Subgroups — C₁₃₇×#21433 #46870819

Order 35072.

668 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2298×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1924×C₁₃₇	C₂	1	4	yes	maximal
3	#1917×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	#1768×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1750×C₁₃₇	C₂	1	2	yes	maximal
6	#1745×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#117×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×#97×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#97×C₁₃₇	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
14	#388×C₁₃₇	C₄	1	4	yes	—
15	#387×C₁₃₇	C₄	1	4	yes	—
16	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
17	C₄×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
18	#376×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
20	#336×C₁₃₇	C₂²	1	4	yes	—
21	#336×C₁₃₇	C₂²	1	4	yes	—
22	#336×C₁₃₇	C₂²	1	4	yes	—
23	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
25	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
26	#117×C₁₃₇	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
33	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
34	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₁₃₇	—	2	8	no	—
37	C₂²⋊C₈×C₁₃₇	—	2	8	no	—
38	#97×C₁₃₇	—	2	8	no	—
39	#97×C₁₃₇	—	2	4	no	—
40	C₄×C₈×C₁₃₇	—	2	4	no	—
41	#95×C₁₃₇	C₂×C₄	1	4	yes	—
42	#95×C₁₃₇	C₂×C₄	1	4	yes	—
43	#95×C₁₃₇	C₂×C₄	1	4	yes	—
44	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₇	—	2	4	no	—
46	C₂²×C₄×C₁₃₇	—	2	2	no	—
47	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₁₃₇	—	2	4	no	—
54	C₂×C₈×C₁₃₇	—	2	4	no	—
55	C₂×C₈×C₁₃₇	—	2	4	no	—
56	C₂×C₈×C₁₃₇	—	2	4	no	—
57	C₂×C₈×C₁₃₇	—	2	4	no	—
58	C₂×C₈×C₁₃₇	#41	1	8	yes	—
59	C₄⋊C₄×C₁₃₇	—	2	4	no	—
60	C₄⋊C₄×C₁₃₇	—	4	4	no	—
61	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
62	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
63	C₄²×C₁₃₇	—	2	4	no	—
64	C₄²×C₁₃₇	—	2	4	no	—
65	C₄²×C₁₃₇	—	2	2	no	—
66	C₄²×C₁₃₇	—	4	4	no	—
67	C₄²×C₁₃₇	#41	1	4	yes	—
68	C₂³×C₁₃₇	—	2	1	no	—
69	C₂³×C₁₃₇	—	2	2	no	—
70	C₂³×C₁₃₇	—	2	4	no	—
71	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃₇	—	2	4	no	—
73	C₂×C₄×C₁₃₇	—	2	2	no	—
74	C₂×C₄×C₁₃₇	—	2	1	no	—
75	C₂×C₄×C₁₃₇	—	2	4	no	—
76	C₂×C₄×C₁₃₇	—	2	4	no	—
77	C₂×C₄×C₁₃₇	—	2	2	no	—
78	C₂×C₄×C₁₃₇	—	2	4	no	—
79	C₂×C₄×C₁₃₇	—	2	4	no	—
80	C₂×C₄×C₁₃₇	—	2	4	no	—
81	C₂×C₄×C₁₃₇	—	2	4	no	—
82	C₂×C₄×C₁₃₇	—	2	4	no	—
83	C₂×C₄×C₁₃₇	—	4	4	no	—
84	C₂×C₄×C₁₃₇	#117	1	4	yes	—
85	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
86	C₂×C₄×C₁₃₇	C₂×#41	1	4	yes	—
87	C₈×C₁₃₇	—	2	8	no	—
88	C₈×C₁₃₇	—	4	4	no	—
89	C₂²×C₁₃₇	—	4	2	no	—
90	C₂²×C₁₃₇	—	4	4	no	—
91	C₂²×C₁₃₇	—	4	1	no	—
92	C₂²×C₁₃₇	C₂×#117	1	1	yes	characteristic
93	C₂²×C₁₃₇	C₄×#41	1	2	yes	—
94	C₂²×C₁₃₇	#568	1	4	yes	—
95	C₄×C₁₃₇	—	2	4	no	—
96	C₄×C₁₃₇	—	4	2	no	—
97	C₄×C₁₃₇	—	4	4	no	—
98	C₄×C₁₃₇	—	4	4	no	—
99	C₄×C₁₃₇	—	4	1	no	—
100	C₄×C₁₃₇	#529	1	4	yes	—
101	C₂×C₁₃₇	—	8	1	no	—
102	C₂×C₁₃₇	#2298	1	1	yes	characteristic
103	C₂×C₁₃₇	#2923	1	2	yes	—
104	C₂×C₁₃₇	#2967	1	4	yes	—
105	#21433	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₁₃₇	#21433	1	1	yes	characteristic, direct product factor, Sylow
107	#2298	C₂×C₁₃₇	1	1	yes	characteristic
108	#1924	C₂×C₁₃₇	1	4	yes	—
109	#1917	C₂×C₁₃₇	1	4	yes	complemented
110	#1768	C₂×C₁₃₇	1	2	yes	complemented
111	#1750	C₂×C₁₃₇	1	2	yes	—
112	#1745	C₂×C₁₃₇	1	1	yes	characteristic, complemented
113	C₄×#97	C₂×C₁₃₇	1	1	yes	characteristic, complemented
114	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
115	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
116	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	4	yes	—
117	C₂×#97	C₂²×C₁₃₇	1	4	yes	—
118	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
120	#388	C₄×C₁₃₇	1	4	yes	—
121	#387	C₄×C₁₃₇	1	4	yes	—
122	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
123	C₄×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
124	#376	C₂²×C₁₃₇	1	1	yes	characteristic
125	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
126	#336	C₂²×C₁₃₇	1	4	yes	—
127	#336	C₂²×C₁₃₇	1	4	yes	—
128	#336	C₂²×C₁₃₇	1	4	yes	—
129	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
131	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
132	#117	—	2	4	no	—
133	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
134	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
135	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
136	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
137	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
138	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
139	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
140	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
141	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
142	C₄⋊C₈	—	2	8	no	—
143	C₂²⋊C₈	—	2	8	no	—
144	#97	—	2	8	no	—
145	#97	—	2	4	no	—
146	C₄×C₈	—	2	4	no	—
147	#95	C₂×C₄×C₁₃₇	1	4	yes	—
148	#95	C₂×C₄×C₁₃₇	1	4	yes	—
149	#95	C₂×C₄×C₁₃₇	1	4	yes	—
150	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
154	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
155	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
156	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
157	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
158	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	#41×C₁₃₇	1	8	yes	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₂²⋊C₄	—	4	4	no	—
169	C₄²	—	2	4	no	—
170	C₄²	—	2	4	no	—
171	C₄²	—	2	2	no	—
172	C₄²	—	4	4	no	—
173	C₄²	#41×C₁₃₇	1	4	yes	—
174	C₂³	—	2	1	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	#117×C₁₃₇	1	4	yes	—
191	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
192	C₂×C₄	C₂×#41×C₁₃₇	1	4	yes	—
193	C₈	—	2	8	no	—
194	C₈	—	4	4	no	—
195	C₂²	—	4	1	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	C₂×#117×C₁₃₇	1	1	yes	characteristic
199	C₂²	C₄×#41×C₁₃₇	1	2	yes	—
200	C₂²	#568×C₁₃₇	1	4	yes	—
201	C₄	—	2	4	no	—
202	C₄	—	4	1	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	4	no	—
205	C₄	—	4	4	no	—
206	C₄	#529×C₁₃₇	1	4	yes	—
207	C₂	—	8	1	no	—
208	C₂	#2298×C₁₃₇	1	1	yes	characteristic
209	C₂	#2923×C₁₃₇	1	2	yes	—
210	C₂	#2967×C₁₃₇	1	4	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
3. #1917×C₁₃₇	C₂	semidirect
4. #1768×C₁₃₇	C₂	semidirect
6. #1745×C₁₃₇	C₂	semidirect
7. C₄×#97×C₁₃₇	C₂	semidirect
105. #21433	C₁₃₇	direct (N × H)
106. C₁₃₇	#21433	direct (N × H)
109. #1917	C₂×C₁₃₇	semidirect
110. #1768	C₂×C₁₃₇	semidirect
112. #1745	C₂×C₁₃₇	semidirect
113. C₄×#97	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2298×C₁₃₇	2
#1924×C₁₃₇	2
#1917×C₁₃₇	2
#1768×C₁₃₇	2
#1750×C₁₃₇	2
#1745×C₁₃₇	2
C₄×#97×C₁₃₇	2
#21433	137

Subgroups — C137×#21433 #46870819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21433 #46870819