Subgroups — C₂×#11744563 #14267695

Order 1792.

59186 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	#11744563	C₂	1	64	yes	maximal, direct product factor
3	C₂×D₇×2₋¹⁺⁴	C₂	1	16	yes	maximal, complemented
4	C₂×#90620	C₂	1	16	yes	maximal, complemented
5	C₂×#90616	C₂	1	20	yes	maximal, complemented
6	C₂²×#6551	C₂	1	10	yes	maximal, complemented
7	C₂×C₇×2₋¹⁺⁴	C₂²	1	16	yes	complemented
8	C₂×C₇×2₋¹⁺⁴	C₂²	1	32	yes	complemented
9	C₂²×C₇×#41	C₂²	1	10	yes	complemented
10	C₂³×C₇×Q₈	C₂²	1	5	yes	complemented
11	#90620	C₂²	1	512	yes	complemented
12	D₇×2₋¹⁺⁴	C₂²	1	512	yes	complemented
13	#90616	C₂²	1	640	yes	complemented
14	C₂×#6554	C₂²	1	80	yes	complemented
15	C₂×#6553	C₂²	1	80	yes	complemented
16	C₂×D₇×#41	C₂²	1	160	yes	complemented
17	C₂×#6551	C₂²	1	40	yes	complemented
18	C₂×#6551	C₂²	1	80	yes	complemented
19	C₂×#6551	C₂²	1	320	yes	complemented
20	C₂²×#1161	C₂²	1	40	yes	complemented
21	C₂²×Q₈×D₇	C₂²	1	40	yes	complemented
22	C₂×#6548	C₂²	1	40	yes	complemented
23	C₂²×#1157	C₂²	1	60	yes	complemented
24	C₂×2₋¹⁺⁶	—	7	1	no	maximal, Sylow
25	C₇×2₋¹⁺⁴	C₂³	1	256	yes	complemented
26	C₂×C₇×#41	C₂³	1	80	yes	—
27	C₂×C₇×#41	C₂³	1	160	yes	complemented
28	C₂²×C₇×Q₈	C₂³	1	10	yes	—
29	C₂²×C₇×Q₈	C₂³	1	40	yes	—
30	C₂²×C₇×Q₈	C₂³	1	80	yes	complemented
31	C₂²×C₇×D₄	C₂³	1	10	yes	—
32	C₂³×C₄×C₇	C₂³	1	15	yes	—
33	C₂²×C₂²⋊D₇	C₂³	1	40	yes	—
34	#6554	C₂³	1	1280	yes	complemented
35	#6553	C₂³	1	1280	yes	complemented
36	D₇×#41	C₂³	1	2560	yes	complemented
37	#6551	C₂³	1	640	yes	complemented
38	#6551	C₂³	1	1280	yes	complemented
39	C₂×#1161	C₂³	1	160	yes	—
40	C₂×#1161	C₂³	1	640	yes	complemented
41	C₂×Q₈×D₇	C₂³	1	160	yes	—
42	C₂×Q₈×D₇	C₂³	1	640	yes	complemented
43	#6548	C₂³	1	640	yes	complemented
44	C₂×#1159	C₂³	1	160	yes	—
45	C₂×D₄×D₇	C₂³	1	160	yes	—
46	C₂×#1157	C₂³	1	80	yes	—
47	C₂×#1157	C₂³	1	240	yes	—
48	C₂×#1157	C₂³	1	960	yes	complemented
49	C₂²×D₂₈	C₂³	1	60	yes	—
50	C₂²×C₄×D₇	C₂³	1	120	yes	—
51	C₂²×Dic₁₄	C₂³	1	60	yes	—
52	2₋¹⁺⁶	—	7	64	no	—
53	C₂×#585	—	7	16	no	—
54	C₂×#585	—	7	20	no	—
55	C₂²×2₋¹⁺⁴	—	7	16	no	—
56	C₂²×2₋¹⁺⁴	—	7	10	no	—
57	C₂²×2₋¹⁺⁴	D₇	1	1	yes	characteristic, complemented
58	C₂⁴×C₇	C₂⁴	1	5	yes	—
59	C₂³×D₇	C₂⁴	1	40	yes	—
60	C₇×#41	C₂⁴	1	640	yes	—
61	C₂×C₇×Q₈	C₂⁴	1	40	yes	—
62	C₂×C₇×Q₈	C₂⁴	1	320	yes	—
63	C₂×C₇×Q₈	C₂⁴	1	80	yes	—
64	C₂×C₇×D₄	C₂⁴	1	40	yes	—
65	C₂×C₇×D₄	C₂⁴	1	80	yes	—
66	C₂²×C₄×C₇	C₂⁴	1	60	yes	—
67	C₂²×C₄×C₇	C₂⁴	1	10	yes	—
68	C₂²×C₄×C₇	C₂⁴	1	120	yes	—
69	C₂×C₂²⋊D₇	C₂⁴	1	16	yes	—
70	C₂×C₂²⋊D₇	C₂⁴	1	80	yes	—
71	C₂×C₂²⋊D₇	C₂⁴	1	320	yes	—
72	C₂²×Dic₇	C₂⁴	1	40	yes	—
73	#1161	C₂⁴	1	1280	yes	—
74	Q₈×D₇	C₂⁴	1	1280	yes	—
75	#1159	C₂⁴	1	1280	yes	—
76	D₄×D₇	C₂⁴	1	1280	yes	—
77	#1157	C₂⁴	1	640	yes	—
78	#1157	C₂⁴	1	1920	yes	—
79	C₂×D₂₈	C₂⁴	1	80	yes	—
80	C₂×D₂₈	C₂⁴	1	480	yes	—
81	C₂×C₄×D₇	C₂⁴	1	160	yes	—
82	C₂×C₄×D₇	C₂⁴	1	960	yes	—
83	C₂×Dic₁₄	C₂⁴	1	80	yes	—
84	C₂×Dic₁₄	C₂⁴	1	480	yes	—
85	#585	—	7	512	no	—
86	#585	—	7	640	no	—
87	C₂×2₋¹⁺⁴	—	7	512	no	—
88	C₂×2₋¹⁺⁴	—	7	80	no	—
89	C₂×2₋¹⁺⁴	—	7	80	no	—
90	C₂×2₋¹⁺⁴	—	7	40	no	—
91	C₂×2₋¹⁺⁴	—	7	320	no	—
92	C₂×2₋¹⁺⁴	C₂×D₇	1	16	yes	complemented
93	C₂×2₋¹⁺⁴	C₂×D₇	1	32	yes	complemented
94	C₂×H(5,𝔽₂)	—	7	80	no	—
95	C₂×H(5,𝔽₂)	—	7	40	no	—
96	C₂²×#41	—	7	160	no	—
97	C₂²×#41	—	7	40	no	—
98	C₂²×#41	—	7	60	no	—
99	C₂²×#41	C₂×D₇	1	10	yes	complemented
100	C₂³×Q₈	—	7	40	no	—
101	C₂³×Q₈	C₂×D₇	1	5	yes	complemented
102	C₂³×C₇	—	4	10	no	—
103	C₂³×C₇	C₂⁵	1	10	yes	—
104	C₂³×C₇	C₂⁵	1	1	yes	characteristic
105	C₂³×C₇	C₂⁵	1	20	yes	—
106	C₂²×D₇	—	4	80	no	—
107	C₂²×D₇	C₂⁵	1	16	yes	—
108	C₂²×D₇	C₂⁵	1	160	yes	—
109	C₇×Q₈	C₂⁵	1	160	yes	—
110	C₇×D₄	C₂⁵	1	160	yes	—
111	C₂×C₄×C₇	C₂⁵	1	20	yes	—
112	C₂×C₄×C₇	C₂⁵	1	240	yes	—
113	C₂×C₄×C₇	C₂⁵	1	40	yes	—
114	C₂²⋊D₇	C₂⁵	1	64	yes	—
115	C₂²⋊D₇	C₂⁵	1	320	yes	—
116	C₂×Dic₇	C₂⁵	1	16	yes	—
117	C₂×Dic₇	C₂⁵	1	160	yes	—
118	D₂₈	C₂⁵	1	320	yes	—
119	C₄×D₇	C₂⁵	1	640	yes	—
120	Dic₁₄	C₂⁵	1	320	yes	—
121	2₋¹⁺⁴	—	7	1280	no	—
122	2₋¹⁺⁴	—	7	1280	no	—
123	2₋¹⁺⁴	—	7	640	no	—
124	2₋¹⁺⁴	C₂²×D₇	1	256	yes	complemented
125	H(5,𝔽₂)	—	7	1280	no	—
126	H(5,𝔽₂)	—	7	640	no	—
127	C₂×#41	—	7	2560	no	—
128	C₂×#41	—	7	160	no	—
129	C₂×#41	—	7	640	no	—
130	C₂×#41	—	7	160	no	—
131	C₂×#41	—	7	240	no	—
132	C₂×#41	—	7	80	no	—
133	C₂×#41	—	7	960	no	—
134	C₂×#41	C₂²×D₇	1	80	yes	—
135	C₂×#41	C₂²×D₇	1	160	yes	complemented
136	C₂²×Q₈	—	7	160	no	—
137	C₂²×Q₈	—	7	640	no	—
138	C₂²×Q₈	—	7	60	no	—
139	C₂²×Q₈	C₂²×D₇	1	40	yes	—
140	C₂²×Q₈	C₂²×D₇	1	10	yes	—
141	C₂²×Q₈	C₂²×D₇	1	80	yes	complemented
142	C₂²×D₄	—	7	40	no	—
143	C₂²×D₄	—	7	160	no	—
144	C₂²×D₄	—	7	60	no	—
145	C₂²×D₄	C₂²×D₇	1	10	yes	—
146	C₂³×C₄	—	7	120	no	—
147	C₂³×C₄	C₂²×D₇	1	15	yes	—
148	C₂²×C₇	—	2	20	no	—
149	C₂²×C₇	—	2	2	no	—
150	C₂²×C₇	—	4	20	no	—
151	C₂²×C₇	C₂⁶	1	20	yes	—
152	C₂²×C₇	C₂⁶	1	2	yes	—
153	C₂²×C₇	C₂⁶	1	1	yes	characteristic
154	C₂×D₇	—	2	32	no	—
155	C₂×D₇	—	4	160	no	—
156	C₂×D₇	C₂⁶	1	32	yes	—
157	C₄×C₇	C₂⁶	1	40	yes	—
158	Dic₇	C₂⁶	1	32	yes	—
159	C₂⁴	—	7	40	no	—
160	C₂⁴	C₂³×D₇	1	5	yes	—
161	#41	—	7	1280	no	—
162	#41	—	7	1280	no	—
163	#41	—	7	1920	no	—
164	#41	—	7	640	no	—
165	#41	C₂³×D₇	1	640	yes	—
166	C₂×Q₈	—	7	1280	no	—
167	C₂×Q₈	—	7	80	no	—
168	C₂×Q₈	—	7	480	no	—
169	C₂×Q₈	C₂³×D₇	1	40	yes	—
170	C₂×Q₈	C₂³×D₇	1	320	yes	—
171	C₂×Q₈	C₂³×D₇	1	80	yes	—
172	C₂×D₄	—	7	80	no	—
173	C₂×D₄	—	7	16	no	—
174	C₂×D₄	—	7	320	no	—
175	C₂×D₄	—	7	1280	no	—
176	C₂×D₄	—	7	80	no	—
177	C₂×D₄	—	7	480	no	—
178	C₂×D₄	C₂³×D₇	1	40	yes	—
179	C₂×D₄	C₂³×D₇	1	80	yes	—
180	C₂²×C₄	—	7	40	no	—
181	C₂²×C₄	—	7	160	no	—
182	C₂²×C₄	—	7	960	no	—
183	C₂²×C₄	C₂³×D₇	1	60	yes	—
184	C₂²×C₄	C₂³×D₇	1	10	yes	—
185	C₂²×C₄	C₂³×D₇	1	120	yes	—
186	C₂×C₇	—	2	20	no	—
187	C₂×C₇	—	2	2	no	—
188	C₂×C₇	2₋¹⁺⁶	1	2	yes	complemented
189	C₂×C₇	C₂⁷	1	1	yes	characteristic
190	D₇	—	2	32	no	—
191	C₂³	—	4	10	no	—
192	C₂³	—	7	16	no	—
193	C₂³	—	7	160	no	—
194	C₂³	—	28	80	no	—
195	C₂³	C₂⁴×D₇	1	10	yes	—
196	C₂³	C₂⁴×D₇	1	1	yes	characteristic
197	C₂³	C₂⁴×D₇	1	20	yes	—
198	Q₈	—	7	320	no	—
199	Q₈	C₂⁴×D₇	1	160	yes	—
200	D₄	—	7	320	no	—
201	D₄	—	7	64	no	—
202	D₄	—	7	320	no	—
203	D₄	C₂⁴×D₇	1	160	yes	—
204	C₂×C₄	—	7	160	no	—
205	C₂×C₄	—	7	16	no	—
206	C₂×C₄	—	7	640	no	—
207	C₂×C₄	C₂⁴×D₇	1	240	yes	—
208	C₂×C₄	C₂⁴×D₇	1	40	yes	—
209	C₂×C₄	C₂⁴×D₇	1	20	yes	—
210	C₇	C₂×2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
211	C₂²	—	2	2	no	—
212	C₂²	—	2	20	no	—
213	C₂²	—	4	20	no	—
214	C₂²	—	7	32	no	—
215	C₂²	—	14	32	no	—
216	C₂²	—	28	160	no	—
217	C₂²	C₂⁵×D₇	1	20	yes	—
218	C₂²	C₂⁵×D₇	1	2	yes	—
219	C₂²	C₂⁵×D₇	1	1	yes	characteristic
220	C₄	—	7	32	no	—
221	C₄	C₂⁵×D₇	1	40	yes	—
222	C₂	—	2	2	no	—
223	C₂	—	2	20	no	—
224	C₂	—	14	32	no	—
225	C₂	#11744563	1	2	yes	direct product factor
226	C₂	C₂⁶×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂⁵×D₇
commutator, cosocle	C₂×C₇	C₂⁷
Frattini	C₂	C₂⁶×D₇
Fitting	C₂²×C₇×2₋¹⁺⁴	C₂
socle	C₂²×C₇	C₂⁶
hypercenter	C₂²×2₋¹⁺⁴	D₇
hypocenter	C₇	C₂×2₋¹⁺⁶
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×2₋¹⁺⁴	C₂	semidirect
2. #11744563	C₂	direct (N × H)
3. C₂×D₇×2₋¹⁺⁴	C₂	semidirect
4. C₂×#90620	C₂	semidirect
5. C₂×#90616	C₂	semidirect
6. C₂²×#6551	C₂	semidirect
7. C₂×C₇×2₋¹⁺⁴	C₂²	semidirect
8. C₂×C₇×2₋¹⁺⁴	C₂²	semidirect
9. C₂²×C₇×#41	C₂²	semidirect
10. C₂³×C₇×Q₈	C₂²	semidirect
11. #90620	C₂²	semidirect
12. D₇×2₋¹⁺⁴	C₂²	semidirect
13. #90616	C₂²	semidirect
14. C₂×#6554	C₂²	semidirect
15. C₂×#6553	C₂²	semidirect
16. C₂×D₇×#41	C₂²	semidirect
17. C₂×#6551	C₂²	semidirect
18. C₂×#6551	C₂²	semidirect
19. C₂×#6551	C₂²	semidirect
20. C₂²×#1161	C₂²	semidirect
21. C₂²×Q₈×D₇	C₂²	semidirect
22. C₂×#6548	C₂²	semidirect
23. C₂²×#1157	C₂²	semidirect
25. C₇×2₋¹⁺⁴	C₂³	semidirect
27. C₂×C₇×#41	C₂³	semidirect
30. C₂²×C₇×Q₈	C₂³	semidirect
34. #6554	C₂³	semidirect
35. #6553	C₂³	semidirect
36. D₇×#41	C₂³	semidirect
37. #6551	C₂³	semidirect
38. #6551	C₂³	semidirect
40. C₂×#1161	C₂³	semidirect
42. C₂×Q₈×D₇	C₂³	semidirect
43. #6548	C₂³	semidirect
48. C₂×#1157	C₂³	semidirect
57. C₂²×2₋¹⁺⁴	D₇	semidirect
92. C₂×2₋¹⁺⁴	C₂×D₇	semidirect
93. C₂×2₋¹⁺⁴	C₂×D₇	semidirect
99. C₂²×#41	C₂×D₇	semidirect
101. C₂³×Q₈	C₂×D₇	semidirect
124. 2₋¹⁺⁴	C₂²×D₇	semidirect
135. C₂×#41	C₂²×D₇	semidirect
141. C₂²×Q₈	C₂²×D₇	semidirect
188. C₂×C₇	2₋¹⁺⁶	semidirect
210. C₇	C₂×2₋¹⁺⁶	semidirect
225. C₂	#11744563	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×C₇×2₋¹⁺⁴	2
#11744563	2
C₂×D₇×2₋¹⁺⁴	2
C₂×#90620	2
C₂×#90616	2
C₂²×#6551	2
C₂×2₋¹⁺⁶	7

Subgroups — C2×#11744563 #14267695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11744563 #14267695