Subgroups — C₂³×#10587578 #17940883

Order 4352.

14334 conjugacy classes of subgroups.

The subgroup lattice has 200 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×#10587578	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#63227	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₁₇	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄⋊C₄×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×#10587578	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊Dic₁₇	C₂²	1	28	yes	complemented
14	C₂²×#63227	C₂²	1	56	yes	—
15	C₂²×C₄×Dic₁₇	C₂²	1	56	yes	complemented
16	C₂⁴×C₄×C₁₇	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₁₇	C₂²	1	28	yes	complemented
19	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
20	C₂⁴×D₁₇	—	2	8	no	—
21	C₂⁴×D₁₇	—	2	8	no	—
22	C₂⁴×D₁₇	C₂×C₄	1	16	yes	—
23	C₂⁴×D₁₇	C₂³	1	14	yes	—
24	C₂³×C₄×C₁₇	—	2	16	no	—
25	C₂³×C₄×C₁₇	C₂×C₄	1	16	yes	—
26	C₂³×C₄×C₁₇	C₂³	1	28	yes	—
27	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
28	C₂³×Dic₁₇	—	2	16	no	—
29	C₂³×Dic₁₇	C₂×C₄	1	16	yes	—
30	C₂³×Dic₁₇	C₂³	1	28	yes	—
31	C₂³×Dic₁₇	C₂³	1	14	yes	—
32	C₂²×C₄×D₁₇	C₂×C₄	1	448	yes	complemented
33	C₂²×C₄×D₁₇	C₂³	1	112	yes	—
34	C₂×C₄⋊C₄×C₁₇	C₂³	1	112	yes	complemented
35	C₂×#63227	C₂³	1	224	yes	—
36	C₂×C₄⋊Dic₁₇	C₂³	1	112	yes	complemented
37	C₂×C₄×Dic₁₇	C₂³	1	224	yes	complemented
38	#10587578	C₂³	1	512	yes	direct product factor
39	C₂⁴×C₁₇	C₂²×C₄	1	8	yes	—
40	C₂⁴×C₁₇	#41	1	8	yes	—
41	C₂⁴×C₁₇	#41	1	8	yes	—
42	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
43	C₂³×D₁₇	—	2	128	no	—
44	C₂³×D₁₇	—	2	56	no	—
45	C₂³×D₁₇	—	2	56	no	—
46	C₂³×D₁₇	C₂²×C₄	1	112	yes	—
47	C₂³×D₁₇	C₂⁴	1	28	yes	—
48	C₂²×C₄×C₁₇	—	2	112	no	—
49	C₂²×C₄×C₁₇	C₂²×C₄	1	112	yes	—
50	C₂²×C₄×C₁₇	C₂⁴	1	56	yes	—
51	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
52	C₂²×Dic₁₇	—	2	112	no	—
53	C₂²×Dic₁₇	C₂²×C₄	1	112	yes	—
54	C₂²×Dic₁₇	C₂⁴	1	56	yes	—
55	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
56	C₂×C₄×D₁₇	C₂²×C₄	1	896	yes	complemented
57	C₂×C₄×D₁₇	C₂⁴	1	64	yes	—
58	C₄⋊C₄×C₁₇	C₂⁴	1	64	yes	complemented
59	#63227	C₂⁴	1	128	yes	—
60	C₄⋊Dic₁₇	C₂⁴	1	64	yes	complemented
61	C₄×Dic₁₇	C₂⁴	1	128	yes	complemented
62	C₂³×#117	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	#117	1	64	yes	complemented
64	C₂³×C₁₇	C₂³×C₄	1	28	yes	—
65	C₂³×C₁₇	C₂×#41	1	28	yes	—
66	C₂³×C₁₇	C₂×#41	1	28	yes	—
67	C₂³×C₁₇	C₂⁵	1	7	yes	—
68	C₂²×D₁₇	—	2	448	no	—
69	C₂²×D₁₇	—	2	56	no	—
70	C₂²×D₁₇	—	2	56	no	—
71	C₂²×D₁₇	C₂³×C₄	1	112	yes	—
72	C₂²×D₁₇	C₂⁵	1	8	yes	—
73	C₂×C₄×C₁₇	—	2	112	no	—
74	C₂×C₄×C₁₇	C₂³×C₄	1	112	yes	—
75	C₂×C₄×C₁₇	C₂⁵	1	16	yes	—
76	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
77	C₂×Dic₁₇	—	2	112	no	—
78	C₂×Dic₁₇	C₂³×C₄	1	112	yes	—
79	C₂×Dic₁₇	C₂⁵	1	16	yes	—
80	C₂×Dic₁₇	C₂⁵	1	8	yes	—
81	C₄×D₁₇	C₂³×C₄	1	256	yes	complemented
82	C₂⁵×C₄	—	17	1	no	—
83	C₂²×#117	—	17	56	no	—
84	C₂³×C₄⋊C₄	—	17	1	no	—
85	C₂³×C₄⋊C₄	D₁₇	1	1	yes	characteristic, complemented
86	C₂³×C₂²⋊C₄	—	17	2	no	—
87	C₂³×C₄²	—	17	2	no	—
88	C₂²×C₁₇	C₂×#117	1	112	yes	complemented
89	C₂²×C₁₇	C₂⁴×C₄	1	14	yes	—
90	C₂²×C₁₇	C₂²×#41	1	14	yes	—
91	C₂²×C₁₇	C₂²×#41	1	14	yes	—
92	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
93	C₂×D₁₇	—	2	8	no	—
94	C₂×D₁₇	—	2	8	no	—
95	C₂×D₁₇	—	2	224	no	—
96	C₂×D₁₇	C₂⁴×C₄	1	16	yes	—
97	C₄×C₁₇	—	2	16	no	—
98	C₄×C₁₇	C₂⁴×C₄	1	16	yes	—
99	Dic₁₇	—	2	16	no	—
100	Dic₁₇	C₂⁴×C₄	1	16	yes	—
101	C₂⁶	—	17	1	no	—
102	C₂⁴×C₄	—	17	1	no	—
103	C₂⁴×C₄	—	17	32	no	—
104	C₂⁴×C₄	—	17	28	no	—
105	C₂⁴×C₄	—	17	2	no	—
106	C₂⁴×C₄	C₂×D₁₇	1	2	yes	—
107	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
108	C₂×#117	—	17	448	no	—
109	C₂²×C₄⋊C₄	—	17	28	no	—
110	C₂²×C₄⋊C₄	C₂×D₁₇	1	28	yes	complemented
111	C₂²×C₂²⋊C₄	—	17	56	no	—
112	C₂²×C₄²	—	17	56	no	—
113	C₂×C₁₇	C₂²×#117	1	28	yes	complemented
114	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
115	C₂×C₁₇	C₂³×#41	1	1	yes	characteristic
116	C₂×C₁₇	C₂³×#41	1	1	yes	characteristic
117	D₁₇	—	2	16	no	—
118	C₂⁵	—	17	16	no	—
119	C₂⁵	—	17	14	no	—
120	C₂⁵	—	34	8	no	—
121	C₂⁵	—	34	8	no	—
122	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
123	C₂³×C₄	—	2	16	no	—
124	C₂³×C₄	—	17	16	no	—
125	C₂³×C₄	—	17	14	no	—
126	C₂³×C₄	—	17	448	no	—
127	C₂³×C₄	—	17	112	no	—
128	C₂³×C₄	—	17	28	no	—
129	C₂³×C₄	—	34	16	no	—
130	C₂³×C₄	C₄×D₁₇	1	16	yes	—
131	C₂³×C₄	C₂²×D₁₇	1	28	yes	—
132	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
133	#117	—	17	512	no	—
134	C₂×C₄⋊C₄	—	17	112	no	—
135	C₂×C₄⋊C₄	C₂²×D₁₇	1	112	yes	complemented
136	C₂×C₂²⋊C₄	—	17	224	no	—
137	C₂×C₄²	—	17	224	no	—
138	C₁₇	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
139	C₂⁴	—	17	112	no	—
140	C₂⁴	—	17	28	no	—
141	C₂⁴	—	34	56	no	—
142	C₂⁴	—	34	56	no	—
143	C₂⁴	—	34	128	no	—
144	C₂⁴	C₂×C₄×D₁₇	1	8	yes	—
145	C₂⁴	#63254	1	8	yes	—
146	C₂⁴	#63256	1	8	yes	—
147	C₂⁴	C₂³×D₁₇	1	7	yes	—
148	C₂²×C₄	—	2	112	no	—
149	C₂²×C₄	—	17	112	no	—
150	C₂²×C₄	—	17	28	no	—
151	C₂²×C₄	—	17	896	no	—
152	C₂²×C₄	—	17	64	no	—
153	C₂²×C₄	—	17	56	no	—
154	C₂²×C₄	—	34	112	no	—
155	C₂²×C₄	C₂×C₄×D₁₇	1	112	yes	—
156	C₂²×C₄	C₂³×D₁₇	1	56	yes	—
157	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
158	C₄⋊C₄	—	17	64	no	—
159	C₄⋊C₄	C₂³×D₁₇	1	64	yes	complemented
160	C₂²⋊C₄	—	17	128	no	—
161	C₄²	—	17	128	no	—
162	C₂³	—	17	112	no	—
163	C₂³	—	17	8	no	—
164	C₂³	—	34	448	no	—
165	C₂³	—	34	56	no	—
166	C₂³	—	34	56	no	—
167	C₂³	#10587578	1	64	yes	direct product factor
168	C₂³	C₂²×C₄×D₁₇	1	28	yes	—
169	C₂³	C₂×#63254	1	28	yes	—
170	C₂³	C₂×#63256	1	28	yes	—
171	C₂³	C₂⁴×D₁₇	1	7	yes	—
172	C₂×C₄	—	2	112	no	—
173	C₂×C₄	—	17	112	no	—
174	C₂×C₄	—	17	8	no	—
175	C₂×C₄	—	17	256	no	—
176	C₂×C₄	—	17	16	no	—
177	C₂×C₄	—	34	112	no	—
178	C₂×C₄	C₂²×C₄×D₁₇	1	112	yes	—
179	C₂×C₄	C₂⁴×D₁₇	1	16	yes	—
180	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
181	C₂²	—	17	16	no	—
182	C₂²	—	34	224	no	—
183	C₂²	—	34	8	no	—
184	C₂²	—	34	8	no	—
185	C₂²	C₂×#10587578	1	112	yes	direct product factor
186	C₂²	C₂³×C₄×D₁₇	1	14	yes	—
187	C₂²	C₂²×#63254	1	14	yes	—
188	C₂²	C₂²×#63256	1	14	yes	—
189	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
190	C₄	—	2	16	no	—
191	C₄	—	17	16	no	—
192	C₄	—	34	16	no	—
193	C₄	C₂³×C₄×D₁₇	1	16	yes	—
194	C₂	—	34	16	no	—
195	C₂	C₂²×#10587578	1	28	yes	direct product factor
196	C₂	C₂⁴×C₄×D₁₇	1	1	yes	characteristic
197	C₂	C₂³×#63254	1	1	yes	characteristic
198	C₂	C₂³×#63256	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂²×#10587578	C₂	direct (N × H)
3. C₂³×C₄⋊Dic₁₇	C₂	semidirect
5. C₂³×C₄×Dic₁₇	C₂	semidirect
6. C₂³×C₄⋊C₄×C₁₇	C₂	semidirect
10. C₂³×C₄×D₁₇	C₄	semidirect
12. C₂×#10587578	C₂²	direct (N × H)
13. C₂²×C₄⋊Dic₁₇	C₂²	semidirect
15. C₂²×C₄×Dic₁₇	C₂²	semidirect
18. C₂²×C₄⋊C₄×C₁₇	C₂²	semidirect
32. C₂²×C₄×D₁₇	C₂×C₄	semidirect
34. C₂×C₄⋊C₄×C₁₇	C₂³	semidirect
36. C₂×C₄⋊Dic₁₇	C₂³	semidirect
37. C₂×C₄×Dic₁₇	C₂³	semidirect
38. #10587578	C₂³	direct (N × H)
56. C₂×C₄×D₁₇	C₂²×C₄	semidirect
58. C₄⋊C₄×C₁₇	C₂⁴	semidirect
60. C₄⋊Dic₁₇	C₂⁴	semidirect
61. C₄×Dic₁₇	C₂⁴	semidirect
63. C₂³×C₁₇	#117	semidirect
81. C₄×D₁₇	C₂³×C₄	semidirect
85. C₂³×C₄⋊C₄	D₁₇	semidirect
88. C₂²×C₁₇	C₂×#117	semidirect
110. C₂²×C₄⋊C₄	C₂×D₁₇	semidirect
113. C₂×C₁₇	C₂²×#117	semidirect
135. C₂×C₄⋊C₄	C₂²×D₁₇	semidirect
138. C₁₇	C₂³×#117	semidirect
159. C₄⋊C₄	C₂³×D₁₇	semidirect
167. C₂³	#10587578	direct (N × H)
185. C₂²	C₂×#10587578	direct (N × H)
195. C₂	C₂²×#10587578	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×#10587578	2
C₂³×C₄⋊Dic₁₇	2
C₂³×#63227	2
C₂³×C₄×Dic₁₇	2
C₂³×C₄⋊C₄×C₁₇	2
C₂³×#117	17

Subgroups — C23×#10587578 #17940883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587578 #17940883