Subgroups — C₁₃₇×#20809 #46870195

Order 35072.

890 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₃₇	C₂	1	2	yes	maximal
2	C₄×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1843×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#1832×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1798×C₁₃₇	C₂	1	4	yes	maximal
6	C₄×C₄⋊C₈×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
9	#431×C₁₃₇	—	2	8	no	—
10	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	8	yes	—
11	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₁₃₇	C₂²	1	4	yes	—
14	C₄×C₄⋊C₄×C₁₃₇	C₄	1	4	yes	—
15	C₄×C₂²⋊C₄×C₁₃₇	C₄	1	4	yes	—
16	#376×C₁₃₇	C₂²	1	1	yes	characteristic
17	#376×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₄³×C₁₃₇	C₂²	1	2	yes	—
19	#336×C₁₃₇	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₁₃₇	C₂³	1	8	yes	—
23	#117×C₁₃₇	—	2	8	no	—
24	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
25	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₃₇	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₃₇	D₄	1	4	yes	—
29	C₂×C₄²×C₁₃₇	Q₈	1	4	yes	—
30	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
31	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
33	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₁₃₇	—	2	8	no	—
35	C₄⋊C₈×C₁₃₇	—	2	16	no	—
36	C₂²⋊C₈×C₁₃₇	—	2	8	no	—
37	C₄×C₈×C₁₃₇	—	2	8	no	—
38	#95×C₁₃₇	C₂×C₄	1	4	yes	—
39	#95×C₁₃₇	C₂×C₄	1	4	yes	—
40	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₇	—	2	8	no	—
42	C₂²×C₄×C₁₃₇	—	2	4	no	—
43	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	8	yes	—
44	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₁₃₇	—	2	8	no	—
53	C₂×C₈×C₁₃₇	—	2	16	no	—
54	C₄⋊C₄×C₁₃₇	—	2	8	no	—
55	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
56	C₄²×C₁₃₇	—	2	16	no	—
57	C₄²×C₁₃₇	—	2	4	no	—
58	C₄²×C₁₃₇	C₄⋊C₄	1	8	yes	—
59	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
60	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
61	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
62	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
63	C₄²×C₁₃₇	#41	1	8	yes	—
64	C₂³×C₁₃₇	—	2	2	no	—
65	C₂³×C₁₃₇	—	2	4	no	—
66	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
67	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃₇	—	2	2	no	—
69	C₂×C₄×C₁₃₇	—	2	8	no	—
70	C₂×C₄×C₁₃₇	—	2	4	no	—
71	C₂×C₄×C₁₃₇	—	2	4	no	—
72	C₂×C₄×C₁₃₇	—	2	4	no	—
73	C₂×C₄×C₁₃₇	—	2	16	no	—
74	C₂×C₄×C₁₃₇	—	2	4	no	—
75	C₂×C₄×C₁₃₇	—	2	8	no	—
76	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
79	C₂×C₄×C₁₃₇	#117	1	8	yes	—
80	C₂×C₄×C₁₃₇	#131	1	8	yes	—
81	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃₇	C₂×#41	1	4	yes	—
84	C₈×C₁₃₇	—	4	8	no	—
85	C₂²×C₁₃₇	—	2	4	no	—
86	C₂²×C₁₃₇	—	2	2	no	—
87	C₂²×C₁₃₇	—	4	4	no	—
88	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₁₃₇	C₂×#117	1	2	yes	—
90	C₂²×C₁₃₇	C₂×#131	1	4	yes	—
91	C₄×C₁₃₇	—	2	4	no	—
92	C₄×C₁₃₇	—	2	8	no	—
93	C₄×C₁₃₇	—	2	2	no	—
94	C₄×C₁₃₇	—	4	4	no	—
95	C₄×C₁₃₇	#424	1	8	yes	—
96	C₄×C₁₃₇	#532	1	2	yes	—
97	C₄×C₁₃₇	#533	1	2	yes	—
98	C₂×C₁₃₇	—	4	2	no	—
99	C₂×C₁₃₇	#2285	1	1	yes	characteristic
100	C₂×C₁₃₇	C₂×#424	1	2	yes	—
101	C₂×C₁₃₇	#2922	1	4	yes	—
102	#20809	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₁₃₇	#20809	1	1	yes	characteristic, direct product factor, Sylow
104	C₂×#431	C₂×C₁₃₇	1	2	yes	—
105	C₄×#117	C₂×C₁₃₇	1	1	yes	characteristic
106	#1843	C₂×C₁₃₇	1	2	yes	complemented
107	#1832	C₂×C₁₃₇	1	2	yes	complemented
108	#1798	C₂×C₁₃₇	1	4	yes	—
109	C₄×C₄⋊C₈	C₂×C₁₃₇	1	4	yes	complemented
110	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
111	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
112	#431	—	2	8	no	—
113	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	8	yes	—
114	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
115	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	4	yes	—
116	C₂×C₄×C₈	C₂²×C₁₃₇	1	4	yes	—
117	C₄×C₄⋊C₄	C₄×C₁₃₇	1	4	yes	—
118	C₄×C₂²⋊C₄	C₄×C₁₃₇	1	4	yes	—
119	#376	C₂²×C₁₃₇	1	1	yes	characteristic
120	#376	C₂²×C₁₃₇	1	1	yes	characteristic
121	C₄³	C₂²×C₁₃₇	1	2	yes	—
122	#336	C₂²×C₁₃₇	1	8	yes	—
123	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
124	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₁₃₇	1	8	yes	—
126	#117	—	2	8	no	—
127	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
128	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
129	C₂×C₄²	C₂×C₄×C₁₃₇	1	8	yes	—
130	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
131	C₂×C₄²	D₄×C₁₃₇	1	4	yes	—
132	C₂×C₄²	Q₈×C₁₃₇	1	4	yes	—
133	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
134	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
135	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
136	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
137	C₄⋊C₈	—	2	8	no	—
138	C₄⋊C₈	—	2	16	no	—
139	C₂²⋊C₈	—	2	8	no	—
140	C₄×C₈	—	2	8	no	—
141	#95	C₂×C₄×C₁₃₇	1	4	yes	—
142	#95	C₂×C₄×C₁₃₇	1	4	yes	—
143	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	8	yes	—
147	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
148	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
149	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
151	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
152	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
153	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
154	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	2	16	no	—
157	C₄⋊C₄	—	2	8	no	—
158	C₂²⋊C₄	—	2	8	no	—
159	C₄²	—	2	16	no	—
160	C₄²	—	2	4	no	—
161	C₄²	C₄⋊C₄×C₁₃₇	1	8	yes	—
162	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
163	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
164	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
165	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
166	C₄²	#41×C₁₃₇	1	8	yes	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	4	no	—
169	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
170	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	16	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
180	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
181	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
182	C₂×C₄	#117×C₁₃₇	1	8	yes	—
183	C₂×C₄	#131×C₁₃₇	1	8	yes	—
184	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
185	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
186	C₂×C₄	C₂×#41×C₁₃₇	1	4	yes	—
187	C₈	—	4	8	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	4	4	no	—
191	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
192	C₂²	C₂×#117×C₁₃₇	1	2	yes	—
193	C₂²	C₂×#131×C₁₃₇	1	4	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	8	no	—
197	C₄	—	4	4	no	—
198	C₄	#424×C₁₃₇	1	8	yes	—
199	C₄	#532×C₁₃₇	1	2	yes	—
200	C₄	#533×C₁₃₇	1	2	yes	—
201	C₂	—	4	2	no	—
202	C₂	#2285×C₁₃₇	1	1	yes	characteristic
203	C₂	C₂×#424×C₁₃₇	1	2	yes	—
204	C₂	#2922×C₁₃₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #1843×C₁₃₇	C₂	semidirect
4. #1832×C₁₃₇	C₂	semidirect
6. C₄×C₄⋊C₈×C₁₃₇	C₂	semidirect
102. #20809	C₁₃₇	direct (N × H)
103. C₁₃₇	#20809	direct (N × H)
106. #1843	C₂×C₁₃₇	semidirect
107. #1832	C₂×C₁₃₇	semidirect
109. C₄×C₄⋊C₈	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₁₃₇	2
C₄×#117×C₁₃₇	2
#1843×C₁₃₇	2
#1832×C₁₃₇	2
#1798×C₁₃₇	2
C₄×C₄⋊C₈×C₁₃₇	2
#20809	137

Subgroups — C137×#20809 #46870195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20809 #46870195