Subgroups — C₂³×#10587612 #17940884

Order 4352.

14334 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587612	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₁₇	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊Dic₁₇	C₂	1	2	yes	maximal
4	C₂³×C₄×Dic₁₇	C₂	1	2	yes	maximal, complemented
5	C₂⁵×C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₁₇	C₂²	1	4	yes	—
7	C₂×#10587612	C₂²	1	448	yes	direct product factor
8	C₂²×C₄⋊Dic₁₇	C₂²	1	56	yes	complemented
9	C₂²×C₂²⋊Dic₁₇	C₂²	1	56	yes	—
10	C₂²×C₄×Dic₁₇	C₂²	1	56	yes	complemented
11	C₂⁶×C₁₇	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₁₇	C₄	1	32	yes	complemented
13	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₁₇	C₂²	1	28	yes	—
16	C₂⁵×C₁₇	—	2	16	no	—
17	C₂⁵×C₁₇	C₂×C₄	1	16	yes	—
18	C₂⁵×C₁₇	C₂³	1	14	yes	—
19	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₁₇	C₂×C₄	1	16	yes	—
21	C₂³×C₄×C₁₇	C₂×C₄	1	16	yes	—
22	C₂³×C₄×C₁₇	C₂×C₄	1	448	yes	complemented
23	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
24	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
25	C₂³×C₄×C₁₇	C₂³	1	112	yes	—
26	C₂³×Dic₁₇	—	2	32	no	—
27	C₂³×Dic₁₇	C₂³	1	56	yes	—
28	C₂×C₂²⋊Dic₁₇	C₂³	1	224	yes	—
29	#10587612	C₂³	1	512	yes	direct product factor
30	C₂×C₄⋊Dic₁₇	C₂³	1	224	yes	complemented
31	C₂×C₄×Dic₁₇	C₂³	1	224	yes	complemented
32	C₂⁴×C₁₇	—	2	128	no	—
33	C₂⁴×C₁₇	—	2	112	no	—
34	C₂⁴×C₁₇	C₂²×C₄	1	112	yes	—
35	C₂⁴×C₁₇	C₂²×C₄	1	8	yes	—
36	C₂⁴×C₁₇	#41	1	16	yes	—
37	C₂⁴×C₁₇	C₂⁴	1	28	yes	—
38	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
39	C₂²×C₄×C₁₇	C₂²×C₄	1	112	yes	—
40	C₂²×C₄×C₁₇	C₂²×C₄	1	112	yes	—
41	C₂²×C₄×C₁₇	C₂²×C₄	1	896	yes	complemented
42	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
43	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
44	C₂²×C₄×C₁₇	C₂⁴	1	64	yes	—
45	C₂²×Dic₁₇	—	2	224	no	—
46	C₂²×Dic₁₇	C₂⁴	1	112	yes	—
47	C₂²⋊Dic₁₇	C₂⁴	1	128	yes	—
48	C₄⋊Dic₁₇	C₂⁴	1	128	yes	complemented
49	C₄×Dic₁₇	C₂⁴	1	128	yes	complemented
50	C₂³×#117	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	—	2	448	no	—
52	C₂³×C₁₇	—	2	112	no	—
53	C₂³×C₁₇	#117	1	64	yes	complemented
54	C₂³×C₁₇	C₂³×C₄	1	112	yes	—
55	C₂³×C₁₇	C₂³×C₄	1	28	yes	—
56	C₂³×C₁₇	C₂×#41	1	56	yes	—
57	C₂³×C₁₇	C₂⁵	1	8	yes	—
58	C₂³×C₁₇	C₂⁵	1	7	yes	—
59	C₂×C₄×C₁₇	C₂³×C₄	1	112	yes	—
60	C₂×C₄×C₁₇	C₂³×C₄	1	112	yes	—
61	C₂×C₄×C₁₇	C₂³×C₄	1	256	yes	complemented
62	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
63	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
64	C₂×Dic₁₇	—	2	224	no	—
65	C₂×Dic₁₇	C₂⁵	1	32	yes	—
66	C₂⁵×C₄	D₁₇	1	1	yes	characteristic
67	C₂²×#117	—	17	56	no	—
68	C₂³×C₄⋊C₄	—	17	2	no	—
69	C₂³×C₂²⋊C₄	—	17	2	no	—
70	C₂³×C₄²	—	17	2	no	—
71	C₂²×C₁₇	—	2	16	no	—
72	C₂²×C₁₇	—	2	224	no	—
73	C₂²×C₁₇	C₂×#117	1	112	yes	complemented
74	C₂²×C₁₇	C₂⁴×C₄	1	16	yes	—
75	C₂²×C₁₇	C₂⁴×C₄	1	14	yes	—
76	C₂²×C₁₇	C₂²×#41	1	28	yes	—
77	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
78	C₄×C₁₇	C₂⁴×C₄	1	16	yes	—
79	C₄×C₁₇	C₂⁴×C₄	1	16	yes	—
80	Dic₁₇	—	2	32	no	—
81	C₂⁶	C₂×D₁₇	1	1	yes	characteristic
82	C₂⁴×C₄	—	17	4	no	—
83	C₂⁴×C₄	Dic₁₇	1	32	yes	complemented
84	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
85	C₂⁴×C₄	C₂×D₁₇	1	28	yes	—
86	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
87	C₂×#117	—	17	448	no	—
88	C₂²×C₄⋊C₄	—	17	56	no	—
89	C₂²×C₂²⋊C₄	—	17	56	no	—
90	C₂²×C₄²	—	17	56	no	—
91	C₂×C₁₇	—	2	16	no	—
92	C₂×C₁₇	C₂²×#117	1	28	yes	complemented
93	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
94	C₂×C₁₇	C₂³×#41	1	2	yes	—
95	C₂⁵	—	2	16	no	—
96	C₂⁵	C₂×Dic₁₇	1	16	yes	—
97	C₂⁵	C₂²×D₁₇	1	14	yes	—
98	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
99	C₂³×C₄	—	17	56	no	—
100	C₂³×C₄	—	34	32	no	—
101	C₂³×C₄	C₂×Dic₁₇	1	16	yes	—
102	C₂³×C₄	C₂×Dic₁₇	1	16	yes	—
103	C₂³×C₄	C₂×Dic₁₇	1	448	yes	complemented
104	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
105	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
106	C₂³×C₄	C₂²×D₁₇	1	112	yes	—
107	#117	—	17	512	no	—
108	C₂×C₄⋊C₄	—	17	224	no	—
109	C₂×C₂²⋊C₄	—	17	224	no	—
110	C₂×C₄²	—	17	224	no	—
111	C₁₇	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
112	C₂⁴	—	2	112	no	—
113	C₂⁴	—	2	128	no	—
114	C₂⁴	#63252	1	16	yes	—
115	C₂⁴	C₂²×Dic₁₇	1	112	yes	—
116	C₂⁴	C₂²×Dic₁₇	1	8	yes	—
117	C₂⁴	C₂³×D₁₇	1	28	yes	—
118	C₂⁴	C₂³×D₁₇	1	7	yes	—
119	C₂²×C₄	—	17	112	no	—
120	C₂²×C₄	—	34	224	no	—
121	C₂²×C₄	C₂²×Dic₁₇	1	112	yes	—
122	C₂²×C₄	C₂²×Dic₁₇	1	112	yes	—
123	C₂²×C₄	C₂²×Dic₁₇	1	896	yes	complemented
124	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
125	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
126	C₂²×C₄	C₂³×D₁₇	1	64	yes	—
127	C₄⋊C₄	—	17	128	no	—
128	C₂²⋊C₄	—	17	128	no	—
129	C₄²	—	17	128	no	—
130	C₂³	—	2	448	no	—
131	C₂³	—	2	112	no	—
132	C₂³	#10587612	1	64	yes	direct product factor
133	C₂³	C₂×#63252	1	56	yes	—
134	C₂³	C₂³×Dic₁₇	1	112	yes	—
135	C₂³	C₂³×Dic₁₇	1	28	yes	—
136	C₂³	C₂⁴×D₁₇	1	8	yes	—
137	C₂³	C₂⁴×D₁₇	1	7	yes	—
138	C₂×C₄	—	17	32	no	—
139	C₂×C₄	—	34	224	no	—
140	C₂×C₄	C₂³×Dic₁₇	1	112	yes	—
141	C₂×C₄	C₂³×Dic₁₇	1	112	yes	—
142	C₂×C₄	C₂³×Dic₁₇	1	256	yes	complemented
143	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
144	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
145	C₂²	—	2	224	no	—
146	C₂²	—	2	16	no	—
147	C₂²	C₂×#10587612	1	112	yes	direct product factor
148	C₂²	C₂⁴×Dic₁₇	1	16	yes	—
149	C₂²	C₂⁴×Dic₁₇	1	14	yes	—
150	C₂²	C₂²×#63252	1	28	yes	—
151	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
152	C₄	—	34	32	no	—
153	C₄	C₂⁴×Dic₁₇	1	16	yes	—
154	C₄	C₂⁴×Dic₁₇	1	16	yes	—
155	C₂	—	2	16	no	—
156	C₂	C₂²×#10587612	1	28	yes	direct product factor
157	C₂	C₂⁵×Dic₁₇	1	1	yes	characteristic
158	C₂	C₂³×#63252	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₁₇
commutator	C₂×C₁₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₇
Fitting	C₂⁵×C₄×C₁₇	C₂
socle	C₂⁵×C₁₇	C₂³
prime radical	C₂⁶×C₁₇	C₂²
cosocle	C₂²×C₁₇	C₂⁶
hypercenter	C₂⁵×C₄	D₁₇
hypocenter	C₁₇	C₂³×#117
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#10587612	C₂	direct (N × H)
2. C₂³×C₄⋊Dic₁₇	C₂	semidirect
4. C₂³×C₄×Dic₁₇	C₂	semidirect
7. C₂×#10587612	C₂²	direct (N × H)
8. C₂²×C₄⋊Dic₁₇	C₂²	semidirect
10. C₂²×C₄×Dic₁₇	C₂²	semidirect
12. C₂⁴×C₄×C₁₇	C₄	semidirect
22. C₂³×C₄×C₁₇	C₂×C₄	semidirect
29. #10587612	C₂³	direct (N × H)
30. C₂×C₄⋊Dic₁₇	C₂³	semidirect
31. C₂×C₄×Dic₁₇	C₂³	semidirect
41. C₂²×C₄×C₁₇	C₂²×C₄	semidirect
48. C₄⋊Dic₁₇	C₂⁴	semidirect
49. C₄×Dic₁₇	C₂⁴	semidirect
53. C₂³×C₁₇	#117	semidirect
61. C₂×C₄×C₁₇	C₂³×C₄	semidirect
73. C₂²×C₁₇	C₂×#117	semidirect
83. C₂⁴×C₄	Dic₁₇	semidirect
92. C₂×C₁₇	C₂²×#117	semidirect
103. C₂³×C₄	C₂×Dic₁₇	semidirect
111. C₁₇	C₂³×#117	semidirect
123. C₂²×C₄	C₂²×Dic₁₇	semidirect
132. C₂³	#10587612	direct (N × H)
142. C₂×C₄	C₂³×Dic₁₇	semidirect
147. C₂²	C₂×#10587612	direct (N × H)
156. C₂	C₂²×#10587612	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×#10587612	2
C₂³×C₄⋊Dic₁₇	2
C₂³×C₂²⋊Dic₁₇	2
C₂³×C₄×Dic₁₇	2
C₂⁵×C₄×C₁₇	2
C₂³×#117	17

Subgroups — C23×#10587612 #17940884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587612 #17940884