Subgroups — C₂³×#10587580 #17940974

Order 4352.

15150 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#10587580	C₂	1	56	yes	maximal, direct product factor
4	C₂³×#63225	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#63227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#63227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#63227	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₁₇	C₂²	1	1	yes	characteristic
10	C₂⁵×D₁₇	C₂²	1	1	yes	characteristic
11	C₂³×D₆₈	—	2	16	no	—
12	C₂³×D₆₈	C₂²	1	28	yes	—
13	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂³×C₄×D₁₇	—	2	16	no	—
16	C₂³×C₄×D₁₇	C₂²	1	28	yes	complemented
17	C₂×#10587580	C₂²	1	448	yes	direct product factor
18	C₂²×#63225	C₂²	1	28	yes	complemented
19	C₂²×#63227	C₂²	1	28	yes	complemented
20	C₂²×#63227	C₂²	1	28	yes	complemented
21	C₂²×#63227	C₂²	1	28	yes	complemented
22	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
23	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
24	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₁₇	C₂²	1	28	yes	complemented
26	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
27	C₂⁴×D₁₇	—	2	8	no	—
28	C₂⁴×D₁₇	—	2	8	no	—
29	C₂⁴×D₁₇	—	2	8	no	—
30	C₂⁴×D₁₇	—	2	8	no	—
31	C₂⁴×D₁₇	—	2	8	no	—
32	C₂⁴×D₁₇	D₄	1	16	yes	—
33	C₂⁴×D₁₇	C₂³	1	14	yes	—
34	C₂⁴×D₁₇	C₂³	1	14	yes	—
35	C₂³×C₄×C₁₇	—	2	8	no	—
36	C₂³×C₄×C₁₇	—	2	8	no	—
37	C₂³×C₄×C₁₇	—	2	8	no	—
38	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
39	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
40	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
41	C₂³×Dic₁₇	—	2	8	no	—
42	C₂³×Dic₁₇	—	2	8	no	—
43	C₂³×Dic₁₇	C₂³	1	14	yes	—
44	C₂³×Dic₁₇	C₂³	1	14	yes	—
45	C₂²×D₆₈	—	2	224	no	—
46	C₂²×D₆₈	C₂³	1	112	yes	—
47	C₂²×C₄×D₁₇	—	2	224	no	—
48	C₂²×C₄×D₁₇	C₂³	1	112	yes	complemented
49	C₂×C₄⋊C₄×C₁₇	C₂³	1	112	yes	complemented
50	C₂×#63227	C₂³	1	112	yes	complemented
51	C₂×#63227	C₂³	1	112	yes	complemented
52	C₂×#63227	C₂³	1	112	yes	complemented
53	C₂×#63225	C₂³	1	112	yes	complemented
54	#10587580	C₂³	1	512	yes	direct product factor
55	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
56	C₂⁴×C₁₇	#41	1	8	yes	—
57	C₂⁴×C₁₇	#41	1	8	yes	—
58	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
59	C₂³×D₁₇	—	2	128	no	—
60	C₂³×D₁₇	—	2	56	no	—
61	C₂³×D₁₇	—	2	56	no	—
62	C₂³×D₁₇	—	2	56	no	—
63	C₂³×D₁₇	—	2	56	no	—
64	C₂³×D₁₇	—	2	56	no	—
65	C₂³×D₁₇	—	4	64	no	—
66	C₂³×D₁₇	C₂×D₄	1	112	yes	—
67	C₂³×D₁₇	C₂⁴	1	28	yes	—
68	C₂³×D₁₇	C₂⁴	1	28	yes	—
69	C₂²×C₄×C₁₇	—	2	56	no	—
70	C₂²×C₄×C₁₇	—	2	56	no	—
71	C₂²×C₄×C₁₇	—	2	56	no	—
72	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
73	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
74	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
75	C₂²×Dic₁₇	—	2	56	no	—
76	C₂²×Dic₁₇	—	2	56	no	—
77	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
78	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
79	C₂×D₆₈	—	2	448	no	—
80	C₂×D₆₈	C₂⁴	1	64	yes	—
81	C₂×C₄×D₁₇	—	2	448	no	—
82	C₂×C₄×D₁₇	C₂⁴	1	64	yes	complemented
83	C₄⋊C₄×C₁₇	C₂⁴	1	64	yes	complemented
84	#63227	C₂⁴	1	64	yes	complemented
85	#63227	C₂⁴	1	64	yes	complemented
86	#63227	C₂⁴	1	64	yes	complemented
87	#63225	C₂⁴	1	64	yes	complemented
88	C₂³×#123	—	17	1	no	maximal, Sylow
89	C₂³×C₁₇	#123	1	64	yes	complemented
90	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
91	C₂³×C₁₇	C₂×#41	1	28	yes	—
92	C₂³×C₁₇	C₂×#41	1	28	yes	—
93	C₂³×C₁₇	C₂⁵	1	7	yes	—
94	C₂²×D₁₇	—	2	448	no	—
95	C₂²×D₁₇	—	2	56	no	—
96	C₂²×D₁₇	—	2	56	no	—
97	C₂²×D₁₇	—	2	56	no	—
98	C₂²×D₁₇	—	2	56	no	—
99	C₂²×D₁₇	—	2	56	no	—
100	C₂²×D₁₇	—	4	224	no	—
101	C₂²×D₁₇	C₂²×D₄	1	112	yes	—
102	C₂²×D₁₇	C₂⁵	1	8	yes	—
103	C₂²×D₁₇	C₂⁵	1	8	yes	—
104	C₂×C₄×C₁₇	—	2	56	no	—
105	C₂×C₄×C₁₇	—	2	56	no	—
106	C₂×C₄×C₁₇	—	2	56	no	—
107	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
108	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
109	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
110	C₂×Dic₁₇	—	2	56	no	—
111	C₂×Dic₁₇	—	2	56	no	—
112	C₂×Dic₁₇	C₂⁵	1	8	yes	—
113	C₂×Dic₁₇	C₂⁵	1	8	yes	—
114	D₆₈	—	2	128	no	—
115	C₄×D₁₇	—	2	128	no	—
116	C₂⁴×D₄	—	17	1	no	—
117	C₂⁵×C₄	—	17	1	no	—
118	C₂²×#123	—	17	56	no	—
119	C₂³×C₄⋊C₄	—	17	1	no	—
120	C₂³×C₄⋊C₄	D₁₇	1	1	yes	characteristic, complemented
121	C₂³×C₂²⋊C₄	—	17	1	no	—
122	C₂³×C₂²⋊C₄	—	17	1	no	—
123	C₂³×C₂²⋊C₄	—	17	1	no	—
124	C₂²×C₁₇	C₂×#123	1	112	yes	complemented
125	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
126	C₂²×C₁₇	C₂²×#41	1	14	yes	—
127	C₂²×C₁₇	C₂²×#41	1	14	yes	—
128	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
129	C₂×D₁₇	—	2	8	no	—
130	C₂×D₁₇	—	2	8	no	—
131	C₂×D₁₇	—	2	224	no	—
132	C₂×D₁₇	—	2	8	no	—
133	C₂×D₁₇	—	2	8	no	—
134	C₂×D₁₇	—	2	8	no	—
135	C₂×D₁₇	—	4	112	no	—
136	C₂×D₁₇	C₂³×D₄	1	16	yes	—
137	C₄×C₁₇	—	2	8	no	—
138	C₄×C₁₇	—	2	8	no	—
139	C₄×C₁₇	—	2	8	no	—
140	Dic₁₇	—	2	8	no	—
141	Dic₁₇	—	2	8	no	—
142	C₂⁶	—	17	1	no	—
143	C₂⁶	—	17	1	no	—
144	C₂³×D₄	—	17	28	no	—
145	C₂³×D₄	—	34	16	no	—
146	C₂⁴×C₄	—	17	1	no	—
147	C₂⁴×C₄	—	17	1	no	—
148	C₂⁴×C₄	—	17	28	no	—
149	C₂⁴×C₄	—	34	16	no	—
150	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
151	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
152	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
153	C₂×#123	—	17	448	no	—
154	C₂²×C₄⋊C₄	—	17	28	no	—
155	C₂²×C₄⋊C₄	C₂×D₁₇	1	28	yes	complemented
156	C₂²×C₂²⋊C₄	—	17	28	no	—
157	C₂²×C₂²⋊C₄	—	17	28	no	—
158	C₂²×C₂²⋊C₄	—	17	28	no	—
159	C₂×C₁₇	C₂²×#123	1	28	yes	complemented
160	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
161	C₂×C₁₇	C₂³×#41	1	1	yes	characteristic
162	C₂×C₁₇	C₂³×#41	1	1	yes	characteristic
163	D₁₇	—	2	16	no	—
164	D₁₇	—	4	8	no	—
165	C₂⁵	—	17	16	no	—
166	C₂⁵	—	17	14	no	—
167	C₂⁵	—	17	14	no	—
168	C₂⁵	—	34	8	no	—
169	C₂⁵	—	34	8	no	—
170	C₂⁵	—	34	8	no	—
171	C₂⁵	—	34	8	no	—
172	C₂⁵	—	34	8	no	—
173	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
174	C₂²×D₄	—	17	112	no	—
175	C₂²×D₄	—	34	224	no	—
176	C₂³×C₄	—	2	8	no	—
177	C₂³×C₄	—	2	8	no	—
178	C₂³×C₄	—	2	8	no	—
179	C₂³×C₄	—	17	14	no	—
180	C₂³×C₄	—	17	14	no	—
181	C₂³×C₄	—	17	112	no	—
182	C₂³×C₄	—	34	224	no	—
183	C₂³×C₄	—	34	8	no	—
184	C₂³×C₄	—	34	8	no	—
185	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
186	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
187	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
188	#123	—	17	512	no	—
189	C₂×C₄⋊C₄	—	17	112	no	—
190	C₂×C₄⋊C₄	C₂²×D₁₇	1	112	yes	complemented
191	C₂×C₂²⋊C₄	—	17	112	no	—
192	C₂×C₂²⋊C₄	—	17	112	no	—
193	C₂×C₂²⋊C₄	—	17	112	no	—
194	C₁₇	C₂³×#123	1	1	yes	characteristic, complemented, Sylow
195	C₂⁴	—	17	112	no	—
196	C₂⁴	—	17	28	no	—
197	C₂⁴	—	17	28	no	—
198	C₂⁴	—	34	56	no	—
199	C₂⁴	—	34	56	no	—
200	C₂⁴	—	34	128	no	—
201	C₂⁴	—	34	56	no	—
202	C₂⁴	—	34	56	no	—
203	C₂⁴	—	34	56	no	—
204	C₂⁴	—	68	64	no	—
205	C₂⁴	#63252	1	8	yes	—
206	C₂⁴	D₄×D₁₇	1	8	yes	—
207	C₂⁴	#63256	1	8	yes	—
208	C₂⁴	C₂³×D₁₇	1	7	yes	—
209	C₂×D₄	—	17	64	no	—
210	C₂×D₄	—	34	448	no	—
211	C₂²×C₄	—	2	56	no	—
212	C₂²×C₄	—	2	56	no	—
213	C₂²×C₄	—	2	56	no	—
214	C₂²×C₄	—	17	28	no	—
215	C₂²×C₄	—	17	28	no	—
216	C₂²×C₄	—	17	64	no	—
217	C₂²×C₄	—	34	448	no	—
218	C₂²×C₄	—	34	56	no	—
219	C₂²×C₄	—	34	56	no	—
220	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
221	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
222	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
223	C₄⋊C₄	—	17	64	no	—
224	C₄⋊C₄	C₂³×D₁₇	1	64	yes	complemented
225	C₂²⋊C₄	—	17	64	no	—
226	C₂²⋊C₄	—	17	64	no	—
227	C₂²⋊C₄	—	17	64	no	—
228	C₂³	—	17	112	no	—
229	C₂³	—	17	8	no	—
230	C₂³	—	17	8	no	—
231	C₂³	—	34	448	no	—
232	C₂³	—	34	56	no	—
233	C₂³	—	34	56	no	—
234	C₂³	—	34	56	no	—
235	C₂³	—	34	56	no	—
236	C₂³	—	34	56	no	—
237	C₂³	—	68	224	no	—
238	C₂³	#10587580	1	64	yes	direct product factor
239	C₂³	C₂×#63252	1	28	yes	—
240	C₂³	C₂×D₄×D₁₇	1	28	yes	—
241	C₂³	C₂×#63256	1	28	yes	—
242	C₂³	C₂⁴×D₁₇	1	7	yes	—
243	D₄	—	34	128	no	—
244	C₂×C₄	—	2	56	no	—
245	C₂×C₄	—	2	56	no	—
246	C₂×C₄	—	2	56	no	—
247	C₂×C₄	—	17	8	no	—
248	C₂×C₄	—	17	8	no	—
249	C₂×C₄	—	34	128	no	—
250	C₂×C₄	—	34	56	no	—
251	C₂×C₄	—	34	56	no	—
252	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
253	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
254	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
255	C₂²	—	17	16	no	—
256	C₂²	—	34	224	no	—
257	C₂²	—	34	8	no	—
258	C₂²	—	34	8	no	—
259	C₂²	—	34	8	no	—
260	C₂²	—	34	8	no	—
261	C₂²	—	34	8	no	—
262	C₂²	—	68	112	no	—
263	C₂²	C₂×#10587580	1	112	yes	direct product factor
264	C₂²	C₂²×D₄×D₁₇	1	14	yes	—
265	C₂²	C₂²×#63252	1	14	yes	—
266	C₂²	C₂²×#63256	1	14	yes	—
267	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
268	C₄	—	2	8	no	—
269	C₄	—	2	8	no	—
270	C₄	—	2	8	no	—
271	C₄	—	34	8	no	—
272	C₄	—	34	8	no	—
273	C₂	—	34	16	no	—
274	C₂	—	68	8	no	—
275	C₂	C₂²×#10587580	1	28	yes	direct product factor
276	C₂	C₂³×D₄×D₁₇	1	1	yes	characteristic
277	C₂	C₂³×#63252	1	1	yes	characteristic
278	C₂	C₂³×#63256	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₇
commutator, cosocle	C₂²×C₁₇	C₂⁶
Frattini	C₂²	C₂⁵×D₁₇
Fitting	C₂³×C₄⋊C₄×C₁₇	C₂
socle	C₂⁵×C₁₇	C₂³
prime radical	C₂⁴×D₆₈	C₂
hypercenter	C₂³×C₄⋊C₄	D₁₇
hypocenter	C₁₇	C₂³×#123
LCS	G ▷ C₂²×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. C₂⁴×C₄×D₁₇	C₂	semidirect
3. C₂²×#10587580	C₂	direct (N × H)
4. C₂³×#63225	C₂	semidirect
5. C₂³×#63227	C₂	semidirect
6. C₂³×#63227	C₂	semidirect
7. C₂³×#63227	C₂	semidirect
8. C₂³×C₄⋊C₄×C₁₇	C₂	semidirect
16. C₂³×C₄×D₁₇	C₂²	semidirect
17. C₂×#10587580	C₂²	direct (N × H)
18. C₂²×#63225	C₂²	semidirect
19. C₂²×#63227	C₂²	semidirect
20. C₂²×#63227	C₂²	semidirect
21. C₂²×#63227	C₂²	semidirect
25. C₂²×C₄⋊C₄×C₁₇	C₂²	semidirect
48. C₂²×C₄×D₁₇	C₂³	semidirect
49. C₂×C₄⋊C₄×C₁₇	C₂³	semidirect
50. C₂×#63227	C₂³	semidirect
51. C₂×#63227	C₂³	semidirect
52. C₂×#63227	C₂³	semidirect
53. C₂×#63225	C₂³	semidirect
54. #10587580	C₂³	direct (N × H)
82. C₂×C₄×D₁₇	C₂⁴	semidirect
83. C₄⋊C₄×C₁₇	C₂⁴	semidirect
84. #63227	C₂⁴	semidirect
85. #63227	C₂⁴	semidirect
86. #63227	C₂⁴	semidirect
87. #63225	C₂⁴	semidirect
89. C₂³×C₁₇	#123	semidirect
120. C₂³×C₄⋊C₄	D₁₇	semidirect
124. C₂²×C₁₇	C₂×#123	semidirect
155. C₂²×C₄⋊C₄	C₂×D₁₇	semidirect
159. C₂×C₁₇	C₂²×#123	semidirect
190. C₂×C₄⋊C₄	C₂²×D₁₇	semidirect
194. C₁₇	C₂³×#123	semidirect
224. C₄⋊C₄	C₂³×D₁₇	semidirect
238. C₂³	#10587580	direct (N × H)
263. C₂²	C₂×#10587580	direct (N × H)
275. C₂	C₂²×#10587580	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×D₆₈	2
C₂⁴×C₄×D₁₇	2
C₂²×#10587580	2
C₂³×#63225	2
C₂³×#63227	2
C₂³×C₄⋊C₄×C₁₇	2
C₂³×#123	17

Subgroups — C23×#10587580 #17940974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587580 #17940974