Subgroups — C₁₃₇×#21913 #46871299

Order 35072.

714 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2296×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1916×C₁₃₇	C₂	1	8	yes	maximal, complemented
3	#1818×C₁₃₇	C₂	1	2	yes	maximal
4	#1798×C₁₃₇	C₂	1	2	yes	maximal
5	C₄×C₂²⋊C₈×C₁₃₇	C₂	1	2	yes	maximal
6	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
10	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₁₃₇	C₂²	1	4	yes	—
12	#389×C₁₃₇	C₄	1	4	yes	—
13	#385×C₁₃₇	C₄	1	4	yes	—
14	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
15	#336×C₁₃₇	C₂²	1	4	yes	—
16	#336×C₁₃₇	C₂²	1	8	yes	—
17	#336×C₁₃₇	C₂²	1	4	yes	—
18	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₁₃₇	C₂³	1	8	yes	—
21	#117×C₁₃₇	—	2	4	no	—
22	#117×C₁₃₇	—	4	2	no	—
23	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
24	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₃₇	—	4	2	no	—
29	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
31	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
32	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
33	C₄⋊C₈×C₁₃₇	—	4	4	no	—
34	C₂²⋊C₈×C₁₃₇	—	2	8	no	—
35	C₄×C₈×C₁₃₇	—	4	4	no	—
36	#95×C₁₃₇	C₂×C₄	1	8	yes	—
37	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃₇	—	2	2	no	—
39	C₂²×C₄×C₁₃₇	—	2	2	no	—
40	C₂²×C₄×C₁₃₇	—	2	8	no	—
41	C₂²×C₄×C₁₃₇	—	4	2	no	—
42	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
44	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
47	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
48	C₂×C₈×C₁₃₇	—	2	8	no	—
49	C₂×C₈×C₁₃₇	—	2	8	no	—
50	C₂×C₈×C₁₃₇	—	2	8	no	—
51	C₄⋊C₄×C₁₃₇	—	2	4	no	—
52	C₄⋊C₄×C₁₃₇	—	2	16	no	—
53	C₄⋊C₄×C₁₃₇	—	4	2	no	—
54	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
55	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
56	C₄²×C₁₃₇	—	2	4	no	—
57	C₄²×C₁₃₇	—	2	4	no	—
58	C₄²×C₁₃₇	—	2	4	no	—
59	C₄²×C₁₃₇	—	4	2	no	—
60	C₂³×C₁₃₇	—	2	1	no	—
61	C₂³×C₁₃₇	—	2	1	no	—
62	C₂³×C₁₃₇	—	2	4	no	—
63	C₂³×C₁₃₇	#117	1	2	yes	—
64	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃₇	—	2	4	no	—
66	C₂×C₄×C₁₃₇	—	2	1	no	—
67	C₂×C₄×C₁₃₇	—	2	1	no	—
68	C₂×C₄×C₁₃₇	—	2	4	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	8	no	—
73	C₂×C₄×C₁₃₇	—	2	8	no	—
74	C₂×C₄×C₁₃₇	—	4	8	no	—
75	C₂×C₄×C₁₃₇	#117	1	2	yes	—
76	C₂×C₄×C₁₃₇	#131	1	8	yes	—
77	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
78	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
79	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
80	C₈×C₁₃₇	—	4	8	no	—
81	C₂²×C₁₃₇	—	2	2	no	—
82	C₂²×C₁₃₇	—	2	4	no	—
83	C₂²×C₁₃₇	—	4	2	no	—
84	C₂²×C₁₃₇	—	4	2	no	—
85	C₂²×C₁₃₇	C₂×#117	1	1	yes	characteristic
86	C₂²×C₁₃₇	#518	1	1	yes	characteristic
87	C₂²×C₁₃₇	#519	1	1	yes	characteristic
88	C₂²×C₁₃₇	C₂×#131	1	4	yes	—
89	C₄×C₁₃₇	—	2	2	no	—
90	C₄×C₁₃₇	—	2	8	no	—
91	C₄×C₁₃₇	—	4	4	no	—
92	C₄×C₁₃₇	—	4	4	no	—
93	C₄×C₁₃₇	#539	1	4	yes	—
94	C₂×C₁₃₇	—	4	2	no	—
95	C₂×C₁₃₇	#2313	1	1	yes	characteristic
96	C₂×C₁₃₇	#2922	1	2	yes	—
97	C₂×C₁₃₇	#2977	1	2	yes	—
98	C₂×C₁₃₇	#2986	1	2	yes	—
99	#21913	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₁₃₇	#21913	1	1	yes	characteristic, direct product factor, Sylow
101	#2296	C₂×C₁₃₇	1	1	yes	characteristic
102	#1916	C₂×C₁₃₇	1	8	yes	complemented
103	#1818	C₂×C₁₃₇	1	2	yes	—
104	#1798	C₂×C₁₃₇	1	2	yes	—
105	C₄×C₂²⋊C₈	C₂×C₁₃₇	1	2	yes	—
106	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
107	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
108	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
109	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
110	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	4	yes	—
111	C₂×C₄×C₈	C₂²×C₁₃₇	1	4	yes	—
112	#389	C₄×C₁₃₇	1	4	yes	—
113	#385	C₄×C₁₃₇	1	4	yes	—
114	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
115	#336	C₂²×C₁₃₇	1	4	yes	—
116	#336	C₂²×C₁₃₇	1	8	yes	—
117	#336	C₂²×C₁₃₇	1	4	yes	—
118	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
119	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
120	C₂²×C₈	C₂³×C₁₃₇	1	8	yes	—
121	#117	—	2	4	no	—
122	#117	—	4	2	no	—
123	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
124	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	—
125	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
126	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
127	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
128	C₂×C₄²	—	4	2	no	—
129	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
130	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
131	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
132	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
133	C₄⋊C₈	—	4	4	no	—
134	C₂²⋊C₈	—	2	8	no	—
135	C₄×C₈	—	4	4	no	—
136	#95	C₂×C₄×C₁₃₇	1	8	yes	—
137	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
143	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
144	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
145	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
146	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
147	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	2	8	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	16	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₂²⋊C₄	—	2	4	no	—
155	C₂²⋊C₄	—	4	2	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	2	4	no	—
159	C₄²	—	4	2	no	—
160	C₂³	—	2	1	no	—
161	C₂³	—	2	1	no	—
162	C₂³	—	2	4	no	—
163	C₂³	#117×C₁₃₇	1	2	yes	—
164	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	4	8	no	—
175	C₂×C₄	#117×C₁₃₇	1	2	yes	—
176	C₂×C₄	#131×C₁₃₇	1	8	yes	—
177	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
178	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
179	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
180	C₈	—	4	8	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	C₂×#117×C₁₃₇	1	1	yes	characteristic
186	C₂²	#518×C₁₃₇	1	1	yes	characteristic
187	C₂²	#519×C₁₃₇	1	1	yes	characteristic
188	C₂²	C₂×#131×C₁₃₇	1	4	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	2	8	no	—
191	C₄	—	4	4	no	—
192	C₄	—	4	4	no	—
193	C₄	#539×C₁₃₇	1	4	yes	—
194	C₂	—	4	2	no	—
195	C₂	#2313×C₁₃₇	1	1	yes	characteristic
196	C₂	#2922×C₁₃₇	1	2	yes	—
197	C₂	#2977×C₁₃₇	1	2	yes	—
198	C₂	#2986×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂⁴
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 ◁ C₂²×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1916×C₁₃₇	C₂	semidirect
99. #21913	C₁₃₇	direct (N × H)
100. C₁₃₇	#21913	direct (N × H)
102. #1916	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2296×C₁₃₇	2
#1916×C₁₃₇	2
#1818×C₁₃₇	2
#1798×C₁₃₇	2
C₄×C₂²⋊C₈×C₁₃₇	2
#21913	137

Subgroups — C137×#21913 #46871299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21913 #46871299