Subgroups — C₂×C₄×#14555494 #31789198

Order 17536.

656 conjugacy classes of subgroups.

The subgroup lattice has 138 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	C₄×#14555494	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17983999	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#11764040	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄²×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555494	C₄	1	16	yes	direct product factor
7	C₂×#14555494	C₂²	1	8	yes	—
8	C₂²×#11764040	C₂²	1	4	yes	—
9	#17983999	C₂²	1	8	yes	complemented
10	C₄×#11764040	C₂²	1	8	yes	complemented
11	C₂³×C₄×C₁₃₇	C₄	1	2	yes	—
12	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₃₇	C₄	1	4	yes	—
14	C₂×C₄²×C₁₃₇	C₄	1	2	yes	—
15	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₃₇	C₂²	1	4	yes	—
17	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
18	#14555494	C₂×C₄	1	32	yes	direct product factor
19	C₂×#11764040	C₂×C₄	1	16	yes	complemented
20	C₂×#11764040	C₂³	1	8	yes	—
21	C₂⁴×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₇	—	2	8	no	—
23	C₂²×C₄×C₁₃₇	C₂×C₄	1	8	yes	—
24	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₃₇	C₂×C₄	1	8	yes	—
26	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
31	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₄²×C₁₃₇	C₂×C₄	1	8	yes	—
33	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
34	C₄²×C₁₃₇	C₂³	1	2	yes	—
35	C₄²×C₁₃₇	C₂³	1	2	yes	—
36	#11764040	C₂²×C₄	1	16	yes	complemented
37	C₂³×C₁₃₇	—	2	4	no	—
38	C₂³×C₁₃₇	C₄²	1	4	yes	—
39	C₂³×C₁₃₇	C₂²×C₄	1	2	yes	—
40	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₇	—	2	4	no	—
42	C₂×C₄×C₁₃₇	—	2	16	no	—
43	C₂×C₄×C₁₃₇	C₄²	1	8	yes	—
44	C₂×C₄×C₁₃₇	C₄²	1	4	yes	—
45	C₂×C₄×C₁₃₇	M₁₆	1	8	yes	complemented
46	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
48	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
53	C₂²×C₁₃₇	—	2	4	no	—
54	C₂²×C₁₃₇	—	2	8	no	—
55	C₂²×C₁₃₇	C₂×C₄²	1	4	yes	—
56	C₂²×C₁₃₇	C₂×C₄²	1	2	yes	—
57	C₂²×C₁₃₇	C₂×M₁₆	1	4	yes	—
58	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₁₃₇	—	2	4	no	—
60	C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
61	C₄×C₁₃₇	C₂×C₄²	1	4	yes	—
62	C₄×C₁₃₇	C₂×M₁₆	1	8	yes	complemented
63	C₄×C₁₃₇	C₂³×C₄	1	2	yes	—
64	C₂×C₁₃₇	—	2	4	no	—
65	C₂×C₁₃₇	C₄×M₁₆	1	4	yes	complemented
66	C₂×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
67	C₂×C₁₃₇	C₂²×M₁₆	1	2	yes	—
68	C₁₃₇	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₄×M₁₆	—	137	1	no	maximal, Sylow
70	C₂²×M₁₆	—	137	2	no	—
71	C₂²×C₄²	D₁₃₇	1	1	yes	characteristic
72	C₄×M₁₆	—	137	8	no	—
73	C₂×#97	—	137	2	no	—
74	C₂×C₄×C₈	—	137	2	no	—
75	C₂³×C₄	Dic₁₃₇	1	2	yes	—
76	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic
77	C₂×M₁₆	—	137	16	no	—
78	C₂×M₁₆	—	137	8	no	—
79	C₂²×C₈	—	137	4	no	—
80	C₂×C₄²	Dic₁₃₇	1	2	yes	—
81	C₂×C₄²	Dic₁₃₇	1	4	yes	—
82	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
83	C₂×C₄²	C₂×D₁₃₇	1	4	yes	—
84	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
85	#97	—	137	8	no	—
86	C₄×C₈	—	137	8	no	—
87	C₂⁴	C₂×Dic₁₃₇	1	1	yes	characteristic
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	C₄×D₁₃₇	1	8	yes	—
90	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
91	C₂²×C₄	C₂×Dic₁₃₇	1	8	yes	—
92	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
93	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
94	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
97	M₁₆	—	137	32	no	—
98	C₂×C₈	—	137	16	no	—
99	C₂×C₈	—	137	8	no	—
100	C₄²	C₂×Dic₁₃₇	1	2	yes	—
101	C₄²	C₂×Dic₁₃₇	1	8	yes	—
102	C₄²	C₂×Dic₁₃₇	1	2	yes	—
103	C₄²	C₂²×D₁₃₇	1	2	yes	—
104	C₄²	C₂²×D₁₃₇	1	2	yes	—
105	C₂³	—	2	4	no	—
106	C₂³	C₄×Dic₁₃₇	1	4	yes	—
107	C₂³	C₂²×Dic₁₃₇	1	2	yes	—
108	C₂³	C₂²×Dic₁₃₇	1	1	yes	characteristic
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	16	no	—
111	C₂×C₄	C₄×Dic₁₃₇	1	8	yes	—
112	C₂×C₄	C₄×Dic₁₃₇	1	4	yes	—
113	C₂×C₄	#14555494	1	8	yes	direct product factor
114	C₂×C₄	C₂×C₄×D₁₃₇	1	4	yes	—
115	C₂×C₄	C₂×C₄×D₁₃₇	1	8	yes	—
116	C₂×C₄	C₂²×Dic₁₃₇	1	4	yes	—
117	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
118	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
119	C₂×C₄	C₂²×Dic₁₃₇	1	4	yes	—
120	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
121	C₈	—	137	16	no	—
122	C₂²	—	2	8	no	—
123	C₂²	—	2	4	no	—
124	C₂²	C₂×C₄×Dic₁₃₇	1	4	yes	—
125	C₂²	C₂×C₄×Dic₁₃₇	1	2	yes	—
126	C₂²	C₂×#14555494	1	4	yes	—
127	C₂²	C₂³×Dic₁₃₇	1	1	yes	characteristic
128	C₄	—	2	4	no	—
129	C₄	C₂×C₄×Dic₁₃₇	1	2	yes	—
130	C₄	C₂×C₄×Dic₁₃₇	1	4	yes	—
131	C₄	C₂×#14555494	1	8	yes	direct product factor
132	C₄	C₂²×C₄×D₁₃₇	1	2	yes	—
133	C₂	—	2	4	no	—
134	C₂	C₄×#14555494	1	4	yes	direct product factor
135	C₂	C₂²×C₄×Dic₁₃₇	1	1	yes	characteristic
136	C₂	C₂²×#14555494	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₃₇
commutator	C₂×C₁₃₇	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂²×C₄²×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂²×C₄
prime radical	C₂⁴×C₁₃₇	C₂×C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂²×C₄²	D₁₃₇
hypocenter	C₁₃₇	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#14555494	C₂	direct (N × H)
3. C₂×#17983999	C₂	semidirect
4. C₂×C₄×#11764040	C₂	semidirect
6. C₂×#14555494	C₄	direct (N × H)
9. #17983999	C₂²	semidirect
10. C₄×#11764040	C₂²	semidirect
18. #14555494	C₂×C₄	direct (N × H)
19. C₂×#11764040	C₂×C₄	semidirect
36. #11764040	C₂²×C₄	semidirect
45. C₂×C₄×C₁₃₇	M₁₆	semidirect
62. C₄×C₁₃₇	C₂×M₁₆	semidirect
65. C₂×C₁₃₇	C₄×M₁₆	semidirect
68. C₁₃₇	C₂×C₄×M₁₆	semidirect
113. C₂×C₄	#14555494	direct (N × H)
131. C₄	C₂×#14555494	direct (N × H)
134. C₂	C₄×#14555494	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
C₄×#14555494	2
C₂×#17983999	2
C₂×C₄×#11764040	2
C₂²×C₄²×C₁₃₇	2
C₂×C₄×M₁₆	137

Subgroups — C2×C4×#14555494 #31789198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14555494 #31789198