Subgroups — C₂³×C₄×Dic₃₄ #17940893

Order 4352.

12446 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₃₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₃₄	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×#63225	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₁₇	C₂	1	2	yes	maximal
6	C₂³×C₄²×C₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₃₄	C₄	1	32	yes	direct product factor
8	C₂³×Dic₃₄	C₂²	1	28	yes	—
9	C₂⁴×Dic₁₇	C₂²	1	2	yes	—
10	C₂⁴×Dic₁₇	C₂²	1	2	yes	—
11	C₂×C₄×Dic₃₄	C₂²	1	448	yes	direct product factor
12	C₂²×C₄⋊Dic₁₇	C₂²	1	28	yes	—
13	C₂²×#63225	C₂²	1	56	yes	—
14	C₂²×C₄×Dic₁₇	C₂²	1	56	yes	—
15	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₁₇	C₂²	1	28	yes	—
19	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₁₇	—	2	8	no	—
21	C₂³×C₄×C₁₇	C₂×C₄	1	16	yes	—
22	C₂³×C₄×C₁₇	Q₈	1	16	yes	complemented
23	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
24	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
25	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
26	C₂³×Dic₁₇	—	2	16	no	—
27	C₂³×Dic₁₇	C₂×C₄	1	32	yes	—
28	C₂³×Dic₁₇	C₂³	1	28	yes	—
29	C₂³×Dic₁₇	C₂³	1	28	yes	—
30	C₂²×Dic₃₄	C₂×C₄	1	448	yes	direct product factor
31	C₂²×Dic₃₄	C₂³	1	112	yes	—
32	C₂×C₄²×C₁₇	C₂³	1	112	yes	—
33	C₂×C₄⋊Dic₁₇	C₂³	1	112	yes	—
34	C₂×#63225	C₂³	1	224	yes	—
35	C₂×C₄×Dic₁₇	C₂³	1	224	yes	—
36	C₄×Dic₃₄	C₂³	1	512	yes	direct product factor
37	C₂⁴×C₁₇	C₂²×C₄	1	8	yes	—
38	C₂⁴×C₁₇	C₂×Q₈	1	8	yes	—
39	C₂⁴×C₁₇	#41	1	8	yes	—
40	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
41	C₂²×C₄×C₁₇	—	2	56	no	—
42	C₂²×C₄×C₁₇	C₂²×C₄	1	112	yes	—
43	C₂²×C₄×C₁₇	C₂×Q₈	1	112	yes	complemented
44	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
45	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
46	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
47	C₂²×Dic₁₇	—	2	112	no	—
48	C₂²×Dic₁₇	C₂²×C₄	1	224	yes	—
49	C₂²×Dic₁₇	C₂⁴	1	56	yes	—
50	C₂²×Dic₁₇	C₂⁴	1	56	yes	—
51	C₂×Dic₃₄	C₂²×C₄	1	896	yes	direct product factor
52	C₂×Dic₃₄	C₂⁴	1	64	yes	—
53	C₄²×C₁₇	C₂⁴	1	64	yes	—
54	C₄⋊Dic₁₇	C₂⁴	1	64	yes	—
55	#63225	C₂⁴	1	128	yes	—
56	C₄×Dic₁₇	C₂⁴	1	128	yes	—
57	C₂³×C₄×Q₈	—	17	1	no	maximal, Sylow
58	C₂³×C₁₇	C₄×Q₈	1	64	yes	complemented
59	C₂³×C₁₇	C₂³×C₄	1	28	yes	—
60	C₂³×C₁₇	C₂²×Q₈	1	28	yes	—
61	C₂³×C₁₇	C₂×#41	1	28	yes	—
62	C₂³×C₁₇	C₂⁵	1	7	yes	—
63	C₂×C₄×C₁₇	—	2	56	no	—
64	C₂×C₄×C₁₇	C₂³×C₄	1	112	yes	—
65	C₂×C₄×C₁₇	C₂²×Q₈	1	112	yes	complemented
66	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
67	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
68	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
69	C₂×Dic₁₇	—	2	112	no	—
70	C₂×Dic₁₇	C₂³×C₄	1	224	yes	—
71	C₂×Dic₁₇	C₂⁵	1	16	yes	—
72	C₂×Dic₁₇	C₂⁵	1	16	yes	—
73	Dic₃₄	C₂³×C₄	1	256	yes	direct product factor
74	C₂⁴×Q₈	—	17	1	no	—
75	C₂²×C₄×Q₈	—	17	56	no	—
76	C₂³×C₄⋊C₄	—	17	2	no	—
77	C₂³×C₄⋊C₄	—	17	1	no	—
78	C₂³×C₄²	—	17	2	no	—
79	C₂³×C₄²	D₁₇	1	1	yes	characteristic
80	C₂²×C₁₇	C₂×C₄×Q₈	1	112	yes	complemented
81	C₂²×C₁₇	C₂⁴×C₄	1	14	yes	—
82	C₂²×C₁₇	C₂³×Q₈	1	14	yes	—
83	C₂²×C₁₇	C₂²×#41	1	14	yes	—
84	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
85	C₄×C₁₇	—	2	8	no	—
86	C₄×C₁₇	C₂⁴×C₄	1	16	yes	—
87	C₄×C₁₇	C₂³×Q₈	1	16	yes	complemented
88	Dic₁₇	—	2	16	no	—
89	Dic₁₇	C₂⁴×C₄	1	32	yes	—
90	C₂³×Q₈	—	17	32	no	—
91	C₂³×Q₈	—	17	28	no	—
92	C₂⁴×C₄	—	17	2	no	—
93	C₂⁴×C₄	—	17	2	no	—
94	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
95	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
96	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
97	C₂×C₄×Q₈	—	17	448	no	—
98	C₂²×C₄⋊C₄	—	17	56	no	—
99	C₂²×C₄⋊C₄	—	17	28	no	—
100	C₂²×C₄²	—	17	56	no	—
101	C₂²×C₄²	C₂×D₁₇	1	28	yes	—
102	C₂×C₁₇	C₂²×C₄×Q₈	1	28	yes	complemented
103	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
104	C₂×C₁₇	C₂⁴×Q₈	1	1	yes	characteristic
105	C₂×C₁₇	C₂³×#41	1	1	yes	characteristic
106	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
107	C₂²×Q₈	—	17	448	no	—
108	C₂²×Q₈	—	17	112	no	—
109	C₂³×C₄	—	2	8	no	—
110	C₂³×C₄	—	17	32	no	—
111	C₂³×C₄	—	17	28	no	—
112	C₂³×C₄	—	17	28	no	—
113	C₂³×C₄	—	34	16	no	—
114	C₂³×C₄	Dic₃₄	1	16	yes	direct product factor
115	C₂³×C₄	C₄×D₁₇	1	16	yes	—
116	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
117	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
118	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
119	C₄×Q₈	—	17	512	no	—
120	C₂×C₄⋊C₄	—	17	224	no	—
121	C₂×C₄⋊C₄	—	17	112	no	—
122	C₂×C₄²	—	17	224	no	—
123	C₂×C₄²	C₂²×D₁₇	1	112	yes	—
124	C₁₇	C₂³×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
125	C₂⁴	C₂×Dic₃₄	1	8	yes	—
126	C₂⁴	C₂×C₄×D₁₇	1	8	yes	—
127	C₂⁴	#63252	1	8	yes	—
128	C₂⁴	C₂³×D₁₇	1	7	yes	—
129	C₂×Q₈	—	17	896	no	—
130	C₂×Q₈	—	17	64	no	—
131	C₂²×C₄	—	2	56	no	—
132	C₂²×C₄	—	17	224	no	—
133	C₂²×C₄	—	17	56	no	—
134	C₂²×C₄	—	17	56	no	—
135	C₂²×C₄	—	34	112	no	—
136	C₂²×C₄	C₂×Dic₃₄	1	112	yes	direct product factor
137	C₂²×C₄	C₂×C₄×D₁₇	1	112	yes	—
138	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
139	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
140	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
141	C₄⋊C₄	—	17	128	no	—
142	C₄⋊C₄	—	17	64	no	—
143	C₄²	—	17	128	no	—
144	C₄²	C₂³×D₁₇	1	64	yes	—
145	C₂³	C₄×Dic₃₄	1	64	yes	direct product factor
146	C₂³	C₂²×Dic₃₄	1	28	yes	—
147	C₂³	C₂²×C₄×D₁₇	1	28	yes	—
148	C₂³	C₂×#63252	1	28	yes	—
149	C₂³	C₂⁴×D₁₇	1	7	yes	—
150	Q₈	—	17	256	no	—
151	C₂×C₄	—	2	56	no	—
152	C₂×C₄	—	17	224	no	—
153	C₂×C₄	—	17	16	no	—
154	C₂×C₄	—	17	16	no	—
155	C₂×C₄	—	34	112	no	—
156	C₂×C₄	C₂²×Dic₃₄	1	112	yes	direct product factor
157	C₂×C₄	C₂²×C₄×D₁₇	1	112	yes	—
158	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
159	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
160	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
161	C₂²	C₂×C₄×Dic₃₄	1	112	yes	direct product factor
162	C₂²	C₂³×C₄×D₁₇	1	14	yes	—
163	C₂²	C₂³×Dic₃₄	1	14	yes	—
164	C₂²	C₂²×#63252	1	14	yes	—
165	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
166	C₄	—	2	8	no	—
167	C₄	—	17	32	no	—
168	C₄	—	34	16	no	—
169	C₄	C₂³×C₄×D₁₇	1	16	yes	—
170	C₄	C₂³×Dic₃₄	1	16	yes	direct product factor
171	C₂	C₂²×C₄×Dic₃₄	1	28	yes	direct product factor
172	C₂	C₂⁴×C₄×D₁₇	1	1	yes	characteristic
173	C₂	C₂⁴×Dic₃₄	1	1	yes	characteristic
174	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, prime radical	C₂⁵×C₁₇	C₂³
cosocle	C₂²×C₁₇	C₂⁶
hypercenter	C₂³×C₄²	D₁₇
hypocenter	C₁₇	C₂³×C₄×Q₈
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. C₂²×C₄×Dic₃₄	C₂	direct (N × H)
7. C₂³×Dic₃₄	C₄	direct (N × H)
11. C₂×C₄×Dic₃₄	C₂²	direct (N × H)
22. C₂³×C₄×C₁₇	Q₈	semidirect
30. C₂²×Dic₃₄	C₂×C₄	direct (N × H)
36. C₄×Dic₃₄	C₂³	direct (N × H)
43. C₂²×C₄×C₁₇	C₂×Q₈	semidirect
51. C₂×Dic₃₄	C₂²×C₄	direct (N × H)
58. C₂³×C₁₇	C₄×Q₈	semidirect
65. C₂×C₄×C₁₇	C₂²×Q₈	semidirect
73. Dic₃₄	C₂³×C₄	direct (N × H)
80. C₂²×C₁₇	C₂×C₄×Q₈	semidirect
87. C₄×C₁₇	C₂³×Q₈	semidirect
102. C₂×C₁₇	C₂²×C₄×Q₈	semidirect
114. C₂³×C₄	Dic₃₄	direct (N × H)
124. C₁₇	C₂³×C₄×Q₈	semidirect
136. C₂²×C₄	C₂×Dic₃₄	direct (N × H)
145. C₂³	C₄×Dic₃₄	direct (N × H)
156. C₂×C₄	C₂²×Dic₃₄	direct (N × H)
161. C₂²	C₂×C₄×Dic₃₄	direct (N × H)
170. C₄	C₂³×Dic₃₄	direct (N × H)
171. C₂	C₂²×C₄×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
C₂⁴×Dic₃₄	2
C₂²×C₄×Dic₃₄	2
C₂³×C₄⋊Dic₁₇	2
C₂³×#63225	2
C₂³×C₄×Dic₁₇	2
C₂³×C₄²×C₁₇	2
C₂³×C₄×Q₈	17

Subgroups — C23×C4×Dic34 #17940893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×Dic₃₄ #17940893