Subgroups — C₂×#10621657 #13062374

Order 1344.

1796 conjugacy classes of subgroups.

The subgroup lattice has 224 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, complemented
2	C₂²×#4268	C₂	1	4	yes	maximal, complemented
3	#10621657	C₂	1	16	yes	maximal, direct product factor
4	C₂×#66730	C₂	1	4	yes	maximal, complemented
5	C₂×D₄×D₂₁	C₂	1	4	yes	maximal, complemented
6	C₂×#66728	C₂	1	2	yes	maximal, complemented
7	C₂×#6548	—	3	1	no	maximal, Hall
8	C₂⁴×C₃×C₇	C₂²	1	2	yes	complemented
9	C₂³×D₂₁	C₂²	1	4	yes	complemented
10	C₂×C₃×C₇×D₄	C₂²	1	4	yes	complemented
11	C₂×C₃×C₇×D₄	C₂²	1	8	yes	complemented
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic, complemented
13	C₂×#4268	C₂²	1	4	yes	complemented
14	C₂×#4268	C₂²	1	8	yes	complemented
15	C₂×#4268	C₂²	1	32	yes	complemented
16	C₂²×Dic₂₁	C₂²	1	4	yes	complemented
17	#66730	C₂²	1	32	yes	complemented
18	D₄×D₂₁	C₂²	1	32	yes	complemented
19	#66728	C₂²	1	16	yes	complemented
20	C₂×D₈₄	C₂²	1	2	yes	complemented
21	C₂×C₄×D₂₁	C₂²	1	4	yes	complemented
22	C₂×Dic₄₂	C₂²	1	2	yes	complemented
23	C₂²×C₇×D₄	D₃	1	1	yes	characteristic, complemented
24	C₂²×C₂²⋊D₇	—	3	4	no	—
25	#6548	—	3	16	no	—
26	C₂×#1159	—	3	4	no	—
27	C₂×D₄×D₇	—	3	4	no	—
28	C₂×#1157	—	3	2	no	—
29	C₂×#1004	—	7	1	no	maximal, Hall
30	C₂³×C₃×C₇	—	4	4	no	—
31	C₂³×C₃×C₇	C₂³	1	4	yes	—
32	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
33	C₂³×C₃×C₇	C₂³	1	8	yes	complemented
34	C₂²×D₂₁	—	4	8	no	—
35	C₂²×D₂₁	C₂³	1	4	yes	—
36	C₂²×D₂₁	C₂³	1	16	yes	complemented
37	C₃×C₇×D₄	C₂³	1	16	yes	complemented
38	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
39	C₂×C₃×C₄×C₇	C₂³	1	4	yes	complemented
40	#4268	C₂³	1	16	yes	complemented
41	#4268	C₂³	1	32	yes	complemented
42	C₂×Dic₂₁	C₂³	1	4	yes	—
43	C₂×Dic₂₁	C₂³	1	16	yes	complemented
44	D₈₄	C₂³	1	8	yes	complemented
45	C₄×D₂₁	C₂³	1	16	yes	complemented
46	Dic₄₂	C₂³	1	8	yes	complemented
47	C₂⁴×C₇	C₂×D₃	1	2	yes	complemented
48	C₂³×D₇	—	3	4	no	—
49	C₂×C₇×D₄	C₂×D₃	1	4	yes	complemented
50	C₂×C₇×D₄	C₂×D₃	1	8	yes	complemented
51	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic, complemented
52	C₂×C₂²⋊D₇	—	3	4	no	—
53	C₂×C₂²⋊D₇	—	3	8	no	—
54	C₂×C₂²⋊D₇	—	3	32	no	—
55	C₂²×Dic₇	—	3	4	no	—
56	#1159	—	3	32	no	—
57	D₄×D₇	—	3	32	no	—
58	#1157	—	3	16	no	—
59	C₂×D₂₈	—	3	2	no	—
60	C₂×C₄×D₇	—	3	4	no	—
61	C₂×Dic₁₄	—	3	2	no	—
62	C₂²×C₃×D₄	D₇	1	1	yes	characteristic, complemented
63	C₂²×#67	—	7	4	no	—
64	#1004	—	7	16	no	—
65	C₂×#237	—	7	4	no	—
66	C₂×D₃×D₄	—	7	4	no	—
67	C₂×#235	—	7	2	no	—
68	C₂²×C₃×C₇	—	2	8	no	—
69	C₂²×C₃×C₇	—	2	2	no	—
70	C₂²×C₃×C₇	—	4	8	no	—
71	C₂²×C₃×C₇	C₂⁴	1	8	yes	—
72	C₂²×C₃×C₇	C₂⁴	1	2	yes	—
73	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
74	C₂×D₂₁	—	2	8	no	—
75	C₂×D₂₁	—	4	16	no	—
76	C₂×D₂₁	C₂⁴	1	8	yes	—
77	C₃×C₄×C₇	C₂⁴	1	4	yes	—
78	Dic₂₁	C₂⁴	1	8	yes	—
79	C₂×H(5,𝔽₂)	—	21	1	no	Sylow
80	C₂³×C₇	—	4	4	no	—
81	C₂³×C₇	C₂²×D₃	1	4	yes	—
82	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
83	C₂³×C₇	C₂²×D₃	1	8	yes	complemented
84	C₂²×D₇	—	3	4	no	—
85	C₂²×D₇	—	3	16	no	—
86	C₂²×D₇	—	12	8	no	—
87	C₇×D₄	C₂²×D₃	1	16	yes	complemented
88	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
89	C₂×C₄×C₇	C₂²×D₃	1	4	yes	complemented
90	C₂²⋊D₇	—	3	16	no	—
91	C₂²⋊D₇	—	3	32	no	—
92	C₂×Dic₇	—	3	4	no	—
93	C₂×Dic₇	—	3	16	no	—
94	D₂₈	—	3	8	no	—
95	C₄×D₇	—	3	16	no	—
96	Dic₁₄	—	3	8	no	—
97	C₂⁴×C₃	C₂×D₇	1	2	yes	complemented
98	C₂³×D₃	—	7	4	no	—
99	C₂×C₃×D₄	C₂×D₇	1	4	yes	complemented
100	C₂×C₃×D₄	C₂×D₇	1	8	yes	complemented
101	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic, complemented
102	C₂×#67	—	7	4	no	—
103	C₂×#67	—	7	8	no	—
104	C₂×#67	—	7	32	no	—
105	C₂²×Dic₃	—	7	4	no	—
106	#237	—	7	32	no	—
107	D₃×D₄	—	7	32	no	—
108	#235	—	7	16	no	—
109	C₂×D₁₂	—	7	2	no	—
110	C₂×C₄×D₃	—	7	4	no	—
111	C₂×Dic₆	—	7	2	no	—
112	C₂×C₃×C₇	—	2	8	no	—
113	C₂×C₃×C₇	—	2	2	no	—
114	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	complemented
115	C₂×C₃×C₇	C₂⁵	1	1	yes	characteristic
116	D₂₁	—	2	8	no	—
117	H(5,𝔽₂)	—	21	16	no	—
118	C₂×#41	—	21	4	no	—
119	C₂×#41	—	21	2	no	—
120	C₂²×D₄	—	21	4	no	—
121	C₂²×D₄	—	21	4	no	—
122	C₂²×D₄	D₂₁	1	1	yes	characteristic, complemented
123	C₂²×C₇	—	2	8	no	—
124	C₂²×C₇	—	2	2	no	—
125	C₂²×C₇	—	4	8	no	—
126	C₂²×C₇	C₂³×D₃	1	8	yes	—
127	C₂²×C₇	C₂³×D₃	1	2	yes	—
128	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
129	C₂×D₇	—	3	8	no	—
130	C₂×D₇	—	6	8	no	—
131	C₂×D₇	—	12	16	no	—
132	C₄×C₇	C₂³×D₃	1	4	yes	—
133	Dic₇	—	3	8	no	—
134	C₂³×C₃	—	4	4	no	—
135	C₂³×C₃	C₂²×D₇	1	4	yes	—
136	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
137	C₂³×C₃	C₂²×D₇	1	8	yes	complemented
138	C₂²×D₃	—	7	4	no	—
139	C₂²×D₃	—	7	16	no	—
140	C₂²×D₃	—	28	8	no	—
141	C₃×D₄	C₂²×D₇	1	16	yes	complemented
142	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
143	C₂×C₃×C₄	C₂²×D₇	1	4	yes	complemented
144	#67	—	7	16	no	—
145	#67	—	7	32	no	—
146	C₂×Dic₃	—	7	4	no	—
147	C₂×Dic₃	—	7	16	no	—
148	D₁₂	—	7	8	no	—
149	C₄×D₃	—	7	16	no	—
150	Dic₆	—	7	8	no	—
151	C₃×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Hall
152	C₂⁴	—	21	4	no	—
153	C₂⁴	C₂×D₂₁	1	2	yes	complemented
154	#41	—	21	32	no	—
155	#41	—	21	16	no	—
156	C₂×Q₈	—	21	2	no	—
157	C₂×D₄	—	21	8	no	—
158	C₂×D₄	—	21	4	no	—
159	C₂×D₄	—	21	32	no	—
160	C₂×D₄	—	21	32	no	—
161	C₂×D₄	—	21	2	no	—
162	C₂×D₄	C₂×D₂₁	1	4	yes	complemented
163	C₂×D₄	C₂×D₂₁	1	8	yes	complemented
164	C₂²×C₄	—	21	4	no	—
165	C₂²×C₄	—	21	4	no	—
166	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic, complemented
167	C₂×C₇	—	2	8	no	—
168	C₂×C₇	—	2	2	no	—
169	C₂×C₇	#1004	1	2	yes	complemented
170	C₂×C₇	C₂⁴×D₃	1	1	yes	characteristic
171	D₇	—	6	8	no	—
172	C₂²×C₃	—	2	8	no	—
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	4	8	no	—
175	C₂²×C₃	C₂³×D₇	1	8	yes	—
176	C₂²×C₃	C₂³×D₇	1	2	yes	—
177	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
178	C₂×D₃	—	7	8	no	—
179	C₂×D₃	—	14	8	no	—
180	C₂×D₃	—	28	16	no	—
181	C₃×C₄	C₂³×D₇	1	4	yes	—
182	Dic₃	—	7	8	no	—
183	C₂³	—	4	4	no	—
184	C₂³	—	21	4	no	—
185	C₂³	—	21	16	no	—
186	C₂³	—	84	8	no	—
187	C₂³	C₂²×D₂₁	1	4	yes	—
188	C₂³	C₂²×D₂₁	1	1	yes	characteristic
189	C₂³	C₂²×D₂₁	1	8	yes	complemented
190	Q₈	—	21	8	no	—
191	D₄	—	21	32	no	—
192	D₄	—	21	16	no	—
193	D₄	—	21	8	no	—
194	D₄	C₂²×D₂₁	1	16	yes	complemented
195	C₂×C₄	—	21	16	no	—
196	C₂×C₄	—	21	4	no	—
197	C₂×C₄	—	21	16	no	—
198	C₂×C₄	C₂²×D₂₁	1	2	yes	—
199	C₂×C₄	C₂²×D₂₁	1	4	yes	complemented
200	C₇	C₂×#1004	1	1	yes	characteristic, complemented, Sylow
201	C₂×C₃	—	2	8	no	—
202	C₂×C₃	—	2	2	no	—
203	C₂×C₃	#6548	1	2	yes	complemented
204	C₂×C₃	C₂⁴×D₇	1	1	yes	characteristic
205	D₃	—	14	8	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	8	no	—
208	C₂²	—	4	8	no	—
209	C₂²	—	21	8	no	—
210	C₂²	—	42	8	no	—
211	C₂²	—	84	16	no	—
212	C₂²	C₂³×D₂₁	1	8	yes	—
213	C₂²	C₂³×D₂₁	1	2	yes	—
214	C₂²	C₂³×D₂₁	1	1	yes	characteristic
215	C₄	—	21	8	no	—
216	C₄	C₂³×D₂₁	1	4	yes	—
217	C₃	C₂×#6548	1	1	yes	characteristic, complemented, Sylow
218	C₂	—	2	2	no	—
219	C₂	—	2	8	no	—
220	C₂	—	42	8	no	—
221	C₂	#10621657	1	2	yes	direct product factor
222	C₂	C₂⁴×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₁
commutator, cosocle	C₂×C₃×C₇	C₂⁵
Frattini	C₂	C₂⁴×D₂₁
Fitting	C₂²×C₃×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂²×D₄	D₂₁
hypocenter	C₃×C₇	C₂×H(5,𝔽₂)
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₇×D₄	C₂	semidirect
2. C₂²×#4268	C₂	semidirect
3. #10621657	C₂	direct (N × H)
4. C₂×#66730	C₂	semidirect
5. C₂×D₄×D₂₁	C₂	semidirect
6. C₂×#66728	C₂	semidirect
8. C₂⁴×C₃×C₇	C₂²	semidirect
9. C₂³×D₂₁	C₂²	semidirect
10. C₂×C₃×C₇×D₄	C₂²	semidirect
11. C₂×C₃×C₇×D₄	C₂²	semidirect
12. C₂²×C₃×C₄×C₇	C₂²	semidirect
13. C₂×#4268	C₂²	semidirect
14. C₂×#4268	C₂²	semidirect
15. C₂×#4268	C₂²	semidirect
16. C₂²×Dic₂₁	C₂²	semidirect
17. #66730	C₂²	semidirect
18. D₄×D₂₁	C₂²	semidirect
19. #66728	C₂²	semidirect
20. C₂×D₈₄	C₂²	semidirect
21. C₂×C₄×D₂₁	C₂²	semidirect
22. C₂×Dic₄₂	C₂²	semidirect
23. C₂²×C₇×D₄	D₃	semidirect
33. C₂³×C₃×C₇	C₂³	semidirect
36. C₂²×D₂₁	C₂³	semidirect
37. C₃×C₇×D₄	C₂³	semidirect
39. C₂×C₃×C₄×C₇	C₂³	semidirect
40. #4268	C₂³	semidirect
41. #4268	C₂³	semidirect
43. C₂×Dic₂₁	C₂³	semidirect
44. D₈₄	C₂³	semidirect
45. C₄×D₂₁	C₂³	semidirect
46. Dic₄₂	C₂³	semidirect
47. C₂⁴×C₇	C₂×D₃	semidirect
49. C₂×C₇×D₄	C₂×D₃	semidirect
50. C₂×C₇×D₄	C₂×D₃	semidirect
51. C₂²×C₄×C₇	C₂×D₃	semidirect
62. C₂²×C₃×D₄	D₇	semidirect
83. C₂³×C₇	C₂²×D₃	semidirect
87. C₇×D₄	C₂²×D₃	semidirect
89. C₂×C₄×C₇	C₂²×D₃	semidirect
97. C₂⁴×C₃	C₂×D₇	semidirect
99. C₂×C₃×D₄	C₂×D₇	semidirect
100. C₂×C₃×D₄	C₂×D₇	semidirect
101. C₂²×C₃×C₄	C₂×D₇	semidirect
114. C₂×C₃×C₇	H(5,𝔽₂)	semidirect
122. C₂²×D₄	D₂₁	semidirect
137. C₂³×C₃	C₂²×D₇	semidirect
141. C₃×D₄	C₂²×D₇	semidirect
143. C₂×C₃×C₄	C₂²×D₇	semidirect
151. C₃×C₇	C₂×H(5,𝔽₂)	semidirect
153. C₂⁴	C₂×D₂₁	semidirect
162. C₂×D₄	C₂×D₂₁	semidirect
163. C₂×D₄	C₂×D₂₁	semidirect
166. C₂²×C₄	C₂×D₂₁	semidirect
169. C₂×C₇	#1004	semidirect
189. C₂³	C₂²×D₂₁	semidirect
194. D₄	C₂²×D₂₁	semidirect
199. C₂×C₄	C₂²×D₂₁	semidirect
200. C₇	C₂×#1004	semidirect
203. C₂×C₃	#6548	semidirect
217. C₃	C₂×#6548	semidirect
221. C₂	#10621657	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
C₂²×#4268	2
#10621657	2
C₂×#66730	2
C₂×D₄×D₂₁	2
C₂×#66728	2
C₂×#6548	3
C₂×#1004	7

Subgroups — C2×#10621657 #13062374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10621657 #13062374