Subgroups — C₂³×#10587491 #17897750

Order 4352.

7646 conjugacy classes of subgroups.

The subgroup lattice has 208 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	C₂⁴×#3622	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#10587491	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×D₁₇	C₄	1	1	yes	characteristic
6	C₂⁴×Dic₁₇	C₄	1	1	yes	characteristic
7	C₂³×C₄×D₁₇	—	2	8	no	—
8	C₂³×C₄×D₁₇	—	2	8	no	—
9	C₂³×C₄×D₁₇	C₄	1	14	yes	—
10	C₂³×C₄×D₁₇	C₂²	1	14	yes	—
11	C₂³×#3622	—	2	8	no	—
12	C₂³×#3622	C₂²	1	14	yes	complemented
13	C₂×#10587491	C₂²	1	112	yes	direct product factor
14	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₁₇	—	2	8	no	—
16	C₂³×C₈×C₁₇	C₂²	1	14	yes	complemented
17	C₂⁵×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂⁴×D₁₇	—	2	8	no	—
19	C₂⁴×D₁₇	—	2	8	no	—
20	C₂⁴×D₁₇	C₈	1	16	yes	complemented
21	C₂⁴×D₁₇	C₂×C₄	1	14	yes	—
22	C₂³×C₄×C₁₇	C₂×C₄	1	7	yes	—
23	C₂³×C₄×C₁₇	D₄	1	8	yes	—
24	C₂³×C₄×C₁₇	D₄	1	8	yes	—
25	C₂³×C₄×C₁₇	C₂³	1	7	yes	—
26	C₂³×Dic₁₇	—	2	8	no	—
27	C₂³×Dic₁₇	C₂×C₄	1	14	yes	—
28	C₂²×C₄×D₁₇	—	2	112	no	—
29	C₂²×C₄×D₁₇	—	2	56	no	—
30	C₂²×C₄×D₁₇	—	2	56	no	—
31	C₂²×C₄×D₁₇	C₂×C₄	1	84	yes	—
32	C₂²×C₄×D₁₇	C₂³	1	28	yes	—
33	C₂²×C₈×C₁₇	—	2	56	no	—
34	C₂²×C₈×C₁₇	C₂³	1	28	yes	complemented
35	C₂²×#3622	—	2	56	no	—
36	C₂²×#3622	C₂³	1	28	yes	complemented
37	#10587491	C₂³	1	64	yes	direct product factor
38	C₂⁴×C₁₇	C₂²⋊C₄	1	8	yes	—
39	C₂⁴×C₁₇	C₂×C₈	1	8	yes	—
40	C₂⁴×C₁₇	M₁₆	1	8	yes	—
41	C₂⁴×C₁₇	C₂²×C₄	1	7	yes	—
42	C₂³×D₁₇	—	2	128	no	—
43	C₂³×D₁₇	—	2	56	no	—
44	C₂³×D₁₇	—	2	56	no	—
45	C₂³×D₁₇	C₂×C₈	1	112	yes	complemented
46	C₂³×D₁₇	C₂²×C₄	1	28	yes	—
47	C₂²×C₄×C₁₇	C₂²⋊C₄	1	56	yes	—
48	C₂²×C₄×C₁₇	C₂²×C₄	1	21	yes	—
49	C₂²×C₄×C₁₇	C₂×D₄	1	28	yes	—
50	C₂²×C₄×C₁₇	C₂×D₄	1	28	yes	—
51	C₂²×C₄×C₁₇	C₂⁴	1	7	yes	—
52	C₂²×Dic₁₇	—	2	56	no	—
53	C₂²×Dic₁₇	C₂²×C₄	1	28	yes	—
54	C₂×C₄×D₁₇	—	2	336	no	—
55	C₂×C₄×D₁₇	—	2	56	no	—
56	C₂×C₄×D₁₇	—	2	56	no	—
57	C₂×C₄×D₁₇	C₂²×C₄	1	56	yes	—
58	C₂×C₄×D₁₇	C₂⁴	1	8	yes	—
59	C₂×C₈×C₁₇	—	2	56	no	—
60	C₂×C₈×C₁₇	C₂⁴	1	8	yes	complemented
61	C₂×#3622	—	2	56	no	—
62	C₂×#3622	C₂⁴	1	8	yes	complemented
63	C₂³×C₂²⋊C₈	—	17	1	no	maximal, Sylow
64	C₂³×C₁₇	C₂²⋊C₈	1	64	yes	complemented
65	C₂³×C₁₇	C₂×C₂²⋊C₄	1	28	yes	—
66	C₂³×C₁₇	C₂²×C₈	1	28	yes	—
67	C₂³×C₁₇	C₂×M₁₆	1	28	yes	—
68	C₂³×C₁₇	C₂³×C₄	1	7	yes	—
69	C₂²×D₁₇	—	2	56	no	—
70	C₂²×D₁₇	—	2	56	no	—
71	C₂²×D₁₇	—	2	448	no	—
72	C₂²×D₁₇	C₂²×C₈	1	112	yes	complemented
73	C₂²×D₁₇	C₂³×C₄	1	8	yes	—
74	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	84	yes	—
75	C₂×C₄×C₁₇	C₂³×C₄	1	7	yes	—
76	C₂×C₄×C₁₇	C₂²×D₄	1	14	yes	—
77	C₂×C₄×C₁₇	C₂²×D₄	1	14	yes	—
78	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
79	C₂×Dic₁₇	—	2	56	no	—
80	C₂×Dic₁₇	C₂³×C₄	1	8	yes	—
81	C₄×D₁₇	—	2	112	no	—
82	C₄×D₁₇	—	2	8	no	—
83	C₄×D₁₇	—	2	8	no	—
84	C₈×C₁₇	—	2	8	no	—
85	#3622	—	2	8	no	—
86	C₂⁵×C₄	—	17	1	no	—
87	C₂⁴×C₈	—	17	1	no	—
88	C₂⁴×C₈	D₁₇	1	1	yes	characteristic, complemented
89	C₂²×C₂²⋊C₈	—	17	28	no	—
90	C₂²×C₁₇	C₂×C₂²⋊C₈	1	112	yes	complemented
91	C₂²×C₁₇	C₂²×C₂²⋊C₄	1	14	yes	—
92	C₂²×C₁₇	C₂³×C₈	1	14	yes	—
93	C₂²×C₁₇	C₂²×M₁₆	1	14	yes	—
94	C₂²×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
95	C₂×D₁₇	—	2	224	no	—
96	C₂×D₁₇	—	2	8	no	—
97	C₂×D₁₇	—	2	8	no	—
98	C₂×D₁₇	C₂³×C₈	1	16	yes	complemented
99	C₄×C₁₇	C₂²×C₂²⋊C₄	1	14	yes	—
100	C₄×C₁₇	C₂³×D₄	1	1	yes	characteristic
101	C₄×C₁₇	C₂³×D₄	1	1	yes	characteristic
102	Dic₁₇	—	2	8	no	—
103	C₂⁶	—	17	1	no	—
104	C₂⁴×C₄	—	17	1	no	—
105	C₂⁴×C₄	—	17	14	no	—
106	C₂⁴×C₄	—	17	14	no	—
107	C₂⁴×C₄	—	34	8	no	—
108	C₂⁴×C₄	—	34	8	no	—
109	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
110	C₂³×C₈	—	2	8	no	—
111	C₂³×C₈	—	17	14	no	—
112	C₂³×C₈	—	34	8	no	—
113	C₂³×C₈	C₂×D₁₇	1	14	yes	complemented
114	C₂×C₂²⋊C₈	—	17	112	no	—
115	C₂×C₁₇	C₂²×C₂²⋊C₈	1	28	yes	complemented
116	C₂×C₁₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₁₇	C₂⁴×C₈	1	1	yes	characteristic
118	C₂×C₁₇	C₂³×M₁₆	1	1	yes	characteristic
119	D₁₇	—	2	16	no	—
120	C₂⁵	—	17	14	no	—
121	C₂⁵	—	17	16	no	—
122	C₂⁵	—	34	8	no	—
123	C₂⁵	—	34	8	no	—
124	C₂⁵	C₄×D₁₇	1	1	yes	characteristic
125	C₂³×C₄	—	17	14	no	—
126	C₂³×C₄	—	17	84	no	—
127	C₂³×C₄	—	17	28	no	—
128	C₂³×C₄	—	34	8	no	—
129	C₂³×C₄	—	34	56	no	—
130	C₂³×C₄	—	34	56	no	—
131	C₂³×C₄	—	34	112	no	—
132	C₂³×C₄	C₄×D₁₇	1	7	yes	—
133	C₂³×C₄	D₆₈	1	8	yes	—
134	C₂³×C₄	C₂²⋊D₁₇	1	8	yes	—
135	C₂³×C₄	C₂²×D₁₇	1	7	yes	—
136	C₂²×C₈	—	2	56	no	—
137	C₂²×C₈	—	17	28	no	—
138	C₂²×C₈	—	34	56	no	—
139	C₂²×C₈	C₂²×D₁₇	1	28	yes	complemented
140	C₂²⋊C₈	—	17	64	no	—
141	C₁₇	C₂³×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
142	C₂⁴	—	17	28	no	—
143	C₂⁴	—	17	112	no	—
144	C₂⁴	—	34	128	no	—
145	C₂⁴	—	34	56	no	—
146	C₂⁴	—	34	56	no	—
147	C₂⁴	C₈×D₁₇	1	8	yes	—
148	C₂⁴	#63218	1	8	yes	—
149	C₂⁴	#63227	1	8	yes	—
150	C₂⁴	C₂×C₄×D₁₇	1	7	yes	—
151	C₂²×C₄	—	17	28	no	—
152	C₂²×C₄	—	17	56	no	—
153	C₂²×C₄	—	17	8	no	—
154	C₂²×C₄	—	34	56	no	—
155	C₂²×C₄	—	34	56	no	—
156	C₂²×C₄	—	34	56	no	—
157	C₂²×C₄	—	34	336	no	—
158	C₂²×C₄	#63227	1	56	yes	—
159	C₂²×C₄	C₂×C₄×D₁₇	1	21	yes	—
160	C₂²×C₄	C₂×D₆₈	1	28	yes	—
161	C₂²×C₄	C₂×C₂²⋊D₁₇	1	28	yes	—
162	C₂²×C₄	C₂³×D₁₇	1	7	yes	—
163	C₂×C₈	—	2	56	no	—
164	C₂×C₈	—	17	8	no	—
165	C₂×C₈	—	34	56	no	—
166	C₂×C₈	C₂³×D₁₇	1	8	yes	complemented
167	C₂³	—	17	112	no	—
168	C₂³	—	17	8	no	—
169	C₂³	—	34	56	no	—
170	C₂³	—	34	448	no	—
171	C₂³	—	34	56	no	—
172	C₂³	#10587491	1	64	yes	direct product factor
173	C₂³	C₂×C₈×D₁₇	1	28	yes	—
174	C₂³	C₂×#63218	1	28	yes	—
175	C₂³	C₂×#63227	1	28	yes	—
176	C₂³	C₂²×C₄×D₁₇	1	7	yes	—
177	C₂×C₄	—	17	8	no	—
178	C₂×C₄	—	34	56	no	—
179	C₂×C₄	—	34	112	no	—
180	C₂×C₄	—	34	8	no	—
181	C₂×C₄	—	34	8	no	—
182	C₂×C₄	C₂×#63227	1	84	yes	—
183	C₂×C₄	C₂²×C₄×D₁₇	1	7	yes	—
184	C₂×C₄	C₂²×D₆₈	1	14	yes	—
185	C₂×C₄	C₂²×C₂²⋊D₁₇	1	14	yes	—
186	C₂×C₄	C₂⁴×D₁₇	1	1	yes	characteristic
187	C₈	—	2	8	no	—
188	C₈	—	34	8	no	—
189	C₂²	—	17	16	no	—
190	C₂²	—	34	8	no	—
191	C₂²	—	34	8	no	—
192	C₂²	—	34	224	no	—
193	C₂²	C₂×#10587491	1	112	yes	direct product factor
194	C₂²	C₂²×#63227	1	14	yes	—
195	C₂²	C₂²×C₈×D₁₇	1	14	yes	—
196	C₂²	C₂²×#63218	1	14	yes	—
197	C₂²	C₂³×C₄×D₁₇	1	1	yes	characteristic
198	C₄	—	34	8	no	—
199	C₄	C₂²×#63227	1	14	yes	—
200	C₄	C₂³×C₂²⋊D₁₇	1	1	yes	characteristic
201	C₄	C₂³×D₆₈	1	1	yes	characteristic
202	C₂	—	34	16	no	—
203	C₂	C₂²×#10587491	1	28	yes	direct product factor
204	C₂	C₂³×#63227	1	1	yes	characteristic
205	C₂	C₂³×C₈×D₁₇	1	1	yes	characteristic
206	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	C₂⁵×C₁₇	C₂×C₄
prime radical	C₂⁵×D₁₇	C₄
cosocle	C₂×C₄×C₁₇	C₂⁵
hypercenter	C₂⁴×C₈	D₁₇
hypocenter	C₁₇	C₂³×C₂²⋊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
2. C₂⁴×#3622	C₂	semidirect
3. C₂²×#10587491	C₂	direct (N × H)
4. C₂⁴×C₈×C₁₇	C₂	semidirect
12. C₂³×#3622	C₂²	semidirect
13. C₂×#10587491	C₂²	direct (N × H)
16. C₂³×C₈×C₁₇	C₂²	semidirect
20. C₂⁴×D₁₇	C₈	semidirect
34. C₂²×C₈×C₁₇	C₂³	semidirect
36. C₂²×#3622	C₂³	semidirect
37. #10587491	C₂³	direct (N × H)
45. C₂³×D₁₇	C₂×C₈	semidirect
60. C₂×C₈×C₁₇	C₂⁴	semidirect
62. C₂×#3622	C₂⁴	semidirect
64. C₂³×C₁₇	C₂²⋊C₈	semidirect
72. C₂²×D₁₇	C₂²×C₈	semidirect
88. C₂⁴×C₈	D₁₇	semidirect
90. C₂²×C₁₇	C₂×C₂²⋊C₈	semidirect
98. C₂×D₁₇	C₂³×C₈	semidirect
113. C₂³×C₈	C₂×D₁₇	semidirect
115. C₂×C₁₇	C₂²×C₂²⋊C₈	semidirect
139. C₂²×C₈	C₂²×D₁₇	semidirect
141. C₁₇	C₂³×C₂²⋊C₈	semidirect
166. C₂×C₈	C₂³×D₁₇	semidirect
172. C₂³	#10587491	direct (N × H)
193. C₂²	C₂×#10587491	direct (N × H)
203. C₂	C₂²×#10587491	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
C₂⁴×#3622	2
C₂²×#10587491	2
C₂⁴×C₈×C₁₇	2
C₂³×C₂²⋊C₈	17

Subgroups — C23×#10587491 #17897750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587491 #17897750