Subgroups — C₂×C₈×Dic₃₄ #14537705

Order 2176.

440 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₄	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₃₄	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587485	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₁₇	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₄	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₇	C₄	1	1	yes	characteristic
12	C₂×#63225	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₇	C₂²	1	2	yes	—
14	C₂²×#3622	C₂²	1	2	yes	—
15	C₄×Dic₃₄	C₄	1	4	yes	—
16	C₄×Dic₃₄	C₂²	1	4	yes	—
17	C₄×C₈×C₁₇	C₂²	1	4	yes	—
18	#10587485	C₂²	1	8	yes	—
19	C₈×Dic₁₇	C₂²	1	8	yes	—
20	C₄⋊#3622	C₂²	1	4	yes	—
21	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
25	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
26	C₂×Dic₃₄	C₈	1	8	yes	direct product factor
27	C₂×Dic₃₄	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₁₇	—	2	2	no	—
29	C₂×C₈×C₁₇	Q₈	1	4	yes	complemented
30	C₂×C₈×C₁₇	C₂³	1	2	yes	—
31	C₂×C₈×C₁₇	C₂³	1	2	yes	—
32	C₄²×C₁₇	C₂×C₄	1	2	yes	—
33	C₄²×C₁₇	C₂³	1	2	yes	—
34	C₄⋊Dic₁₇	C₂×C₄	1	4	yes	—
35	#63225	C₂×C₄	1	8	yes	—
36	C₄×Dic₁₇	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₇	C₂³	1	4	yes	—
38	C₂×#3622	—	2	4	no	—
39	C₂×#3622	C₂³	1	4	yes	—
40	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₇	#41	1	2	yes	—
48	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₁₇	—	2	4	no	—
50	C₂×Dic₁₇	C₂×C₈	1	8	yes	—
51	C₂×Dic₁₇	C₂²×C₄	1	4	yes	—
52	C₂×Dic₁₇	C₂²×C₄	1	4	yes	—
53	Dic₃₄	C₂×C₈	1	16	yes	direct product factor
54	C₈×C₁₇	—	2	2	no	—
55	C₈×C₁₇	C₂×Q₈	1	4	yes	complemented
56	#3622	—	2	4	no	—
57	C₂×C₈×Q₈	—	17	1	no	maximal, Sylow
58	C₂²×C₁₇	C₄×Q₈	1	2	yes	—
59	C₂²×C₁₇	C₂²×C₈	1	2	yes	—
60	C₂²×C₁₇	#131	1	2	yes	—
61	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₁₇	—	2	2	no	—
63	C₄×C₁₇	C₄×Q₈	1	2	yes	—
64	C₄×C₁₇	C₂²×C₈	1	4	yes	—
65	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
66	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
67	Dic₁₇	—	2	4	no	—
68	Dic₁₇	C₂²×C₈	1	8	yes	—
69	C₂×C₄×Q₈	—	17	1	no	—
70	C₈×Q₈	—	17	8	no	—
71	C₂×C₄⋊C₈	—	17	2	no	—
72	C₂×C₄⋊C₈	—	17	1	no	—
73	C₂×C₄×C₈	—	17	2	no	—
74	C₂×C₄×C₈	D₁₇	1	1	yes	characteristic
75	C₂×C₁₇	C₈×Q₈	1	4	yes	complemented
76	C₂×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
77	C₂×C₁₇	C₂³×C₈	1	1	yes	characteristic
78	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
79	C₂²×Q₈	—	17	1	no	—
80	C₂²×C₈	—	17	2	no	—
81	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
83	C₄×Q₈	—	17	4	no	—
84	C₄×Q₈	—	17	4	no	—
85	C₂×C₄⋊C₄	—	17	2	no	—
86	C₂×C₄⋊C₄	—	17	1	no	—
87	C₂×C₄²	—	17	2	no	—
88	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
89	C₄⋊C₈	—	17	8	no	—
90	C₄⋊C₈	—	17	4	no	—
91	C₄×C₈	—	17	8	no	—
92	C₄×C₈	C₂×D₁₇	1	4	yes	—
93	C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	17	4	no	—
95	C₂×Q₈	—	17	8	no	—
96	C₂²×C₄	—	17	2	no	—
97	C₂²×C₄	—	17	2	no	—
98	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	17	4	no	—
103	C₂×C₈	—	34	4	no	—
104	C₂×C₈	Dic₃₄	1	4	yes	direct product factor
105	C₂×C₈	C₂²×D₁₇	1	2	yes	—
106	C₂×C₈	C₂²×D₁₇	1	2	yes	—
107	C₄⋊C₄	—	17	8	no	—
108	C₄⋊C₄	—	17	4	no	—
109	C₄²	—	17	4	no	—
110	C₄²	—	17	4	no	—
111	C₄²	C₄×D₁₇	1	2	yes	—
112	C₄²	C₂²×D₁₇	1	2	yes	—
113	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
114	Q₈	—	17	16	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	17	4	no	—
117	C₂×C₄	—	17	8	no	—
118	C₂×C₄	—	17	4	no	—
119	C₂×C₄	—	34	4	no	—
120	C₂×C₄	C₈×D₁₇	1	4	yes	—
121	C₂×C₄	C₂×Dic₃₄	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
123	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
125	C₂×C₄	#63252	1	2	yes	—
126	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	34	4	no	—
129	C₈	C₂×Dic₃₄	1	4	yes	direct product factor
130	C₂²	C₄×Dic₃₄	1	2	yes	—
131	C₂²	C₂×C₈×D₁₇	1	2	yes	—
132	C₂²	#10587587	1	2	yes	—
133	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	17	8	no	—
136	C₄	—	34	4	no	—
137	C₄	C₄×Dic₃₄	1	2	yes	—
138	C₄	C₂×C₈×D₁₇	1	4	yes	—
139	C₄	C₂²×Dic₃₄	1	1	yes	characteristic
140	C₄	C₂×#63252	1	1	yes	characteristic
141	C₂	C₈×Dic₃₄	1	4	yes	direct product factor
142	C₂	C₂×C₄×Dic₃₄	1	1	yes	characteristic
143	C₂	C₂²×C₈×D₁₇	1	1	yes	characteristic
144	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, prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₇
hypocenter	C₁₇	C₂×C₈×Q₈
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
2. C₈×Dic₃₄	C₂	direct (N × H)
26. C₂×Dic₃₄	C₈	direct (N × H)
29. C₂×C₈×C₁₇	Q₈	semidirect
53. Dic₃₄	C₂×C₈	direct (N × H)
55. C₈×C₁₇	C₂×Q₈	semidirect
75. C₂×C₁₇	C₈×Q₈	semidirect
93. C₁₇	C₂×C₈×Q₈	semidirect
104. C₂×C₈	Dic₃₄	direct (N × H)
129. C₈	C₂×Dic₃₄	direct (N × H)
141. C₂	C₈×Dic₃₄	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×Dic₃₄	2
C₈×Dic₃₄	2
C₂×C₄⋊#3622	2
C₂×#10587485	2
C₂×C₈×Dic₁₇	2
C₂×C₄×C₈×C₁₇	2
C₂×C₈×Q₈	17

Subgroups — C2×C8×Dic34 #14537705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×Dic₃₄ #14537705