Subgroups — C₂×C₃×D₄×Dic₇ #13061134

Order 1344.

852 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×C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×Dic₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₂²⋊Dic₇	C₂	1	2	yes	maximal, complemented
4	C₃×D₄×Dic₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₄×Dic₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂⁴×C₃×C₇	C₂²	1	2	yes	—
9	C₂×C₃×C₇×D₄	C₄	1	8	yes	complemented
10	C₂×C₃×C₇×D₄	C₂²	1	4	yes	—
11	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Dic₇	—	2	8	no	—
13	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
15	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Dic₇	C₂²	1	8	yes	complemented
17	C₃×C₂²⋊Dic₇	C₂²	1	8	yes	complemented
18	C₃×C₄⋊Dic₇	C₂²	1	4	yes	complemented
19	C₃×C₄×Dic₇	C₂²	1	4	yes	complemented
20	C₂²×C₇×D₄	C₂×C₃	1	1	yes	characteristic
21	C₂³×Dic₇	C₂×C₃	1	2	yes	complemented
22	C₂×C₂²⋊Dic₇	C₂×C₃	1	2	yes	complemented
23	D₄×Dic₇	C₂×C₃	1	8	yes	direct product factor
24	C₂×C₄⋊Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×C₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×C₃×C₄×D₄	—	7	1	no	maximal, Hall
27	C₂³×C₃×C₇	—	2	4	no	—
28	C₂³×C₃×C₇	—	2	4	no	—
29	C₂³×C₃×C₇	C₂×C₄	1	8	yes	complemented
30	C₂³×C₃×C₇	C₂³	1	4	yes	—
31	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
32	C₃×C₇×D₄	C₂×C₄	1	16	yes	complemented
33	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	complemented
34	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
35	C₂×C₃×Dic₇	—	2	4	no	—
36	C₂×C₃×Dic₇	—	2	2	no	—
37	C₂×C₃×Dic₇	—	2	16	no	—
38	C₂×C₃×Dic₇	D₄	1	4	yes	direct product factor
39	C₂×C₃×Dic₇	C₂³	1	2	yes	—
40	C₂×C₃×Dic₇	C₂³	1	4	yes	—
41	C₂×C₃×Dic₇	C₂³	1	2	yes	—
42	C₂⁴×C₇	C₂²×C₃	1	2	yes	—
43	C₂×C₇×D₄	C₃×C₄	1	8	yes	complemented
44	C₂×C₇×D₄	C₂²×C₃	1	4	yes	—
45	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
46	C₂²×Dic₇	—	2	8	no	—
47	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
48	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
49	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
50	C₂²×Dic₇	C₂²×C₃	1	8	yes	complemented
51	C₂²⋊Dic₇	C₂²×C₃	1	8	yes	complemented
52	C₄⋊Dic₇	C₂²×C₃	1	4	yes	complemented
53	C₄×Dic₇	C₂²×C₃	1	4	yes	complemented
54	C₂²×C₃×D₄	D₇	1	1	yes	characteristic
55	C₂³×C₃×C₄	—	7	2	no	—
56	C₃×C₄×D₄	—	7	8	no	—
57	C₂×C₃×C₄⋊C₄	—	7	1	no	—
58	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
59	C₂×C₃×C₄²	—	7	1	no	—
60	C₂²×C₃×C₇	—	2	16	no	—
61	C₂²×C₃×C₇	—	2	4	no	—
62	C₂²×C₃×C₇	—	2	4	no	—
63	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₃×C₇	C₂²×C₄	1	8	yes	complemented
65	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
66	C₂²×C₃×C₇	#41	1	2	yes	—
67	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
68	C₃×C₄×C₇	C₂²×C₄	1	4	yes	complemented
69	C₃×Dic₇	—	2	4	no	—
70	C₃×Dic₇	—	2	2	no	—
71	C₃×Dic₇	C₂×D₄	1	4	yes	direct product factor
72	C₂×C₄×D₄	—	7	1	no	Sylow
73	C₂³×C₇	—	2	4	no	—
74	C₂³×C₇	—	2	4	no	—
75	C₂³×C₇	C₂×C₃×C₄	1	8	yes	complemented
76	C₂³×C₇	C₂³×C₃	1	4	yes	—
77	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
78	C₇×D₄	C₂×C₃×C₄	1	16	yes	complemented
79	C₂×C₄×C₇	C₂×C₃×C₄	1	4	yes	complemented
80	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
81	C₂×Dic₇	—	2	4	no	—
82	C₂×Dic₇	—	2	2	no	—
83	C₂×Dic₇	—	2	16	no	—
84	C₂×Dic₇	C₃×D₄	1	4	yes	direct product factor
85	C₂×Dic₇	C₂³×C₃	1	2	yes	—
86	C₂×Dic₇	C₂³×C₃	1	4	yes	—
87	C₂×Dic₇	C₂³×C₃	1	2	yes	—
88	C₂⁴×C₃	C₂×D₇	1	2	yes	—
89	C₂×C₃×D₄	Dic₇	1	8	yes	direct product factor
90	C₂×C₃×D₄	C₂×D₇	1	4	yes	—
91	C₂²×C₃×C₄	—	7	1	no	—
92	C₂²×C₃×C₄	—	7	8	no	—
93	C₂²×C₃×C₄	—	7	1	no	—
94	C₂²×C₃×C₄	—	7	2	no	—
95	C₂²×C₃×C₄	—	14	8	no	—
96	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
97	C₃×C₄⋊C₄	—	7	4	no	—
98	C₃×C₂²⋊C₄	—	7	8	no	—
99	C₃×C₄²	—	7	4	no	—
100	C₂×C₃×C₇	—	2	8	no	—
101	C₂×C₃×C₇	C₄×D₄	1	4	yes	complemented
102	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
105	C₂²×D₄	C₃×D₇	1	1	yes	characteristic
106	C₂³×C₄	—	7	2	no	—
107	C₄×D₄	—	7	8	no	—
108	C₂×C₄⋊C₄	—	7	1	no	—
109	C₂×C₂²⋊C₄	—	7	2	no	—
110	C₂×C₄²	—	7	1	no	—
111	C₂²×C₇	—	2	16	no	—
112	C₂²×C₇	—	2	4	no	—
113	C₂²×C₇	—	2	4	no	—
114	C₂²×C₇	C₂²×C₃×C₄	1	2	yes	—
115	C₂²×C₇	C₂²×C₃×C₄	1	8	yes	complemented
116	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
117	C₂²×C₇	C₃×#41	1	2	yes	—
118	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
119	C₄×C₇	C₂²×C₃×C₄	1	4	yes	complemented
120	Dic₇	—	2	4	no	—
121	Dic₇	—	2	2	no	—
122	Dic₇	C₂×C₃×D₄	1	4	yes	direct product factor
123	C₂³×C₃	—	2	4	no	—
124	C₂³×C₃	—	2	4	no	—
125	C₂³×C₃	C₂×Dic₇	1	8	yes	complemented
126	C₂³×C₃	C₂²×D₇	1	4	yes	—
127	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
128	C₃×D₄	C₂×Dic₇	1	16	yes	direct product factor
129	C₂×C₃×C₄	—	7	4	no	—
130	C₂×C₃×C₄	—	7	2	no	—
131	C₂×C₃×C₄	—	7	2	no	—
132	C₂×C₃×C₄	—	7	4	no	—
133	C₂×C₃×C₄	—	14	16	no	—
134	C₂×C₃×C₄	—	14	4	no	—
135	C₂×C₃×C₄	—	14	2	no	—
136	C₂×C₃×C₄	C₂×Dic₇	1	4	yes	complemented
137	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
138	C₃×C₇	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Hall
139	C₂⁴	C₂×C₃×D₇	1	2	yes	—
140	C₂×D₄	C₃×Dic₇	1	8	yes	direct product factor
141	C₂×D₄	C₂×C₃×D₇	1	4	yes	—
142	C₂²×C₄	—	7	1	no	—
143	C₂²×C₄	—	7	1	no	—
144	C₂²×C₄	—	7	2	no	—
145	C₂²×C₄	—	7	8	no	—
146	C₂²×C₄	—	14	8	no	—
147	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
148	C₄⋊C₄	—	7	4	no	—
149	C₂²⋊C₄	—	7	8	no	—
150	C₄²	—	7	4	no	—
151	C₂×C₇	—	2	8	no	—
152	C₂×C₇	C₃×C₄×D₄	1	4	yes	complemented
153	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
154	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
155	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
156	C₂²×C₃	—	2	16	no	—
157	C₂²×C₃	—	2	4	no	—
158	C₂²×C₃	—	2	4	no	—
159	C₂²×C₃	D₄×D₇	1	2	yes	—
160	C₂²×C₃	#1159	1	2	yes	—
161	C₂²×C₃	C₂²×Dic₇	1	2	yes	—
162	C₂²×C₃	C₂²×Dic₇	1	8	yes	complemented
163	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
164	C₃×C₄	—	7	4	no	—
165	C₃×C₄	—	14	4	no	—
166	C₃×C₄	—	14	2	no	—
167	C₃×C₄	C₂²×Dic₇	1	4	yes	complemented
168	C₂³	—	2	4	no	—
169	C₂³	—	2	4	no	—
170	C₂³	C₂×C₃×Dic₇	1	8	yes	complemented
171	C₂³	C₂²×C₃×D₇	1	4	yes	—
172	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
173	D₄	C₂×C₃×Dic₇	1	16	yes	direct product factor
174	C₂×C₄	—	7	4	no	—
175	C₂×C₄	—	7	2	no	—
176	C₂×C₄	—	7	2	no	—
177	C₂×C₄	—	7	4	no	—
178	C₂×C₄	—	14	16	no	—
179	C₂×C₄	—	14	4	no	—
180	C₂×C₄	—	14	2	no	—
181	C₂×C₄	C₂×C₃×Dic₇	1	4	yes	complemented
182	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
183	C₇	C₂×C₃×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₃	—	2	8	no	—
185	C₂×C₃	D₄×Dic₇	1	4	yes	direct product factor
186	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
187	C₂×C₃	C₂×#1159	1	1	yes	characteristic
188	C₂×C₃	C₂³×Dic₇	1	1	yes	characteristic
189	C₂²	—	2	16	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	C₃×D₄×D₇	1	2	yes	—
193	C₂²	C₃×#1159	1	2	yes	—
194	C₂²	C₂²×C₃×Dic₇	1	2	yes	—
195	C₂²	C₂²×C₃×Dic₇	1	8	yes	complemented
196	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
197	C₄	—	7	4	no	—
198	C₄	—	14	4	no	—
199	C₄	—	14	2	no	—
200	C₄	C₂²×C₃×Dic₇	1	4	yes	complemented
201	C₃	C₂×D₄×Dic₇	1	1	yes	characteristic, direct product factor, Sylow
202	C₂	—	2	8	no	—
203	C₂	C₃×D₄×Dic₇	1	4	yes	direct product factor
204	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
205	C₂	C₂×C₃×#1159	1	1	yes	characteristic
206	C₂	C₂³×C₃×Dic₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×Dic₇	C₂	semidirect
3. C₂×C₃×C₂²⋊Dic₇	C₂	semidirect
4. C₃×D₄×Dic₇	C₂	direct (N × H)
5. C₂×C₃×C₄⋊Dic₇	C₂	semidirect
6. C₂×C₃×C₄×Dic₇	C₂	semidirect
7. C₂×D₄×Dic₇	C₃	direct (N × H)
9. C₂×C₃×C₇×D₄	C₄	semidirect
16. C₂²×C₃×Dic₇	C₂²	semidirect
17. C₃×C₂²⋊Dic₇	C₂²	semidirect
18. C₃×C₄⋊Dic₇	C₂²	semidirect
19. C₃×C₄×Dic₇	C₂²	semidirect
21. C₂³×Dic₇	C₂×C₃	semidirect
22. C₂×C₂²⋊Dic₇	C₂×C₃	semidirect
23. D₄×Dic₇	C₂×C₃	direct (N × H)
24. C₂×C₄⋊Dic₇	C₂×C₃	semidirect
25. C₂×C₄×Dic₇	C₂×C₃	semidirect
29. C₂³×C₃×C₇	C₂×C₄	semidirect
32. C₃×C₇×D₄	C₂×C₄	semidirect
33. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
38. C₂×C₃×Dic₇	D₄	direct (N × H)
43. C₂×C₇×D₄	C₃×C₄	semidirect
50. C₂²×Dic₇	C₂²×C₃	semidirect
51. C₂²⋊Dic₇	C₂²×C₃	semidirect
52. C₄⋊Dic₇	C₂²×C₃	semidirect
53. C₄×Dic₇	C₂²×C₃	semidirect
64. C₂²×C₃×C₇	C₂²×C₄	semidirect
68. C₃×C₄×C₇	C₂²×C₄	semidirect
71. C₃×Dic₇	C₂×D₄	direct (N × H)
75. C₂³×C₇	C₂×C₃×C₄	semidirect
78. C₇×D₄	C₂×C₃×C₄	semidirect
79. C₂×C₄×C₇	C₂×C₃×C₄	semidirect
84. C₂×Dic₇	C₃×D₄	direct (N × H)
89. C₂×C₃×D₄	Dic₇	direct (N × H)
101. C₂×C₃×C₇	C₄×D₄	semidirect
115. C₂²×C₇	C₂²×C₃×C₄	semidirect
119. C₄×C₇	C₂²×C₃×C₄	semidirect
122. Dic₇	C₂×C₃×D₄	direct (N × H)
125. C₂³×C₃	C₂×Dic₇	semidirect
128. C₃×D₄	C₂×Dic₇	direct (N × H)
136. C₂×C₃×C₄	C₂×Dic₇	semidirect
138. C₃×C₇	C₂×C₄×D₄	semidirect
140. C₂×D₄	C₃×Dic₇	direct (N × H)
152. C₂×C₇	C₃×C₄×D₄	semidirect
162. C₂²×C₃	C₂²×Dic₇	semidirect
167. C₃×C₄	C₂²×Dic₇	semidirect
170. C₂³	C₂×C₃×Dic₇	semidirect
173. D₄	C₂×C₃×Dic₇	direct (N × H)
181. C₂×C₄	C₂×C₃×Dic₇	semidirect
183. C₇	C₂×C₃×C₄×D₄	semidirect
185. C₂×C₃	D₄×Dic₇	direct (N × H)
195. C₂²	C₂²×C₃×Dic₇	semidirect
200. C₄	C₂²×C₃×Dic₇	semidirect
201. C₃	C₂×D₄×Dic₇	direct (N × H)
203. C₂	C₃×D₄×Dic₇	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×C₇×D₄	2
C₂³×C₃×Dic₇	2
C₂×C₃×C₂²⋊Dic₇	2
C₃×D₄×Dic₇	2
C₂×C₃×C₄⋊Dic₇	2
C₂×C₃×C₄×Dic₇	2
C₂×D₄×Dic₇	3
C₂×C₃×C₄×D₄	7

Subgroups — C2×C3×D4×Dic7 #13061134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₄×Dic₇ #13061134