Subgroups — C₂×#10620848 #13059419

Order 1344.

532 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	C₂×C₇×#235	C₂	1	1	yes	maximal, characteristic
2	C₂×#66627	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#66558	C₂	1	2	yes	maximal, complemented
5	#10620848	C₂	1	8	yes	maximal, direct product factor
6	C₂×D₃×C₇⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₂×#6511	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×#67	C₄	1	2	yes	—
10	C₇×#235	C₄	1	4	yes	—
11	C₇×#235	C₂²	1	4	yes	—
12	C₂×C₇×D₁₂	C₄	1	1	yes	characteristic
13	C₂×C₄×D₃×C₇	C₂²	1	2	yes	—
14	C₂×C₇×Dic₆	C₄	1	1	yes	characteristic
15	#66627	C₂²	1	4	yes	complemented
16	C₂×C₃⋊(C₇⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₃×C₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₇⋊C₈	C₂²	1	4	yes	complemented
20	#66558	C₂²	1	8	yes	complemented
21	D₃×C₇⋊C₈	C₂²	1	8	yes	complemented
22	C₂×C₇×#41	—	3	1	no	—
23	#6511	—	3	8	no	—
24	C₂×#1136	—	3	1	no	—
25	C₂×#1136	—	3	2	no	—
26	C₂²×C₇⋊C₈	—	3	2	no	—
27	C₂²×C₇⋊C₈	D₃	1	1	yes	characteristic, complemented
28	C₂×#901	—	7	1	no	maximal, Hall
29	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₃×C₇	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
35	C₇×#67	C₂×C₄	1	8	yes	—
36	C₂×C₇×Dic₃	C₂×C₄	1	2	yes	—
37	C₇×D₁₂	C₂×C₄	1	4	yes	—
38	C₄×D₃×C₇	C₂×C₄	1	4	yes	—
39	C₄×D₃×C₇	C₂³	1	4	yes	—
40	C₇×Dic₆	C₂×C₄	1	4	yes	—
41	C₃⋊(C₇⋊C₈)	C₂³	1	4	yes	—
42	C₃×C₇⋊C₈	C₂³	1	2	yes	—
43	C₃×C₇⋊C₈	C₂³	1	2	yes	—
44	C₇×#41	—	3	4	no	—
45	C₇×#41	—	3	4	no	—
46	C₂×C₇×Q₈	—	3	1	no	—
47	C₂×C₇×D₄	—	3	2	no	—
48	C₂×C₇×D₄	—	3	1	no	—
49	C₂²×C₄×C₇	—	3	2	no	—
50	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
51	#1136	—	3	4	no	—
52	#1136	—	3	8	no	—
53	C₂×C₇⋊C₈	—	3	2	no	—
54	C₂×C₇⋊C₈	—	3	8	no	—
55	C₂×C₇⋊C₈	C₂×D₃	1	1	yes	characteristic
56	C₂×C₇⋊C₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₇⋊C₈	C₂×D₃	1	4	yes	complemented
58	C₂×#235	D₇	1	1	yes	characteristic
59	C₂²×C₃×C₈	—	7	1	no	—
60	C₂×#208	—	7	1	no	—
61	#901	—	7	8	no	—
62	C₂×ΓL(1,𝔽₂₅)	—	7	2	no	—
63	C₂×D₃×C₈	—	7	2	no	—
64	C₂²×C₃×C₇	—	2	2	no	—
65	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×D₃×C₇	—	2	4	no	—
68	C₂×D₃×C₇	C₂²×C₄	1	4	yes	—
69	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
70	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
72	C₇×Dic₃	C₂²×C₄	1	4	yes	—
73	C₂×#131	—	21	1	no	Sylow
74	C₂³×C₇	—	3	2	no	—
75	C₂³×C₇	C₄×D₃	1	1	yes	characteristic
76	C₇×Q₈	—	3	4	no	—
77	C₇×D₄	—	3	8	no	—
78	C₇×D₄	—	3	4	no	—
79	C₂×C₄×C₇	—	3	2	no	—
80	C₂×C₄×C₇	—	3	4	no	—
81	C₂×C₄×C₇	—	3	4	no	—
82	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
83	C₂×C₄×C₇	C₄×D₃	1	2	yes	—
84	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
85	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
86	C₇⋊C₈	—	3	4	no	—
87	C₇⋊C₈	C₂²×D₃	1	2	yes	—
88	C₇⋊C₈	C₂²×D₃	1	2	yes	—
89	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
90	C₂×#67	Dic₇	1	2	yes	—
91	#235	Dic₇	1	4	yes	—
92	#235	C₂×D₇	1	4	yes	—
93	C₂×D₁₂	Dic₇	1	1	yes	characteristic
94	C₂×C₄×D₃	C₂×D₇	1	2	yes	—
95	C₂×Dic₆	Dic₇	1	1	yes	characteristic
96	C₂×C₃×C₈	—	7	1	no	—
97	C₂×C₃×C₈	—	7	1	no	—
98	C₂×C₃×C₈	—	7	4	no	—
99	#208	—	7	4	no	—
100	C₂×C₃⋊C₈	—	7	2	no	—
101	ΓL(1,𝔽₂₅)	—	7	8	no	—
102	D₃×C₈	—	7	8	no	—
103	C₂×C₃×C₇	—	2	2	no	—
104	C₂×C₃×C₇	#131	1	2	yes	complemented
105	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
106	D₃×C₇	—	2	4	no	—
107	C₂×#41	—	3	1	no	—
108	#131	—	21	8	no	—
109	C₂×M₁₆	—	21	2	no	—
110	C₂×M₁₆	—	21	1	no	—
111	C₂²×C₈	—	7	1	no	—
112	C₂²×C₈	—	21	2	no	—
113	C₂²×C₇	—	2	2	no	—
114	C₂²×C₇	—	3	4	no	—
115	C₂²×C₇	—	6	4	no	—
116	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
117	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
118	C₄×C₇	—	3	4	no	—
119	C₄×C₇	C₂×C₄×D₃	1	1	yes	characteristic
120	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
121	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
122	C₂³×C₃	C₂×Dic₇	1	1	yes	characteristic
123	C₂²×D₃	C₂×Dic₇	1	2	yes	—
124	C₂×C₃×C₄	C₂×Dic₇	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×Dic₇	1	2	yes	—
126	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
127	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
128	#67	C₂×Dic₇	1	8	yes	—
129	C₂×Dic₃	C₂×Dic₇	1	2	yes	—
130	D₁₂	C₂×Dic₇	1	4	yes	—
131	C₄×D₃	C₂×Dic₇	1	4	yes	—
132	C₄×D₃	C₂²×D₇	1	4	yes	—
133	Dic₆	C₂×Dic₇	1	4	yes	—
134	C₃×C₈	—	7	2	no	—
135	C₃×C₈	—	7	2	no	—
136	C₃⋊C₈	—	7	4	no	—
137	C₃×C₇	C₂×#131	1	1	yes	characteristic, complemented, Hall
138	#41	—	3	4	no	—
139	#41	—	3	4	no	—
140	C₂×Q₈	—	3	1	no	—
141	C₂×D₄	—	3	2	no	—
142	C₂×D₄	—	3	1	no	—
143	C₂²×C₄	—	3	2	no	—
144	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
145	M₁₆	—	21	8	no	—
146	M₁₆	—	21	4	no	—
147	C₂×C₈	—	7	4	no	—
148	C₂×C₈	—	7	1	no	—
149	C₂×C₈	—	7	1	no	—
150	C₂×C₈	—	21	8	no	—
151	C₂×C₈	—	21	2	no	—
152	C₂×C₇	—	2	2	no	—
153	C₂×C₇	—	6	4	no	—
154	C₂×C₇	#901	1	2	yes	complemented
155	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	C₂²×Dic₇	1	2	yes	—
158	C₂²×C₃	C₂²×Dic₇	1	1	yes	characteristic
159	C₂×D₃	—	2	4	no	—
160	C₂×D₃	C₂²×Dic₇	1	4	yes	—
161	C₃×C₄	C₂²×Dic₇	1	2	yes	—
162	C₃×C₄	C₂²×Dic₇	1	1	yes	characteristic
163	C₃×C₄	C₂³×D₇	1	1	yes	characteristic
164	Dic₃	C₂²×Dic₇	1	4	yes	—
165	C₂³	—	3	2	no	—
166	C₂³	D₃×Dic₇	1	1	yes	characteristic
167	Q₈	—	3	4	no	—
168	D₄	—	3	8	no	—
169	D₄	—	3	4	no	—
170	C₂×C₄	—	3	2	no	—
171	C₂×C₄	—	3	4	no	—
172	C₂×C₄	—	3	4	no	—
173	C₂×C₄	D₃×Dic₇	1	1	yes	characteristic
174	C₂×C₄	D₃×Dic₇	1	2	yes	—
175	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
176	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
177	C₈	—	7	2	no	—
178	C₈	—	7	2	no	—
179	C₈	—	21	4	no	—
180	C₇	C₂×#901	1	1	yes	characteristic, complemented, Sylow
181	C₂×C₃	—	2	2	no	—
182	C₂×C₃	#6511	1	2	yes	complemented
183	C₂×C₃	C₂³×Dic₇	1	1	yes	characteristic
184	D₃	—	2	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	3	4	no	—
187	C₂²	—	6	4	no	—
188	C₂²	C₂×D₃×Dic₇	1	2	yes	—
189	C₂²	C₂×D₃×Dic₇	1	1	yes	characteristic
190	C₄	—	3	4	no	—
191	C₄	C₂×D₃×Dic₇	1	1	yes	characteristic
192	C₄	C₂×D₃×Dic₇	1	2	yes	—
193	C₄	C₂²×D₃×D₇	1	1	yes	characteristic
194	C₃	C₂×#6511	1	1	yes	characteristic, complemented, Sylow
195	C₂	—	2	2	no	—
196	C₂	—	6	4	no	—
197	C₂	#10620848	1	2	yes	direct product factor
198	C₂	C₂²×D₃×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₇
commutator	C₂×C₃×C₇	C₂³×C₄
Frattini	C₄	C₂²×D₃×D₇
Fitting	C₂²×C₃×C₄×C₇	C₂²
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂×C₇×#235	C₂
cosocle	C₃×C₄×C₇	C₂⁴
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	C₂×#131
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#66627	C₂	semidirect
3. C₂²×C₃×C₇⋊C₈	C₂	semidirect
4. C₂×#66558	C₂	semidirect
5. #10620848	C₂	direct (N × H)
6. C₂×D₃×C₇⋊C₈	C₂	semidirect
15. #66627	C₂²	semidirect
19. C₂×C₃×C₇⋊C₈	C₂²	semidirect
20. #66558	C₂²	semidirect
21. D₃×C₇⋊C₈	C₂²	semidirect
27. C₂²×C₇⋊C₈	D₃	semidirect
57. C₂×C₇⋊C₈	C₂×D₃	semidirect
104. C₂×C₃×C₇	#131	semidirect
137. C₃×C₇	C₂×#131	semidirect
154. C₂×C₇	#901	semidirect
180. C₇	C₂×#901	semidirect
182. C₂×C₃	#6511	semidirect
194. C₃	C₂×#6511	semidirect
197. C₂	#10620848	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×#235	2
C₂×#66627	2
C₂²×C₃×C₇⋊C₈	2
C₂×#66558	2
#10620848	2
C₂×D₃×C₇⋊C₈	2
C₂×#6511	3
C₂×#901	7

Subgroups — C2×#10620848 #13059419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10620848 #13059419