Subgroups — C₂³×#10587485 #17897931

Order 4352.

3870 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#3622	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#10587485	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₁₇	C₄	1	1	yes	characteristic
6	C₂⁴×Dic₁₇	C₄	1	1	yes	characteristic
7	C₂³×#3622	—	2	8	no	—
8	C₂³×#3622	C₂²	1	14	yes	—
9	C₂²×C₄×Dic₁₇	C₄	1	14	yes	—
10	C₂²×C₄×Dic₁₇	C₂²	1	14	yes	—
11	C₂×#10587485	C₂²	1	112	yes	direct product factor
12	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₁₇	—	2	8	no	—
14	C₂³×C₈×C₁₇	C₂²	1	14	yes	—
15	C₂⁵×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₇	C₂×C₄	1	7	yes	—
17	C₂³×C₄×C₁₇	D₄	1	8	yes	—
18	C₂³×C₄×C₁₇	Q₈	1	8	yes	—
19	C₂³×C₄×C₁₇	C₂³	1	7	yes	—
20	C₂³×Dic₁₇	—	2	8	no	—
21	C₂³×Dic₁₇	C₈	1	16	yes	complemented
22	C₂³×Dic₁₇	C₂×C₄	1	14	yes	—
23	C₂³×Dic₁₇	C₂×C₄	1	14	yes	—
24	C₂²×C₈×C₁₇	—	2	56	no	—
25	C₂²×C₈×C₁₇	C₂³	1	28	yes	—
26	C₂×C₄×Dic₁₇	C₂×C₄	1	84	yes	—
27	C₂×C₄×Dic₁₇	C₂³	1	28	yes	—
28	C₂²×#3622	—	2	56	no	—
29	C₂²×#3622	C₂³	1	28	yes	—
30	#10587485	C₂³	1	64	yes	direct product factor
31	C₂⁴×C₁₇	C₄⋊C₄	1	8	yes	—
32	C₂⁴×C₁₇	C₂×C₈	1	8	yes	—
33	C₂⁴×C₁₇	M₁₆	1	8	yes	—
34	C₂⁴×C₁₇	C₂²×C₄	1	7	yes	—
35	C₂²×C₄×C₁₇	C₄⋊C₄	1	56	yes	—
36	C₂²×C₄×C₁₇	C₂²×C₄	1	21	yes	—
37	C₂²×C₄×C₁₇	C₂×D₄	1	28	yes	—
38	C₂²×C₄×C₁₇	C₂×Q₈	1	28	yes	—
39	C₂²×C₄×C₁₇	C₂⁴	1	7	yes	—
40	C₂²×Dic₁₇	—	2	56	no	—
41	C₂²×Dic₁₇	C₂×C₈	1	112	yes	complemented
42	C₂²×Dic₁₇	C₂²×C₄	1	28	yes	—
43	C₂²×Dic₁₇	C₂²×C₄	1	28	yes	—
44	C₂×C₈×C₁₇	—	2	56	no	—
45	C₂×C₈×C₁₇	C₂⁴	1	8	yes	—
46	C₄×Dic₁₇	C₂²×C₄	1	56	yes	—
47	C₄×Dic₁₇	C₂⁴	1	8	yes	—
48	C₂×#3622	—	2	56	no	—
49	C₂×#3622	C₂⁴	1	8	yes	—
50	C₂³×C₄⋊C₈	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	C₄⋊C₈	1	64	yes	complemented
52	C₂³×C₁₇	C₂×C₄⋊C₄	1	28	yes	—
53	C₂³×C₁₇	C₂²×C₈	1	28	yes	—
54	C₂³×C₁₇	C₂×M₁₆	1	28	yes	—
55	C₂³×C₁₇	C₂³×C₄	1	7	yes	—
56	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	84	yes	—
57	C₂×C₄×C₁₇	C₂³×C₄	1	7	yes	—
58	C₂×C₄×C₁₇	C₂²×D₄	1	14	yes	—
59	C₂×C₄×C₁₇	C₂²×Q₈	1	14	yes	—
60	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
61	C₂×Dic₁₇	—	2	56	no	—
62	C₂×Dic₁₇	C₂²×C₈	1	112	yes	complemented
63	C₂×Dic₁₇	C₂³×C₄	1	8	yes	—
64	C₂×Dic₁₇	C₂³×C₄	1	8	yes	—
65	C₈×C₁₇	—	2	8	no	—
66	#3622	—	2	8	no	—
67	C₂⁴×C₈	—	17	1	no	—
68	C₂⁴×C₈	D₁₇	1	1	yes	characteristic
69	C₂³×C₄²	—	17	1	no	—
70	C₂²×C₄⋊C₈	—	17	28	no	—
71	C₂²×C₁₇	C₂×C₄⋊C₈	1	112	yes	complemented
72	C₂²×C₁₇	C₂²×C₄⋊C₄	1	14	yes	—
73	C₂²×C₁₇	C₂³×C₈	1	14	yes	—
74	C₂²×C₁₇	C₂²×M₁₆	1	14	yes	—
75	C₂²×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
76	C₄×C₁₇	C₂²×C₄⋊C₄	1	14	yes	—
77	C₄×C₁₇	C₂³×D₄	1	1	yes	characteristic
78	C₄×C₁₇	C₂³×Q₈	1	1	yes	characteristic
79	Dic₁₇	—	2	8	no	—
80	Dic₁₇	C₂³×C₈	1	16	yes	complemented
81	C₂⁴×C₄	—	17	1	no	—
82	C₂⁴×C₄	—	17	1	no	—
83	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
84	C₂³×C₈	—	2	8	no	—
85	C₂³×C₈	—	17	14	no	—
86	C₂³×C₈	—	34	8	no	—
87	C₂³×C₈	C₂×D₁₇	1	14	yes	—
88	C₂²×C₄²	—	17	14	no	—
89	C₂²×C₄²	—	17	14	no	—
90	C₂×C₄⋊C₈	—	17	112	no	—
91	C₂×C₁₇	C₂²×C₄⋊C₈	1	28	yes	complemented
92	C₂×C₁₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₁₇	C₂⁴×C₈	1	1	yes	characteristic
94	C₂×C₁₇	C₂³×M₁₆	1	1	yes	characteristic
95	C₂⁵	C₄×D₁₇	1	1	yes	characteristic
96	C₂³×C₄	—	17	14	no	—
97	C₂³×C₄	—	17	16	no	—
98	C₂³×C₄	—	17	14	no	—
99	C₂³×C₄	—	34	8	no	—
100	C₂³×C₄	Dic₃₄	1	8	yes	—
101	C₂³×C₄	C₄×D₁₇	1	7	yes	—
102	C₂³×C₄	C₂²⋊D₁₇	1	8	yes	—
103	C₂³×C₄	C₂²×D₁₇	1	7	yes	—
104	C₂²×C₈	—	2	56	no	—
105	C₂²×C₈	—	17	28	no	—
106	C₂²×C₈	—	34	56	no	—
107	C₂²×C₈	C₂²×D₁₇	1	28	yes	—
108	C₂×C₄²	—	17	84	no	—
109	C₂×C₄²	—	17	28	no	—
110	C₄⋊C₈	—	17	64	no	—
111	C₁₇	C₂³×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
112	C₂⁴	C₈×D₁₇	1	8	yes	—
113	C₂⁴	#63218	1	8	yes	—
114	C₂⁴	#63225	1	8	yes	—
115	C₂⁴	C₂×C₄×D₁₇	1	7	yes	—
116	C₂²×C₄	—	17	28	no	—
117	C₂²×C₄	—	17	112	no	—
118	C₂²×C₄	—	17	28	no	—
119	C₂²×C₄	—	34	56	no	—
120	C₂²×C₄	#63225	1	56	yes	—
121	C₂²×C₄	C₂×Dic₃₄	1	28	yes	—
122	C₂²×C₄	C₂×C₄×D₁₇	1	21	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₁₇	1	28	yes	—
124	C₂²×C₄	C₂³×D₁₇	1	7	yes	—
125	C₂×C₈	—	2	56	no	—
126	C₂×C₈	—	17	8	no	—
127	C₂×C₈	—	34	56	no	—
128	C₂×C₈	C₂³×D₁₇	1	8	yes	—
129	C₄²	—	17	56	no	—
130	C₄²	—	17	8	no	—
131	C₂³	#10587485	1	64	yes	direct product factor
132	C₂³	C₂×C₈×D₁₇	1	28	yes	—
133	C₂³	C₂×#63218	1	28	yes	—
134	C₂³	C₂×#63225	1	28	yes	—
135	C₂³	C₂²×C₄×D₁₇	1	7	yes	—
136	C₂×C₄	—	17	112	no	—
137	C₂×C₄	—	17	8	no	—
138	C₂×C₄	—	17	8	no	—
139	C₂×C₄	—	34	56	no	—
140	C₂×C₄	C₂×#63225	1	84	yes	—
141	C₂×C₄	C₂²×Dic₃₄	1	14	yes	—
142	C₂×C₄	C₂²×C₄×D₁₇	1	7	yes	—
143	C₂×C₄	C₂²×C₂²⋊D₁₇	1	14	yes	—
144	C₂×C₄	C₂⁴×D₁₇	1	1	yes	characteristic
145	C₈	—	2	8	no	—
146	C₈	—	34	8	no	—
147	C₂²	C₂×#10587485	1	112	yes	direct product factor
148	C₂²	C₂²×#63225	1	14	yes	—
149	C₂²	C₂²×C₈×D₁₇	1	14	yes	—
150	C₂²	C₂²×#63218	1	14	yes	—
151	C₂²	C₂³×C₄×D₁₇	1	1	yes	characteristic
152	C₄	—	17	16	no	—
153	C₄	—	34	8	no	—
154	C₄	C₂²×#63225	1	14	yes	—
155	C₄	C₂³×C₂²⋊D₁₇	1	1	yes	characteristic
156	C₄	C₂³×Dic₃₄	1	1	yes	characteristic
157	C₂	C₂²×#10587485	1	28	yes	direct product factor
158	C₂	C₂³×#63225	1	1	yes	characteristic
159	C₂	C₂³×C₈×D₁₇	1	1	yes	characteristic
160	C₂	C₂³×#63218	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₁₇
commutator	C₂×C₁₇	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₇
Fitting	C₂⁴×C₈×C₁₇	C₂
socle, prime radical	C₂⁵×C₁₇	C₂×C₄
cosocle	C₂×C₄×C₁₇	C₂⁵
hypercenter	C₂⁴×C₈	D₁₇
hypocenter	C₁₇	C₂³×C₄⋊C₈
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#10587485	C₂	direct (N × H)
11. C₂×#10587485	C₂²	direct (N × H)
21. C₂³×Dic₁₇	C₈	semidirect
30. #10587485	C₂³	direct (N × H)
41. C₂²×Dic₁₇	C₂×C₈	semidirect
51. C₂³×C₁₇	C₄⋊C₈	semidirect
62. C₂×Dic₁₇	C₂²×C₈	semidirect
71. C₂²×C₁₇	C₂×C₄⋊C₈	semidirect
80. Dic₁₇	C₂³×C₈	semidirect
91. C₂×C₁₇	C₂²×C₄⋊C₈	semidirect
111. C₁₇	C₂³×C₄⋊C₈	semidirect
131. C₂³	#10587485	direct (N × H)
147. C₂²	C₂×#10587485	direct (N × H)
157. C₂	C₂²×#10587485	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#3622	2
C₂³×C₄×Dic₁₇	2
C₂²×#10587485	2
C₂⁴×C₈×C₁₇	2
C₂³×C₄⋊C₈	17

Subgroups — C23×#10587485 #17897931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587485 #17897931