Subgroups — C₂³×C₈×#595 #17979398

Order 4352.

4638 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63242	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×#595	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×#595	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×#595	C₄	1	2	yes	—
6	C₂²×#63242	C₄	1	14	yes	complemented
7	C₂²×#63242	C₂²	1	14	yes	—
8	C₂³×#3624	C₄	1	2	yes	complemented
9	C₂²×C₄×#595	C₄	1	14	yes	—
10	C₂²×C₄×#595	C₂²	1	14	yes	—
11	C₂×C₈×#595	C₂²	1	112	yes	direct product factor
12	C₂³×C₄×D₁₇	C₂²	1	1	yes	characteristic
13	C₂³×#3622	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₈×D₁₇	C₄	1	14	yes	complemented
15	C₂²×C₈×D₁₇	C₂²	1	14	yes	—
16	C₂³×C₈×C₁₇	C₄	1	1	yes	characteristic, complemented
17	C₂⁴×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×#595	C₈	1	32	yes	direct product factor
19	C₂³×#595	C₂×C₄	1	28	yes	—
20	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×D₁₇	C₂×C₄	1	7	yes	—
23	C₂²×C₄×D₁₇	C₂×C₄	1	7	yes	—
24	C₂²×C₄×D₁₇	C₂×C₄	1	7	yes	—
25	C₂²×C₄×D₁₇	C₂³	1	7	yes	—
26	C₂²×#3624	C₂×C₄	1	28	yes	complemented
27	C₂×C₄×#595	C₂×C₄	1	84	yes	—
28	C₂×C₄×#595	C₂³	1	28	yes	—
29	C₂×#63242	C₂×C₄	1	84	yes	complemented
30	C₂×#63242	C₂³	1	28	yes	—
31	C₂²×C₈×C₁₇	C₂×C₄	1	14	yes	complemented
32	C₂²×#3622	C₂×C₄	1	14	yes	complemented
33	C₂×C₈×D₁₇	C₂×C₄	1	84	yes	complemented
34	C₂×C₈×D₁₇	C₂³	1	28	yes	—
35	C₈×#595	C₂³	1	64	yes	direct product factor
36	C₂⁴×C₁₇	C₄²	1	1	yes	characteristic
37	C₂³×D₁₇	C₄²	1	7	yes	—
38	C₂³×D₁₇	C₂×C₈	1	16	yes	—
39	C₂³×D₁₇	C₂²×C₄	1	7	yes	—
40	C₂²×#595	C₂×C₈	1	224	yes	direct product factor
41	C₂²×#595	C₂²×C₄	1	56	yes	—
42	C₂²×C₄×C₁₇	C₄²	1	7	yes	—
43	C₂²×C₄×C₁₇	C₂²×C₄	1	7	yes	—
44	C₂²×Dic₁₇	C₄²	1	7	yes	—
45	C₂²×Dic₁₇	C₂²×C₄	1	7	yes	—
46	C₂×C₄×D₁₇	C₄²	1	42	yes	—
47	C₂×C₄×D₁₇	C₂²×C₄	1	21	yes	—
48	C₂×C₄×D₁₇	C₂²×C₄	1	21	yes	—
49	C₂×C₄×D₁₇	C₂²×C₄	1	21	yes	—
50	C₂×C₄×D₁₇	C₂⁴	1	7	yes	—
51	C₂×#3624	C₂²×C₄	1	56	yes	complemented
52	C₄×#595	C₂²×C₄	1	56	yes	—
53	C₄×#595	C₂⁴	1	8	yes	—
54	#63242	C₂²×C₄	1	56	yes	complemented
55	#63242	C₂⁴	1	8	yes	—
56	C₂×C₈×C₁₇	C₂²×C₄	1	28	yes	complemented
57	C₂×#3622	C₂²×C₄	1	28	yes	complemented
58	C₈×D₁₇	C₂²×C₄	1	56	yes	complemented
59	C₈×D₁₇	C₂⁴	1	8	yes	—
60	C₂³×C₄×C₈	—	17	1	no	maximal, Sylow
61	C₂³×C₁₇	C₄×C₈	1	8	yes	complemented
62	C₂³×C₁₇	C₂×C₄²	1	7	yes	—
63	C₂²×D₁₇	C₄×C₈	1	56	yes	complemented
64	C₂²×D₁₇	C₂×C₄²	1	21	yes	—
65	C₂²×D₁₇	C₂²×C₈	1	56	yes	—
66	C₂²×D₁₇	C₂³×C₄	1	7	yes	—
67	C₂×#595	C₂²×C₈	1	224	yes	direct product factor
68	C₂×#595	C₂³×C₄	1	16	yes	—
69	C₂×C₄×C₁₇	C₂×C₄²	1	21	yes	—
70	C₂×C₄×C₁₇	C₂³×C₄	1	7	yes	—
71	C₂×Dic₁₇	C₂×C₄²	1	21	yes	—
72	C₂×Dic₁₇	C₂³×C₄	1	7	yes	—
73	C₄×D₁₇	C₂×C₄²	1	42	yes	—
74	C₄×D₁₇	C₂³×C₄	1	7	yes	—
75	C₄×D₁₇	C₂³×C₄	1	7	yes	—
76	C₄×D₁₇	C₂³×C₄	1	7	yes	—
77	C₄×D₁₇	C₂⁵	1	1	yes	characteristic
78	#3624	C₂³×C₄	1	16	yes	complemented
79	C₈×C₁₇	C₂³×C₄	1	8	yes	complemented
80	#3622	C₂³×C₄	1	8	yes	complemented
81	C₂⁴×C₈	—	17	1	no	—
82	C₂⁴×C₈	—	17	1	no	—
83	C₂³×C₄²	—	17	1	no	—
84	C₂²×C₄×C₈	—	17	28	no	—
85	C₂²×C₁₇	C₂×C₄×C₈	1	28	yes	complemented
86	C₂²×C₁₇	C₂²×C₄²	1	7	yes	—
87	C₂×D₁₇	C₂×C₄×C₈	1	84	yes	complemented
88	C₂×D₁₇	C₂²×C₄²	1	7	yes	—
89	C₂×D₁₇	C₂³×C₈	1	28	yes	—
90	C₂×D₁₇	C₂⁴×C₄	1	1	yes	characteristic
91	#595	C₂³×C₈	1	32	yes	direct product factor
92	C₄×C₁₇	C₂²×C₄²	1	7	yes	—
93	C₄×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
94	Dic₁₇	C₂²×C₄²	1	7	yes	—
95	Dic₁₇	C₂⁴×C₄	1	1	yes	characteristic
96	C₂⁴×C₄	—	17	2	no	—
97	C₂⁴×C₄	—	17	1	no	—
98	C₂³×C₈	—	17	14	no	—
99	C₂³×C₈	—	17	2	no	—
100	C₂³×C₈	—	17	14	no	—
101	C₂³×C₈	—	17	1	no	—
102	C₂³×C₈	—	17	14	no	—
103	C₂³×C₈	—	17	14	no	—
104	C₂³×C₈	#595	1	1	yes	characteristic, direct product factor
105	C₂²×C₄²	—	17	14	no	—
106	C₂²×C₄²	—	17	14	no	—
107	C₂×C₄×C₈	—	17	112	no	—
108	C₂×C₁₇	C₂²×C₄×C₈	1	14	yes	complemented
109	C₂×C₁₇	C₂³×C₄²	1	1	yes	characteristic
110	D₁₇	C₂²×C₄×C₈	1	14	yes	complemented
111	D₁₇	C₂⁴×C₈	1	2	yes	—
112	C₂⁵	—	17	1	no	—
113	C₂³×C₄	—	17	32	no	—
114	C₂³×C₄	—	17	28	no	—
115	C₂³×C₄	—	17	1	no	—
116	C₂³×C₄	—	17	7	no	—
117	C₂³×C₄	—	17	7	no	—
118	C₂³×C₄	—	17	7	no	—
119	C₂³×C₄	—	17	7	no	—
120	C₂³×C₄	C₂×#595	1	1	yes	characteristic
121	C₂²×C₈	—	17	84	no	—
122	C₂²×C₈	—	17	28	no	—
123	C₂²×C₈	—	17	28	no	—
124	C₂²×C₈	—	17	14	no	—
125	C₂²×C₈	—	17	28	no	—
126	C₂²×C₈	—	17	84	no	—
127	C₂²×C₈	C₂×#595	1	14	yes	direct product factor
128	C₂×C₄²	—	17	84	no	—
129	C₂×C₄²	—	17	28	no	—
130	C₄×C₈	—	17	64	no	—
131	C₁₇	C₂³×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
132	C₂⁴	—	17	7	no	—
133	C₂⁴	—	17	7	no	—
134	C₂⁴	—	17	16	no	—
135	C₂⁴	C₄×#595	1	1	yes	characteristic
136	C₂²×C₄	—	17	224	no	—
137	C₂²×C₄	—	17	56	no	—
138	C₂²×C₄	—	17	7	no	—
139	C₂²×C₄	—	17	7	no	—
140	C₂²×C₄	—	17	42	no	—
141	C₂²×C₄	—	17	21	no	—
142	C₂²×C₄	—	17	21	no	—
143	C₂²×C₄	—	17	21	no	—
144	C₂²×C₄	—	17	7	no	—
145	C₂²×C₄	C₄×#595	1	7	yes	—
146	C₂²×C₄	C₂²×#595	1	7	yes	—
147	C₂×C₈	—	17	56	no	—
148	C₂×C₈	—	17	8	no	—
149	C₂×C₈	—	17	56	no	—
150	C₂×C₈	—	17	28	no	—
151	C₂×C₈	—	17	8	no	—
152	C₂×C₈	—	17	56	no	—
153	C₂×C₈	C₂²×#595	1	28	yes	direct product factor
154	C₄²	—	17	56	no	—
155	C₄²	—	17	8	no	—
156	C₂³	—	17	56	no	—
157	C₂³	—	17	56	no	—
158	C₂³	—	17	7	no	—
159	C₂³	—	17	21	no	—
160	C₂³	C₈×#595	1	8	yes	direct product factor
161	C₂³	C₂×C₄×#595	1	7	yes	—
162	C₂×C₄	—	17	16	no	—
163	C₂×C₄	—	17	224	no	—
164	C₂×C₄	—	17	7	no	—
165	C₂×C₄	—	17	21	no	—
166	C₂×C₄	—	17	7	no	—
167	C₂×C₄	—	17	7	no	—
168	C₂×C₄	—	17	42	no	—
169	C₂×C₄	—	17	1	no	—
170	C₂×C₄	—	17	7	no	—
171	C₂×C₄	C₂×C₄×#595	1	21	yes	—
172	C₂×C₄	C₂³×#595	1	7	yes	—
173	C₈	—	17	16	no	—
174	C₈	—	17	8	no	—
175	C₈	C₂³×#595	1	8	yes	direct product factor
176	C₂²	—	17	1	no	—
177	C₂²	—	17	7	no	—
178	C₂²	—	17	28	no	—
179	C₂²	—	17	84	no	—
180	C₂²	C₂×C₈×#595	1	28	yes	direct product factor
181	C₂²	C₂²×C₄×#595	1	7	yes	—
182	C₄	—	17	32	no	—
183	C₄	—	17	1	no	—
184	C₄	—	17	7	no	—
185	C₄	C₂²×C₄×#595	1	7	yes	—
186	C₄	C₂⁴×#595	1	1	yes	characteristic
187	C₂	—	17	2	no	—
188	C₂	—	17	14	no	—
189	C₂	C₂²×C₈×#595	1	14	yes	direct product factor
190	C₂	C₂³×C₄×#595	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₈	#595
commutator, hypocenter	C₁₇	C₂³×C₄×C₈
Frattini	C₄	C₂⁴×#595
Fitting	C₂³×C₈×C₁₇	C₄
socle	C₂⁴×C₁₇	C₄²
prime radical	C₂⁴×D₁₇	C₂×C₄
cosocle	C₄×D₁₇	C₂⁵
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×C₈×#595	C₂	direct (N × H)
6. C₂²×#63242	C₄	semidirect
8. C₂³×#3624	C₄	semidirect
11. C₂×C₈×#595	C₂²	direct (N × H)
13. C₂³×#3622	C₄	semidirect
14. C₂²×C₈×D₁₇	C₄	semidirect
16. C₂³×C₈×C₁₇	C₄	semidirect
18. C₂³×#595	C₈	direct (N × H)
26. C₂²×#3624	C₂×C₄	semidirect
29. C₂×#63242	C₂×C₄	semidirect
31. C₂²×C₈×C₁₇	C₂×C₄	semidirect
32. C₂²×#3622	C₂×C₄	semidirect
33. C₂×C₈×D₁₇	C₂×C₄	semidirect
35. C₈×#595	C₂³	direct (N × H)
40. C₂²×#595	C₂×C₈	direct (N × H)
51. C₂×#3624	C₂²×C₄	semidirect
54. #63242	C₂²×C₄	semidirect
56. C₂×C₈×C₁₇	C₂²×C₄	semidirect
57. C₂×#3622	C₂²×C₄	semidirect
58. C₈×D₁₇	C₂²×C₄	semidirect
61. C₂³×C₁₇	C₄×C₈	semidirect
63. C₂²×D₁₇	C₄×C₈	semidirect
67. C₂×#595	C₂²×C₈	direct (N × H)
78. #3624	C₂³×C₄	semidirect
79. C₈×C₁₇	C₂³×C₄	semidirect
80. #3622	C₂³×C₄	semidirect
85. C₂²×C₁₇	C₂×C₄×C₈	semidirect
87. C₂×D₁₇	C₂×C₄×C₈	semidirect
91. #595	C₂³×C₈	direct (N × H)
104. C₂³×C₈	#595	direct (N × H)
108. C₂×C₁₇	C₂²×C₄×C₈	semidirect
110. D₁₇	C₂²×C₄×C₈	semidirect
127. C₂²×C₈	C₂×#595	direct (N × H)
131. C₁₇	C₂³×C₄×C₈	semidirect
153. C₂×C₈	C₂²×#595	direct (N × H)
160. C₂³	C₈×#595	direct (N × H)
175. C₈	C₂³×#595	direct (N × H)
180. C₂²	C₂×C₈×#595	direct (N × H)
189. C₂	C₂²×C₈×#595	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#63242	2
C₂³×C₄×#595	2
C₂²×C₈×#595	2
C₂³×C₈×D₁₇	2
C₂³×C₄×C₈	17

Subgroups — C23×C8×#595 #17979398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₈×#595 #17979398