Subgroups — C₂×#343×C₁₃₇ #31776752

Order 17536.

432 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₁₃₇	C₂	1	2	yes	maximal
2	C₂²×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#343×C₁₃₇	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₄×C₁₃₇	C₄	1	2	yes	—
5	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₁₃₇	—	2	4	no	—
7	C₂×M₁₆×C₁₃₇	—	2	2	no	—
8	C₂×M₁₆×C₁₃₇	C₄	1	8	yes	complemented
9	C₂×M₁₆×C₁₃₇	C₂²	1	4	yes	—
10	C₂²×C₈×C₁₃₇	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₃₇	C₄	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
15	C₂⁴×C₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₃₇	—	2	2	no	—
17	C₂²×C₄×C₁₃₇	—	2	2	no	—
18	C₂²×C₄×C₁₃₇	—	4	4	no	—
19	C₂²×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
23	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
24	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₃₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₃₇	—	2	8	no	—
28	M₁₆×C₁₃₇	—	2	4	no	—
29	M₁₆×C₁₃₇	C₂×C₄	1	8	yes	complemented
30	C₂×C₈×C₁₃₇	—	2	8	no	—
31	C₂×C₈×C₁₃₇	—	2	4	no	—
32	C₄⋊C₄×C₁₃₇	—	2	4	no	—
33	C₄⋊C₄×C₁₃₇	—	2	4	no	—
34	C₂³×C₁₃₇	—	2	2	no	—
35	C₂³×C₁₃₇	—	2	2	no	—
36	C₂³×C₁₃₇	C₄²	1	2	yes	—
37	C₂³×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂³×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂³×C₁₃₇	C₄⋊C₄	1	2	yes	—
40	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₇	—	2	4	no	—
42	C₂×C₄×C₁₃₇	—	4	4	no	—
43	C₂×C₄×C₁₃₇	—	4	4	no	—
44	C₂×C₄×C₁₃₇	—	4	4	no	—
45	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
46	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
48	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
49	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
51	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
52	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
56	C₈×C₁₃₇	—	2	8	no	—
57	C₂²×C₁₃₇	—	2	4	no	—
58	C₂²×C₁₃₇	—	2	4	no	—
59	C₂²×C₁₃₇	—	2	1	no	—
60	C₂²×C₁₃₇	—	2	1	no	—
61	C₂²×C₁₃₇	—	2	1	no	—
62	C₂²×C₁₃₇	—	2	1	no	—
63	C₂²×C₁₃₇	#95	1	2	yes	—
64	C₂²×C₁₃₇	#95	1	2	yes	—
65	C₂²×C₁₃₇	#100	1	2	yes	—
66	C₂²×C₁₃₇	#101	1	2	yes	—
67	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₄×C₁₃₇	—	4	4	no	—
71	C₄×C₁₃₇	#95	1	4	yes	—
72	C₄×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
73	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₁₃₇	—	2	2	no	—
75	C₂×C₁₃₇	—	2	1	no	—
76	C₂×C₁₃₇	—	2	1	no	—
77	C₂×C₁₃₇	#343	1	4	yes	direct product factor
78	C₂×C₁₃₇	C₂×#95	1	1	yes	characteristic
79	C₂×C₁₃₇	C₂×#100	1	1	yes	characteristic
80	C₂×C₁₃₇	C₂×#101	1	1	yes	characteristic
81	C₁₃₇	C₂×#343	1	1	yes	characteristic, direct product factor, Sylow
82	C₂×#343	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂²×M₁₆	C₂×C₁₃₇	1	2	yes	—
84	C₂²×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
85	#343	C₂×C₁₃₇	1	4	yes	direct product factor
86	C₂³×C₄	C₄×C₁₃₇	1	2	yes	—
87	C₂³×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
88	C₂×M₁₆	—	2	4	no	—
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	C₄×C₁₃₇	1	8	yes	complemented
91	C₂×M₁₆	C₂²×C₁₃₇	1	4	yes	—
92	C₂²×C₈	—	2	2	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	2	2	no	—
95	C₂×C₄⋊C₄	C₄×C₁₃₇	1	2	yes	—
96	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
97	C₂⁴	C₂×C₄×C₁₃₇	1	1	yes	characteristic
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	4	4	no	—
101	C₂²×C₄	C₂×C₄×C₁₃₇	1	4	yes	—
102	C₂²×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
103	C₂²×C₄	C₂×C₄×C₁₃₇	1	4	yes	—
104	C₂²×C₄	D₄×C₁₃₇	1	2	yes	—
105	C₂²×C₄	D₄×C₁₃₇	1	2	yes	—
106	C₂²×C₄	D₄×C₁₃₇	1	2	yes	—
107	C₂²×C₄	Q₈×C₁₃₇	1	2	yes	—
108	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
109	M₁₆	—	2	8	no	—
110	M₁₆	—	2	4	no	—
111	M₁₆	C₂×C₄×C₁₃₇	1	8	yes	complemented
112	C₂×C₈	—	2	8	no	—
113	C₂×C₈	—	2	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₂³	—	2	2	no	—
117	C₂³	—	2	2	no	—
118	C₂³	C₄²×C₁₃₇	1	2	yes	—
119	C₂³	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
120	C₂³	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
121	C₂³	C₄⋊C₄×C₁₃₇	1	2	yes	—
122	C₂³	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	C₄²×C₁₃₇	1	2	yes	—
128	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
129	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	4	yes	—
130	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	4	yes	—
131	C₂×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
132	C₂×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
133	C₂×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
134	C₂×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
135	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
136	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
137	C₂×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
138	C₈	—	2	8	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	4	no	—
144	C₂²	—	2	4	no	—
145	C₂²	#95×C₁₃₇	1	2	yes	—
146	C₂²	#95×C₁₃₇	1	2	yes	—
147	C₂²	#100×C₁₃₇	1	2	yes	—
148	C₂²	#101×C₁₃₇	1	2	yes	—
149	C₂²	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
150	C₂²	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
151	C₂²	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
152	C₄	—	4	4	no	—
153	C₄	#95×C₁₃₇	1	4	yes	—
154	C₄	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
155	C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	#343×C₁₃₇	1	4	yes	direct product factor
160	C₂	C₂×#95×C₁₃₇	1	1	yes	characteristic
161	C₂	C₂×#100×C₁₃₇	1	1	yes	characteristic
162	C₂	C₂×#101×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²⋊C₄
commutator	C₂²	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₃₇
prime radical	C₂⁴×C₁₃₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #343×C₁₃₇	C₂	direct (N × H)
8. C₂×M₁₆×C₁₃₇	C₄	semidirect
29. M₁₆×C₁₃₇	C₂×C₄	semidirect
77. C₂×C₁₃₇	#343	direct (N × H)
81. C₁₃₇	C₂×#343	direct (N × H)
82. C₂×#343	C₁₃₇	direct (N × H)
85. #343	C₂×C₁₃₇	direct (N × H)
90. C₂×M₁₆	C₄×C₁₃₇	semidirect
111. M₁₆	C₂×C₄×C₁₃₇	semidirect
159. C₂	#343×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₁₃₇	2
C₂²×C₄⋊C₄×C₁₃₇	2
#343×C₁₃₇	2
C₂×#343	137

Subgroups — C2×#343×C137 #31776752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#343×C₁₃₇ #31776752