Subgroups — #3383×C₁₃₇ #31778394

Order 17536.

324 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557×C₁₃₇	C₂	1	2	yes	maximal
2	#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#499×C₁₃₇	C₂	1	2	yes	maximal
4	#479×C₁₃₇	C₂	1	4	yes	maximal
5	#475×C₁₃₇	C₂	1	2	yes	maximal
6	C₈×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₁₆×C₁₃₇	C₂	1	2	yes	maximal
8	C₄×Q₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
11	C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
12	#125×C₁₃₇	—	2	4	no	—
13	#125×C₁₃₇	C₂²	1	4	yes	—
14	C₄×Q₈×C₁₃₇	—	2	2	no	—
15	C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
16	C₄×Q₈×C₁₃₇	C₂²	1	4	yes	—
17	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	#107×C₁₃₇	C₂²	1	1	yes	characteristic
19	#107×C₁₃₇	C₂²	1	2	yes	—
20	#107×C₁₃₇	C₂²	1	2	yes	—
21	#106×C₁₃₇	C₂²	1	4	yes	—
22	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
24	#103×C₁₃₇	C₂²	1	4	yes	—
25	#103×C₁₃₇	C₂²	1	2	yes	—
26	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
27	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₁₃₇	—	2	2	no	—
29	C₂×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
30	C₂×Q₈×C₁₃₇	C₂³	1	2	yes	—
31	Q₁₆×C₁₃₇	Q₈	1	4	yes	complemented
32	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
33	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₃₇	—	2	4	no	—
36	C₄⋊C₄×C₁₃₇	—	2	2	no	—
37	C₄⋊C₄×C₁₃₇	—	2	4	no	—
38	C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₄⋊C₄×C₁₃₇	—	2	4	no	—
40	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
42	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
43	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
44	C₄²×C₁₃₇	—	2	4	no	—
45	C₄²×C₁₃₇	—	2	2	no	—
46	C₄²×C₁₃₇	C₂³	1	2	yes	—
47	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
48	Q₈×C₁₃₇	—	2	2	no	—
49	Q₈×C₁₃₇	—	2	2	no	—
50	Q₈×C₁₃₇	—	4	4	no	—
51	Q₈×C₁₃₇	C₂×Q₈	1	4	yes	—
52	C₂×C₄×C₁₃₇	—	2	2	no	—
53	C₂×C₄×C₁₃₇	—	2	2	no	—
54	C₂×C₄×C₁₃₇	—	2	4	no	—
55	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
60	C₈×C₁₃₇	—	2	2	no	—
61	C₈×C₁₃₇	—	2	1	no	—
62	C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₁₃₇	—	2	4	no	—
65	C₄×C₁₃₇	—	2	2	no	—
66	C₄×C₁₃₇	—	2	2	no	—
67	C₄×C₁₃₇	—	2	1	no	—
68	C₄×C₁₃₇	—	4	4	no	—
69	C₄×C₁₃₇	—	4	2	no	—
70	C₄×C₁₃₇	#135	1	2	yes	—
71	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
72	C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
73	C₂×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
74	C₂×C₁₃₇	C₂×#135	1	1	yes	characteristic
75	C₂×C₁₃₇	#578	1	1	yes	characteristic
76	C₁₃₇	#3383	1	1	yes	characteristic, direct product factor, Sylow
77	#3383	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	#557	C₂×C₁₃₇	1	2	yes	—
79	#500	C₂×C₁₃₇	1	1	yes	characteristic
80	#499	C₂×C₁₃₇	1	2	yes	—
81	#479	C₂×C₁₃₇	1	4	yes	—
82	#475	C₂×C₁₃₇	1	2	yes	—
83	C₈×Q₈	C₂×C₁₃₇	1	1	yes	characteristic
84	C₄×Q₁₆	C₂×C₁₃₇	1	2	yes	—
85	C₄×Q₁₆	C₂×C₁₃₇	1	1	yes	characteristic
86	C₂×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
87	C₄⋊Q₈	—	2	2	no	—
88	C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
89	#125	—	2	4	no	—
90	#125	C₂²×C₁₃₇	1	4	yes	—
91	C₄×Q₈	—	2	2	no	—
92	C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
93	C₄×Q₈	C₂²×C₁₃₇	1	4	yes	—
94	C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
95	#107	C₂²×C₁₃₇	1	1	yes	characteristic
96	#107	C₂²×C₁₃₇	1	2	yes	—
97	#107	C₂²×C₁₃₇	1	2	yes	—
98	#106	C₂²×C₁₃₇	1	4	yes	—
99	C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
100	C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
101	#103	C₂²×C₁₃₇	1	4	yes	—
102	#103	C₂²×C₁₃₇	1	2	yes	—
103	C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
104	C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
107	C₂×Q₈	C₂³×C₁₃₇	1	2	yes	—
108	Q₁₆	Q₈×C₁₃₇	1	4	yes	complemented
109	C₂×C₈	C₂³×C₁₃₇	1	2	yes	—
110	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
111	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
118	C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
119	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
120	C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
121	C₄²	—	2	4	no	—
122	C₄²	—	2	2	no	—
123	C₄²	C₂³×C₁₃₇	1	2	yes	—
124	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	2	2	no	—
127	Q₈	—	4	4	no	—
128	Q₈	C₂×Q₈×C₁₃₇	1	4	yes	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
133	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
134	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
135	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
136	C₂×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	1	no	—
139	C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
140	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	2	2	no	—
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	4	no	—
146	C₄	—	4	2	no	—
147	C₄	#135×C₁₃₇	1	2	yes	—
148	C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
149	C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
150	C₂	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
151	C₂	C₂×#135×C₁₃₇	1	1	yes	characteristic
152	C₂	#578×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄×Q₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
31. Q₁₆×C₁₃₇	Q₈	semidirect
76. C₁₃₇	#3383	direct (N × H)
77. #3383	C₁₃₇	direct (N × H)
108. Q₁₆	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#557×C₁₃₇	2
#500×C₁₃₇	2
#499×C₁₃₇	2
#479×C₁₃₇	2
#475×C₁₃₇	2
C₈×Q₈×C₁₃₇	2
C₄×Q₁₆×C₁₃₇	2
#3383	137

Subgroups — #3383×C137 #31778394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3383×C₁₃₇ #31778394