Subgroups — C₁₃₇×#20827 #46870213

Order 35072.

758 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	#2920×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1849×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#1838×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1811×C₁₃₇	C₂	1	4	yes	maximal
6	C₄×#108×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
9	#432×C₁₃₇	—	2	4	no	—
10	#431×C₁₃₇	—	2	4	no	—
11	C₂×#108×C₁₃₇	C₂²	1	8	yes	—
12	#423×C₁₃₇	C₂²	1	4	yes	—
13	#407×C₁₃₇	C₂²	1	4	yes	—
14	C₄×M₁₆×C₁₃₇	C₄	1	8	yes	—
15	C₄×M₁₆×C₁₃₇	C₂²	1	4	yes	—
16	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
19	#341×C₁₃₇	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₃₇	Q₈	1	2	yes	—
21	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₃₇	—	2	8	no	—
23	C₂×M₁₆×C₁₃₇	C₂×C₄	1	8	yes	—
24	C₂×M₁₆×C₁₃₇	C₂³	1	8	yes	—
25	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
26	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
27	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
29	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
30	C₂×C₄²×C₁₃₇	Q₈	1	2	yes	—
31	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
33	#108×C₁₃₇	—	2	16	no	—
34	C₄⋊C₈×C₁₃₇	—	2	8	no	—
35	C₂²⋊C₈×C₁₃₇	—	2	8	no	—
36	#97×C₁₃₇	—	2	4	no	—
37	#97×C₁₃₇	C₂×C₄	1	4	yes	—
38	#97×C₁₃₇	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₁₃₇	—	2	4	no	—
40	C₄×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
41	C₂⁴×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₇	—	2	4	no	—
43	C₂²×C₄×C₁₃₇	—	4	4	no	—
44	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
52	M₁₆×C₁₃₇	—	2	16	no	—
53	M₁₆×C₁₃₇	—	2	16	no	—
54	C₂×C₈×C₁₃₇	—	2	4	no	—
55	C₂×C₈×C₁₃₇	—	2	8	no	—
56	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₁₃₇	#41	1	8	yes	—
59	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
60	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
61	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
62	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
63	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
65	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂³×C₁₃₇	—	2	2	no	—
67	C₂³×C₁₃₇	—	4	2	no	—
68	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
69	C₂³×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₄×C₁₃₇	—	2	4	no	—
72	C₂×C₄×C₁₃₇	—	2	4	no	—
73	C₂×C₄×C₁₃₇	—	2	8	no	—
74	C₂×C₄×C₁₃₇	—	2	4	no	—
75	C₂×C₄×C₁₃₇	—	4	4	no	—
76	C₂×C₄×C₁₃₇	—	4	2	no	—
77	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₇	C₂×#41	1	4	yes	—
84	C₈×C₁₃₇	—	2	4	no	—
85	C₈×C₁₃₇	—	4	8	no	—
86	C₈×C₁₃₇	#117	1	8	yes	—
87	C₂²×C₁₃₇	—	2	2	no	—
88	C₂²×C₁₃₇	—	2	4	no	—
89	C₂²×C₁₃₇	—	2	2	no	—
90	C₂²×C₁₃₇	—	4	4	no	—
91	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₁₃₇	#533	1	2	yes	—
93	C₄×C₁₃₇	—	2	4	no	—
94	C₄×C₁₃₇	—	2	2	no	—
95	C₄×C₁₃₇	—	2	2	no	—
96	C₄×C₁₃₇	#430	1	4	yes	—
97	C₄×C₁₃₇	C₂×#117	1	2	yes	—
98	C₄×C₁₃₇	#532	1	2	yes	—
99	C₂×C₁₃₇	—	2	2	no	—
100	C₂×C₁₃₇	—	4	2	no	—
101	C₂×C₁₃₇	#2285	1	1	yes	characteristic
102	C₂×C₁₃₇	C₂×#430	1	2	yes	—
103	#20827	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₁₃₇	#20827	1	1	yes	characteristic, direct product factor, Sylow
105	#2920	C₂×C₁₃₇	1	2	yes	—
106	C₂×C₄×M₁₆	C₂×C₁₃₇	1	1	yes	characteristic
107	#1849	C₂×C₁₃₇	1	2	yes	complemented
108	#1838	C₂×C₁₃₇	1	2	yes	complemented
109	#1811	C₂×C₁₃₇	1	4	yes	—
110	C₄×#108	C₂×C₁₃₇	1	4	yes	complemented
111	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
112	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
113	#432	—	2	4	no	—
114	#431	—	2	4	no	—
115	C₂×#108	C₂²×C₁₃₇	1	8	yes	—
116	#423	C₂²×C₁₃₇	1	4	yes	—
117	#407	C₂²×C₁₃₇	1	4	yes	—
118	C₄×M₁₆	C₄×C₁₃₇	1	8	yes	—
119	C₄×M₁₆	C₂²×C₁₃₇	1	4	yes	—
120	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
121	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
123	#341	C₂²×C₁₃₇	1	8	yes	—
124	C₂³×C₄	Q₈×C₁₃₇	1	2	yes	—
125	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
126	C₂×M₁₆	—	2	8	no	—
127	C₂×M₁₆	C₂×C₄×C₁₃₇	1	8	yes	—
128	C₂×M₁₆	C₂³×C₁₃₇	1	8	yes	—
129	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
130	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
131	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
132	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
133	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
134	C₂×C₄²	Q₈×C₁₃₇	1	2	yes	—
135	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
136	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
137	#108	—	2	16	no	—
138	C₄⋊C₈	—	2	8	no	—
139	C₂²⋊C₈	—	2	8	no	—
140	#97	—	2	4	no	—
141	#97	C₂×C₄×C₁₃₇	1	4	yes	—
142	#97	C₂×C₄×C₁₃₇	1	4	yes	—
143	C₄×C₈	—	2	4	no	—
144	C₄×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
145	C₂⁴	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	4	4	no	—
148	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
149	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
150	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
151	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
152	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
153	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
154	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
155	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
156	M₁₆	—	2	16	no	—
157	M₁₆	—	2	16	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
161	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
162	C₂×C₈	#41×C₁₃₇	1	8	yes	—
163	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
164	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
165	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
166	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
167	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
168	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
169	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
170	C₂³	—	2	2	no	—
171	C₂³	—	4	2	no	—
172	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
173	C₂³	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
182	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
183	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
185	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
186	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
187	C₂×C₄	C₂×#41×C₁₃₇	1	4	yes	—
188	C₈	—	2	4	no	—
189	C₈	—	4	8	no	—
190	C₈	#117×C₁₃₇	1	8	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	4	4	no	—
195	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
196	C₂²	#533×C₁₃₇	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	#430×C₁₃₇	1	4	yes	—
201	C₄	C₂×#117×C₁₃₇	1	2	yes	—
202	C₄	#532×C₁₃₇	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	4	2	no	—
205	C₂	#2285×C₁₃₇	1	1	yes	characteristic
206	C₂	C₂×#430×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃₇ ◁ C₂²×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1849×C₁₃₇	C₂	semidirect
4. #1838×C₁₃₇	C₂	semidirect
6. C₄×#108×C₁₃₇	C₂	semidirect
103. #20827	C₁₃₇	direct (N × H)
104. C₁₃₇	#20827	direct (N × H)
107. #1849	C₂×C₁₃₇	semidirect
108. #1838	C₂×C₁₃₇	semidirect
110. C₄×#108	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2920×C₁₃₇	2
C₂×C₄×M₁₆×C₁₃₇	2
#1849×C₁₃₇	2
#1838×C₁₃₇	2
#1811×C₁₃₇	2
C₄×#108×C₁₃₇	2
#20827	137

Subgroups — C137×#20827 #46870213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20827 #46870213