇ #46863009