Subgroups — C₂³×D₄×#595 #17981033

Order 4352.

19998 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×#595	C₂	1	2	yes	maximal, complemented
2	C₂²×D₄×#595	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊#595	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₂²⋊#595	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×#595	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×D₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×#595	—	2	32	no	—
8	C₂⁴×#595	C₂²	1	1	yes	characteristic
9	C₂⁴×#595	C₂²	1	2	yes	—
10	C₂⁴×#595	C₂²	1	1	yes	characteristic
11	C₂⁴×#595	C₂²	1	56	yes	complemented
12	C₂×D₄×#595	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊#595	C₂²	1	28	yes	complemented
14	C₂²×C₂²⋊#595	C₂²	1	56	yes	complemented
15	C₂²×C₄×#595	C₂²	1	28	yes	complemented
16	C₂⁵×D₁₇	C₂²	1	2	yes	—
17	C₂³×D₆₈	C₄	1	1	yes	characteristic, complemented
18	C₂³×C₂²⋊D₁₇	C₄	1	2	yes	complemented
19	C₂²×D₄×D₁₇	C₄	1	28	yes	complemented
20	C₂²×D₄×D₁₇	C₂²	1	28	yes	—
21	C₂³×C₄×D₁₇	C₂²	1	1	yes	characteristic
22	C₂³×D₄×C₁₇	C₄	1	1	yes	characteristic, complemented
23	C₂⁵×C₁₇	C₂×C₄	1	2	yes	complemented
24	C₂⁴×D₁₇	—	2	16	no	—
25	C₂⁴×D₁₇	—	2	16	no	—
26	C₂⁴×D₁₇	C₂×C₄	1	28	yes	complemented
27	C₂⁴×D₁₇	C₂×C₄	1	2	yes	complemented
28	C₂⁴×D₁₇	C₂³	1	28	yes	—
29	C₂⁴×D₁₇	C₂³	1	1	yes	characteristic
30	C₂³×#595	—	2	16	no	—
31	C₂³×#595	—	2	8	no	—
32	C₂³×#595	—	2	448	no	—
33	C₂³×#595	D₄	1	16	yes	direct product factor
34	C₂³×#595	C₂³	1	14	yes	—
35	C₂³×#595	C₂³	1	28	yes	—
36	C₂³×#595	C₂³	1	14	yes	—
37	C₂³×#595	C₂³	1	224	yes	complemented
38	C₂²×D₄×C₁₇	C₂×C₄	1	28	yes	complemented
39	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic, complemented
40	C₂²×C₂²⋊D₁₇	C₂×C₄	1	56	yes	complemented
41	C₂³×Dic₁₇	C₂×C₄	1	1	yes	characteristic, complemented
42	C₂×D₄×D₁₇	C₂×C₄	1	336	yes	complemented
43	C₂×D₄×D₁₇	C₂³	1	112	yes	—
44	C₂²×D₆₈	C₂×C₄	1	28	yes	complemented
45	C₂²×C₄×D₁₇	C₂×C₄	1	14	yes	complemented
46	C₂²×C₄×D₁₇	C₂³	1	14	yes	—
47	C₂×C₂²⋊#595	C₂³	1	224	yes	complemented
48	D₄×#595	C₂³	1	512	yes	direct product factor
49	C₂×C₄⋊#595	C₂³	1	112	yes	complemented
50	C₂×C₄×#595	C₂³	1	112	yes	complemented
51	C₂⁴×C₁₇	—	2	16	no	—
52	C₂⁴×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂⁴×C₁₇	C₂²×C₄	1	28	yes	complemented
54	C₂³×D₁₇	—	2	16	no	—
55	C₂³×D₁₇	—	2	224	no	—
56	C₂³×D₁₇	—	2	112	no	—
57	C₂³×D₁₇	—	2	112	no	—
58	C₂³×D₁₇	C₂²×C₄	1	7	yes	—
59	C₂³×D₁₇	C₂²×C₄	1	168	yes	complemented
60	C₂³×D₁₇	C₂²×C₄	1	28	yes	complemented
61	C₂³×D₁₇	C₂×D₄	1	8	yes	—
62	C₂³×D₁₇	#41	1	8	yes	—
63	C₂³×D₁₇	C₂⁴	1	56	yes	—
64	C₂³×D₁₇	C₂⁴	1	7	yes	—
65	C₂²×#595	—	2	112	no	—
66	C₂²×#595	—	2	56	no	—
67	C₂²×#595	—	2	896	no	—
68	C₂²×#595	C₂×D₄	1	112	yes	direct product factor
69	C₂²×#595	C₂⁴	1	28	yes	—
70	C₂²×#595	C₂⁴	1	56	yes	—
71	C₂²×#595	C₂⁴	1	28	yes	—
72	C₂²×#595	C₂⁴	1	128	yes	complemented
73	C₂×D₄×C₁₇	C₂²×C₄	1	112	yes	complemented
74	C₂²×C₄×C₁₇	C₂²×C₄	1	14	yes	complemented
75	C₂×C₂²⋊D₁₇	C₂²×C₄	1	224	yes	complemented
76	C₂²×Dic₁₇	C₂²×C₄	1	14	yes	complemented
77	D₄×D₁₇	C₂²×C₄	1	448	yes	complemented
78	D₄×D₁₇	C₂⁴	1	64	yes	—
79	C₂×D₆₈	C₂²×C₄	1	112	yes	complemented
80	C₂×C₄×D₁₇	C₂²×C₄	1	84	yes	complemented
81	C₂×C₄×D₁₇	C₂⁴	1	28	yes	—
82	C₂²⋊#595	C₂⁴	1	128	yes	complemented
83	C₄⋊#595	C₂⁴	1	64	yes	complemented
84	C₄×#595	C₂⁴	1	64	yes	complemented
85	C₂³×C₄×D₄	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	—	2	112	no	—
87	C₂³×C₁₇	C₄×D₄	1	8	yes	complemented
88	C₂³×C₁₇	C₂³×C₄	1	7	yes	—
89	C₂³×C₁₇	C₂³×C₄	1	56	yes	complemented
90	C₂²×D₁₇	—	2	112	no	—
91	C₂²×D₁₇	—	2	672	no	—
92	C₂²×D₁₇	—	2	112	no	—
93	C₂²×D₁₇	—	2	112	no	—
94	C₂²×D₁₇	C₄×D₄	1	56	yes	complemented
95	C₂²×D₁₇	C₂³×C₄	1	21	yes	—
96	C₂²×D₁₇	C₂³×C₄	1	112	yes	complemented
97	C₂²×D₁₇	C₂³×C₄	1	56	yes	complemented
98	C₂²×D₁₇	C₂²×D₄	1	28	yes	—
99	C₂²×D₁₇	C₂×#41	1	28	yes	—
100	C₂²×D₁₇	C₂⁵	1	16	yes	—
101	C₂²×D₁₇	C₂⁵	1	7	yes	—
102	C₂×#595	—	2	112	no	—
103	C₂×#595	—	2	56	no	—
104	C₂×#595	—	2	256	no	—
105	C₂×#595	C₂²×D₄	1	112	yes	direct product factor
106	C₂×#595	C₂⁵	1	8	yes	—
107	C₂×#595	C₂⁵	1	16	yes	—
108	C₂×#595	C₂⁵	1	8	yes	—
109	D₄×C₁₇	C₂³×C₄	1	64	yes	complemented
110	C₂×C₄×C₁₇	C₂³×C₄	1	28	yes	complemented
111	C₂²⋊D₁₇	C₂³×C₄	1	128	yes	complemented
112	C₂×Dic₁₇	C₂³×C₄	1	28	yes	complemented
113	D₆₈	C₂³×C₄	1	64	yes	complemented
114	C₄×D₁₇	C₂³×C₄	1	56	yes	complemented
115	C₄×D₁₇	C₂⁵	1	8	yes	—
116	C₂⁴×D₄	—	17	1	no	—
117	C₂⁵×C₄	—	17	2	no	—
118	C₂²×C₄×D₄	—	17	56	no	—
119	C₂³×C₄⋊C₄	—	17	1	no	—
120	C₂³×C₂²⋊C₄	—	17	2	no	—
121	C₂³×C₄²	—	17	1	no	—
122	C₂²×C₁₇	—	2	112	no	—
123	C₂²×C₁₇	C₂×C₄×D₄	1	28	yes	complemented
124	C₂²×C₁₇	C₂⁴×C₄	1	7	yes	—
125	C₂²×C₁₇	C₂⁴×C₄	1	16	yes	complemented
126	C₂×D₁₇	—	2	112	no	—
127	C₂×D₁₇	—	2	16	no	—
128	C₂×D₁₇	—	2	16	no	—
129	C₂×D₁₇	—	2	224	no	—
130	C₂×D₁₇	C₂×C₄×D₄	1	84	yes	complemented
131	C₂×D₁₇	C₂⁴×C₄	1	7	yes	—
132	C₂×D₁₇	C₂⁴×C₄	1	16	yes	complemented
133	C₂×D₁₇	C₂³×D₄	1	14	yes	—
134	C₂×D₁₇	C₂²×#41	1	14	yes	—
135	C₂×D₁₇	C₂⁶	1	1	yes	characteristic
136	#595	—	2	16	no	—
137	#595	—	2	8	no	—
138	#595	C₂³×D₄	1	16	yes	direct product factor
139	C₄×C₁₇	C₂⁴×C₄	1	8	yes	complemented
140	Dic₁₇	C₂⁴×C₄	1	8	yes	complemented
141	C₂⁶	—	17	2	no	—
142	C₂³×D₄	—	17	2	no	—
143	C₂³×D₄	—	17	28	no	—
144	C₂³×D₄	—	17	28	no	—
145	C₂³×D₄	—	17	1	no	—
146	C₂³×D₄	#595	1	1	yes	characteristic, direct product factor
147	C₂⁴×C₄	—	17	1	no	—
148	C₂⁴×C₄	—	17	1	no	—
149	C₂⁴×C₄	—	17	2	no	—
150	C₂⁴×C₄	—	17	56	no	—
151	C₂⁴×C₄	—	17	1	no	—
152	C₂⁴×C₄	—	34	32	no	—
153	C₂×C₄×D₄	—	17	448	no	—
154	C₂²×C₄⋊C₄	—	17	28	no	—
155	C₂²×C₂²⋊C₄	—	17	56	no	—
156	C₂²×C₄²	—	17	28	no	—
157	C₂×C₁₇	—	2	16	no	—
158	C₂×C₁₇	C₂²×C₄×D₄	1	14	yes	complemented
159	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
160	D₁₇	—	2	16	no	—
161	D₁₇	C₂²×C₄×D₄	1	14	yes	complemented
162	D₁₇	C₂⁴×D₄	1	1	yes	characteristic
163	D₁₇	C₂³×#41	1	1	yes	characteristic
164	C₂⁵	—	17	1	no	—
165	C₂⁵	—	17	2	no	—
166	C₂⁵	—	17	28	no	—
167	C₂⁵	—	17	28	no	—
168	C₂⁵	—	34	16	no	—
169	C₂⁵	—	34	16	no	—
170	C₂⁵	C₂×#595	1	2	yes	complemented
171	C₂²×D₄	—	17	56	no	—
172	C₂²×D₄	—	17	112	no	—
173	C₂²×D₄	—	17	336	no	—
174	C₂²×D₄	—	17	28	no	—
175	C₂²×D₄	C₂×#595	1	28	yes	direct product factor
176	C₂³×C₄	—	17	16	no	—
177	C₂³×C₄	—	17	14	no	—
178	C₂³×C₄	—	17	1	no	—
179	C₂³×C₄	—	17	14	no	—
180	C₂³×C₄	—	17	14	no	—
181	C₂³×C₄	—	17	14	no	—
182	C₂³×C₄	—	17	28	no	—
183	C₂³×C₄	—	17	224	no	—
184	C₂³×C₄	—	34	448	no	—
185	C₂³×C₄	—	34	8	no	—
186	C₂³×C₄	—	34	16	no	—
187	C₂³×C₄	C₂×#595	1	1	yes	characteristic, complemented
188	C₄×D₄	—	17	512	no	—
189	C₂×C₄⋊C₄	—	17	112	no	—
190	C₂×C₂²⋊C₄	—	17	224	no	—
191	C₂×C₄²	—	17	112	no	—
192	C₁₇	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	—	2	16	no	—
194	C₂⁴	—	17	8	no	—
195	C₂⁴	—	17	8	no	—
196	C₂⁴	—	17	7	no	—
197	C₂⁴	—	17	7	no	—
198	C₂⁴	—	17	28	no	—
199	C₂⁴	—	17	168	no	—
200	C₂⁴	—	17	56	no	—
201	C₂⁴	—	34	112	no	—
202	C₂⁴	—	34	112	no	—
203	C₂⁴	—	34	224	no	—
204	C₂⁴	—	34	16	no	—
205	C₂⁴	C₂²×#595	1	1	yes	characteristic
206	C₂⁴	C₂²×#595	1	28	yes	complemented
207	C₂×D₄	—	17	224	no	—
208	C₂×D₄	—	17	448	no	—
209	C₂×D₄	—	17	64	no	—
210	C₂×D₄	—	17	112	no	—
211	C₂×D₄	C₂²×#595	1	112	yes	direct product factor
212	C₂²×C₄	—	17	112	no	—
213	C₂²×C₄	—	17	28	no	—
214	C₂²×C₄	—	17	14	no	—
215	C₂²×C₄	—	17	84	no	—
216	C₂²×C₄	—	17	28	no	—
217	C₂²×C₄	—	17	28	no	—
218	C₂²×C₄	—	17	56	no	—
219	C₂²×C₄	—	17	128	no	—
220	C₂²×C₄	—	34	896	no	—
221	C₂²×C₄	—	34	56	no	—
222	C₂²×C₄	—	34	112	no	—
223	C₂²×C₄	C₂²×#595	1	14	yes	complemented
224	C₄⋊C₄	—	17	64	no	—
225	C₂²⋊C₄	—	17	128	no	—
226	C₄²	—	17	64	no	—
227	C₂³	—	2	112	no	—
228	C₂³	—	17	28	no	—
229	C₂³	—	17	28	no	—
230	C₂³	—	17	56	no	—
231	C₂³	—	17	7	no	—
232	C₂³	—	17	21	no	—
233	C₂³	—	17	56	no	—
234	C₂³	—	17	16	no	—
235	C₂³	—	17	112	no	—
236	C₂³	—	34	112	no	—
237	C₂³	—	34	112	no	—
238	C₂³	—	34	672	no	—
239	C₂³	—	34	112	no	—
240	C₂³	D₄×#595	1	8	yes	direct product factor
241	C₂³	C₂³×#595	1	7	yes	—
242	C₂³	C₂³×#595	1	56	yes	complemented
243	D₄	—	17	128	no	—
244	D₄	—	17	64	no	—
245	D₄	C₂³×#595	1	64	yes	direct product factor
246	C₂×C₄	—	17	8	no	—
247	C₂×C₄	—	17	112	no	—
248	C₂×C₄	—	17	28	no	—
249	C₂×C₄	—	17	8	no	—
250	C₂×C₄	—	17	56	no	—
251	C₂×C₄	—	17	8	no	—
252	C₂×C₄	—	17	16	no	—
253	C₂×C₄	—	34	256	no	—
254	C₂×C₄	—	34	112	no	—
255	C₂×C₄	—	34	56	no	—
256	C₂×C₄	C₂³×#595	1	28	yes	complemented
257	C₂²	—	2	112	no	—
258	C₂²	—	17	14	no	—
259	C₂²	—	17	14	no	—
260	C₂²	—	17	84	no	—
261	C₂²	—	17	1	no	—
262	C₂²	—	17	7	no	—
263	C₂²	—	17	16	no	—
264	C₂²	—	34	16	no	—
265	C₂²	—	34	16	no	—
266	C₂²	—	34	224	no	—
267	C₂²	—	34	112	no	—
268	C₂²	C₂×D₄×#595	1	28	yes	direct product factor
269	C₂²	C₂⁴×#595	1	7	yes	—
270	C₂²	C₂⁴×#595	1	16	yes	complemented
271	C₄	—	17	16	no	—
272	C₄	—	17	8	no	—
273	C₄	—	34	16	no	—
274	C₄	—	34	8	no	—
275	C₄	C₂⁴×#595	1	8	yes	complemented
276	C₂	—	2	16	no	—
277	C₂	—	17	1	no	—
278	C₂	—	17	1	no	—
279	C₂	—	17	14	no	—
280	C₂	—	34	16	no	—
281	C₂	C₂²×D₄×#595	1	14	yes	direct product factor
282	C₂	C₂⁵×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×#595
commutator	C₂×C₁₇	C₂⁵×C₄
Frattini	C₂	C₂⁵×#595
Fitting	C₂³×D₄×C₁₇	C₄
socle	C₂⁴×C₁₇	C₂²×C₄
prime radical	C₂³×D₄×D₁₇	C₂
cosocle	C₂×D₁₇	C₂⁶
hypercenter	C₂³×D₄	#595
hypocenter	C₁₇	C₂³×C₄×D₄
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×#595	C₂	semidirect
2. C₂²×D₄×#595	C₂	direct (N × H)
3. C₂³×C₄⋊#595	C₂	semidirect
4. C₂³×C₂²⋊#595	C₂	semidirect
5. C₂³×C₄×#595	C₂	semidirect
11. C₂⁴×#595	C₂²	semidirect
12. C₂×D₄×#595	C₂²	direct (N × H)
13. C₂²×C₄⋊#595	C₂²	semidirect
14. C₂²×C₂²⋊#595	C₂²	semidirect
15. C₂²×C₄×#595	C₂²	semidirect
17. C₂³×D₆₈	C₄	semidirect
18. C₂³×C₂²⋊D₁₇	C₄	semidirect
19. C₂²×D₄×D₁₇	C₄	semidirect
22. C₂³×D₄×C₁₇	C₄	semidirect
23. C₂⁵×C₁₇	C₂×C₄	semidirect
26. C₂⁴×D₁₇	C₂×C₄	semidirect
27. C₂⁴×D₁₇	C₂×C₄	semidirect
33. C₂³×#595	D₄	direct (N × H)
37. C₂³×#595	C₂³	semidirect
38. C₂²×D₄×C₁₇	C₂×C₄	semidirect
39. C₂³×C₄×C₁₇	C₂×C₄	semidirect
40. C₂²×C₂²⋊D₁₇	C₂×C₄	semidirect
41. C₂³×Dic₁₇	C₂×C₄	semidirect
42. C₂×D₄×D₁₇	C₂×C₄	semidirect
44. C₂²×D₆₈	C₂×C₄	semidirect
45. C₂²×C₄×D₁₇	C₂×C₄	semidirect
47. C₂×C₂²⋊#595	C₂³	semidirect
48. D₄×#595	C₂³	direct (N × H)
49. C₂×C₄⋊#595	C₂³	semidirect
50. C₂×C₄×#595	C₂³	semidirect
53. C₂⁴×C₁₇	C₂²×C₄	semidirect
59. C₂³×D₁₇	C₂²×C₄	semidirect
60. C₂³×D₁₇	C₂²×C₄	semidirect
68. C₂²×#595	C₂×D₄	direct (N × H)
72. C₂²×#595	C₂⁴	semidirect
73. C₂×D₄×C₁₇	C₂²×C₄	semidirect
74. C₂²×C₄×C₁₇	C₂²×C₄	semidirect
75. C₂×C₂²⋊D₁₇	C₂²×C₄	semidirect
76. C₂²×Dic₁₇	C₂²×C₄	semidirect
77. D₄×D₁₇	C₂²×C₄	semidirect
79. C₂×D₆₈	C₂²×C₄	semidirect
80. C₂×C₄×D₁₇	C₂²×C₄	semidirect
82. C₂²⋊#595	C₂⁴	semidirect
83. C₄⋊#595	C₂⁴	semidirect
84. C₄×#595	C₂⁴	semidirect
87. C₂³×C₁₇	C₄×D₄	semidirect
89. C₂³×C₁₇	C₂³×C₄	semidirect
94. C₂²×D₁₇	C₄×D₄	semidirect
96. C₂²×D₁₇	C₂³×C₄	semidirect
97. C₂²×D₁₇	C₂³×C₄	semidirect
105. C₂×#595	C₂²×D₄	direct (N × H)
109. D₄×C₁₇	C₂³×C₄	semidirect
110. C₂×C₄×C₁₇	C₂³×C₄	semidirect
111. C₂²⋊D₁₇	C₂³×C₄	semidirect
112. C₂×Dic₁₇	C₂³×C₄	semidirect
113. D₆₈	C₂³×C₄	semidirect
114. C₄×D₁₇	C₂³×C₄	semidirect
123. C₂²×C₁₇	C₂×C₄×D₄	semidirect
125. C₂²×C₁₇	C₂⁴×C₄	semidirect
130. C₂×D₁₇	C₂×C₄×D₄	semidirect
132. C₂×D₁₇	C₂⁴×C₄	semidirect
138. #595	C₂³×D₄	direct (N × H)
139. C₄×C₁₇	C₂⁴×C₄	semidirect
140. Dic₁₇	C₂⁴×C₄	semidirect
146. C₂³×D₄	#595	direct (N × H)
158. C₂×C₁₇	C₂²×C₄×D₄	semidirect
161. D₁₇	C₂²×C₄×D₄	semidirect
170. C₂⁵	C₂×#595	semidirect
175. C₂²×D₄	C₂×#595	direct (N × H)
187. C₂³×C₄	C₂×#595	semidirect
192. C₁₇	C₂³×C₄×D₄	semidirect
206. C₂⁴	C₂²×#595	semidirect
211. C₂×D₄	C₂²×#595	direct (N × H)
223. C₂²×C₄	C₂²×#595	semidirect
240. C₂³	D₄×#595	direct (N × H)
242. C₂³	C₂³×#595	semidirect
245. D₄	C₂³×#595	direct (N × H)
256. C₂×C₄	C₂³×#595	semidirect
268. C₂²	C₂×D₄×#595	direct (N × H)
270. C₂²	C₂⁴×#595	semidirect
275. C₄	C₂⁴×#595	semidirect
281. C₂	C₂²×D₄×#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
C₂⁵×#595	2
C₂²×D₄×#595	2
C₂³×C₄⋊#595	2
C₂³×C₂²⋊#595	2
C₂³×C₄×#595	2
C₂³×D₄×D₁₇	2
C₂³×C₄×D₄	17

Subgroups — C23×D4×#595 #17981033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₄×#595 #17981033