Subgroups — C₂×Dic₇×2₋¹⁺⁴ #14260406

Order 1792.

11210 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	C₂²×C₇×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	Dic₇×2₋¹⁺⁴	C₂	1	32	yes	maximal, direct product factor
3	C₂×#90518	C₂	1	10	yes	maximal, complemented
4	C₂×#41×Dic₇	C₂	1	10	yes	maximal, complemented
5	C₂×#90503	C₂	1	5	yes	maximal, complemented
6	C₂²×Q₈×Dic₇	C₂	1	5	yes	maximal, complemented
7	C₂×C₇×2₋¹⁺⁴	C₄	1	32	yes	complemented
8	C₂×C₇×2₋¹⁺⁴	C₂²	1	16	yes	—
9	C₂²×C₇×#41	C₂²	1	10	yes	—
10	C₂³×C₇×Q₈	C₂²	1	5	yes	—
11	#90518	C₂²	1	160	yes	complemented
12	#41×Dic₇	C₂²	1	160	yes	complemented
13	#90503	C₂²	1	80	yes	complemented
14	C₂×Q₈×Dic₇	C₂²	1	10	yes	—
15	C₂×Q₈×Dic₇	C₂²	1	20	yes	—
16	C₂×Q₈×Dic₇	C₂²	1	10	yes	—
17	C₂×Q₈×Dic₇	C₂²	1	80	yes	complemented
18	C₂×D₄×Dic₇	C₂²	1	10	yes	—
19	C₂×D₄×Dic₇	C₂²	1	30	yes	—
20	C₂×#6489	C₂²	1	30	yes	—
21	C₂²×C₄⋊Dic₇	C₂²	1	15	yes	—
22	C₂²×C₄×Dic₇	C₂²	1	15	yes	—
23	C₂×C₄×2₋¹⁺⁴	—	7	1	no	maximal, Sylow
24	C₇×2₋¹⁺⁴	C₂×C₄	1	256	yes	complemented
25	C₂×C₇×#41	C₂×C₄	1	160	yes	complemented
26	C₂×C₇×#41	C₂³	1	80	yes	—
27	C₂²×C₇×Q₈	C₂×C₄	1	80	yes	complemented
28	C₂²×C₇×Q₈	C₂³	1	10	yes	—
29	C₂²×C₇×Q₈	C₂³	1	40	yes	—
30	C₂²×C₇×D₄	C₂³	1	10	yes	—
31	C₂³×C₄×C₇	C₂³	1	15	yes	—
32	C₂³×Dic₇	C₂³	1	5	yes	—
33	C₂³×Dic₇	C₂³	1	15	yes	—
34	C₂×C₂²⋊Dic₇	C₂³	1	20	yes	—
35	Q₈×Dic₇	C₂³	1	80	yes	—
36	Q₈×Dic₇	C₂³	1	160	yes	—
37	Q₈×Dic₇	C₂³	1	80	yes	—
38	D₄×Dic₇	C₂³	1	80	yes	—
39	D₄×Dic₇	C₂³	1	240	yes	—
40	#6489	C₂³	1	240	yes	—
41	C₂×C₄⋊Dic₇	C₂³	1	30	yes	—
42	C₂×C₄⋊Dic₇	C₂³	1	120	yes	—
43	C₂×C₄⋊Dic₇	C₂³	1	10	yes	—
44	C₂×C₄×Dic₇	C₂³	1	10	yes	—
45	C₂×C₄×Dic₇	C₂³	1	30	yes	—
46	C₂×C₄×Dic₇	C₂³	1	120	yes	—
47	C₂²×2₋¹⁺⁴	D₇	1	1	yes	characteristic
48	C₄×2₋¹⁺⁴	—	7	32	no	—
49	C₂×#520	—	7	10	no	—
50	C₂×#519	—	7	5	no	—
51	C₂×C₄×#41	—	7	10	no	—
52	C₂²×C₄×Q₈	—	7	5	no	—
53	C₂⁴×C₇	C₂⁴	1	5	yes	—
54	C₇×#41	C₂²×C₄	1	640	yes	complemented
55	C₂×C₇×Q₈	C₂²×C₄	1	80	yes	—
56	C₂×C₇×Q₈	C₂²×C₄	1	320	yes	complemented
57	C₂×C₇×Q₈	C₂⁴	1	40	yes	—
58	C₂×C₇×D₄	C₂²×C₄	1	80	yes	—
59	C₂×C₇×D₄	C₂⁴	1	40	yes	—
60	C₂²×C₄×C₇	C₂²×C₄	1	120	yes	—
61	C₂²×C₄×C₇	C₂⁴	1	10	yes	—
62	C₂²×C₄×C₇	C₂⁴	1	60	yes	—
63	C₂²×Dic₇	—	2	20	no	—
64	C₂²×Dic₇	—	4	30	no	—
65	C₂²×Dic₇	C₂⁴	1	5	yes	—
66	C₂²×Dic₇	C₂⁴	1	20	yes	—
67	C₂²×Dic₇	C₂⁴	1	10	yes	—
68	C₂²×Dic₇	C₂⁴	1	60	yes	—
69	C₂²×Dic₇	C₂⁴	1	1	yes	characteristic
70	C₂²⋊Dic₇	C₂⁴	1	80	yes	—
71	C₄⋊Dic₇	C₂⁴	1	120	yes	—
72	C₄⋊Dic₇	C₂⁴	1	40	yes	—
73	C₄×Dic₇	C₂⁴	1	40	yes	—
74	C₄×Dic₇	C₂⁴	1	120	yes	—
75	C₂×2₋¹⁺⁴	Dic₇	1	32	yes	direct product factor
76	C₂×2₋¹⁺⁴	C₂×D₇	1	16	yes	—
77	C₂²×#41	C₂×D₇	1	10	yes	—
78	C₂³×Q₈	C₂×D₇	1	5	yes	—
79	#520	—	7	160	no	—
80	#519	—	7	80	no	—
81	C₄×#41	—	7	160	no	—
82	C₂×C₄×Q₈	—	7	10	no	—
83	C₂×C₄×Q₈	—	7	20	no	—
84	C₂×C₄×Q₈	—	7	80	no	—
85	C₂×C₄×Q₈	—	7	10	no	—
86	C₂×C₄×D₄	—	7	10	no	—
87	C₂×C₄×D₄	—	7	30	no	—
88	C₂×#117	—	7	30	no	—
89	C₂²×C₄⋊C₄	—	7	15	no	—
90	C₂²×C₄²	—	7	15	no	—
91	C₂³×C₇	—	2	10	no	—
92	C₂³×C₇	—	2	10	no	—
93	C₂³×C₇	C₂³×C₄	1	20	yes	—
94	C₂³×C₇	C₂⁵	1	10	yes	—
95	C₂³×C₇	C₂⁵	1	1	yes	characteristic
96	C₇×Q₈	C₂³×C₄	1	160	yes	—
97	C₇×D₄	C₂³×C₄	1	160	yes	—
98	C₂×C₄×C₇	C₂³×C₄	1	40	yes	—
99	C₂×C₄×C₇	C₂³×C₄	1	240	yes	—
100	C₂×C₄×C₇	C₂⁵	1	20	yes	—
101	C₂×Dic₇	—	2	10	no	—
102	C₂×Dic₇	—	2	20	no	—
103	C₂×Dic₇	—	2	40	no	—
104	C₂×Dic₇	—	4	60	no	—
105	C₂×Dic₇	2₋¹⁺⁴	1	4	yes	direct product factor
106	C₂×Dic₇	C₂⁵	1	10	yes	—
107	C₂×Dic₇	C₂⁵	1	20	yes	—
108	C₂×Dic₇	C₂⁵	1	2	yes	—
109	2₋¹⁺⁴	C₂×Dic₇	1	256	yes	direct product factor
110	C₂×#41	C₂×Dic₇	1	160	yes	complemented
111	C₂×#41	C₂²×D₇	1	80	yes	—
112	C₂²×Q₈	C₂×Dic₇	1	80	yes	complemented
113	C₂²×Q₈	C₂²×D₇	1	10	yes	—
114	C₂²×Q₈	C₂²×D₇	1	40	yes	—
115	C₂²×D₄	C₂²×D₇	1	10	yes	—
116	C₂³×C₄	—	7	5	no	—
117	C₂³×C₄	—	7	15	no	—
118	C₂³×C₄	C₂²×D₇	1	15	yes	—
119	C₄×Q₈	—	7	80	no	—
120	C₄×Q₈	—	7	160	no	—
121	C₄×Q₈	—	7	80	no	—
122	C₄×D₄	—	7	80	no	—
123	C₄×D₄	—	7	240	no	—
124	#117	—	7	240	no	—
125	C₂×C₄⋊C₄	—	7	30	no	—
126	C₂×C₄⋊C₄	—	7	120	no	—
127	C₂×C₄⋊C₄	—	7	10	no	—
128	C₂×C₂²⋊C₄	—	7	20	no	—
129	C₂×C₄²	—	7	30	no	—
130	C₂×C₄²	—	7	10	no	—
131	C₂×C₄²	—	7	120	no	—
132	C₂²×C₇	—	2	10	no	—
133	C₂²×C₇	—	2	10	no	—
134	C₂²×C₇	—	2	40	no	—
135	C₂²×C₇	C₂⁴×C₄	1	20	yes	—
136	C₂²×C₇	C₂⁴×C₄	1	2	yes	—
137	C₂²×C₇	C₂×2₋¹⁺⁴	1	2	yes	—
138	C₂²×C₇	#585	1	2	yes	—
139	C₂²×C₇	C₂⁶	1	1	yes	characteristic
140	C₄×C₇	C₂⁴×C₄	1	40	yes	—
141	Dic₇	—	2	10	no	—
142	Dic₇	—	2	20	no	—
143	Dic₇	C₂×2₋¹⁺⁴	1	4	yes	direct product factor
144	C₂⁴	C₂³×D₇	1	5	yes	—
145	#41	C₂²×Dic₇	1	640	yes	complemented
146	C₂×Q₈	C₂²×Dic₇	1	80	yes	—
147	C₂×Q₈	C₂²×Dic₇	1	320	yes	complemented
148	C₂×Q₈	C₂³×D₇	1	40	yes	—
149	C₂×D₄	C₂²×Dic₇	1	80	yes	—
150	C₂×D₄	C₂³×D₇	1	40	yes	—
151	C₂²×C₄	—	7	1	no	—
152	C₂²×C₄	—	7	20	no	—
153	C₂²×C₄	—	7	5	no	—
154	C₂²×C₄	—	7	10	no	—
155	C₂²×C₄	—	7	60	no	—
156	C₂²×C₄	—	14	20	no	—
157	C₂²×C₄	—	28	30	no	—
158	C₂²×C₄	C₂²×Dic₇	1	120	yes	—
159	C₂²×C₄	C₂³×D₇	1	60	yes	—
160	C₂²×C₄	C₂³×D₇	1	10	yes	—
161	C₄⋊C₄	—	7	120	no	—
162	C₄⋊C₄	—	7	40	no	—
163	C₂²⋊C₄	—	7	80	no	—
164	C₄²	—	7	40	no	—
165	C₄²	—	7	120	no	—
166	C₂×C₇	—	2	20	no	—
167	C₂×C₇	C₄×2₋¹⁺⁴	1	4	yes	complemented
168	C₂×C₇	C₂⁵×C₄	1	1	yes	characteristic
169	C₂×C₇	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
170	C₂×C₇	C₂×#585	1	1	yes	characteristic
171	C₂³	—	2	10	no	—
172	C₂³	—	2	10	no	—
173	C₂³	C₂³×Dic₇	1	20	yes	—
174	C₂³	C₂⁴×D₇	1	10	yes	—
175	C₂³	C₂⁴×D₇	1	1	yes	characteristic
176	Q₈	C₂³×Dic₇	1	160	yes	—
177	D₄	C₂³×Dic₇	1	160	yes	—
178	C₂×C₄	—	7	2	no	—
179	C₂×C₄	—	7	4	no	—
180	C₂×C₄	—	7	10	no	—
181	C₂×C₄	—	7	20	no	—
182	C₂×C₄	—	14	40	no	—
183	C₂×C₄	—	14	10	no	—
184	C₂×C₄	—	14	20	no	—
185	C₂×C₄	—	28	60	no	—
186	C₂×C₄	C₂³×Dic₇	1	40	yes	—
187	C₂×C₄	C₂³×Dic₇	1	240	yes	—
188	C₂×C₄	C₂⁴×D₇	1	20	yes	—
189	C₇	C₂×C₄×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
190	C₂²	—	2	10	no	—
191	C₂²	—	2	10	no	—
192	C₂²	—	2	40	no	—
193	C₂²	D₇×2₋¹⁺⁴	1	2	yes	—
194	C₂²	#90620	1	2	yes	—
195	C₂²	C₂⁴×Dic₇	1	20	yes	—
196	C₂²	C₂⁴×Dic₇	1	2	yes	—
197	C₂²	C₂⁵×D₇	1	1	yes	characteristic
198	C₄	—	7	4	no	—
199	C₄	—	14	20	no	—
200	C₄	—	14	10	no	—
201	C₄	C₂⁴×Dic₇	1	40	yes	—
202	C₂	—	2	20	no	—
203	C₂	Dic₇×2₋¹⁺⁴	1	4	yes	direct product factor
204	C₂	C₂×#90620	1	1	yes	characteristic
205	C₂	C₂×D₇×2₋¹⁺⁴	1	1	yes	characteristic
206	C₂	C₂⁵×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₇
commutator	C₂×C₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₇
Fitting, prime radical	C₂²×C₇×2₋¹⁺⁴	C₂
socle	C₂³×C₇	C₂⁵
cosocle	C₂²×C₇	C₂⁶
hypercenter	C₂²×2₋¹⁺⁴	D₇
hypocenter	C₇	C₂×C₄×2₋¹⁺⁴
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
2. Dic₇×2₋¹⁺⁴	C₂	direct (N × H)
3. C₂×#90518	C₂	semidirect
4. C₂×#41×Dic₇	C₂	semidirect
5. C₂×#90503	C₂	semidirect
6. C₂²×Q₈×Dic₇	C₂	semidirect
7. C₂×C₇×2₋¹⁺⁴	C₄	semidirect
11. #90518	C₂²	semidirect
12. #41×Dic₇	C₂²	semidirect
13. #90503	C₂²	semidirect
17. C₂×Q₈×Dic₇	C₂²	semidirect
24. C₇×2₋¹⁺⁴	C₂×C₄	semidirect
25. C₂×C₇×#41	C₂×C₄	semidirect
27. C₂²×C₇×Q₈	C₂×C₄	semidirect
54. C₇×#41	C₂²×C₄	semidirect
56. C₂×C₇×Q₈	C₂²×C₄	semidirect
75. C₂×2₋¹⁺⁴	Dic₇	direct (N × H)
105. C₂×Dic₇	2₋¹⁺⁴	direct (N × H)
109. 2₋¹⁺⁴	C₂×Dic₇	direct (N × H)
110. C₂×#41	C₂×Dic₇	semidirect
112. C₂²×Q₈	C₂×Dic₇	semidirect
143. Dic₇	C₂×2₋¹⁺⁴	direct (N × H)
145. #41	C₂²×Dic₇	semidirect
147. C₂×Q₈	C₂²×Dic₇	semidirect
167. C₂×C₇	C₄×2₋¹⁺⁴	semidirect
189. C₇	C₂×C₄×2₋¹⁺⁴	semidirect
203. C₂	Dic₇×2₋¹⁺⁴	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×C₇×2₋¹⁺⁴	2
Dic₇×2₋¹⁺⁴	2
C₂×#90518	2
C₂×#41×Dic₇	2
C₂×#90503	2
C₂²×Q₈×Dic₇	2
C₂×C₄×2₋¹⁺⁴	7

Subgroups — C2×Dic7×2−1+4 #14260406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₇×2₋¹⁺⁴ #14260406