Subgroups — C₁₃₇×#33401 #46882787

Order 35072.

684 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3407×C₁₃₇	C₂	1	2	yes	maximal
2	C₄×#131×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2153×C₁₃₇	C₂	1	6	yes	maximal, complemented
4	C₂×#364×C₁₃₇	C₂	1	6	yes	maximal, complemented
5	C₂×#131×C₁₃₇	C₄	1	1	yes	characteristic
6	C₂×#131×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₄×#41×C₁₃₇	C₄	1	1	yes	characteristic
8	#504×C₁₃₇	—	2	8	no	—
9	C₂×M₃₂×C₁₃₇	C₂²	1	12	yes	—
10	#405×C₁₃₇	C₄	1	3	yes	—
11	#405×C₁₃₇	C₂²	1	3	yes	—
12	C₄×M₁₆×C₁₃₇	C₄	1	3	yes	—
13	C₂×C₄×C₈×C₁₃₇	C₂²	1	3	yes	—
14	#364×C₁₃₇	C₂²	1	4	yes	—
15	#364×C₁₃₇	C₂²	1	12	yes	—
16	C₂×#41×C₁₃₇	C₂×C₄	1	1	yes	characteristic
17	#131×C₁₃₇	C₈	1	8	yes	—
18	#131×C₁₃₇	D₄	1	4	yes	—
19	#131×C₁₃₇	Q₈	1	4	yes	—
20	C₂×M₁₆×C₁₃₇	C₂×C₄	1	3	yes	—
21	C₂×M₁₆×C₁₃₇	C₂×C₄	1	3	yes	—
22	C₂²×C₈×C₁₃₇	C₂×C₄	1	3	yes	—
23	C₂²×C₈×C₁₃₇	C₂³	1	3	yes	—
24	C₄×Q₈×C₁₃₇	—	2	1	no	—
25	C₄×D₄×C₁₃₇	—	2	3	no	—
26	#117×C₁₃₇	C₂×C₄	1	3	yes	—
27	C₂×C₄²×C₁₃₇	C₂×C₄	1	3	yes	—
28	M₃₂×C₁₃₇	—	2	12	no	—
29	M₃₂×C₁₃₇	C₂³	1	8	yes	—
30	C₂×C₁₆×C₁₃₇	—	2	12	no	—
31	#97×C₁₃₇	C₂×C₄	1	6	yes	—
32	C₄×C₈×C₁₃₇	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₁₃₇	C₂×C₄	1	3	yes	—
34	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₁₃₇	C₂³	1	3	yes	—
36	#41×C₁₃₇	C₄⋊C₄	1	4	yes	—
37	#41×C₁₃₇	C₂×C₈	1	4	yes	—
38	C₂×Q₈×C₁₃₇	M₁₆	1	1	yes	characteristic
39	C₂×D₄×C₁₃₇	M₁₆	1	3	yes	—
40	C₂²×C₄×C₁₃₇	—	2	3	no	—
41	C₂²×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
42	M₁₆×C₁₃₇	C₄⋊C₄	1	12	yes	—
43	M₁₆×C₁₃₇	C₂×C₈	1	12	yes	—
44	C₂×C₈×C₁₃₇	C₂×C₈	1	12	yes	—
45	C₂×C₈×C₁₃₇	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃₇	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₁₃₇	C₂×D₄	1	6	yes	—
49	C₂×C₈×C₁₃₇	C₂×Q₈	1	6	yes	—
50	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄×C₁₃₇	—	2	3	no	—
52	C₂²⋊C₄×C₁₃₇	—	2	3	no	—
53	C₄²×C₁₃₇	—	2	3	no	—
54	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₁₃₇	C₂²×C₄	1	3	yes	—
56	C₁₆×C₁₃₇	—	2	8	no	—
57	C₂³×C₁₃₇	C₂×M₁₆	1	3	yes	—
58	Q₈×C₁₃₇	C₄⋊C₈	1	4	yes	—
59	D₄×C₁₃₇	C₄⋊C₈	1	12	yes	—
60	C₂×C₄×C₁₃₇	—	2	1	no	—
61	C₂×C₄×C₁₃₇	—	2	3	no	—
62	C₂×C₄×C₁₃₇	—	4	6	no	—
63	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	6	yes	—
64	C₂×C₄×C₁₃₇	C₂²×C₈	1	6	yes	—
65	C₂×C₄×C₁₃₇	C₂×M₁₆	1	3	yes	—
66	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
67	C₈×C₁₃₇	C₂×C₄⋊C₄	1	6	yes	—
68	C₈×C₁₃₇	C₂²×C₈	1	2	yes	—
69	C₈×C₁₃₇	C₂²×C₈	1	6	yes	—
70	C₈×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
71	C₈×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
72	C₂²×C₁₃₇	—	4	3	no	—
73	C₂²×C₁₃₇	C₂×C₄⋊C₈	1	6	yes	—
74	C₂²×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
75	C₄×C₁₃₇	—	2	2	no	—
76	C₄×C₁₃₇	—	4	3	no	—
77	C₄×C₁₃₇	C₂×C₄⋊C₈	1	6	yes	—
78	C₄×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
80	C₂×C₁₃₇	—	2	1	no	—
81	C₂×C₁₃₇	—	2	6	no	—
82	C₂×C₁₃₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
83	#33401	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₁₃₇	#33401	1	1	yes	characteristic, direct product factor, Sylow
85	#3407	C₂×C₁₃₇	1	2	yes	—
86	C₄×#131	C₂×C₁₃₇	1	1	yes	characteristic
87	#2153	C₂×C₁₃₇	1	6	yes	complemented
88	C₂×#364	C₂×C₁₃₇	1	6	yes	complemented
89	C₂×#131	C₄×C₁₃₇	1	1	yes	characteristic
90	C₂×#131	C₂²×C₁₃₇	1	1	yes	characteristic
91	C₄×#41	C₄×C₁₃₇	1	1	yes	characteristic
92	#504	—	2	8	no	—
93	C₂×M₃₂	C₂²×C₁₃₇	1	12	yes	—
94	#405	C₄×C₁₃₇	1	3	yes	—
95	#405	C₂²×C₁₃₇	1	3	yes	—
96	C₄×M₁₆	C₄×C₁₃₇	1	3	yes	—
97	C₂×C₄×C₈	C₂²×C₁₃₇	1	3	yes	—
98	#364	C₂²×C₁₃₇	1	4	yes	—
99	#364	C₂²×C₁₃₇	1	12	yes	—
100	C₂×#41	C₂×C₄×C₁₃₇	1	1	yes	characteristic
101	#131	C₈×C₁₃₇	1	8	yes	—
102	#131	D₄×C₁₃₇	1	4	yes	—
103	#131	Q₈×C₁₃₇	1	4	yes	—
104	C₂×M₁₆	C₂×C₄×C₁₃₇	1	3	yes	—
105	C₂×M₁₆	C₂×C₄×C₁₃₇	1	3	yes	—
106	C₂²×C₈	C₂×C₄×C₁₃₇	1	3	yes	—
107	C₂²×C₈	C₂³×C₁₃₇	1	3	yes	—
108	C₄×Q₈	—	2	1	no	—
109	C₄×D₄	—	2	3	no	—
110	#117	C₂×C₄×C₁₃₇	1	3	yes	—
111	C₂×C₄²	C₂×C₄×C₁₃₇	1	3	yes	—
112	M₃₂	—	2	12	no	—
113	M₃₂	C₂³×C₁₃₇	1	8	yes	—
114	C₂×C₁₆	—	2	12	no	—
115	#97	C₂×C₄×C₁₃₇	1	6	yes	—
116	C₄×C₈	C₂×C₄×C₁₃₇	1	3	yes	—
117	C₄×C₈	C₂×C₄×C₁₃₇	1	3	yes	—
118	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
119	C₄×C₈	C₂³×C₁₃₇	1	3	yes	—
120	#41	C₄⋊C₄×C₁₃₇	1	4	yes	—
121	#41	C₂×C₈×C₁₃₇	1	4	yes	—
122	C₂×Q₈	M₁₆×C₁₃₇	1	1	yes	characteristic
123	C₂×D₄	M₁₆×C₁₃₇	1	3	yes	—
124	C₂²×C₄	—	2	3	no	—
125	C₂²×C₄	C₂²×C₄×C₁₃₇	1	3	yes	—
126	M₁₆	C₄⋊C₄×C₁₃₇	1	12	yes	—
127	M₁₆	C₂×C₈×C₁₃₇	1	12	yes	—
128	C₂×C₈	C₂×C₈×C₁₃₇	1	12	yes	—
129	C₂×C₈	C₂²×C₄×C₁₃₇	1	3	yes	—
130	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
131	C₂×C₈	C₂²×C₄×C₁₃₇	1	3	yes	—
132	C₂×C₈	C₂×D₄×C₁₃₇	1	6	yes	—
133	C₂×C₈	C₂×Q₈×C₁₃₇	1	6	yes	—
134	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
135	C₄⋊C₄	—	2	3	no	—
136	C₂²⋊C₄	—	2	3	no	—
137	C₄²	—	2	3	no	—
138	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
139	C₄²	C₂²×C₄×C₁₃₇	1	3	yes	—
140	C₁₆	—	2	8	no	—
141	C₂³	C₂×M₁₆×C₁₃₇	1	3	yes	—
142	Q₈	C₄⋊C₈×C₁₃₇	1	4	yes	—
143	D₄	C₄⋊C₈×C₁₃₇	1	12	yes	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	3	no	—
146	C₂×C₄	—	4	6	no	—
147	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	6	yes	—
148	C₂×C₄	C₂²×C₈×C₁₃₇	1	6	yes	—
149	C₂×C₄	C₂×M₁₆×C₁₃₇	1	3	yes	—
150	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
151	C₈	C₂×C₄⋊C₄×C₁₃₇	1	6	yes	—
152	C₈	C₂²×C₈×C₁₃₇	1	2	yes	—
153	C₈	C₂²×C₈×C₁₃₇	1	6	yes	—
154	C₈	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
155	C₈	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
156	C₂²	—	4	3	no	—
157	C₂²	C₂×C₄⋊C₈×C₁₃₇	1	6	yes	—
158	C₂²	C₂²×M₁₆×C₁₃₇	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	3	no	—
161	C₄	C₂×C₄⋊C₈×C₁₃₇	1	6	yes	—
162	C₄	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
163	C₄	C₂³×C₈×C₁₃₇	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	6	no	—
166	C₂	C₂²×C₄⋊C₈×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃₇	C₂²×D₄
commutator	C₄	C₂³×C₈×C₁₃₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
socle	C₂×C₁₃₇	C₂²×C₄⋊C₈
prime radical	C₂×#41×C₁₃₇	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃₇ ◁ C₂×#131×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #2153×C₁₃₇	C₂	semidirect
4. C₂×#364×C₁₃₇	C₂	semidirect
83. #33401	C₁₃₇	direct (N × H)
84. C₁₃₇	#33401	direct (N × H)
87. #2153	C₂×C₁₃₇	semidirect
88. C₂×#364	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3407×C₁₃₇	2
C₄×#131×C₁₃₇	2
#2153×C₁₃₇	2
C₂×#364×C₁₃₇	2
#33401	137

Subgroups — C137×#33401 #46882787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33401 #46882787