Subgroups — C₂³×#10587558 #17940882

Order 4352.

14334 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#10587558	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#63225	C₂	1	2	yes	maximal, complemented
4	C₂³×#63227	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₇	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₁₇	C₂²	1	2	yes	—
9	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic
10	C₂³×C₄×D₁₇	C₄	1	32	yes	complemented
11	C₂³×C₄×D₁₇	C₂²	1	28	yes	—
12	C₂×#10587558	C₂²	1	448	yes	direct product factor
13	C₂²×#63225	C₂²	1	56	yes	complemented
14	C₂²×#63227	C₂²	1	56	yes	—
15	C₂²×C₄×Dic₁₇	C₂²	1	28	yes	complemented
16	C₂⁴×C₄×C₁₇	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₁₇	C₂²	1	28	yes	complemented
19	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
20	C₂⁴×D₁₇	—	2	16	no	—
21	C₂⁴×D₁₇	C₂×C₄	1	16	yes	—
22	C₂⁴×D₁₇	C₂³	1	14	yes	—
23	C₂³×C₄×C₁₇	—	2	16	no	—
24	C₂³×C₄×C₁₇	C₂×C₄	1	16	yes	—
25	C₂³×C₄×C₁₇	C₂³	1	28	yes	—
26	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
27	C₂³×Dic₁₇	—	2	16	no	—
28	C₂³×Dic₁₇	C₂×C₄	1	16	yes	—
29	C₂³×Dic₁₇	C₂³	1	28	yes	—
30	C₂³×Dic₁₇	C₂³	1	14	yes	—
31	C₂²×C₄×D₁₇	C₂×C₄	1	448	yes	complemented
32	C₂²×C₄×D₁₇	C₂³	1	112	yes	—
33	C₂×C₄²×C₁₇	C₂³	1	112	yes	complemented
34	C₂×#63227	C₂³	1	224	yes	—
35	C₂×#63225	C₂³	1	224	yes	complemented
36	C₂×C₄×Dic₁₇	C₂³	1	112	yes	complemented
37	#10587558	C₂³	1	512	yes	direct product factor
38	C₂⁴×C₁₇	C₂²×C₄	1	8	yes	—
39	C₂⁴×C₁₇	#41	1	16	yes	—
40	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
41	C₂³×D₁₇	—	2	128	no	—
42	C₂³×D₁₇	—	2	112	no	—
43	C₂³×D₁₇	C₂²×C₄	1	112	yes	—
44	C₂³×D₁₇	C₂⁴	1	28	yes	—
45	C₂²×C₄×C₁₇	—	2	112	no	—
46	C₂²×C₄×C₁₇	C₂²×C₄	1	112	yes	—
47	C₂²×C₄×C₁₇	C₂⁴	1	56	yes	—
48	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
49	C₂²×Dic₁₇	—	2	112	no	—
50	C₂²×Dic₁₇	C₂²×C₄	1	112	yes	—
51	C₂²×Dic₁₇	C₂⁴	1	56	yes	—
52	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
53	C₂×C₄×D₁₇	C₂²×C₄	1	896	yes	complemented
54	C₂×C₄×D₁₇	C₂⁴	1	64	yes	—
55	C₄²×C₁₇	C₂⁴	1	64	yes	complemented
56	#63227	C₂⁴	1	128	yes	—
57	#63225	C₂⁴	1	128	yes	complemented
58	C₄×Dic₁₇	C₂⁴	1	64	yes	complemented
59	C₂³×#117	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	#117	1	64	yes	complemented
61	C₂³×C₁₇	C₂³×C₄	1	28	yes	—
62	C₂³×C₁₇	C₂×#41	1	56	yes	—
63	C₂³×C₁₇	C₂⁵	1	7	yes	—
64	C₂²×D₁₇	—	2	448	no	—
65	C₂²×D₁₇	—	2	112	no	—
66	C₂²×D₁₇	C₂³×C₄	1	112	yes	—
67	C₂²×D₁₇	C₂⁵	1	8	yes	—
68	C₂×C₄×C₁₇	—	2	112	no	—
69	C₂×C₄×C₁₇	C₂³×C₄	1	112	yes	—
70	C₂×C₄×C₁₇	C₂⁵	1	16	yes	—
71	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
72	C₂×Dic₁₇	—	2	112	no	—
73	C₂×Dic₁₇	C₂³×C₄	1	112	yes	—
74	C₂×Dic₁₇	C₂⁵	1	16	yes	—
75	C₂×Dic₁₇	C₂⁵	1	8	yes	—
76	C₄×D₁₇	C₂³×C₄	1	256	yes	complemented
77	C₂⁵×C₄	—	17	1	no	—
78	C₂²×#117	—	17	56	no	—
79	C₂³×C₄⋊C₄	—	17	2	no	—
80	C₂³×C₂²⋊C₄	—	17	2	no	—
81	C₂³×C₄²	—	17	1	no	—
82	C₂³×C₄²	D₁₇	1	1	yes	characteristic, complemented
83	C₂²×C₁₇	C₂×#117	1	112	yes	complemented
84	C₂²×C₁₇	C₂⁴×C₄	1	14	yes	—
85	C₂²×C₁₇	C₂²×#41	1	28	yes	—
86	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
87	C₂×D₁₇	—	2	16	no	—
88	C₂×D₁₇	—	2	224	no	—
89	C₂×D₁₇	C₂⁴×C₄	1	16	yes	—
90	C₄×C₁₇	—	2	16	no	—
91	C₄×C₁₇	C₂⁴×C₄	1	16	yes	—
92	Dic₁₇	—	2	16	no	—
93	Dic₁₇	C₂⁴×C₄	1	16	yes	—
94	C₂⁶	—	17	1	no	—
95	C₂⁴×C₄	—	17	1	no	—
96	C₂⁴×C₄	—	17	32	no	—
97	C₂⁴×C₄	—	17	28	no	—
98	C₂⁴×C₄	—	17	2	no	—
99	C₂⁴×C₄	C₂×D₁₇	1	2	yes	—
100	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
101	C₂×#117	—	17	448	no	—
102	C₂²×C₄⋊C₄	—	17	56	no	—
103	C₂²×C₂²⋊C₄	—	17	56	no	—
104	C₂²×C₄²	—	17	28	no	—
105	C₂²×C₄²	C₂×D₁₇	1	28	yes	complemented
106	C₂×C₁₇	C₂²×#117	1	28	yes	complemented
107	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
108	C₂×C₁₇	C₂³×#41	1	2	yes	—
109	D₁₇	—	2	16	no	—
110	C₂⁵	—	17	16	no	—
111	C₂⁵	—	17	14	no	—
112	C₂⁵	—	34	16	no	—
113	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
114	C₂³×C₄	—	2	16	no	—
115	C₂³×C₄	—	17	16	no	—
116	C₂³×C₄	—	17	14	no	—
117	C₂³×C₄	—	17	448	no	—
118	C₂³×C₄	—	17	112	no	—
119	C₂³×C₄	—	17	28	no	—
120	C₂³×C₄	—	34	16	no	—
121	C₂³×C₄	C₄×D₁₇	1	16	yes	—
122	C₂³×C₄	C₂²×D₁₇	1	28	yes	—
123	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
124	#117	—	17	512	no	—
125	C₂×C₄⋊C₄	—	17	224	no	—
126	C₂×C₂²⋊C₄	—	17	224	no	—
127	C₂×C₄²	—	17	112	no	—
128	C₂×C₄²	C₂²×D₁₇	1	112	yes	complemented
129	C₁₇	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
130	C₂⁴	—	17	112	no	—
131	C₂⁴	—	17	28	no	—
132	C₂⁴	—	34	112	no	—
133	C₂⁴	—	34	128	no	—
134	C₂⁴	C₂×C₄×D₁₇	1	8	yes	—
135	C₂⁴	#63252	1	16	yes	—
136	C₂⁴	C₂³×D₁₇	1	7	yes	—
137	C₂²×C₄	—	2	112	no	—
138	C₂²×C₄	—	17	112	no	—
139	C₂²×C₄	—	17	28	no	—
140	C₂²×C₄	—	17	896	no	—
141	C₂²×C₄	—	17	64	no	—
142	C₂²×C₄	—	17	56	no	—
143	C₂²×C₄	—	34	112	no	—
144	C₂²×C₄	C₂×C₄×D₁₇	1	112	yes	—
145	C₂²×C₄	C₂³×D₁₇	1	56	yes	—
146	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
147	C₄⋊C₄	—	17	128	no	—
148	C₂²⋊C₄	—	17	128	no	—
149	C₄²	—	17	64	no	—
150	C₄²	C₂³×D₁₇	1	64	yes	complemented
151	C₂³	—	17	112	no	—
152	C₂³	—	17	8	no	—
153	C₂³	—	34	448	no	—
154	C₂³	—	34	112	no	—
155	C₂³	#10587558	1	64	yes	direct product factor
156	C₂³	C₂²×C₄×D₁₇	1	28	yes	—
157	C₂³	C₂×#63252	1	56	yes	—
158	C₂³	C₂⁴×D₁₇	1	7	yes	—
159	C₂×C₄	—	2	112	no	—
160	C₂×C₄	—	17	112	no	—
161	C₂×C₄	—	17	8	no	—
162	C₂×C₄	—	17	256	no	—
163	C₂×C₄	—	17	16	no	—
164	C₂×C₄	—	34	112	no	—
165	C₂×C₄	C₂²×C₄×D₁₇	1	112	yes	—
166	C₂×C₄	C₂⁴×D₁₇	1	16	yes	—
167	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
168	C₂²	—	17	16	no	—
169	C₂²	—	34	224	no	—
170	C₂²	—	34	16	no	—
171	C₂²	C₂×#10587558	1	112	yes	direct product factor
172	C₂²	C₂³×C₄×D₁₇	1	14	yes	—
173	C₂²	C₂²×#63252	1	28	yes	—
174	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
175	C₄	—	2	16	no	—
176	C₄	—	17	16	no	—
177	C₄	—	34	16	no	—
178	C₄	C₂³×C₄×D₁₇	1	16	yes	—
179	C₂	—	34	16	no	—
180	C₂	C₂²×#10587558	1	28	yes	direct product factor
181	C₂	C₂⁴×C₄×D₁₇	1	1	yes	characteristic
182	C₂	C₂³×#63252	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₁₇
commutator	C₂×C₁₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₇
Fitting	C₂³×C₄²×C₁₇	C₂
socle	C₂⁵×C₁₇	C₂³
prime radical	C₂⁵×D₁₇	C₂²
cosocle	C₂²×C₁₇	C₂⁶
hypercenter	C₂³×C₄²	D₁₇
hypocenter	C₁₇	C₂³×#117
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×#10587558	C₂	direct (N × H)
3. C₂³×#63225	C₂	semidirect
5. C₂³×C₄×Dic₁₇	C₂	semidirect
6. C₂³×C₄²×C₁₇	C₂	semidirect
10. C₂³×C₄×D₁₇	C₄	semidirect
12. C₂×#10587558	C₂²	direct (N × H)
13. C₂²×#63225	C₂²	semidirect
15. C₂²×C₄×Dic₁₇	C₂²	semidirect
18. C₂²×C₄²×C₁₇	C₂²	semidirect
31. C₂²×C₄×D₁₇	C₂×C₄	semidirect
33. C₂×C₄²×C₁₇	C₂³	semidirect
35. C₂×#63225	C₂³	semidirect
36. C₂×C₄×Dic₁₇	C₂³	semidirect
37. #10587558	C₂³	direct (N × H)
53. C₂×C₄×D₁₇	C₂²×C₄	semidirect
55. C₄²×C₁₇	C₂⁴	semidirect
57. #63225	C₂⁴	semidirect
58. C₄×Dic₁₇	C₂⁴	semidirect
60. C₂³×C₁₇	#117	semidirect
76. C₄×D₁₇	C₂³×C₄	semidirect
82. C₂³×C₄²	D₁₇	semidirect
83. C₂²×C₁₇	C₂×#117	semidirect
105. C₂²×C₄²	C₂×D₁₇	semidirect
106. C₂×C₁₇	C₂²×#117	semidirect
128. C₂×C₄²	C₂²×D₁₇	semidirect
129. C₁₇	C₂³×#117	semidirect
150. C₄²	C₂³×D₁₇	semidirect
155. C₂³	#10587558	direct (N × H)
171. C₂²	C₂×#10587558	direct (N × H)
180. C₂	C₂²×#10587558	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×C₄×D₁₇	2
C₂²×#10587558	2
C₂³×#63225	2
C₂³×#63227	2
C₂³×C₄×Dic₁₇	2
C₂³×C₄²×C₁₇	2
C₂³×#117	17

Subgroups — C23×#10587558 #17940882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587558 #17940882