Subgroups — C₂×C₄×C₈×#595 #17968120

Order 4352.

2046 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#63242	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×#595	C₂	1	4	yes	maximal
3	C₂×C₄²×#595	C₂	1	1	yes	maximal, characteristic
4	C₄×C₈×#595	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63242	C₂²	1	2	yes	—
7	C₂²×C₄×#595	C₄	1	4	yes	—
8	C₂²×C₄×#595	C₂²	1	2	yes	—
9	C₄×#63242	C₄	1	4	yes	complemented
10	C₄×#63242	C₂²	1	4	yes	—
11	C₂×C₄×#3624	C₄	1	4	yes	complemented
12	C₂×C₈×#595	C₄	1	32	yes	direct product factor
13	C₂×C₈×#595	C₂²	1	16	yes	—
14	C₄²×#595	C₄	1	4	yes	—
15	C₄²×#595	C₂²	1	4	yes	—
16	C₂²×C₈×D₁₇	C₂²	1	2	yes	—
17	C₂×C₄²×D₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×#3622	C₄	1	1	yes	characteristic, complemented
19	C₂×C₈×Dic₁₇	C₄	1	2	yes	complemented
20	C₄×C₈×D₁₇	C₄	1	4	yes	complemented
21	C₄×C₈×D₁₇	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic, complemented
23	C₂³×#595	C₂×C₄	1	4	yes	—
24	C₂²×C₄×D₁₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₁₇	C₂³	1	1	yes	characteristic
26	C₂²×#3624	C₂×C₄	1	4	yes	—
27	C₂×C₄×#595	C₈	1	32	yes	direct product factor
28	C₂×C₄×#595	C₂×C₄	1	4	yes	—
29	C₂×C₄×#595	C₂×C₄	1	16	yes	—
30	C₂×C₄×#595	C₂×C₄	1	16	yes	—
31	C₂×C₄×#595	C₂³	1	4	yes	—
32	C₂×#63242	C₂×C₄	1	4	yes	—
33	C₂×#63242	C₂×C₄	1	16	yes	—
34	C₂×#63242	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
39	C₂²×#3622	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₇	C₂×C₄	1	16	yes	—
41	C₂×C₈×D₁₇	C₂×C₄	1	4	yes	—
42	C₂×C₈×D₁₇	C₂³	1	4	yes	—
43	C₄²×D₁₇	C₂×C₄	1	2	yes	—
44	C₄²×D₁₇	C₂×C₄	1	2	yes	—
45	C₄²×D₁₇	C₂×C₄	1	2	yes	—
46	C₄²×D₁₇	C₂³	1	2	yes	—
47	C₄×#3624	C₂×C₄	1	16	yes	complemented
48	C₈×#595	C₂×C₄	1	64	yes	direct product factor
49	C₄×C₈×C₁₇	C₂×C₄	1	4	yes	complemented
50	C₈×Dic₁₇	C₂×C₄	1	8	yes	complemented
51	C₄×#3622	C₂×C₄	1	4	yes	complemented
52	C₂³×D₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×#595	C₄²	1	16	yes	—
54	C₂²×#595	C₂×C₈	1	32	yes	—
55	C₂²×#595	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
57	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₁₇	C₄²	1	2	yes	—
59	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×D₁₇	C₄²	1	4	yes	—
61	C₂×C₄×D₁₇	C₂×C₈	1	16	yes	—
62	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×D₁₇	C₂²×C₄	1	8	yes	—
66	C₂×C₄×D₁₇	C₂²×C₄	1	4	yes	—
67	C₂×C₄×D₁₇	C₂⁴	1	1	yes	characteristic
68	C₂×#3624	C₄²	1	16	yes	complemented
69	C₂×#3624	C₂²×C₄	1	8	yes	—
70	C₄×#595	C₄²	1	16	yes	—
71	C₄×#595	C₂×C₈	1	64	yes	direct product factor
72	C₄×#595	C₂²×C₄	1	16	yes	—
73	#63242	C₄²	1	16	yes	complemented
74	#63242	C₂²×C₄	1	16	yes	—
75	C₂×C₈×C₁₇	C₄²	1	8	yes	complemented
76	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
77	C₄²×C₁₇	C₄²	1	2	yes	—
78	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
79	C₄×Dic₁₇	C₄²	1	4	yes	—
80	C₄×Dic₁₇	C₄²	1	2	yes	—
81	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
82	C₄×Dic₁₇	C₂²×C₄	1	4	yes	—
83	C₂×#3622	C₄²	1	8	yes	complemented
84	C₂×#3622	C₂²×C₄	1	4	yes	—
85	C₈×D₁₇	C₄²	1	16	yes	complemented
86	C₈×D₁₇	C₂²×C₄	1	16	yes	—
87	C₂×C₄²×C₈	—	17	1	no	maximal, Sylow
88	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
89	C₂²×D₁₇	C₂×C₄²	1	1	yes	characteristic
90	C₂²×D₁₇	C₂×C₄²	1	4	yes	—
91	C₂²×D₁₇	C₂²×C₈	1	8	yes	—
92	C₂²×D₁₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×#595	C₄×C₈	1	64	yes	direct product factor
94	C₂×#595	C₂×C₄²	1	16	yes	—
95	C₂×#595	C₂²×C₈	1	32	yes	—
96	C₂×C₄×C₁₇	C₄×C₈	1	8	yes	complemented
97	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂×C₄²	1	4	yes	—
99	C₂×C₄×C₁₇	C₂×C₄²	1	4	yes	—
100	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×Dic₁₇	C₄×C₈	1	8	yes	complemented
102	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
103	C₂×Dic₁₇	C₂×C₄²	1	4	yes	—
104	C₂×Dic₁₇	C₂×C₄²	1	4	yes	—
105	C₂×Dic₁₇	C₂³×C₄	1	1	yes	characteristic
106	C₄×D₁₇	C₄×C₈	1	16	yes	complemented
107	C₄×D₁₇	C₂×C₄²	1	4	yes	—
108	C₄×D₁₇	C₂×C₄²	1	4	yes	—
109	C₄×D₁₇	C₂×C₄²	1	4	yes	—
110	C₄×D₁₇	C₂²×C₈	1	16	yes	—
111	C₄×D₁₇	C₂³×C₄	1	4	yes	—
112	#3624	C₂×C₄²	1	16	yes	complemented
113	C₈×C₁₇	C₂×C₄²	1	8	yes	complemented
114	#3622	C₂×C₄²	1	8	yes	complemented
115	C₂²×C₄×C₈	—	17	4	no	—
116	C₂²×C₄×C₈	—	17	1	no	—
117	C₂²×C₄×C₈	—	17	1	no	—
118	C₂×C₄³	—	17	1	no	—
119	C₄²×C₈	—	17	8	no	—
120	C₂²×C₁₇	C₄³	1	2	yes	—
121	C₂²×C₁₇	C₂×C₄×C₈	1	4	yes	—
122	C₂²×C₁₇	C₂²×C₄²	1	1	yes	characteristic
123	C₂×D₁₇	C₄³	1	2	yes	—
124	C₂×D₁₇	C₂×C₄×C₈	1	16	yes	—
125	C₂×D₁₇	C₂×C₄×C₈	1	4	yes	—
126	C₂×D₁₇	C₂²×C₄²	1	2	yes	—
127	C₂×D₁₇	C₂³×C₈	1	4	yes	—
128	#595	C₂×C₄×C₈	1	32	yes	direct product factor
129	C₄×C₁₇	C₄³	1	2	yes	—
130	C₄×C₁₇	C₂×C₄×C₈	1	8	yes	complemented
131	C₄×C₁₇	C₂²×C₄²	1	2	yes	—
132	Dic₁₇	C₄³	1	2	yes	—
133	Dic₁₇	C₂×C₄×C₈	1	8	yes	complemented
134	Dic₁₇	C₂²×C₄²	1	2	yes	—
135	C₂³×C₈	—	17	2	no	—
136	C₂³×C₈	—	17	2	no	—
137	C₂²×C₄²	—	17	2	no	—
138	C₂²×C₄²	—	17	4	no	—
139	C₂²×C₄²	—	17	1	no	—
140	C₂×C₄×C₈	—	17	32	no	—
141	C₂×C₄×C₈	—	17	16	no	—
142	C₂×C₄×C₈	—	17	4	no	—
143	C₂×C₄×C₈	—	17	4	no	—
144	C₂×C₄×C₈	—	17	4	no	—
145	C₂×C₄×C₈	—	17	2	no	—
146	C₂×C₄×C₈	—	17	1	no	—
147	C₂×C₄×C₈	—	17	4	no	—
148	C₂×C₄×C₈	—	17	4	no	—
149	C₂×C₄×C₈	#595	1	1	yes	characteristic, direct product factor
150	C₄³	—	17	4	no	—
151	C₄³	—	17	4	no	—
152	C₂×C₁₇	C₄²×C₈	1	4	yes	complemented
153	C₂×C₁₇	C₂×C₄³	1	1	yes	characteristic
154	C₂×C₁₇	C₂²×C₄×C₈	1	2	yes	—
155	D₁₇	C₄²×C₈	1	4	yes	complemented
156	D₁₇	C₂²×C₄×C₈	1	4	yes	—
157	C₂³×C₄	—	17	4	no	—
158	C₂³×C₄	—	17	2	no	—
159	C₂³×C₄	—	17	1	no	—
160	C₂²×C₈	—	17	4	no	—
161	C₂²×C₈	—	17	16	no	—
162	C₂²×C₈	—	17	4	no	—
163	C₂²×C₈	—	17	4	no	—
164	C₂²×C₈	—	17	2	no	—
165	C₂²×C₈	—	17	16	no	—
166	C₂²×C₈	—	17	4	no	—
167	C₂²×C₈	—	17	4	no	—
168	C₂²×C₈	C₂×#595	1	2	yes	—
169	C₂×C₄²	—	17	32	no	—
170	C₂×C₄²	—	17	4	no	—
171	C₂×C₄²	—	17	16	no	—
172	C₂×C₄²	—	17	16	no	—
173	C₂×C₄²	—	17	4	no	—
174	C₂×C₄²	—	17	2	no	—
175	C₂×C₄²	—	17	1	no	—
176	C₂×C₄²	—	17	2	no	—
177	C₂×C₄²	—	17	2	no	—
178	C₂×C₄²	—	17	2	no	—
179	C₂×C₄²	—	17	2	no	—
180	C₂×C₄²	C₂×#595	1	1	yes	characteristic
181	C₄×C₈	—	17	64	no	—
182	C₄×C₈	—	17	16	no	—
183	C₄×C₈	—	17	8	no	—
184	C₄×C₈	—	17	4	no	—
185	C₄×C₈	C₂×#595	1	4	yes	direct product factor
186	C₁₇	C₂×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
187	C₂⁴	—	17	1	no	—
188	C₂²×C₄	—	17	16	no	—
189	C₂²×C₄	—	17	32	no	—
190	C₂²×C₄	—	17	8	no	—
191	C₂²×C₄	—	17	2	no	—
192	C₂²×C₄	—	17	1	no	—
193	C₂²×C₄	—	17	4	no	—
194	C₂²×C₄	—	17	1	no	—
195	C₂²×C₄	—	17	4	no	—
196	C₂²×C₄	—	17	8	no	—
197	C₂²×C₄	—	17	1	no	—
198	C₂²×C₄	—	17	1	no	—
199	C₂²×C₄	—	17	1	no	—
200	C₂²×C₄	—	17	16	no	—
201	C₂²×C₄	C₄×#595	1	2	yes	—
202	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
203	C₂×C₈	—	17	16	no	—
204	C₂×C₈	—	17	16	no	—
205	C₂×C₈	—	17	16	no	—
206	C₂×C₈	—	17	8	no	—
207	C₂×C₈	—	17	8	no	—
208	C₂×C₈	—	17	4	no	—
209	C₂×C₈	—	17	16	no	—
210	C₂×C₈	—	17	16	no	—
211	C₂×C₈	C₄×#595	1	8	yes	direct product factor
212	C₂×C₈	C₂²×#595	1	4	yes	—
213	C₄²	—	17	16	no	—
214	C₄²	—	17	64	no	—
215	C₄²	—	17	16	no	—
216	C₄²	—	17	4	no	—
217	C₄²	—	17	2	no	—
218	C₄²	—	17	4	no	—
219	C₄²	—	17	2	no	—
220	C₄²	C₄×#595	1	2	yes	—
221	C₄²	C₂²×#595	1	2	yes	—
222	C₂³	—	17	4	no	—
223	C₂³	—	17	8	no	—
224	C₂³	—	17	1	no	—
225	C₂³	—	17	1	no	—
226	C₂³	C₂×C₄×#595	1	1	yes	characteristic
227	C₂×C₄	—	17	16	no	—
228	C₂×C₄	—	17	32	no	—
229	C₂×C₄	—	17	64	no	—
230	C₂×C₄	—	17	4	no	—
231	C₂×C₄	—	17	8	no	—
232	C₂×C₄	—	17	1	no	—
233	C₂×C₄	—	17	1	no	—
234	C₂×C₄	—	17	4	no	—
235	C₂×C₄	—	17	4	no	—
236	C₂×C₄	—	17	4	no	—
237	C₂×C₄	—	17	16	no	—
238	C₂×C₄	—	17	16	no	—
239	C₂×C₄	—	17	4	no	—
240	C₂×C₄	—	17	4	no	—
241	C₂×C₄	C₈×#595	1	8	yes	direct product factor
242	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
243	C₂×C₄	C₂×C₄×#595	1	4	yes	—
244	C₂×C₄	C₂×C₄×#595	1	4	yes	—
245	C₂×C₄	C₂³×#595	1	1	yes	characteristic
246	C₈	—	17	16	no	—
247	C₈	—	17	8	no	—
248	C₈	C₂×C₄×#595	1	8	yes	direct product factor
249	C₂²	—	17	16	no	—
250	C₂²	—	17	2	no	—
251	C₂²	—	17	2	no	—
252	C₂²	—	17	4	no	—
253	C₂²	—	17	4	no	—
254	C₂²	C₄²×#595	1	2	yes	—
255	C₂²	C₂×C₈×#595	1	4	yes	—
256	C₂²	C₂²×C₄×#595	1	1	yes	characteristic
257	C₄	—	17	32	no	—
258	C₄	—	17	2	no	—
259	C₄	—	17	2	no	—
260	C₄	—	17	8	no	—
261	C₄	C₄²×#595	1	2	yes	—
262	C₄	C₂×C₈×#595	1	8	yes	direct product factor
263	C₄	C₂²×C₄×#595	1	2	yes	—
264	C₂	—	17	4	no	—
265	C₂	—	17	4	no	—
266	C₂	C₄×C₈×#595	1	4	yes	direct product factor
267	C₂	C₂×C₄²×#595	1	1	yes	characteristic
268	C₂	C₂²×C₈×#595	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	#595
commutator, hypocenter	C₁₇	C₂×C₄²×C₈
Frattini	C₂×C₄	C₂³×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂³×C₁₇	C₂×C₄²
prime radical	C₂³×D₁₇	C₂²×C₄
cosocle	C₂×C₄×D₁₇	C₂⁴
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₄×C₈×#595	C₂	direct (N × H)
9. C₄×#63242	C₄	semidirect
11. C₂×C₄×#3624	C₄	semidirect
12. C₂×C₈×#595	C₄	direct (N × H)
18. C₂×C₄×#3622	C₄	semidirect
19. C₂×C₈×Dic₁₇	C₄	semidirect
20. C₄×C₈×D₁₇	C₄	semidirect
22. C₂×C₄×C₈×C₁₇	C₄	semidirect
27. C₂×C₄×#595	C₈	direct (N × H)
47. C₄×#3624	C₂×C₄	semidirect
48. C₈×#595	C₂×C₄	direct (N × H)
49. C₄×C₈×C₁₇	C₂×C₄	semidirect
50. C₈×Dic₁₇	C₂×C₄	semidirect
51. C₄×#3622	C₂×C₄	semidirect
68. C₂×#3624	C₄²	semidirect
71. C₄×#595	C₂×C₈	direct (N × H)
73. #63242	C₄²	semidirect
75. C₂×C₈×C₁₇	C₄²	semidirect
83. C₂×#3622	C₄²	semidirect
85. C₈×D₁₇	C₄²	semidirect
93. C₂×#595	C₄×C₈	direct (N × H)
96. C₂×C₄×C₁₇	C₄×C₈	semidirect
101. C₂×Dic₁₇	C₄×C₈	semidirect
106. C₄×D₁₇	C₄×C₈	semidirect
112. #3624	C₂×C₄²	semidirect
113. C₈×C₁₇	C₂×C₄²	semidirect
114. #3622	C₂×C₄²	semidirect
128. #595	C₂×C₄×C₈	direct (N × H)
130. C₄×C₁₇	C₂×C₄×C₈	semidirect
133. Dic₁₇	C₂×C₄×C₈	semidirect
149. C₂×C₄×C₈	#595	direct (N × H)
152. C₂×C₁₇	C₄²×C₈	semidirect
155. D₁₇	C₄²×C₈	semidirect
185. C₄×C₈	C₂×#595	direct (N × H)
186. C₁₇	C₂×C₄²×C₈	semidirect
211. C₂×C₈	C₄×#595	direct (N × H)
241. C₂×C₄	C₈×#595	direct (N × H)
248. C₈	C₂×C₄×#595	direct (N × H)
262. C₄	C₂×C₈×#595	direct (N × H)
266. C₂	C₄×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
C₂×C₄×#63242	2
C₂²×C₈×#595	2
C₂×C₄²×#595	2
C₄×C₈×#595	2
C₂×C₄×C₈×D₁₇	2
C₂×C₄²×C₈	17

Subgroups — C2×C4×C8×#595 #17968120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₈×#595 #17968120