Subgroups — C₂×C₇×(C₂×D₃)⋊C₈ #13056142

Order 1344.

404 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₃×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×(C₂×D₃)⋊C₈	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₇×C₂²⋊C₈	—	3	1	no	maximal, Hall
6	C₂³×D₃×C₇	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂²×C₇×Dic₃	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃×C₇	—	2	2	no	—
10	C₂×C₄×D₃×C₇	—	2	2	no	—
11	C₂×C₄×D₃×C₇	C₄	1	2	yes	—
12	C₂×C₄×D₃×C₇	C₂²	1	2	yes	—
13	C₂×C₃×C₇×C₈	—	2	2	no	—
14	C₂×C₃×C₇×C₈	C₂²	1	2	yes	complemented
15	C₂×C₇×C₃⋊C₈	—	2	2	no	—
16	C₂×C₇×C₃⋊C₈	C₂²	1	2	yes	complemented
17	C₂³×C₄×C₇	—	3	1	no	—
18	C₂²×C₇×C₈	—	3	1	no	—
19	C₂²×C₇×C₈	D₃	1	1	yes	characteristic, complemented
20	C₇×C₂²⋊C₈	—	3	4	no	—
21	C₂×(C₂×D₃)⋊C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×D₃×C₇	—	2	2	no	—
24	C₂²×D₃×C₇	—	2	2	no	—
25	C₂²×D₃×C₇	C₈	1	4	yes	complemented
26	C₂²×D₃×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
29	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
30	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂×C₇×Dic₃	—	2	2	no	—
32	C₂×C₇×Dic₃	C₂×C₄	1	2	yes	—
33	C₄×D₃×C₇	—	2	4	no	—
34	C₄×D₃×C₇	—	2	2	no	—
35	C₄×D₃×C₇	—	2	2	no	—
36	C₃×C₇×C₈	—	2	2	no	—
37	C₇×C₃⋊C₈	—	2	2	no	—
38	C₂⁴×C₇	—	3	1	no	—
39	C₂²×C₄×C₇	—	3	1	no	—
40	C₂²×C₄×C₇	—	3	2	no	—
41	C₂²×C₄×C₇	—	3	2	no	—
42	C₂²×C₄×C₇	—	6	2	no	—
43	C₂²×C₄×C₇	—	6	2	no	—
44	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
45	C₂×C₇×C₈	—	2	2	no	—
46	C₂×C₇×C₈	—	3	2	no	—
47	C₂×C₇×C₈	—	6	2	no	—
48	C₂×C₇×C₈	C₂×D₃	1	2	yes	complemented
49	C₂²×C₄×D₃	C₂×C₇	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂×C₇	1	1	yes	characteristic, complemented
51	C₂²×C₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
52	(C₂×D₃)⋊C₈	C₂×C₇	1	4	yes	direct product factor
53	C₂²×C₃×C₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₃×C₇	C₂×C₈	1	2	yes	—
55	C₂²×C₃×C₇	M₁₆	1	2	yes	—
56	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×D₃×C₇	—	2	8	no	—
58	C₂×D₃×C₇	—	2	2	no	—
59	C₂×D₃×C₇	—	2	2	no	—
60	C₂×D₃×C₇	C₂×C₈	1	4	yes	complemented
61	C₃×C₄×C₇	C₂²⋊C₄	1	2	yes	—
62	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
63	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₇×Dic₃	—	2	2	no	—
65	C₂×C₂²⋊C₈	—	3	1	no	Sylow
66	C₂³×C₇	—	3	4	no	—
67	C₂³×C₇	—	3	2	no	—
68	C₂³×C₇	—	6	2	no	—
69	C₂³×C₇	—	6	2	no	—
70	C₂³×C₇	C₄×D₃	1	1	yes	characteristic
71	C₂×C₄×C₇	—	3	2	no	—
72	C₂×C₄×C₇	—	6	2	no	—
73	C₂×C₄×C₇	—	6	4	no	—
74	C₂×C₄×C₇	—	6	2	no	—
75	C₂×C₄×C₇	—	6	2	no	—
76	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
77	C₂×C₄×C₇	D₁₂	1	2	yes	—
78	C₂×C₄×C₇	#67	1	2	yes	—
79	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
80	C₇×C₈	—	2	2	no	—
81	C₇×C₈	—	6	2	no	—
82	C₂³×D₃	C₄×C₇	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
84	C₂²×Dic₃	C₄×C₇	1	1	yes	characteristic
85	C₂×C₄×D₃	—	2	2	no	—
86	C₂×C₄×D₃	—	2	2	no	—
87	C₂×C₄×D₃	C₄×C₇	1	2	yes	—
88	C₂×C₄×D₃	C₂²×C₇	1	2	yes	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₂×C₃×C₈	C₂²×C₇	1	2	yes	complemented
91	C₂×C₃⋊C₈	—	2	2	no	—
92	C₂×C₃⋊C₈	C₂²×C₇	1	2	yes	complemented
93	C₂×C₃×C₇	C₂²⋊C₈	1	4	yes	complemented
94	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₃×C₇	C₂×M₁₆	1	1	yes	characteristic
97	D₃×C₇	—	2	4	no	—
98	C₂³×C₄	—	3	1	no	—
99	C₂²×C₈	—	3	1	no	—
100	C₂²×C₈	D₃×C₇	1	1	yes	characteristic, complemented
101	C₂²⋊C₈	—	3	4	no	—
102	C₂²×C₇	—	3	4	no	—
103	C₂²×C₇	—	6	8	no	—
104	C₂²×C₇	—	6	2	no	—
105	C₂²×C₇	—	6	2	no	—
106	C₂²×C₇	D₃×C₈	1	2	yes	—
107	C₂²×C₇	ΓL(1,𝔽₂₅)	1	2	yes	—
108	C₂²×C₇	(C₂×D₃)⋊C₄	1	2	yes	—
109	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
110	C₄×C₇	—	6	2	no	—
111	C₄×C₇	(C₂×D₃)⋊C₄	1	2	yes	—
112	C₄×C₇	C₂×D₁₂	1	1	yes	characteristic
113	C₄×C₇	C₂×#67	1	1	yes	characteristic
114	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
115	C₂²×D₃	—	2	2	no	—
116	C₂²×D₃	—	2	2	no	—
117	C₂²×D₃	C₇×C₈	1	4	yes	complemented
118	C₂²×D₃	C₂×C₄×C₇	1	2	yes	—
119	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
120	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
121	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
122	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
123	C₂×Dic₃	—	2	2	no	—
124	C₂×Dic₃	C₂×C₄×C₇	1	2	yes	—
125	C₄×D₃	—	2	4	no	—
126	C₄×D₃	—	2	2	no	—
127	C₄×D₃	—	2	2	no	—
128	C₃×C₈	—	2	2	no	—
129	C₃⋊C₈	—	2	2	no	—
130	C₃×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
131	C₂⁴	—	3	1	no	—
132	C₂²×C₄	—	3	1	no	—
133	C₂²×C₄	—	3	2	no	—
134	C₂²×C₄	—	3	2	no	—
135	C₂²×C₄	—	6	2	no	—
136	C₂²×C₄	—	6	2	no	—
137	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	3	2	no	—
140	C₂×C₈	—	6	2	no	—
141	C₂×C₈	C₂×D₃×C₇	1	2	yes	complemented
142	C₂×C₇	—	6	4	no	—
143	C₂×C₇	(C₂×D₃)⋊C₈	1	4	yes	direct product factor
144	C₂×C₇	C₂×D₃×C₈	1	1	yes	characteristic
145	C₂×C₇	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
146	C₂×C₇	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
147	C₂²×C₃	C₇×C₂²⋊C₄	1	2	yes	—
148	C₂²×C₃	C₂×C₇×C₈	1	2	yes	—
149	C₂²×C₃	C₇×M₁₆	1	2	yes	—
150	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
151	C₂×D₃	—	2	8	no	—
152	C₂×D₃	—	2	2	no	—
153	C₂×D₃	—	2	2	no	—
154	C₂×D₃	C₂×C₇×C₈	1	4	yes	complemented
155	C₃×C₄	C₇×C₂²⋊C₄	1	2	yes	—
156	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
157	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
158	Dic₃	—	2	2	no	—
159	C₂³	—	3	4	no	—
160	C₂³	—	3	2	no	—
161	C₂³	—	6	2	no	—
162	C₂³	—	6	2	no	—
163	C₂³	C₄×D₃×C₇	1	1	yes	characteristic
164	C₂×C₄	—	3	2	no	—
165	C₂×C₄	—	6	2	no	—
166	C₂×C₄	—	6	4	no	—
167	C₂×C₄	—	6	2	no	—
168	C₂×C₄	—	6	2	no	—
169	C₂×C₄	C₄×D₃×C₇	1	1	yes	characteristic
170	C₂×C₄	C₇×D₁₂	1	2	yes	—
171	C₂×C₄	C₇×#67	1	2	yes	—
172	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	6	2	no	—
175	C₇	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
176	C₂×C₃	C₇×C₂²⋊C₈	1	4	yes	complemented
177	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₃	C₂²×C₇×C₈	1	1	yes	characteristic
179	C₂×C₃	C₂×C₇×M₁₆	1	1	yes	characteristic
180	D₃	—	2	4	no	—
181	C₂²	—	3	4	no	—
182	C₂²	—	6	2	no	—
183	C₂²	—	6	2	no	—
184	C₂²	—	6	8	no	—
185	C₂²	D₃×C₇×C₈	1	2	yes	—
186	C₂²	C₇×ΓL(1,𝔽₂₅)	1	2	yes	—
187	C₂²	C₇×(C₂×D₃)⋊C₄	1	2	yes	—
188	C₂²	C₂×C₄×D₃×C₇	1	1	yes	characteristic
189	C₄	—	6	2	no	—
190	C₄	C₇×(C₂×D₃)⋊C₄	1	2	yes	—
191	C₄	C₂×C₇×D₁₂	1	1	yes	characteristic
192	C₄	C₂×C₇×#67	1	1	yes	characteristic
193	C₃	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
194	C₂	—	6	4	no	—
195	C₂	C₇×(C₂×D₃)⋊C₈	1	4	yes	direct product factor
196	C₂	C₂×D₃×C₇×C₈	1	1	yes	characteristic
197	C₂	C₂×C₇×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
198	C₂	C₂×C₇×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₃
commutator	C₂×C₃	C₂²×C₇×C₈
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₂²×C₃×C₇×C₈	C₂
socle	C₂³×C₃×C₇	C₂×C₄
prime radical	C₂³×D₃×C₇	C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₂²×C₇×C₈	D₃
hypocenter	C₃	C₂×C₇×C₂²⋊C₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₇×C₈	C₂	semidirect
3. C₂²×C₇×C₃⋊C₈	C₂	semidirect
4. C₇×(C₂×D₃)⋊C₈	C₂	direct (N × H)
14. C₂×C₃×C₇×C₈	C₂²	semidirect
16. C₂×C₇×C₃⋊C₈	C₂²	semidirect
19. C₂²×C₇×C₈	D₃	semidirect
21. C₂×(C₂×D₃)⋊C₈	C₇	direct (N × H)
25. C₂²×D₃×C₇	C₈	semidirect
48. C₂×C₇×C₈	C₂×D₃	semidirect
50. C₂²×C₃×C₈	C₂×C₇	semidirect
51. C₂²×C₃⋊C₈	C₂×C₇	semidirect
52. (C₂×D₃)⋊C₈	C₂×C₇	direct (N × H)
60. C₂×D₃×C₇	C₂×C₈	semidirect
90. C₂×C₃×C₈	C₂²×C₇	semidirect
92. C₂×C₃⋊C₈	C₂²×C₇	semidirect
93. C₂×C₃×C₇	C₂²⋊C₈	semidirect
100. C₂²×C₈	D₃×C₇	semidirect
117. C₂²×D₃	C₇×C₈	semidirect
130. C₃×C₇	C₂×C₂²⋊C₈	semidirect
141. C₂×C₈	C₂×D₃×C₇	semidirect
143. C₂×C₇	(C₂×D₃)⋊C₈	direct (N × H)
154. C₂×D₃	C₂×C₇×C₈	semidirect
175. C₇	C₂×(C₂×D₃)⋊C₈	direct (N × H)
176. C₂×C₃	C₇×C₂²⋊C₈	semidirect
193. C₃	C₂×C₇×C₂²⋊C₈	semidirect
195. C₂	C₇×(C₂×D₃)⋊C₈	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×C₄×D₃×C₇	2
C₂²×C₃×C₇×C₈	2
C₂²×C₇×C₃⋊C₈	2
C₇×(C₂×D₃)⋊C₈	2
C₂×C₇×C₂²⋊C₈	3
C₂×(C₂×D₃)⋊C₈	7

Subgroups — C2×C7×(C2×D3)⋊C8 #13056142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×(C₂×D₃)⋊C₈ #13056142