Subgroups — C₁₃₇×#9825 #46859211

Order 35072.

558 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1990×C₁₃₇	C₂	1	4	yes	maximal
3	#1845×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₁₃₇	—	2	2	no	—
6	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
7	#494×C₁₃₇	—	2	2	no	—
8	#494×C₁₃₇	C₄	1	4	yes	complemented
9	C₂×#107×C₁₃₇	—	2	2	no	—
10	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	#384×C₁₃₇	C₂²	1	2	yes	—
13	#340×C₁₃₇	—	2	8	no	—
14	C₈²×C₁₃₇	—	2	2	no	—
15	C₂²×Q₈×C₁₃₇	—	2	2	no	—
16	C₂×Q₁₆×C₁₃₇	—	2	8	no	—
17	C₂×Q₁₆×C₁₃₇	—	4	4	no	—
18	C₂×Q₁₆×C₁₃₇	—	4	4	no	—
19	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
20	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
21	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
22	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
23	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
27	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
28	#107×C₁₃₇	—	4	4	no	—
29	C₄×C₈×C₁₃₇	—	2	2	no	—
30	C₄×C₈×C₁₃₇	—	2	2	no	—
31	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
33	C₂×Q₈×C₁₃₇	—	2	4	no	—
34	C₂×Q₈×C₁₃₇	—	4	2	no	—
35	C₂×Q₈×C₁₃₇	—	4	4	no	—
36	C₂×Q₈×C₁₃₇	—	4	4	no	—
37	C₂²×C₄×C₁₃₇	—	4	2	no	—
38	C₂²×C₄×C₁₃₇	—	4	2	no	—
39	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
41	Q₁₆×C₁₃₇	—	4	16	no	—
42	C₂×C₈×C₁₃₇	—	2	4	no	—
43	C₂×C₈×C₁₃₇	—	2	4	no	—
44	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₁₃₇	D₈	1	4	yes	—
46	C₂×C₈×C₁₃₇	SD₁₆	1	4	yes	—
47	C₂×C₈×C₁₃₇	Q₁₆	1	8	yes	complemented
48	C₄⋊C₄×C₁₃₇	—	2	8	no	—
49	C₄⋊C₄×C₁₃₇	—	4	4	no	—
50	C₄⋊C₄×C₁₃₇	—	4	4	no	—
51	C₄⋊C₄×C₁₃₇	—	4	8	no	—
52	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₁₃₇	#41	1	2	yes	—
55	C₂³×C₁₃₇	#127	1	1	yes	characteristic
56	Q₈×C₁₃₇	—	4	8	no	—
57	Q₈×C₁₃₇	—	8	4	no	—
58	C₂×C₄×C₁₃₇	—	4	4	no	—
59	C₂×C₄×C₁₃₇	—	8	2	no	—
60	C₂×C₄×C₁₃₇	—	8	2	no	—
61	C₂×C₄×C₁₃₇	—	8	2	no	—
62	C₂×C₄×C₁₃₇	—	8	4	no	—
63	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
65	C₂×C₄×C₁₃₇	#121	1	2	yes	—
66	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
67	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
69	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	4	yes	—
70	C₂×C₄×C₁₃₇	#135	1	4	yes	—
71	C₈×C₁₃₇	—	2	8	no	—
72	C₈×C₁₃₇	#102	1	8	yes	—
73	C₂²×C₁₃₇	#390	1	1	yes	characteristic
74	C₂²×C₁₃₇	#492	1	1	yes	characteristic
75	C₂²×C₁₃₇	#493	1	1	yes	characteristic
76	C₂²×C₁₃₇	#494	1	2	yes	—
77	C₂²×C₁₃₇	#495	1	2	yes	—
78	C₄×C₁₃₇	—	8	4	no	—
79	C₄×C₁₃₇	—	16	2	no	—
80	C₄×C₁₃₇	C₂×#102	1	2	yes	—
81	C₄×C₁₃₇	C₄×Q₁₆	1	4	yes	—
82	C₄×C₁₃₇	#467	1	4	yes	—
83	C₄×C₁₃₇	#486	1	2	yes	—
84	C₂×C₁₃₇	#1707	1	2	yes	—
85	C₂×C₁₃₇	#1709	1	1	yes	characteristic
86	C₂×C₁₃₇	#1713	1	1	yes	characteristic
87	C₂×C₁₃₇	#1957	1	1	yes	characteristic
88	C₂×C₁₃₇	#1969	1	2	yes	—
89	#9825	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₁₃₇	#9825	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×#494	C₂×C₁₃₇	1	1	yes	characteristic
92	#1990	C₂×C₁₃₇	1	4	yes	—
93	#1845	C₂×C₁₃₇	1	1	yes	characteristic
94	C₂×C₈²	C₂×C₁₃₇	1	1	yes	characteristic
95	C₂²×Q₁₆	—	2	2	no	—
96	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
97	#494	—	2	2	no	—
98	#494	C₄×C₁₃₇	1	4	yes	complemented
99	C₂×#107	—	2	2	no	—
100	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
101	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
102	#384	C₂²×C₁₃₇	1	2	yes	—
103	#340	—	2	8	no	—
104	C₈²	—	2	2	no	—
105	C₂²×Q₈	—	2	2	no	—
106	C₂×Q₁₆	—	2	8	no	—
107	C₂×Q₁₆	—	4	4	no	—
108	C₂×Q₁₆	—	4	4	no	—
109	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
110	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
111	C₄⋊Q₈	—	2	2	no	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	4	yes	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
118	#107	—	4	4	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
122	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂×Q₈	—	4	4	no	—
126	C₂×Q₈	—	4	4	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
130	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
131	Q₁₆	—	4	16	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	4	yes	—
135	C₂×C₈	D₈×C₁₃₇	1	4	yes	—
136	C₂×C₈	SD₁₆×C₁₃₇	1	4	yes	—
137	C₂×C₈	Q₁₆×C₁₃₇	1	8	yes	complemented
138	C₄⋊C₄	—	2	8	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	8	no	—
142	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
143	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
144	C₄²	#41×C₁₃₇	1	2	yes	—
145	C₂³	#127×C₁₃₇	1	1	yes	characteristic
146	Q₈	—	4	8	no	—
147	Q₈	—	8	4	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
154	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
155	C₂×C₄	#121×C₁₃₇	1	2	yes	—
156	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
157	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
158	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
159	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	4	yes	—
160	C₂×C₄	#135×C₁₃₇	1	4	yes	—
161	C₈	—	2	8	no	—
162	C₈	#102×C₁₃₇	1	8	yes	—
163	C₂²	#390×C₁₃₇	1	1	yes	characteristic
164	C₂²	#492×C₁₃₇	1	1	yes	characteristic
165	C₂²	#493×C₁₃₇	1	1	yes	characteristic
166	C₂²	#494×C₁₃₇	1	2	yes	—
167	C₂²	#495×C₁₃₇	1	2	yes	—
168	C₄	—	8	4	no	—
169	C₄	—	16	2	no	—
170	C₄	C₂×#102×C₁₃₇	1	2	yes	—
171	C₄	C₄×Q₁₆×C₁₃₇	1	4	yes	—
172	C₄	#467×C₁₃₇	1	4	yes	—
173	C₄	#486×C₁₃₇	1	2	yes	—
174	C₂	#1707×C₁₃₇	1	2	yes	—
175	C₂	#1709×C₁₃₇	1	1	yes	characteristic
176	C₂	#1713×C₁₃₇	1	1	yes	characteristic
177	C₂	#1957×C₁₃₇	1	1	yes	characteristic
178	C₂	#1969×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	#127
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. #494×C₁₃₇	C₄	semidirect
47. C₂×C₈×C₁₃₇	Q₁₆	semidirect
89. #9825	C₁₃₇	direct (N × H)
90. C₁₃₇	#9825	direct (N × H)
98. #494	C₄×C₁₃₇	semidirect
137. C₂×C₈	Q₁₆×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₁₃₇	2
#1990×C₁₃₇	2
#1845×C₁₃₇	2
C₂×C₈²×C₁₃₇	2
#9825	137

Subgroups — C137×#9825 #46859211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9825 #46859211