Subgroups — C₂×C₈×D₆₈ #14537458

Order 2176.

616 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₁₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₆₈	C₂	1	1	yes	maximal, characteristic
3	C₈×D₆₈	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10587491	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₆₈	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₁₇	C₂²	1	2	yes	—
9	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×#63227	C₄	1	2	yes	—
13	C₂×C₄⋊Dic₁₇	C₄	1	1	yes	characteristic
14	C₂²×#3622	C₂²	1	2	yes	—
15	C₂×C₈×D₁₇	—	2	8	no	—
16	C₂×C₈×D₁₇	C₂²	1	8	yes	complemented
17	C₄×D₆₈	C₄	1	4	yes	—
18	C₄×D₆₈	C₂²	1	4	yes	—
19	C₄×C₈×C₁₇	C₂²	1	4	yes	complemented
20	#10587491	C₂²	1	8	yes	complemented
21	C₄⋊#3622	C₂²	1	4	yes	complemented
22	C₂³×D₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
27	C₂×D₆₈	C₈	1	8	yes	direct product factor
28	C₂×D₆₈	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₇	—	2	4	no	—
30	C₂×C₄×D₁₇	—	2	4	no	—
31	C₂×C₄×D₁₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×D₁₇	C₂³	1	4	yes	—
33	C₂×C₈×C₁₇	—	2	2	no	—
34	C₂×C₈×C₁₇	D₄	1	4	yes	complemented
35	C₂×C₈×C₁₇	C₂³	1	2	yes	—
36	C₂×C₈×C₁₇	C₂³	1	2	yes	—
37	C₄²×C₁₇	C₂×C₄	1	2	yes	—
38	C₄²×C₁₇	C₂³	1	2	yes	—
39	#63227	C₂×C₄	1	8	yes	—
40	C₄⋊Dic₁₇	C₂×C₄	1	4	yes	—
41	C₂×#3622	—	2	4	no	—
42	C₂×#3622	C₂³	1	4	yes	—
43	C₈×D₁₇	—	2	16	no	—
44	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×D₁₇	—	2	4	no	—
46	C₂²×D₁₇	—	2	4	no	—
47	C₂²×D₁₇	C₂×C₈	1	8	yes	complemented
48	C₂²×D₁₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₁₇	—	2	2	no	—
50	C₂×C₄×C₁₇	C₂×C₈	1	4	yes	complemented
51	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₇	#41	1	2	yes	—
56	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₁₇	—	2	4	no	—
58	C₂×Dic₁₇	C₂²×C₄	1	4	yes	—
59	D₆₈	C₂×C₈	1	16	yes	direct product factor
60	C₄×D₁₇	—	2	8	no	—
61	C₄×D₁₇	—	2	4	no	—
62	C₄×D₁₇	—	2	4	no	—
63	C₈×C₁₇	—	2	2	no	—
64	C₈×C₁₇	C₂×D₄	1	4	yes	complemented
65	#3622	—	2	4	no	—
66	C₂×C₈×D₄	—	17	1	no	maximal, Sylow
67	C₂²×C₁₇	C₄×D₄	1	2	yes	—
68	C₂²×C₁₇	C₂²×C₈	1	2	yes	—
69	C₂²×C₁₇	#131	1	2	yes	—
70	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
71	C₂×D₁₇	—	2	4	no	—
72	C₂×D₁₇	—	2	4	no	—
73	C₂×D₁₇	—	2	16	no	—
74	C₂×D₁₇	C₂²×C₈	1	8	yes	complemented
75	C₄×C₁₇	—	2	2	no	—
76	C₄×C₁₇	C₄×D₄	1	2	yes	—
77	C₄×C₁₇	C₂²×C₈	1	4	yes	complemented
78	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
80	Dic₁₇	—	2	4	no	—
81	C₂³×C₈	—	17	2	no	—
82	C₂×C₄×D₄	—	17	1	no	—
83	C₈×D₄	—	17	8	no	—
84	C₂×C₄⋊C₈	—	17	1	no	—
85	C₂×C₂²⋊C₈	—	17	2	no	—
86	C₂×C₄×C₈	D₁₇	1	1	yes	characteristic, complemented
87	C₂×C₁₇	C₈×D₄	1	4	yes	complemented
88	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
89	C₂×C₁₇	C₂³×C₈	1	1	yes	characteristic
90	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
91	D₁₇	—	2	8	no	—
92	C₂²×D₄	—	17	1	no	—
93	C₂³×C₄	—	17	2	no	—
94	C₂²×C₈	—	17	2	no	—
95	C₂²×C₈	—	17	8	no	—
96	C₂²×C₈	—	34	8	no	—
97	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
99	C₄×D₄	—	17	4	no	—
100	C₄×D₄	—	17	4	no	—
101	C₂×C₄⋊C₄	—	17	1	no	—
102	C₂×C₂²⋊C₄	—	17	2	no	—
103	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
104	C₄⋊C₈	—	17	4	no	—
105	C₂²⋊C₈	—	17	8	no	—
106	C₄×C₈	C₂×D₁₇	1	4	yes	complemented
107	C₁₇	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
108	C₂⁴	—	17	2	no	—
109	C₂×D₄	—	17	4	no	—
110	C₂×D₄	—	17	8	no	—
111	C₂²×C₄	—	17	2	no	—
112	C₂²×C₄	—	17	4	no	—
113	C₂²×C₄	—	17	4	no	—
114	C₂²×C₄	—	34	4	no	—
115	C₂²×C₄	—	34	4	no	—
116	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
117	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	17	4	no	—
121	C₂×C₈	—	34	4	no	—
122	C₂×C₈	—	34	16	no	—
123	C₂×C₈	D₆₈	1	4	yes	direct product factor
124	C₂×C₈	C₂²×D₁₇	1	2	yes	—
125	C₂×C₈	C₂²×D₁₇	1	2	yes	—
126	C₄⋊C₄	—	17	4	no	—
127	C₂²⋊C₄	—	17	8	no	—
128	C₄²	C₄×D₁₇	1	2	yes	—
129	C₄²	C₂²×D₁₇	1	2	yes	—
130	C₂³	—	17	4	no	—
131	C₂³	—	17	8	no	—
132	C₂³	—	34	4	no	—
133	C₂³	—	34	4	no	—
134	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
135	D₄	—	17	16	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	17	4	no	—
138	C₂×C₄	—	34	4	no	—
139	C₂×C₄	—	34	4	no	—
140	C₂×C₄	—	34	4	no	—
141	C₂×C₄	—	34	8	no	—
142	C₂×C₄	C₈×D₁₇	1	4	yes	complemented
143	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
144	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
146	C₂×C₄	C₂×D₆₈	1	2	yes	—
147	C₂×C₄	#63252	1	2	yes	—
148	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	34	4	no	—
151	C₈	C₂×D₆₈	1	4	yes	direct product factor
152	C₂²	—	17	8	no	—
153	C₂²	—	34	4	no	—
154	C₂²	—	34	4	no	—
155	C₂²	—	34	16	no	—
156	C₂²	C₄×D₆₈	1	2	yes	—
157	C₂²	C₂×C₈×D₁₇	1	2	yes	—
158	C₂²	#10587587	1	2	yes	—
159	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	34	4	no	—
162	C₄	C₄×D₆₈	1	2	yes	—
163	C₄	C₂×C₈×D₁₇	1	4	yes	complemented
164	C₄	C₂²×D₆₈	1	1	yes	characteristic
165	C₄	C₂×#63252	1	1	yes	characteristic
166	C₂	—	34	8	no	—
167	C₂	C₈×D₆₈	1	4	yes	direct product factor
168	C₂	C₂×C₄×D₆₈	1	1	yes	characteristic
169	C₂	C₂²×C₈×D₁₇	1	1	yes	characteristic
170	C₂	C₂×#10587587	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₇
commutator	C₂×C₁₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₂×C₄×C₈×C₁₇	C₂
socle	C₂³×C₁₇	C₂²×C₄
prime radical	C₂²×D₆₈	C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₇
hypocenter	C₁₇	C₂×C₈×D₄
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₈×D₁₇	C₂	semidirect
3. C₈×D₆₈	C₂	direct (N × H)
4. C₂×C₄⋊#3622	C₂	semidirect
5. C₂×#10587491	C₂	semidirect
6. C₂×C₄×C₈×C₁₇	C₂	semidirect
16. C₂×C₈×D₁₇	C₂²	semidirect
19. C₄×C₈×C₁₇	C₂²	semidirect
20. #10587491	C₂²	semidirect
21. C₄⋊#3622	C₂²	semidirect
27. C₂×D₆₈	C₈	direct (N × H)
34. C₂×C₈×C₁₇	D₄	semidirect
47. C₂²×D₁₇	C₂×C₈	semidirect
50. C₂×C₄×C₁₇	C₂×C₈	semidirect
59. D₆₈	C₂×C₈	direct (N × H)
64. C₈×C₁₇	C₂×D₄	semidirect
74. C₂×D₁₇	C₂²×C₈	semidirect
77. C₄×C₁₇	C₂²×C₈	semidirect
86. C₂×C₄×C₈	D₁₇	semidirect
87. C₂×C₁₇	C₈×D₄	semidirect
106. C₄×C₈	C₂×D₁₇	semidirect
107. C₁₇	C₂×C₈×D₄	semidirect
123. C₂×C₈	D₆₈	direct (N × H)
142. C₂×C₄	C₈×D₁₇	semidirect
151. C₈	C₂×D₆₈	direct (N × H)
163. C₄	C₂×C₈×D₁₇	semidirect
167. C₂	C₈×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
C₂²×C₈×D₁₇	2
C₂×C₄×D₆₈	2
C₈×D₆₈	2
C₂×C₄⋊#3622	2
C₂×#10587491	2
C₂×C₄×C₈×C₁₇	2
C₂×C₈×D₄	17

Subgroups — C2×C8×D68 #14537458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×D₆₈ #14537458