Subgroups — C₂×C₇×#2244 #13210915

Order 1792.

686 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
3	C₇×#2244	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×#357	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×D₈	—	2	1	no	—
7	C₂²×C₇×D₈	C₂²	1	2	yes	—
8	C₂×C₇×#127	—	2	1	no	—
9	C₂×C₇×#125	—	2	1	no	—
10	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
11	C₇×#499	—	2	2	no	—
12	C₇×#499	C₂²	1	4	yes	complemented
13	C₇×#493	—	2	2	no	—
14	C₇×#493	C₂²	1	4	yes	—
15	C₂×C₇×#107	C₂²	1	2	yes	—
16	C₂×C₇×#106	—	2	1	no	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇×#357	C₂²	1	16	yes	complemented
19	C₇×#346	C₂²	1	4	yes	complemented
20	C₂×#2244	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×D₄	—	2	2	no	—
22	C₂²×C₇×D₄	—	4	2	no	—
23	C₂×C₇×D₈	—	2	2	no	—
24	C₂×C₇×D₈	—	2	4	no	—
25	C₂×C₇×D₈	—	2	8	no	—
26	C₂×C₇×D₈	—	4	4	no	—
27	C₂×C₇×D₈	C₂³	1	4	yes	—
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₇×#127	—	2	4	no	—
31	C₇×#127	—	4	2	no	—
32	C₇×#125	—	2	4	no	—
33	C₇×#125	—	4	2	no	—
34	C₂×C₇×C₄⋊C₄	—	2	2	no	—
35	C₂×C₇×C₄⋊C₄	—	4	2	no	—
36	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
37	C₂×C₇×C₁₆	—	2	4	no	—
38	C₂×C₇×C₁₆	C₂³	1	4	yes	—
39	C₇×#107	—	2	2	no	—
40	C₇×#107	C₂³	1	4	yes	—
41	C₇×#106	—	2	4	no	—
42	C₄×C₇×C₈	D₄	1	2	yes	—
43	C₄×C₇×C₈	C₂³	1	2	yes	—
44	C₂×#499	C₂×C₇	1	1	yes	characteristic, complemented
45	C₂×#493	C₂×C₇	1	1	yes	characteristic
46	#2244	C₂×C₇	1	8	yes	direct product factor
47	C₂×#357	C₂×C₇	1	4	yes	complemented
48	C₂×#346	C₂×C₇	1	1	yes	characteristic, complemented
49	C₂⁴×C₇	—	4	2	no	—
50	C₂×C₇×D₄	—	2	4	no	—
51	C₂×C₇×D₄	—	4	4	no	—
52	C₂×C₇×D₄	—	4	8	no	—
53	C₂×C₇×D₄	—	4	2	no	—
54	C₂×C₇×D₄	—	4	4	no	—
55	C₂×C₇×D₄	—	8	8	no	—
56	C₂²×C₄×C₇	—	4	2	no	—
57	C₂²×C₄×C₇	D₈	1	2	yes	—
58	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₇×D₈	—	2	8	no	—
60	C₇×D₈	—	4	4	no	—
61	C₇×D₈	—	4	8	no	—
62	C₂×C₇×C₈	SD₁₆	1	4	yes	—
63	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
65	C₂×C₇×C₈	#41	1	4	yes	—
66	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
67	C₇×C₄⋊C₄	—	2	4	no	—
68	C₇×C₄⋊C₄	—	4	2	no	—
69	C₇×C₄⋊C₄	—	4	8	no	—
70	C₄²×C₇	D₈	1	2	yes	—
71	C₄²×C₇	C₂×D₄	1	2	yes	—
72	C₇×C₁₆	—	2	4	no	—
73	C₂²×D₈	—	2	1	no	—
74	C₂²×D₈	C₂²×C₇	1	2	yes	—
75	C₂×#127	—	2	1	no	—
76	C₂×#125	—	2	1	no	—
77	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
78	#499	—	2	2	no	—
79	#499	C₂²×C₇	1	4	yes	complemented
80	#493	—	2	2	no	—
81	#493	C₂²×C₇	1	4	yes	—
82	C₂×#107	C₂²×C₇	1	2	yes	—
83	C₂×#106	—	2	1	no	—
84	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
85	#357	C₂²×C₇	1	16	yes	complemented
86	#346	C₂²×C₇	1	4	yes	complemented
87	C₂³×C₇	—	4	4	no	—
88	C₂³×C₇	—	8	4	no	—
89	C₂³×C₇	—	8	8	no	—
90	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
91	C₇×D₄	—	4	4	no	—
92	C₇×D₄	—	4	4	no	—
93	C₇×D₄	—	8	16	no	—
94	C₇×D₄	—	8	4	no	—
95	C₂×C₄×C₇	—	2	4	no	—
96	C₂×C₄×C₇	—	4	4	no	—
97	C₂×C₄×C₇	—	8	4	no	—
98	C₂×C₄×C₇	#124	1	2	yes	—
99	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
100	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
102	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
103	C₇×C₈	#124	1	2	yes	—
104	C₇×C₈	C₂×SD₁₆	1	4	yes	—
105	C₇×C₈	C₂×#41	1	2	yes	—
106	C₂²×D₄	—	2	2	no	—
107	C₂²×D₄	—	4	2	no	—
108	C₂×D₈	—	2	4	no	—
109	C₂×D₈	—	2	2	no	—
110	C₂×D₈	—	2	8	no	—
111	C₂×D₈	—	4	4	no	—
112	C₂×D₈	C₂³×C₇	1	4	yes	—
113	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
115	#127	—	2	4	no	—
116	#127	—	4	2	no	—
117	#125	—	2	4	no	—
118	#125	—	4	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	2	no	—
121	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
122	C₂×C₁₆	—	2	4	no	—
123	C₂×C₁₆	C₂³×C₇	1	4	yes	—
124	#107	—	2	2	no	—
125	#107	C₂³×C₇	1	4	yes	—
126	#106	—	2	4	no	—
127	C₄×C₈	C₇×D₄	1	2	yes	—
128	C₄×C₈	C₂³×C₇	1	2	yes	—
129	C₂²×C₇	—	8	4	no	—
130	C₂²×C₇	—	8	8	no	—
131	C₂²×C₇	—	16	8	no	—
132	C₂²×C₇	#486	1	2	yes	—
133	C₂²×C₇	#509	1	4	yes	—
134	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₇	—	2	4	no	—
136	C₄×C₇	—	8	4	no	—
137	C₄×C₇	#486	1	2	yes	—
138	C₄×C₇	C₂×#124	1	1	yes	characteristic
139	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
140	C₂⁴	—	4	2	no	—
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	—	4	4	no	—
143	C₂×D₄	—	4	8	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂×D₄	—	4	4	no	—
146	C₂×D₄	—	8	8	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	C₇×D₈	1	2	yes	—
149	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
150	D₈	—	2	8	no	—
151	D₈	—	4	4	no	—
152	D₈	—	4	8	no	—
153	C₂×C₈	C₇×SD₁₆	1	4	yes	—
154	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
155	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
156	C₂×C₈	C₇×#41	1	4	yes	—
157	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	8	no	—
161	C₄²	C₇×D₈	1	2	yes	—
162	C₄²	C₂×C₇×D₄	1	2	yes	—
163	C₁₆	—	2	4	no	—
164	C₂×C₇	—	16	4	no	—
165	C₂×C₇	#2244	1	4	yes	direct product factor
166	C₂×C₇	C₂×#486	1	1	yes	characteristic
167	C₂×C₇	C₂×#509	1	2	yes	—
168	C₂³	—	4	4	no	—
169	C₂³	—	8	4	no	—
170	C₂³	—	8	8	no	—
171	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
172	D₄	—	4	4	no	—
173	D₄	—	4	4	no	—
174	D₄	—	8	16	no	—
175	D₄	—	8	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	8	4	no	—
179	C₂×C₄	C₇×#124	1	2	yes	—
180	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
181	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
182	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
183	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
184	C₈	C₇×#124	1	2	yes	—
185	C₈	C₂×C₇×SD₁₆	1	4	yes	—
186	C₈	C₂×C₇×#41	1	2	yes	—
187	C₇	C₂×#2244	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²	—	8	8	no	—
189	C₂²	—	8	4	no	—
190	C₂²	—	16	8	no	—
191	C₂²	C₇×#486	1	2	yes	—
192	C₂²	C₇×#509	1	4	yes	—
193	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	8	4	no	—
196	C₄	C₇×#486	1	2	yes	—
197	C₄	C₂×C₇×#124	1	1	yes	characteristic
198	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
199	C₂	—	16	4	no	—
200	C₂	C₇×#2244	1	4	yes	direct product factor
201	C₂	C₂×C₇×#486	1	1	yes	characteristic
202	C₂	C₂×C₇×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
prime radical	C₂×C₇×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#499	C₂	semidirect
3. C₇×#2244	C₂	direct (N × H)
4. C₂×C₇×#357	C₂	semidirect
5. C₂×C₇×#346	C₂	semidirect
12. C₇×#499	C₂²	semidirect
18. C₇×#357	C₂²	semidirect
19. C₇×#346	C₂²	semidirect
20. C₂×#2244	C₇	direct (N × H)
44. C₂×#499	C₂×C₇	semidirect
46. #2244	C₂×C₇	direct (N × H)
47. C₂×#357	C₂×C₇	semidirect
48. C₂×#346	C₂×C₇	semidirect
79. #499	C₂²×C₇	semidirect
85. #357	C₂²×C₇	semidirect
86. #346	C₂²×C₇	semidirect
165. C₂×C₇	#2244	direct (N × H)
187. C₇	C₂×#2244	direct (N × H)
200. C₂	C₇×#2244	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#499	2
C₂×C₇×#493	2
C₇×#2244	2
C₂×C₇×#357	2
C₂×C₇×#346	2
C₂×#2244	7

Subgroups — C2×C7×#2244 #13210915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2244 #13210915