Subgroups — C₁₃₇×#19668 #46869054

Order 35072.

644 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3231×C₁₃₇	C₂	1	4	yes	maximal
2	#3170×C₁₃₇	C₂	1	2	yes	maximal
3	#1718×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	#1713×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1710×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#1707×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1450×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#533×C₁₃₇	C₂²	1	4	yes	—
9	#500×C₁₃₇	C₂²	1	2	yes	—
10	#500×C₁₃₇	C₂²	1	4	yes	—
11	#498×C₁₃₇	C₂²	1	4	yes	—
12	#494×C₁₃₇	C₂²	1	2	yes	—
13	#487×C₁₃₇	C₂²	1	4	yes	—
14	#487×C₁₃₇	C₂²	1	4	yes	—
15	#487×C₁₃₇	C₂²	1	4	yes	—
16	#485×C₁₃₇	—	2	8	no	—
17	#479×C₁₃₇	—	2	8	no	—
18	#467×C₁₃₇	—	2	4	no	—
19	C₄×Q₁₆×C₁₃₇	—	2	4	no	—
20	#431×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	#322×C₁₃₇	C₂²	1	2	yes	—
23	C₈²×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₈²×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
26	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
27	C₄⋊Q₈×C₁₃₇	C₂³	1	4	yes	—
28	C₄⋊Q₈×C₁₃₇	C₂³	1	4	yes	—
29	#125×C₁₃₇	—	2	4	no	—
30	C₂²⋊Q₈×C₁₃₇	—	2	4	no	—
31	C₂²⋊Q₈×C₁₃₇	—	2	4	no	—
32	C₄×Q₈×C₁₃₇	—	2	4	no	—
33	C₄×Q₈×C₁₃₇	—	2	4	no	—
34	#117×C₁₃₇	—	2	4	no	—
35	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
36	#107×C₁₃₇	—	2	4	no	—
37	#107×C₁₃₇	—	2	8	no	—
38	#106×C₁₃₇	—	2	8	no	—
39	C₄⋊C₈×C₁₃₇	D₄	1	4	yes	—
40	#103×C₁₃₇	—	2	8	no	—
41	#103×C₁₃₇	—	2	8	no	—
42	#103×C₁₃₇	—	2	8	no	—
43	C₂²⋊C₈×C₁₃₇	D₄	1	4	yes	—
44	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
45	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
48	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
49	C₂×Q₈×C₁₃₇	—	2	4	no	—
50	C₂×Q₈×C₁₃₇	—	2	4	no	—
51	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₁₃₇	—	4	8	no	—
54	C₂×C₈×C₁₃₇	—	2	4	no	—
55	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
57	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
58	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
59	C₄⋊C₄×C₁₃₇	—	2	4	no	—
60	C₄⋊C₄×C₁₃₇	—	2	4	no	—
61	C₄⋊C₄×C₁₃₇	—	2	4	no	—
62	C₄⋊C₄×C₁₃₇	—	2	4	no	—
63	C₄⋊C₄×C₁₃₇	—	4	4	no	—
64	C₄⋊C₄×C₁₃₇	—	4	4	no	—
65	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
66	C₄²×C₁₃₇	—	4	4	no	—
67	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
69	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
70	C₂³×C₁₃₇	#127	1	1	yes	characteristic
71	Q₈×C₁₃₇	—	4	8	no	—
72	Q₈×C₁₃₇	—	4	8	no	—
73	C₂×C₄×C₁₃₇	—	2	2	no	—
74	C₂×C₄×C₁₃₇	—	2	4	no	—
75	C₂×C₄×C₁₃₇	—	4	4	no	—
76	C₂×C₄×C₁₃₇	—	4	4	no	—
77	C₂×C₄×C₁₃₇	#127	1	2	yes	—
78	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
81	C₈×C₁₃₇	—	2	2	no	—
82	C₈×C₁₃₇	—	2	4	no	—
83	C₈×C₁₃₇	—	2	4	no	—
84	C₈×C₁₃₇	#135	1	4	yes	—
85	C₂²×C₁₃₇	—	2	1	no	—
86	C₂²×C₁₃₇	—	2	2	no	—
87	C₂²×C₁₃₇	C₂×#127	1	1	yes	characteristic
88	C₄×C₁₃₇	—	2	1	no	—
89	C₄×C₁₃₇	—	2	2	no	—
90	C₄×C₁₃₇	—	8	4	no	—
91	C₄×C₁₃₇	—	8	4	no	—
92	C₄×C₁₃₇	C₂×#135	1	2	yes	—
93	C₄×C₁₃₇	#578	1	4	yes	—
94	C₂×C₁₃₇	—	4	1	no	—
95	C₂×C₁₃₇	#3147	1	1	yes	characteristic
96	C₂×C₁₃₇	#3156	1	2	yes	—
97	#19668	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₁₃₇	#19668	1	1	yes	characteristic, direct product factor, Sylow
99	#3231	C₂×C₁₃₇	1	4	yes	—
100	#3170	C₂×C₁₃₇	1	2	yes	—
101	#1718	C₂×C₁₃₇	1	4	yes	complemented
102	#1713	C₂×C₁₃₇	1	1	yes	characteristic, complemented
103	#1710	C₂×C₁₃₇	1	1	yes	characteristic, complemented
104	#1707	C₂×C₁₃₇	1	2	yes	complemented
105	#1450	C₂×C₁₃₇	1	1	yes	characteristic
106	#533	C₂²×C₁₃₇	1	4	yes	—
107	#500	C₂²×C₁₃₇	1	2	yes	—
108	#500	C₂²×C₁₃₇	1	4	yes	—
109	#498	C₂²×C₁₃₇	1	4	yes	—
110	#494	C₂²×C₁₃₇	1	2	yes	—
111	#487	C₂²×C₁₃₇	1	4	yes	—
112	#487	C₂²×C₁₃₇	1	4	yes	—
113	#487	C₂²×C₁₃₇	1	4	yes	—
114	#485	—	2	8	no	—
115	#479	—	2	8	no	—
116	#467	—	2	4	no	—
117	C₄×Q₁₆	—	2	4	no	—
118	#431	C₂²×C₁₃₇	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
120	#322	C₂²×C₁₃₇	1	2	yes	—
121	C₈²	C₂²×C₁₃₇	1	1	yes	characteristic
122	C₈²	C₂²×C₁₃₇	1	1	yes	characteristic
123	C₂×Q₁₆	—	2	4	no	—
124	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
125	C₄⋊Q₈	C₂³×C₁₃₇	1	4	yes	—
126	C₄⋊Q₈	C₂³×C₁₃₇	1	4	yes	—
127	#125	—	2	4	no	—
128	C₂²⋊Q₈	—	2	4	no	—
129	C₂²⋊Q₈	—	2	4	no	—
130	C₄×Q₈	—	2	4	no	—
131	C₄×Q₈	—	2	4	no	—
132	#117	—	2	4	no	—
133	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
134	#107	—	2	4	no	—
135	#107	—	2	8	no	—
136	#106	—	2	8	no	—
137	C₄⋊C₈	D₄×C₁₃₇	1	4	yes	—
138	#103	—	2	8	no	—
139	#103	—	2	8	no	—
140	#103	—	2	8	no	—
141	C₂²⋊C₈	D₄×C₁₃₇	1	4	yes	—
142	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
143	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
144	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
145	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
146	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	2	4	no	—
149	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
150	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
151	Q₁₆	—	4	8	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
154	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
155	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
156	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₂²⋊C₄	—	4	4	no	—
164	C₄²	—	4	4	no	—
165	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
166	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
167	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
168	C₂³	#127×C₁₃₇	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	4	8	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	#127×C₁₃₇	1	2	yes	—
176	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
179	C₈	—	2	2	no	—
180	C₈	—	2	4	no	—
181	C₈	—	2	4	no	—
182	C₈	#135×C₁₃₇	1	4	yes	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	C₂×#127×C₁₃₇	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	2	no	—
188	C₄	—	8	4	no	—
189	C₄	—	8	4	no	—
190	C₄	C₂×#135×C₁₃₇	1	2	yes	—
191	C₄	#578×C₁₃₇	1	4	yes	—
192	C₂	—	4	1	no	—
193	C₂	#3147×C₁₃₇	1	1	yes	characteristic
194	C₂	#3156×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
prime radical	C₂³×C₁₃₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1718×C₁₃₇	C₂	semidirect
4. #1713×C₁₃₇	C₂	semidirect
5. #1710×C₁₃₇	C₂	semidirect
6. #1707×C₁₃₇	C₂	semidirect
97. #19668	C₁₃₇	direct (N × H)
98. C₁₃₇	#19668	direct (N × H)
101. #1718	C₂×C₁₃₇	semidirect
102. #1713	C₂×C₁₃₇	semidirect
103. #1710	C₂×C₁₃₇	semidirect
104. #1707	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3231×C₁₃₇	2
#3170×C₁₃₇	2
#1718×C₁₃₇	2
#1713×C₁₃₇	2
#1710×C₁₃₇	2
#1707×C₁₃₇	2
#1450×C₁₃₇	2
#19668	137

Subgroups — C137×#19668 #46869054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#19668 #46869054