Subgroups — C₂×#22577452 #31789396

Order 17536.

460 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555494	C₂	1	2	yes	maximal
2	#22577452	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17984024	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555494	—	2	2	no	—
6	C₂×#14555494	—	2	4	no	—
7	C₂×#14555494	C₂²	1	4	yes	—
8	C₂×#14555494	C₂²	1	8	yes	—
9	C₂²×#11764040	—	2	2	no	—
10	#17984024	C₂²	1	16	yes	complemented
11	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₁₃₇	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₃₇	C₄	1	4	yes	—
14	C₂×M₁₆×C₁₃₇	C₂²	1	4	yes	—
15	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
16	#14555494	—	2	8	no	—
17	#14555494	—	2	4	no	—
18	#14555494	C₂³	1	8	yes	—
19	C₂×#11764040	—	2	8	no	—
20	C₂×#11764040	—	2	4	no	—
21	C₂⁴×C₁₃₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₃₇	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₃₇	Q₈	1	2	yes	—
28	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
30	M₁₆×C₁₃₇	C₂×C₄	1	16	yes	—
31	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
33	C₂×C₈×C₁₃₇	C₂³	1	4	yes	—
34	#11764040	—	2	8	no	—
35	C₂³×C₁₃₇	—	4	2	no	—
36	C₂³×C₁₃₇	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₃₇	C₄⋊C₄	1	2	yes	—
38	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
39	C₂³×C₁₃₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
42	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
53	C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
54	C₂²×C₁₃₇	—	2	4	no	—
55	C₂²×C₁₃₇	—	2	2	no	—
56	C₂²×C₁₃₇	—	4	2	no	—
57	C₂²×C₁₃₇	—	4	2	no	—
58	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
60	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
64	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
67	C₂×C₁₃₇	—	2	4	no	—
68	C₂×C₁₃₇	—	2	1	no	—
69	C₂×C₁₃₇	—	2	1	no	—
70	C₂×C₁₃₇	#430	1	2	yes	complemented
71	C₂×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₁₃₇	C₂×#430	1	1	yes	characteristic, complemented, Sylow
73	C₂×#430	—	137	1	no	maximal, Sylow
74	C₂²×M₁₆	—	137	2	no	—
75	C₂²×M₁₆	D₁₃₇	1	1	yes	characteristic
76	#430	—	137	8	no	—
77	C₂×#108	—	137	4	no	—
78	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic
79	C₂×M₁₆	—	274	4	no	—
80	C₂×M₁₆	—	274	2	no	—
81	C₂×M₁₆	—	137	4	no	—
82	C₂×M₁₆	—	137	8	no	—
83	C₂×M₁₆	Dic₁₃₇	1	4	yes	—
84	C₂×M₁₆	Dic₁₃₇	1	4	yes	—
85	C₂×M₁₆	C₂×D₁₃₇	1	4	yes	—
86	C₂²×C₈	—	274	2	no	—
87	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
88	#108	—	137	16	no	—
89	C₂⁴	Dic₂₇₄	1	1	yes	characteristic
90	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
91	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
92	C₂²×C₄	D₅₄₈	1	2	yes	—
93	C₂²×C₄	D₅₄₈	1	2	yes	—
94	C₂²×C₄	Dic₂₇₄	1	1	yes	characteristic
95	C₂²×C₄	Dic₂₇₄	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
97	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
98	M₁₆	—	274	8	no	—
99	M₁₆	—	274	4	no	—
100	M₁₆	—	137	8	no	—
101	M₁₆	C₂×Dic₁₃₇	1	16	yes	—
102	C₂×C₈	—	274	8	no	—
103	C₂×C₈	—	274	4	no	—
104	C₂×C₈	C₂×Dic₁₃₇	1	4	yes	—
105	C₂×C₈	C₂×Dic₁₃₇	1	4	yes	—
106	C₂×C₈	C₂²×D₁₃₇	1	4	yes	—
107	C₂³	—	4	2	no	—
108	C₂³	C₄⋊Dic₁₃₇	1	2	yes	—
109	C₂³	C₄⋊Dic₁₃₇	1	2	yes	—
110	C₂³	C₂×Dic₂₇₄	1	1	yes	characteristic
111	C₂³	C₂×Dic₂₇₄	1	2	yes	—
112	C₂×C₄	C₄⋊Dic₁₃₇	1	4	yes	—
113	C₂×C₄	C₄⋊Dic₁₃₇	1	2	yes	—
114	C₂×C₄	C₄⋊Dic₁₃₇	1	2	yes	—
115	C₂×C₄	C₄⋊Dic₁₃₇	1	4	yes	—
116	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
118	C₂×C₄	C₂²×Dic₁₃₇	1	4	yes	—
119	C₂×C₄	C₂×D₅₄₈	1	4	yes	—
120	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
121	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
122	C₂×C₄	C₂×Dic₂₇₄	1	2	yes	—
123	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
124	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
125	C₈	—	274	8	no	—
126	C₈	C₂²×Dic₁₃₇	1	8	yes	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
132	C₂²	C₂×C₄⋊Dic₁₃₇	1	4	yes	—
133	C₂²	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
134	C₂²	C₂²×Dic₂₇₄	1	1	yes	characteristic
135	C₄	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
136	C₄	C₂×C₄⋊Dic₁₃₇	1	4	yes	—
137	C₄	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
138	C₄	C₂³×Dic₁₃₇	1	1	yes	characteristic
139	C₄	C₂²×D₅₄₈	1	1	yes	characteristic
140	C₂	—	2	4	no	—
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#22577452	1	2	yes	direct product factor
144	C₂	C₂²×C₄⋊Dic₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₄₈
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂²×M₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃₇	Q₈
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂²×M₁₆	D₁₃₇
hypocenter	C₁₃₇	C₂×#430
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22577452	C₂	direct (N × H)
3. C₂×#17984024	C₂	semidirect
10. #17984024	C₂²	semidirect
70. C₂×C₁₃₇	#430	semidirect
72. C₁₃₇	C₂×#430	semidirect
143. C₂	#22577452	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×#14555494	2
#22577452	2
C₂×#17984024	2
C₂²×M₁₆×C₁₃₇	2
C₂×#430	137

Subgroups — C2×#22577452 #31789396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577452 #31789396