Subgroups — C₂³×C₄×D₆₈ #17940890

Order 4352.

19998 conjugacy classes of subgroups.

The subgroup lattice has 208 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	2	yes	maximal, complemented
3	C₂²×C₄×D₆₈	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	C₂³×C₄²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₇	C₂²	1	2	yes	—
8	C₂³×D₆₈	C₄	1	32	yes	direct product factor
9	C₂³×D₆₈	C₂²	1	28	yes	—
10	C₂⁴×Dic₁₇	C₂²	1	2	yes	—
11	C₂³×C₄×D₁₇	—	2	32	no	—
12	C₂³×C₄×D₁₇	C₂²	1	56	yes	complemented
13	C₂×C₄×D₆₈	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	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₁₇	C₂²	1	28	yes	complemented
20	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
21	C₂⁴×D₁₇	—	2	16	no	—
22	C₂⁴×D₁₇	—	2	16	no	—
23	C₂⁴×D₁₇	C₂×C₄	1	32	yes	complemented
24	C₂⁴×D₁₇	C₂³	1	28	yes	—
25	C₂³×C₄×C₁₇	—	2	8	no	—
26	C₂³×C₄×C₁₇	C₂×C₄	1	16	yes	complemented
27	C₂³×C₄×C₁₇	D₄	1	16	yes	complemented
28	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
29	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
30	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
31	C₂³×Dic₁₇	—	2	16	no	—
32	C₂³×Dic₁₇	C₂³	1	28	yes	—
33	C₂²×D₆₈	C₂×C₄	1	448	yes	direct product factor
34	C₂²×D₆₈	C₂³	1	112	yes	—
35	C₂²×C₄×D₁₇	—	2	448	no	—
36	C₂²×C₄×D₁₇	C₂³	1	224	yes	complemented
37	C₂×C₄²×C₁₇	C₂³	1	112	yes	complemented
38	C₂×#63227	C₂³	1	224	yes	complemented
39	C₂×C₄⋊Dic₁₇	C₂³	1	112	yes	complemented
40	C₄×D₆₈	C₂³	1	512	yes	direct product factor
41	C₂⁴×C₁₇	C₂²×C₄	1	8	yes	—
42	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
43	C₂⁴×C₁₇	#41	1	8	yes	—
44	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
45	C₂³×D₁₇	—	2	256	no	—
46	C₂³×D₁₇	—	2	112	no	—
47	C₂³×D₁₇	—	2	112	no	—
48	C₂³×D₁₇	C₂²×C₄	1	224	yes	complemented
49	C₂³×D₁₇	C₂⁴	1	56	yes	—
50	C₂²×C₄×C₁₇	—	2	56	no	—
51	C₂²×C₄×C₁₇	C₂²×C₄	1	112	yes	complemented
52	C₂²×C₄×C₁₇	C₂×D₄	1	112	yes	complemented
53	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
54	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
55	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
56	C₂²×Dic₁₇	—	2	112	no	—
57	C₂²×Dic₁₇	C₂⁴	1	56	yes	—
58	C₂×D₆₈	C₂²×C₄	1	896	yes	direct product factor
59	C₂×D₆₈	C₂⁴	1	64	yes	—
60	C₂×C₄×D₁₇	—	2	896	no	—
61	C₂×C₄×D₁₇	C₂⁴	1	128	yes	complemented
62	C₄²×C₁₇	C₂⁴	1	64	yes	complemented
63	#63227	C₂⁴	1	128	yes	complemented
64	C₄⋊Dic₁₇	C₂⁴	1	64	yes	complemented
65	C₂³×C₄×D₄	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₄×D₄	1	64	yes	complemented
67	C₂³×C₁₇	C₂³×C₄	1	28	yes	—
68	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
69	C₂³×C₁₇	C₂×#41	1	28	yes	—
70	C₂³×C₁₇	C₂⁵	1	7	yes	—
71	C₂²×D₁₇	—	2	896	no	—
72	C₂²×D₁₇	—	2	112	no	—
73	C₂²×D₁₇	—	2	112	no	—
74	C₂²×D₁₇	C₂³×C₄	1	224	yes	complemented
75	C₂²×D₁₇	C₂⁵	1	16	yes	—
76	C₂×C₄×C₁₇	—	2	56	no	—
77	C₂×C₄×C₁₇	C₂³×C₄	1	112	yes	complemented
78	C₂×C₄×C₁₇	C₂²×D₄	1	112	yes	complemented
79	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
80	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
81	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
82	C₂×Dic₁₇	—	2	112	no	—
83	C₂×Dic₁₇	C₂⁵	1	16	yes	—
84	D₆₈	C₂³×C₄	1	256	yes	direct product factor
85	C₄×D₁₇	—	2	256	no	—
86	C₂⁴×D₄	—	17	1	no	—
87	C₂⁵×C₄	—	17	2	no	—
88	C₂²×C₄×D₄	—	17	56	no	—
89	C₂³×C₄⋊C₄	—	17	1	no	—
90	C₂³×C₂²⋊C₄	—	17	2	no	—
91	C₂³×C₄²	D₁₇	1	1	yes	characteristic, complemented
92	C₂²×C₁₇	C₂×C₄×D₄	1	112	yes	complemented
93	C₂²×C₁₇	C₂⁴×C₄	1	14	yes	—
94	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
95	C₂²×C₁₇	C₂²×#41	1	14	yes	—
96	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
97	C₂×D₁₇	—	2	16	no	—
98	C₂×D₁₇	—	2	16	no	—
99	C₂×D₁₇	—	2	448	no	—
100	C₂×D₁₇	C₂⁴×C₄	1	32	yes	complemented
101	C₄×C₁₇	—	2	8	no	—
102	C₄×C₁₇	C₂⁴×C₄	1	16	yes	complemented
103	C₄×C₁₇	C₂³×D₄	1	16	yes	complemented
104	Dic₁₇	—	2	16	no	—
105	C₂⁶	—	17	2	no	—
106	C₂³×D₄	—	17	32	no	—
107	C₂³×D₄	—	17	28	no	—
108	C₂⁴×C₄	—	17	2	no	—
109	C₂⁴×C₄	—	17	56	no	—
110	C₂⁴×C₄	—	34	32	no	—
111	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
112	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
113	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
114	C₂×C₄×D₄	—	17	448	no	—
115	C₂²×C₄⋊C₄	—	17	28	no	—
116	C₂²×C₂²⋊C₄	—	17	56	no	—
117	C₂²×C₄²	C₂×D₁₇	1	28	yes	complemented
118	C₂×C₁₇	C₂²×C₄×D₄	1	28	yes	complemented
119	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
120	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
121	C₂×C₁₇	C₂³×#41	1	1	yes	characteristic
122	D₁₇	—	2	32	no	—
123	C₂⁵	—	17	32	no	—
124	C₂⁵	—	17	28	no	—
125	C₂⁵	—	34	16	no	—
126	C₂⁵	—	34	16	no	—
127	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
128	C₂²×D₄	—	17	448	no	—
129	C₂²×D₄	—	17	112	no	—
130	C₂³×C₄	—	2	8	no	—
131	C₂³×C₄	—	17	28	no	—
132	C₂³×C₄	—	17	224	no	—
133	C₂³×C₄	—	34	16	no	—
134	C₂³×C₄	—	34	448	no	—
135	C₂³×C₄	C₄×D₁₇	1	16	yes	complemented
136	C₂³×C₄	D₆₈	1	16	yes	direct product factor
137	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
138	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
139	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
140	C₄×D₄	—	17	512	no	—
141	C₂×C₄⋊C₄	—	17	112	no	—
142	C₂×C₂²⋊C₄	—	17	224	no	—
143	C₂×C₄²	C₂²×D₁₇	1	112	yes	complemented
144	C₁₇	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
145	C₂⁴	—	17	224	no	—
146	C₂⁴	—	17	56	no	—
147	C₂⁴	—	34	112	no	—
148	C₂⁴	—	34	112	no	—
149	C₂⁴	—	34	256	no	—
150	C₂⁴	C₂×C₄×D₁₇	1	8	yes	—
151	C₂⁴	C₂×D₆₈	1	8	yes	—
152	C₂⁴	#63252	1	8	yes	—
153	C₂⁴	C₂³×D₁₇	1	7	yes	—
154	C₂×D₄	—	17	896	no	—
155	C₂×D₄	—	17	64	no	—
156	C₂²×C₄	—	2	56	no	—
157	C₂²×C₄	—	17	56	no	—
158	C₂²×C₄	—	17	128	no	—
159	C₂²×C₄	—	34	112	no	—
160	C₂²×C₄	—	34	896	no	—
161	C₂²×C₄	C₂×C₄×D₁₇	1	112	yes	complemented
162	C₂²×C₄	C₂×D₆₈	1	112	yes	direct product factor
163	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
164	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
165	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
166	C₄⋊C₄	—	17	64	no	—
167	C₂²⋊C₄	—	17	128	no	—
168	C₄²	C₂³×D₁₇	1	64	yes	complemented
169	C₂³	—	17	224	no	—
170	C₂³	—	17	16	no	—
171	C₂³	—	34	896	no	—
172	C₂³	—	34	112	no	—
173	C₂³	—	34	112	no	—
174	C₂³	C₄×D₆₈	1	64	yes	direct product factor
175	C₂³	C₂²×C₄×D₁₇	1	28	yes	—
176	C₂³	C₂²×D₆₈	1	28	yes	—
177	C₂³	C₂×#63252	1	28	yes	—
178	C₂³	C₂⁴×D₁₇	1	7	yes	—
179	D₄	—	17	256	no	—
180	C₂×C₄	—	2	56	no	—
181	C₂×C₄	—	17	16	no	—
182	C₂×C₄	—	34	112	no	—
183	C₂×C₄	—	34	256	no	—
184	C₂×C₄	C₂²×C₄×D₁₇	1	112	yes	complemented
185	C₂×C₄	C₂²×D₆₈	1	112	yes	direct product factor
186	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
187	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
188	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
189	C₂²	—	17	32	no	—
190	C₂²	—	34	448	no	—
191	C₂²	—	34	16	no	—
192	C₂²	—	34	16	no	—
193	C₂²	C₂×C₄×D₆₈	1	112	yes	direct product factor
194	C₂²	C₂³×C₄×D₁₇	1	14	yes	—
195	C₂²	C₂³×D₆₈	1	14	yes	—
196	C₂²	C₂²×#63252	1	14	yes	—
197	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
198	C₄	—	2	8	no	—
199	C₄	—	34	16	no	—
200	C₄	C₂³×C₄×D₁₇	1	16	yes	complemented
201	C₄	C₂³×D₆₈	1	16	yes	direct product factor
202	C₂	—	34	32	no	—
203	C₂	C₂²×C₄×D₆₈	1	28	yes	direct product factor
204	C₂	C₂⁴×C₄×D₁₇	1	1	yes	characteristic
205	C₂	C₂⁴×D₆₈	1	1	yes	characteristic
206	C₂	C₂³×#63252	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₁₇
commutator	C₂×C₁₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₇
Fitting	C₂³×C₄²×C₁₇	C₂
socle	C₂⁵×C₁₇	C₂³
prime radical	C₂⁴×D₆₈	C₂
cosocle	C₂²×C₁₇	C₂⁶
hypercenter	C₂³×C₄²	D₁₇
hypocenter	C₁₇	C₂³×C₄×D₄
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₁₇	C₂	semidirect
3. C₂²×C₄×D₆₈	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₁₇	C₂	semidirect
5. C₂³×#63227	C₂	semidirect
6. C₂³×C₄²×C₁₇	C₂	semidirect
8. C₂³×D₆₈	C₄	direct (N × H)
12. C₂³×C₄×D₁₇	C₂²	semidirect
13. C₂×C₄×D₆₈	C₂²	direct (N × H)
14. C₂²×C₄⋊Dic₁₇	C₂²	semidirect
15. C₂²×#63227	C₂²	semidirect
19. C₂²×C₄²×C₁₇	C₂²	semidirect
23. C₂⁴×D₁₇	C₂×C₄	semidirect
26. C₂³×C₄×C₁₇	C₂×C₄	semidirect
27. C₂³×C₄×C₁₇	D₄	semidirect
33. C₂²×D₆₈	C₂×C₄	direct (N × H)
36. C₂²×C₄×D₁₇	C₂³	semidirect
37. C₂×C₄²×C₁₇	C₂³	semidirect
38. C₂×#63227	C₂³	semidirect
39. C₂×C₄⋊Dic₁₇	C₂³	semidirect
40. C₄×D₆₈	C₂³	direct (N × H)
48. C₂³×D₁₇	C₂²×C₄	semidirect
51. C₂²×C₄×C₁₇	C₂²×C₄	semidirect
52. C₂²×C₄×C₁₇	C₂×D₄	semidirect
58. C₂×D₆₈	C₂²×C₄	direct (N × H)
61. C₂×C₄×D₁₇	C₂⁴	semidirect
62. C₄²×C₁₇	C₂⁴	semidirect
63. #63227	C₂⁴	semidirect
64. C₄⋊Dic₁₇	C₂⁴	semidirect
66. C₂³×C₁₇	C₄×D₄	semidirect
74. C₂²×D₁₇	C₂³×C₄	semidirect
77. C₂×C₄×C₁₇	C₂³×C₄	semidirect
78. C₂×C₄×C₁₇	C₂²×D₄	semidirect
84. D₆₈	C₂³×C₄	direct (N × H)
91. C₂³×C₄²	D₁₇	semidirect
92. C₂²×C₁₇	C₂×C₄×D₄	semidirect
100. C₂×D₁₇	C₂⁴×C₄	semidirect
102. C₄×C₁₇	C₂⁴×C₄	semidirect
103. C₄×C₁₇	C₂³×D₄	semidirect
117. C₂²×C₄²	C₂×D₁₇	semidirect
118. C₂×C₁₇	C₂²×C₄×D₄	semidirect
135. C₂³×C₄	C₄×D₁₇	semidirect
136. C₂³×C₄	D₆₈	direct (N × H)
143. C₂×C₄²	C₂²×D₁₇	semidirect
144. C₁₇	C₂³×C₄×D₄	semidirect
161. C₂²×C₄	C₂×C₄×D₁₇	semidirect
162. C₂²×C₄	C₂×D₆₈	direct (N × H)
168. C₄²	C₂³×D₁₇	semidirect
174. C₂³	C₄×D₆₈	direct (N × H)
184. C₂×C₄	C₂²×C₄×D₁₇	semidirect
185. C₂×C₄	C₂²×D₆₈	direct (N × H)
193. C₂²	C₂×C₄×D₆₈	direct (N × H)
200. C₄	C₂³×C₄×D₁₇	semidirect
201. C₄	C₂³×D₆₈	direct (N × H)
203. C₂	C₂²×C₄×D₆₈	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×C₄×D₆₈	2
C₂³×C₄⋊Dic₁₇	2
C₂³×#63227	2
C₂³×C₄²×C₁₇	2
C₂³×C₄×D₄	17

Subgroups — C23×C4×D68 #17940890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×D₆₈ #17940890