Subgroups — C₂³×#10587616 #17940961

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	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₂²⋊D₁₇	C₂	1	2	yes	maximal, complemented
3	C₂²×#10587616	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#63227	C₂	1	2	yes	maximal, complemented
6	C₂⁵×C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₇	C₂²	1	2	yes	—
8	C₂³×D₆₈	—	2	16	no	—
9	C₂³×D₆₈	C₂²	1	28	yes	complemented
10	C₂³×C₂²⋊D₁₇	—	2	32	no	—
11	C₂³×C₂²⋊D₁₇	C₂²	1	56	yes	complemented
12	C₂⁴×Dic₁₇	C₂²	1	2	yes	—
13	C₂×#10587616	C₂²	1	448	yes	direct product factor
14	C₂²×C₄⋊Dic₁₇	C₂²	1	28	yes	complemented
15	C₂²×#63227	C₂²	1	56	yes	complemented
16	C₂⁶×C₁₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₁₇	—	2	16	no	—
18	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄×C₁₇	C₂²	1	28	yes	complemented
21	C₂⁵×C₁₇	—	2	8	no	—
22	C₂⁵×C₁₇	—	2	8	no	—
23	C₂⁵×C₁₇	D₄	1	16	yes	complemented
24	C₂⁵×C₁₇	C₂³	1	14	yes	—
25	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
26	C₂⁴×D₁₇	—	2	16	no	—
27	C₂⁴×D₁₇	—	2	16	no	—
28	C₂⁴×D₁₇	—	2	16	no	—
29	C₂⁴×D₁₇	C₂³	1	28	yes	—
30	C₂³×C₄×C₁₇	—	2	8	no	—
31	C₂³×C₄×C₁₇	—	2	224	no	—
32	C₂³×C₄×C₁₇	D₄	1	16	yes	complemented
33	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
34	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
35	C₂³×C₄×C₁₇	C₂³	1	112	yes	complemented
36	C₂²×C₂²⋊D₁₇	—	2	448	no	—
37	C₂²×C₂²⋊D₁₇	C₂³	1	224	yes	complemented
38	C₂³×Dic₁₇	—	2	16	no	—
39	C₂³×Dic₁₇	C₂³	1	28	yes	—
40	C₂²×D₆₈	—	2	224	no	—
41	C₂²×D₆₈	C₂³	1	112	yes	complemented
42	#10587616	C₂³	1	512	yes	direct product factor
43	C₂×#63227	C₂³	1	224	yes	complemented
44	C₂×C₄⋊Dic₁₇	C₂³	1	112	yes	complemented
45	C₂⁴×C₁₇	—	2	128	no	—
46	C₂⁴×C₁₇	—	2	56	no	—
47	C₂⁴×C₁₇	—	2	56	no	—
48	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
49	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
50	C₂⁴×C₁₇	C₂×D₄	1	112	yes	complemented
51	C₂⁴×C₁₇	#41	1	8	yes	—
52	C₂⁴×C₁₇	C₂⁴	1	28	yes	—
53	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
54	C₂³×D₁₇	—	2	112	no	—
55	C₂³×D₁₇	—	2	112	no	—
56	C₂³×D₁₇	—	2	112	no	—
57	C₂³×D₁₇	—	4	128	no	—
58	C₂³×D₁₇	C₂⁴	1	56	yes	—
59	C₂²×C₄×C₁₇	—	2	56	no	—
60	C₂²×C₄×C₁₇	—	2	448	no	—
61	C₂²×C₄×C₁₇	C₂×D₄	1	112	yes	complemented
62	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
63	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
64	C₂²×C₄×C₁₇	C₂⁴	1	64	yes	complemented
65	C₂×C₂²⋊D₁₇	—	2	896	no	—
66	C₂×C₂²⋊D₁₇	C₂⁴	1	128	yes	complemented
67	C₂²×Dic₁₇	—	2	112	no	—
68	C₂²×Dic₁₇	C₂⁴	1	56	yes	—
69	C₂×D₆₈	—	2	448	no	—
70	C₂×D₆₈	C₂⁴	1	64	yes	complemented
71	#63227	C₂⁴	1	128	yes	complemented
72	C₄⋊Dic₁₇	C₂⁴	1	64	yes	complemented
73	C₂³×#121	—	17	1	no	maximal, Sylow
74	C₂³×C₁₇	—	2	448	no	—
75	C₂³×C₁₇	—	2	56	no	—
76	C₂³×C₁₇	—	2	56	no	—
77	C₂³×C₁₇	#121	1	64	yes	complemented
78	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
79	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
80	C₂³×C₁₇	C₂²×D₄	1	112	yes	complemented
81	C₂³×C₁₇	C₂×#41	1	28	yes	—
82	C₂³×C₁₇	C₂⁵	1	8	yes	—
83	C₂³×C₁₇	C₂⁵	1	7	yes	—
84	C₂²×D₁₇	—	2	112	no	—
85	C₂²×D₁₇	—	2	112	no	—
86	C₂²×D₁₇	—	2	112	no	—
87	C₂²×D₁₇	—	4	448	no	—
88	C₂²×D₁₇	C₂⁵	1	16	yes	—
89	C₂×C₄×C₁₇	—	2	56	no	—
90	C₂×C₄×C₁₇	—	2	128	no	—
91	C₂×C₄×C₁₇	C₂²×D₄	1	112	yes	complemented
92	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
93	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
94	C₂²⋊D₁₇	—	2	256	no	—
95	C₂×Dic₁₇	—	2	112	no	—
96	C₂×Dic₁₇	C₂⁵	1	16	yes	—
97	D₆₈	—	2	128	no	—
98	C₂⁴×D₄	—	17	2	no	—
99	C₂⁴×D₄	—	17	1	no	—
100	C₂⁵×C₄	D₁₇	1	1	yes	characteristic, complemented
101	C₂²×#121	—	17	56	no	—
102	C₂³×C₄⋊C₄	—	17	1	no	—
103	C₂³×C₂²⋊C₄	—	17	2	no	—
104	C₂²×C₁₇	—	2	8	no	—
105	C₂²×C₁₇	—	2	8	no	—
106	C₂²×C₁₇	—	2	224	no	—
107	C₂²×C₁₇	C₂×#121	1	112	yes	complemented
108	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
109	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
110	C₂²×C₁₇	C₂³×D₄	1	16	yes	complemented
111	C₂²×C₁₇	C₂²×#41	1	14	yes	—
112	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
113	C₂×D₁₇	—	2	16	no	—
114	C₂×D₁₇	—	2	16	no	—
115	C₂×D₁₇	—	2	16	no	—
116	C₂×D₁₇	—	4	224	no	—
117	C₄×C₁₇	—	2	8	no	—
118	C₄×C₁₇	C₂³×D₄	1	16	yes	complemented
119	Dic₁₇	—	2	16	no	—
120	C₂⁶	—	17	2	no	—
121	C₂⁶	C₂×D₁₇	1	1	yes	characteristic
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	—	2	16	no	—
127	C₂⁴×C₄	—	17	2	no	—
128	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
129	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
130	C₂⁴×C₄	C₂×D₁₇	1	28	yes	complemented
131	C₂×#121	—	17	448	no	—
132	C₂²×C₄⋊C₄	—	17	28	no	—
133	C₂²×C₂²⋊C₄	—	17	56	no	—
134	C₂×C₁₇	—	2	16	no	—
135	C₂×C₁₇	C₂²×#121	1	28	yes	complemented
136	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
137	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
138	C₂×C₁₇	C₂³×#41	1	1	yes	characteristic
139	D₁₇	—	4	16	no	—
140	C₂⁵	—	2	8	no	—
141	C₂⁵	—	2	8	no	—
142	C₂⁵	—	17	28	no	—
143	C₂⁵	—	34	16	no	—
144	C₂⁵	—	34	16	no	—
145	C₂⁵	—	34	16	no	—
146	C₂⁵	D₆₈	1	16	yes	complemented
147	C₂⁵	C₂²×D₁₇	1	14	yes	—
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	8	no	—
154	C₂³×C₄	—	2	224	no	—
155	C₂³×C₄	—	17	28	no	—
156	C₂³×C₄	—	34	16	no	—
157	C₂³×C₄	C₂²⋊D₁₇	1	16	yes	complemented
158	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
159	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
160	C₂³×C₄	C₂²×D₁₇	1	112	yes	complemented
161	#121	—	17	512	no	—
162	C₂×C₄⋊C₄	—	17	112	no	—
163	C₂×C₂²⋊C₄	—	17	224	no	—
164	C₁₇	C₂³×#121	1	1	yes	characteristic, complemented, Sylow
165	C₂⁴	—	2	56	no	—
166	C₂⁴	—	2	56	no	—
167	C₂⁴	—	2	128	no	—
168	C₂⁴	—	17	56	no	—
169	C₂⁴	—	34	112	no	—
170	C₂⁴	—	34	112	no	—
171	C₂⁴	—	34	112	no	—
172	C₂⁴	—	68	128	no	—
173	C₂⁴	C₂×D₆₈	1	8	yes	—
174	C₂⁴	C₂×D₆₈	1	112	yes	complemented
175	C₂⁴	#63252	1	8	yes	—
176	C₂⁴	C₂×C₂²⋊D₁₇	1	8	yes	—
177	C₂⁴	C₂³×D₁₇	1	28	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	56	no	—
184	C₂²×C₄	—	2	448	no	—
185	C₂²×C₄	—	17	56	no	—
186	C₂²×C₄	—	34	112	no	—
187	C₂²×C₄	C₂×C₂²⋊D₁₇	1	112	yes	complemented
188	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
189	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
190	C₂²×C₄	C₂³×D₁₇	1	64	yes	complemented
191	C₄⋊C₄	—	17	64	no	—
192	C₂²⋊C₄	—	17	128	no	—
193	C₂³	—	2	448	no	—
194	C₂³	—	2	56	no	—
195	C₂³	—	2	56	no	—
196	C₂³	—	17	16	no	—
197	C₂³	—	34	112	no	—
198	C₂³	—	34	112	no	—
199	C₂³	—	34	112	no	—
200	C₂³	—	68	448	no	—
201	C₂³	#10587616	1	64	yes	direct product factor
202	C₂³	C₂²×D₆₈	1	28	yes	—
203	C₂³	C₂²×D₆₈	1	112	yes	complemented
204	C₂³	C₂×#63252	1	28	yes	—
205	C₂³	C₂²×C₂²⋊D₁₇	1	28	yes	—
206	C₂³	C₂⁴×D₁₇	1	8	yes	—
207	C₂³	C₂⁴×D₁₇	1	7	yes	—
208	D₄	—	34	256	no	—
209	D₄	—	34	128	no	—
210	C₂×C₄	—	2	56	no	—
211	C₂×C₄	—	2	128	no	—
212	C₂×C₄	—	17	16	no	—
213	C₂×C₄	—	34	112	no	—
214	C₂×C₄	C₂²×C₂²⋊D₁₇	1	112	yes	complemented
215	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
216	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
217	C₂²	—	2	224	no	—
218	C₂²	—	2	8	no	—
219	C₂²	—	2	8	no	—
220	C₂²	—	34	16	no	—
221	C₂²	—	34	16	no	—
222	C₂²	—	34	16	no	—
223	C₂²	—	68	224	no	—
224	C₂²	C₂×#10587616	1	112	yes	direct product factor
225	C₂²	C₂³×C₂²⋊D₁₇	1	14	yes	—
226	C₂²	C₂³×D₆₈	1	14	yes	—
227	C₂²	C₂³×D₆₈	1	16	yes	complemented
228	C₂²	C₂²×#63252	1	14	yes	—
229	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
230	C₄	—	2	8	no	—
231	C₄	—	34	16	no	—
232	C₄	C₂³×C₂²⋊D₁₇	1	16	yes	complemented
233	C₂	—	2	16	no	—
234	C₂	—	68	16	no	—
235	C₂	C₂²×#10587616	1	28	yes	direct product factor
236	C₂	C₂⁴×C₂²⋊D₁₇	1	1	yes	characteristic
237	C₂	C₂⁴×D₆₈	1	1	yes	characteristic
238	C₂	C₂³×#63252	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₇
commutator, cosocle	C₂²×C₁₇	C₂⁶
Frattini	C₂²	C₂⁵×D₁₇
Fitting	C₂⁵×C₄×C₁₇	C₂
socle	C₂⁵×C₁₇	C₂³
hypercenter	C₂⁵×C₄	D₁₇
hypocenter	C₁₇	C₂³×#121
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₆₈	C₂	semidirect
2. C₂⁴×C₂²⋊D₁₇	C₂	semidirect
3. C₂²×#10587616	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₁₇	C₂	semidirect
5. C₂³×#63227	C₂	semidirect
6. C₂⁵×C₄×C₁₇	C₂	semidirect
9. C₂³×D₆₈	C₂²	semidirect
11. C₂³×C₂²⋊D₁₇	C₂²	semidirect
13. C₂×#10587616	C₂²	direct (N × H)
14. C₂²×C₄⋊Dic₁₇	C₂²	semidirect
15. C₂²×#63227	C₂²	semidirect
20. C₂⁴×C₄×C₁₇	C₂²	semidirect
23. C₂⁵×C₁₇	D₄	semidirect
32. C₂³×C₄×C₁₇	D₄	semidirect
35. C₂³×C₄×C₁₇	C₂³	semidirect
37. C₂²×C₂²⋊D₁₇	C₂³	semidirect
41. C₂²×D₆₈	C₂³	semidirect
42. #10587616	C₂³	direct (N × H)
43. C₂×#63227	C₂³	semidirect
44. C₂×C₄⋊Dic₁₇	C₂³	semidirect
50. C₂⁴×C₁₇	C₂×D₄	semidirect
61. C₂²×C₄×C₁₇	C₂×D₄	semidirect
64. C₂²×C₄×C₁₇	C₂⁴	semidirect
66. C₂×C₂²⋊D₁₇	C₂⁴	semidirect
70. C₂×D₆₈	C₂⁴	semidirect
71. #63227	C₂⁴	semidirect
72. C₄⋊Dic₁₇	C₂⁴	semidirect
77. C₂³×C₁₇	#121	semidirect
80. C₂³×C₁₇	C₂²×D₄	semidirect
91. C₂×C₄×C₁₇	C₂²×D₄	semidirect
100. C₂⁵×C₄	D₁₇	semidirect
107. C₂²×C₁₇	C₂×#121	semidirect
110. C₂²×C₁₇	C₂³×D₄	semidirect
118. C₄×C₁₇	C₂³×D₄	semidirect
130. C₂⁴×C₄	C₂×D₁₇	semidirect
135. C₂×C₁₇	C₂²×#121	semidirect
146. C₂⁵	D₆₈	semidirect
157. C₂³×C₄	C₂²⋊D₁₇	semidirect
160. C₂³×C₄	C₂²×D₁₇	semidirect
164. C₁₇	C₂³×#121	semidirect
174. C₂⁴	C₂×D₆₈	semidirect
187. C₂²×C₄	C₂×C₂²⋊D₁₇	semidirect
190. C₂²×C₄	C₂³×D₁₇	semidirect
201. C₂³	#10587616	direct (N × H)
203. C₂³	C₂²×D₆₈	semidirect
214. C₂×C₄	C₂²×C₂²⋊D₁₇	semidirect
224. C₂²	C₂×#10587616	direct (N × H)
227. C₂²	C₂³×D₆₈	semidirect
232. C₄	C₂³×C₂²⋊D₁₇	semidirect
235. C₂	C₂²×#10587616	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×C₂²⋊D₁₇	2
C₂²×#10587616	2
C₂³×C₄⋊Dic₁₇	2
C₂³×#63227	2
C₂⁵×C₄×C₁₇	2
C₂³×#121	17

Subgroups — C23×#10587616 #17940961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587616 #17940961