Subgroups — C₂³×#10587581 #17940962

Order 4352.

19998 conjugacy classes of subgroups.

The subgroup lattice has 240 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	2	yes	maximal, complemented
2	C₂⁴×D₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₄×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#10587581	C₂	1	56	yes	maximal, direct product factor
5	C₂³×#63227	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₇	C₂²	1	2	yes	—
8	C₂⁵×D₁₇	C₂²	1	1	yes	characteristic
9	C₂³×D₆₈	—	2	32	no	—
10	C₂³×D₆₈	—	2	16	no	—
11	C₂³×D₆₈	C₂²	1	56	yes	complemented
12	C₂³×D₆₈	C₂²	1	28	yes	complemented
13	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×D₁₇	—	2	16	no	—
15	C₂³×C₄×D₁₇	C₂²	1	28	yes	complemented
16	C₂×#10587581	C₂²	1	448	yes	direct product factor
17	C₂²×#63227	C₂²	1	56	yes	complemented
18	C₂⁴×C₄×C₁₇	C₂²	1	2	yes	—
19	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₁₇	C₂²	1	28	yes	complemented
21	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
22	C₂⁴×D₁₇	—	2	8	no	—
23	C₂⁴×D₁₇	—	2	8	no	—
24	C₂⁴×D₁₇	—	2	16	no	—
25	C₂⁴×D₁₇	—	2	16	no	—
26	C₂⁴×D₁₇	—	2	16	no	—
27	C₂⁴×D₁₇	D₄	1	16	yes	complemented
28	C₂⁴×D₁₇	C₂³	1	28	yes	—
29	C₂⁴×D₁₇	C₂³	1	14	yes	—
30	C₂³×C₄×C₁₇	—	2	16	no	—
31	C₂³×C₄×C₁₇	D₄	1	16	yes	complemented
32	C₂³×C₄×C₁₇	C₂³	1	28	yes	—
33	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
34	C₂³×Dic₁₇	—	2	8	no	—
35	C₂³×Dic₁₇	C₂³	1	14	yes	—
36	C₂²×D₆₈	—	2	448	no	—
37	C₂²×D₆₈	—	2	224	no	—
38	C₂²×D₆₈	C₂³	1	224	yes	complemented
39	C₂²×D₆₈	C₂³	1	112	yes	complemented
40	C₂²×C₄×D₁₇	—	2	224	no	—
41	C₂²×C₄×D₁₇	C₂³	1	112	yes	complemented
42	C₂×C₄⋊C₄×C₁₇	C₂³	1	112	yes	complemented
43	C₂×#63227	C₂³	1	224	yes	complemented
44	#10587581	C₂³	1	512	yes	direct product factor
45	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
46	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
47	C₂⁴×C₁₇	#41	1	8	yes	—
48	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
49	C₂³×D₁₇	—	2	128	no	—
50	C₂³×D₁₇	—	2	56	no	—
51	C₂³×D₁₇	—	2	56	no	—
52	C₂³×D₁₇	—	2	112	no	—
53	C₂³×D₁₇	—	2	112	no	—
54	C₂³×D₁₇	—	2	112	no	—
55	C₂³×D₁₇	—	4	128	no	—
56	C₂³×D₁₇	C₂×D₄	1	112	yes	complemented
57	C₂³×D₁₇	C₂⁴	1	56	yes	—
58	C₂³×D₁₇	C₂⁴	1	28	yes	—
59	C₂²×C₄×C₁₇	—	2	112	no	—
60	C₂²×C₄×C₁₇	C₂×D₄	1	112	yes	complemented
61	C₂²×C₄×C₁₇	C₂⁴	1	56	yes	—
62	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
63	C₂²×Dic₁₇	—	2	56	no	—
64	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
65	C₂×D₆₈	—	2	896	no	—
66	C₂×D₆₈	—	2	448	no	—
67	C₂×D₆₈	C₂⁴	1	128	yes	complemented
68	C₂×D₆₈	C₂⁴	1	64	yes	complemented
69	C₂×C₄×D₁₇	—	2	448	no	—
70	C₂×C₄×D₁₇	C₂⁴	1	64	yes	complemented
71	C₄⋊C₄×C₁₇	C₂⁴	1	64	yes	complemented
72	#63227	C₂⁴	1	128	yes	complemented
73	C₂³×#121	—	17	1	no	maximal, Sylow
74	C₂³×C₁₇	#121	1	64	yes	complemented
75	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
76	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
77	C₂³×C₁₇	C₂×#41	1	28	yes	—
78	C₂³×C₁₇	C₂⁵	1	7	yes	—
79	C₂²×D₁₇	—	2	448	no	—
80	C₂²×D₁₇	—	2	56	no	—
81	C₂²×D₁₇	—	2	56	no	—
82	C₂²×D₁₇	—	2	112	no	—
83	C₂²×D₁₇	—	2	112	no	—
84	C₂²×D₁₇	—	2	112	no	—
85	C₂²×D₁₇	—	4	448	no	—
86	C₂²×D₁₇	C₂²×D₄	1	112	yes	complemented
87	C₂²×D₁₇	C₂⁵	1	16	yes	—
88	C₂²×D₁₇	C₂⁵	1	8	yes	—
89	C₂×C₄×C₁₇	—	2	112	no	—
90	C₂×C₄×C₁₇	C₂²×D₄	1	112	yes	complemented
91	C₂×C₄×C₁₇	C₂⁵	1	16	yes	—
92	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
93	C₂×Dic₁₇	—	2	56	no	—
94	C₂×Dic₁₇	C₂⁵	1	8	yes	—
95	D₆₈	—	2	256	no	—
96	D₆₈	—	2	128	no	—
97	C₄×D₁₇	—	2	128	no	—
98	C₂⁴×D₄	—	17	2	no	—
99	C₂⁴×D₄	—	17	1	no	—
100	C₂⁵×C₄	—	17	1	no	—
101	C₂²×#121	—	17	56	no	—
102	C₂³×C₄⋊C₄	D₁₇	1	1	yes	characteristic, complemented
103	C₂³×C₂²⋊C₄	—	17	2	no	—
104	C₂²×C₁₇	C₂×#121	1	112	yes	complemented
105	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
106	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
107	C₂²×C₁₇	C₂²×#41	1	14	yes	—
108	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
109	C₂×D₁₇	—	2	8	no	—
110	C₂×D₁₇	—	2	8	no	—
111	C₂×D₁₇	—	2	224	no	—
112	C₂×D₁₇	—	2	16	no	—
113	C₂×D₁₇	—	2	16	no	—
114	C₂×D₁₇	—	2	16	no	—
115	C₂×D₁₇	—	4	224	no	—
116	C₂×D₁₇	C₂³×D₄	1	16	yes	complemented
117	C₄×C₁₇	—	2	16	no	—
118	C₄×C₁₇	C₂³×D₄	1	16	yes	complemented
119	Dic₁₇	—	2	8	no	—
120	C₂⁶	—	17	1	no	—
121	C₂⁶	—	17	2	no	—
122	C₂³×D₄	—	17	56	no	—
123	C₂³×D₄	—	17	28	no	—
124	C₂³×D₄	—	34	32	no	—
125	C₂³×D₄	—	34	16	no	—
126	C₂⁴×C₄	—	17	1	no	—
127	C₂⁴×C₄	—	17	28	no	—
128	C₂⁴×C₄	—	34	16	no	—
129	C₂⁴×C₄	C₂×D₁₇	1	2	yes	—
130	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
131	C₂×#121	—	17	448	no	—
132	C₂²×C₄⋊C₄	C₂×D₁₇	1	28	yes	complemented
133	C₂²×C₂²⋊C₄	—	17	56	no	—
134	C₂×C₁₇	C₂²×#121	1	28	yes	complemented
135	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
136	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
137	C₂×C₁₇	C₂³×#41	1	1	yes	characteristic
138	D₁₇	—	2	16	no	—
139	D₁₇	—	4	16	no	—
140	C₂⁵	—	17	16	no	—
141	C₂⁵	—	17	14	no	—
142	C₂⁵	—	17	28	no	—
143	C₂⁵	—	34	8	no	—
144	C₂⁵	—	34	8	no	—
145	C₂⁵	—	34	16	no	—
146	C₂⁵	—	34	16	no	—
147	C₂⁵	—	34	16	no	—
148	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
149	C₂²×D₄	—	17	224	no	—
150	C₂²×D₄	—	17	112	no	—
151	C₂²×D₄	—	34	448	no	—
152	C₂²×D₄	—	34	224	no	—
153	C₂³×C₄	—	2	16	no	—
154	C₂³×C₄	—	17	14	no	—
155	C₂³×C₄	—	17	112	no	—
156	C₂³×C₄	—	34	8	no	—
157	C₂³×C₄	—	34	224	no	—
158	C₂³×C₄	D₆₈	1	16	yes	complemented
159	C₂³×C₄	C₂²×D₁₇	1	28	yes	—
160	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
161	#121	—	17	512	no	—
162	C₂×C₄⋊C₄	C₂²×D₁₇	1	112	yes	complemented
163	C₂×C₂²⋊C₄	—	17	224	no	—
164	C₁₇	C₂³×#121	1	1	yes	characteristic, complemented, Sylow
165	C₂⁴	—	17	112	no	—
166	C₂⁴	—	17	28	no	—
167	C₂⁴	—	17	56	no	—
168	C₂⁴	—	34	56	no	—
169	C₂⁴	—	34	56	no	—
170	C₂⁴	—	34	128	no	—
171	C₂⁴	—	34	112	no	—
172	C₂⁴	—	34	112	no	—
173	C₂⁴	—	34	112	no	—
174	C₂⁴	—	68	128	no	—
175	C₂⁴	C₂×D₆₈	1	8	yes	—
176	C₂⁴	D₄×D₁₇	1	8	yes	—
177	C₂⁴	#63256	1	8	yes	—
178	C₂⁴	C₂³×D₁₇	1	7	yes	—
179	C₂×D₄	—	17	128	no	—
180	C₂×D₄	—	17	64	no	—
181	C₂×D₄	—	34	896	no	—
182	C₂×D₄	—	34	448	no	—
183	C₂²×C₄	—	2	112	no	—
184	C₂²×C₄	—	17	28	no	—
185	C₂²×C₄	—	17	64	no	—
186	C₂²×C₄	—	34	56	no	—
187	C₂²×C₄	—	34	448	no	—
188	C₂²×C₄	C₂×D₆₈	1	112	yes	complemented
189	C₂²×C₄	C₂³×D₁₇	1	56	yes	—
190	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
191	C₄⋊C₄	C₂³×D₁₇	1	64	yes	complemented
192	C₂²⋊C₄	—	17	128	no	—
193	C₂³	—	17	112	no	—
194	C₂³	—	17	8	no	—
195	C₂³	—	17	16	no	—
196	C₂³	—	34	448	no	—
197	C₂³	—	34	56	no	—
198	C₂³	—	34	56	no	—
199	C₂³	—	34	112	no	—
200	C₂³	—	34	112	no	—
201	C₂³	—	34	112	no	—
202	C₂³	—	68	448	no	—
203	C₂³	#10587581	1	64	yes	direct product factor
204	C₂³	C₂²×D₆₈	1	28	yes	—
205	C₂³	C₂×D₄×D₁₇	1	28	yes	—
206	C₂³	C₂×#63256	1	28	yes	—
207	C₂³	C₂⁴×D₁₇	1	7	yes	—
208	D₄	—	34	256	no	—
209	D₄	—	34	128	no	—
210	C₂×C₄	—	2	112	no	—
211	C₂×C₄	—	17	8	no	—
212	C₂×C₄	—	34	56	no	—
213	C₂×C₄	—	34	128	no	—
214	C₂×C₄	C₂²×D₆₈	1	112	yes	complemented
215	C₂×C₄	C₂⁴×D₁₇	1	16	yes	—
216	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
217	C₂²	—	17	16	no	—
218	C₂²	—	34	224	no	—
219	C₂²	—	34	8	no	—
220	C₂²	—	34	8	no	—
221	C₂²	—	34	16	no	—
222	C₂²	—	34	16	no	—
223	C₂²	—	34	16	no	—
224	C₂²	—	68	224	no	—
225	C₂²	C₂×#10587581	1	112	yes	direct product factor
226	C₂²	C₂²×D₄×D₁₇	1	14	yes	—
227	C₂²	C₂³×D₆₈	1	14	yes	—
228	C₂²	C₂²×#63256	1	14	yes	—
229	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
230	C₄	—	2	16	no	—
231	C₄	—	34	8	no	—
232	C₄	C₂³×D₆₈	1	16	yes	complemented
233	C₂	—	34	16	no	—
234	C₂	—	68	16	no	—
235	C₂	C₂²×#10587581	1	28	yes	direct product factor
236	C₂	C₂³×D₄×D₁₇	1	1	yes	characteristic
237	C₂	C₂⁴×D₆₈	1	1	yes	characteristic
238	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
hypercenter	C₂³×C₄⋊C₄	D₁₇
hypocenter	C₁₇	C₂³×#121
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
1. C₂⁴×D₆₈	C₂	semidirect
2. C₂⁴×D₆₈	C₂	semidirect
3. C₂⁴×C₄×D₁₇	C₂	semidirect
4. C₂²×#10587581	C₂	direct (N × H)
5. C₂³×#63227	C₂	semidirect
6. C₂³×C₄⋊C₄×C₁₇	C₂	semidirect
11. C₂³×D₆₈	C₂²	semidirect
12. C₂³×D₆₈	C₂²	semidirect
15. C₂³×C₄×D₁₇	C₂²	semidirect
16. C₂×#10587581	C₂²	direct (N × H)
17. C₂²×#63227	C₂²	semidirect
20. C₂²×C₄⋊C₄×C₁₇	C₂²	semidirect
27. C₂⁴×D₁₇	D₄	semidirect
31. C₂³×C₄×C₁₇	D₄	semidirect
38. C₂²×D₆₈	C₂³	semidirect
39. C₂²×D₆₈	C₂³	semidirect
41. C₂²×C₄×D₁₇	C₂³	semidirect
42. C₂×C₄⋊C₄×C₁₇	C₂³	semidirect
43. C₂×#63227	C₂³	semidirect
44. #10587581	C₂³	direct (N × H)
56. C₂³×D₁₇	C₂×D₄	semidirect
60. C₂²×C₄×C₁₇	C₂×D₄	semidirect
67. C₂×D₆₈	C₂⁴	semidirect
68. C₂×D₆₈	C₂⁴	semidirect
70. C₂×C₄×D₁₇	C₂⁴	semidirect
71. C₄⋊C₄×C₁₇	C₂⁴	semidirect
72. #63227	C₂⁴	semidirect
74. C₂³×C₁₇	#121	semidirect
86. C₂²×D₁₇	C₂²×D₄	semidirect
90. C₂×C₄×C₁₇	C₂²×D₄	semidirect
102. C₂³×C₄⋊C₄	D₁₇	semidirect
104. C₂²×C₁₇	C₂×#121	semidirect
116. C₂×D₁₇	C₂³×D₄	semidirect
118. C₄×C₁₇	C₂³×D₄	semidirect
132. C₂²×C₄⋊C₄	C₂×D₁₇	semidirect
134. C₂×C₁₇	C₂²×#121	semidirect
158. C₂³×C₄	D₆₈	semidirect
162. C₂×C₄⋊C₄	C₂²×D₁₇	semidirect
164. C₁₇	C₂³×#121	semidirect
188. C₂²×C₄	C₂×D₆₈	semidirect
191. C₄⋊C₄	C₂³×D₁₇	semidirect
203. C₂³	#10587581	direct (N × H)
214. C₂×C₄	C₂²×D₆₈	semidirect
225. C₂²	C₂×#10587581	direct (N × H)
232. C₄	C₂³×D₆₈	semidirect
235. C₂	C₂²×#10587581	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×#10587581	2
C₂³×#63227	2
C₂³×C₄⋊C₄×C₁₇	2
C₂³×#121	17

Subgroups — C23×#10587581 #17940962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587581 #17940962