Subgroups — C₃×D₄×Dic₁₄ #13060878

Order 1344.

560 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₁₄	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃×D₄×Dic₇	C₂	1	2	yes	maximal
4	C₃×#6487	C₂	1	2	yes	maximal, complemented
5	C₃×#6451	C₂	1	2	yes	maximal, complemented
6	C₃×#6441	C₂	1	4	yes	maximal, complemented
7	C₃×#6432	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×Dic₁₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
12	C₂²×C₃×Dic₇	C₂²	1	4	yes	—
13	C₂×C₃×Dic₁₄	—	2	8	no	—
14	C₂×C₃×Dic₁₄	C₂²	1	4	yes	—
15	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
16	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
17	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₃×C₂²⋊Dic₇	C₂²	1	4	yes	—
21	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
22	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
23	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
24	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
25	C₃×Dic₇⋊C₄	C₂²	1	4	yes	—
26	C₃×C₄×Dic₇	C₂²	1	2	yes	—
27	C₂²×Dic₁₄	C₂×C₃	1	2	yes	complemented
28	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic
29	D₄×Dic₇	C₂×C₃	1	2	yes	—
30	#6487	C₂×C₃	1	2	yes	complemented
31	#6451	C₂×C₃	1	2	yes	complemented
32	#6441	C₂×C₃	1	4	yes	complemented
33	#6432	C₂×C₃	1	1	yes	characteristic, complemented
34	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
35	C₃×D₄×Q₈	—	7	1	no	maximal, Hall
36	C₂³×C₃×C₇	C₂³	1	2	yes	—
37	C₃×C₇×D₄	Q₈	1	4	yes	complemented
38	C₂×C₃×C₄×C₇	—	2	4	no	—
39	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂×C₃×Dic₇	—	2	8	no	—
44	C₂×C₃×Dic₇	C₂³	1	2	yes	—
45	C₂×C₃×Dic₇	C₂³	1	4	yes	—
46	C₂×C₃×Dic₇	C₂³	1	2	yes	—
47	C₃×Dic₁₄	—	2	4	no	—
48	C₃×Dic₁₄	—	2	8	no	—
49	C₃×Dic₁₄	D₄	1	4	yes	direct product factor
50	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
52	C₂²×Dic₇	C₂²×C₃	1	4	yes	—
53	C₂×Dic₁₄	—	2	8	no	—
54	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
55	C₂×Dic₁₄	C₂²×C₃	1	4	yes	—
56	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
57	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
58	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
59	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
60	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	—
61	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
62	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
63	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
64	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
65	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
66	C₄×Dic₇	C₂²×C₃	1	2	yes	—
67	C₂²×C₃×Q₈	—	7	2	no	—
68	C₃×C₄⋊Q₈	—	7	2	no	—
69	C₃×C₄⋊Q₈	—	7	1	no	—
70	C₃×C₂²⋊Q₈	—	7	2	no	—
71	C₃×C₂²⋊Q₈	—	7	4	no	—
72	C₃×C₄×Q₈	—	7	1	no	—
73	C₃×C₄×D₄	—	7	2	no	—
74	C₃×C₄×D₄	D₇	1	1	yes	characteristic
75	C₂²×C₃×C₇	—	2	2	no	—
76	C₂²×C₃×C₇	—	2	2	no	—
77	C₂²×C₃×C₇	C₂×Q₈	1	4	yes	complemented
78	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
79	C₃×C₄×C₇	—	2	2	no	—
80	C₃×C₄×C₇	—	2	1	no	—
81	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
82	C₃×C₄×C₇	C₂×Q₈	1	2	yes	complemented
83	C₃×Dic₇	—	2	4	no	—
84	C₃×Dic₇	—	2	2	no	—
85	C₃×Dic₇	C₂×D₄	1	4	yes	—
86	D₄×Q₈	—	7	1	no	Sylow
87	C₂³×C₇	C₂³×C₃	1	2	yes	—
88	C₇×D₄	C₃×Q₈	1	4	yes	complemented
89	C₂×C₄×C₇	—	2	4	no	—
90	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
92	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
94	C₂×Dic₇	—	2	8	no	—
95	C₂×Dic₇	C₂³×C₃	1	2	yes	—
96	C₂×Dic₇	C₂³×C₃	1	4	yes	—
97	C₂×Dic₇	C₂³×C₃	1	2	yes	—
98	Dic₁₄	—	2	4	no	—
99	Dic₁₄	—	2	8	no	—
100	Dic₁₄	C₃×D₄	1	4	yes	direct product factor
101	C₂×C₃×Q₈	—	7	1	no	—
102	C₂×C₃×Q₈	—	7	4	no	—
103	C₂×C₃×Q₈	—	7	2	no	—
104	C₂×C₃×Q₈	—	14	8	no	—
105	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
106	C₂²×C₃×C₄	—	7	4	no	—
107	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
108	C₃×C₄⋊C₄	—	7	2	no	—
109	C₃×C₄⋊C₄	—	7	4	no	—
110	C₃×C₄⋊C₄	—	7	2	no	—
111	C₃×C₄⋊C₄	—	7	1	no	—
112	C₃×C₄⋊C₄	—	7	2	no	—
113	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
114	C₃×C₂²⋊C₄	—	7	4	no	—
115	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
116	C₃×C₄²	—	7	2	no	—
117	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
118	C₂×C₃×C₇	—	2	4	no	—
119	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂²×Q₈	—	7	2	no	—
123	C₄⋊Q₈	—	7	2	no	—
124	C₄⋊Q₈	—	7	1	no	—
125	C₂²⋊Q₈	—	7	2	no	—
126	C₂²⋊Q₈	—	7	4	no	—
127	C₄×Q₈	—	7	1	no	—
128	C₄×D₄	—	7	2	no	—
129	C₄×D₄	C₃×D₇	1	1	yes	characteristic
130	C₂²×C₇	—	2	2	no	—
131	C₂²×C₇	—	2	2	no	—
132	C₂²×C₇	C₂×C₃×Q₈	1	4	yes	complemented
133	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
134	C₄×C₇	—	2	2	no	—
135	C₄×C₇	—	2	1	no	—
136	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
137	C₄×C₇	C₂×C₃×Q₈	1	2	yes	complemented
138	Dic₇	—	2	4	no	—
139	Dic₇	—	2	2	no	—
140	Dic₇	C₂×C₃×D₄	1	4	yes	—
141	C₂³×C₃	C₂²×D₇	1	2	yes	—
142	C₃×Q₈	—	7	4	no	—
143	C₃×Q₈	—	14	8	no	—
144	C₃×Q₈	—	14	4	no	—
145	C₃×D₄	Dic₁₄	1	4	yes	direct product factor
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	7	2	no	—
148	C₂×C₃×C₄	—	7	2	no	—
149	C₂×C₃×C₄	—	7	4	no	—
150	C₂×C₃×C₄	—	14	8	no	—
151	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
153	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
155	C₃×C₇	D₄×Q₈	1	1	yes	characteristic, complemented, Hall
156	C₂×Q₈	—	7	1	no	—
157	C₂×Q₈	—	7	4	no	—
158	C₂×Q₈	—	7	2	no	—
159	C₂×Q₈	—	14	8	no	—
160	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
161	C₂²×C₄	—	7	4	no	—
162	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
163	C₄⋊C₄	—	7	2	no	—
164	C₄⋊C₄	—	7	4	no	—
165	C₄⋊C₄	—	7	2	no	—
166	C₄⋊C₄	—	7	1	no	—
167	C₄⋊C₄	—	7	2	no	—
168	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
169	C₂²⋊C₄	—	7	4	no	—
170	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
171	C₄²	—	7	2	no	—
172	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
173	C₂×C₇	—	2	4	no	—
174	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
175	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
176	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	C₂×Dic₁₄	1	4	yes	complemented
180	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	7	4	no	—
184	C₃×C₄	—	14	4	no	—
185	C₃×C₄	—	14	2	no	—
186	C₃×C₄	C₂×Dic₁₄	1	2	yes	complemented
187	C₃×C₄	D₄×D₇	1	2	yes	—
188	C₂³	C₂²×C₃×D₇	1	2	yes	—
189	Q₈	—	7	4	no	—
190	Q₈	—	14	8	no	—
191	Q₈	—	14	4	no	—
192	D₄	C₃×Dic₁₄	1	4	yes	direct product factor
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	7	2	no	—
195	C₂×C₄	—	7	2	no	—
196	C₂×C₄	—	7	4	no	—
197	C₂×C₄	—	14	8	no	—
198	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
200	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
201	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
202	C₇	C₃×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
203	C₂×C₃	—	2	4	no	—
204	C₂×C₃	C₂²×Dic₁₄	1	1	yes	characteristic
205	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
206	C₂×C₃	#6554	1	1	yes	characteristic
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	C₂×C₃×Dic₁₄	1	4	yes	complemented
210	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	2	1	no	—
213	C₄	—	7	4	no	—
214	C₄	—	14	4	no	—
215	C₄	—	14	2	no	—
216	C₄	C₂×C₃×Dic₁₄	1	2	yes	complemented
217	C₄	C₃×D₄×D₇	1	2	yes	—
218	C₃	D₄×Dic₁₄	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	2	4	no	—
220	C₂	C₂²×C₃×Dic₁₄	1	1	yes	characteristic
221	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
222	C₂	C₃×#6554	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₄×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×C₃×C₇×D₄	C₂²
hypercenter	C₃×C₄×D₄	D₇
hypocenter	C₇	C₃×D₄×Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Dic₁₄	C₂	semidirect
4. C₃×#6487	C₂	semidirect
5. C₃×#6451	C₂	semidirect
6. C₃×#6441	C₂	semidirect
7. C₃×#6432	C₂	semidirect
8. C₃×C₄×Dic₁₄	C₂	semidirect
9. D₄×Dic₁₄	C₃	direct (N × H)
27. C₂²×Dic₁₄	C₂×C₃	semidirect
30. #6487	C₂×C₃	semidirect
31. #6451	C₂×C₃	semidirect
32. #6441	C₂×C₃	semidirect
33. #6432	C₂×C₃	semidirect
34. C₄×Dic₁₄	C₂×C₃	semidirect
37. C₃×C₇×D₄	Q₈	semidirect
49. C₃×Dic₁₄	D₄	direct (N × H)
77. C₂²×C₃×C₇	C₂×Q₈	semidirect
82. C₃×C₄×C₇	C₂×Q₈	semidirect
88. C₇×D₄	C₃×Q₈	semidirect
100. Dic₁₄	C₃×D₄	direct (N × H)
132. C₂²×C₇	C₂×C₃×Q₈	semidirect
137. C₄×C₇	C₂×C₃×Q₈	semidirect
145. C₃×D₄	Dic₁₄	direct (N × H)
155. C₃×C₇	D₄×Q₈	semidirect
179. C₂²×C₃	C₂×Dic₁₄	semidirect
186. C₃×C₄	C₂×Dic₁₄	semidirect
192. D₄	C₃×Dic₁₄	direct (N × H)
202. C₇	C₃×D₄×Q₈	semidirect
209. C₂²	C₂×C₃×Dic₁₄	semidirect
216. C₄	C₂×C₃×Dic₁₄	semidirect
218. C₃	D₄×Dic₁₄	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×C₃×Dic₁₄	2
C₃×C₄×C₇×D₄	2
C₃×D₄×Dic₇	2
C₃×#6487	2
C₃×#6451	2
C₃×#6441	2
C₃×#6432	2
C₃×C₄×Dic₁₄	2
D₄×Dic₁₄	3
C₃×D₄×Q₈	7

Subgroups — C3×D4×Dic14 #13060878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×Dic₁₄ #13060878