Subgroups — C₂³×#6553 #14267683

Order 1792.

65726 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	C₂³×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#6553	C₂	1	112	yes	maximal, direct product factor
3	C₂³×#1161	C₂	1	2	yes	maximal, complemented
4	C₂³×D₄×D₇	C₂	1	6	yes	maximal, complemented
5	C₂³×#1157	C₂	1	3	yes	maximal, complemented
6	C₂⁴×D₂₈	C₂	1	3	yes	maximal, complemented
7	C₂⁵×D₇	C₂²	1	6	yes	complemented
8	C₂²×C₇×#41	C₂²	1	56	yes	complemented
9	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
10	C₂³×C₇×D₄	C₂²	1	3	yes	complemented
11	C₂⁴×C₄×C₇	C₂²	1	3	yes	complemented
12	C₂³×C₂²⋊D₇	C₂²	1	6	yes	complemented
13	C₂×#6553	C₂²	1	1792	yes	direct product factor
14	C₂²×#1161	C₂²	1	112	yes	complemented
15	C₂²×D₄×D₇	C₂²	1	336	yes	complemented
16	C₂²×#1157	C₂²	1	168	yes	complemented
17	C₂³×D₂₈	C₂²	1	3	yes	complemented
18	C₂³×D₂₈	C₂²	1	6	yes	complemented
19	C₂³×D₂₈	C₂²	1	168	yes	complemented
20	C₂³×C₄×D₇	C₂²	1	6	yes	complemented
21	C₂³×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
22	C₂³×H(5,𝔽₂)	—	7	1	no	maximal, Sylow
23	C₂⁵×C₇	C₂³	1	3	yes	—
24	C₂⁴×D₇	—	4	48	no	—
25	C₂⁴×D₇	C₂³	1	6	yes	—
26	C₂⁴×D₇	C₂³	1	168	yes	complemented
27	C₂×C₇×#41	C₂³	1	448	yes	complemented
28	C₂²×C₇×Q₈	C₂³	1	28	yes	complemented
29	C₂²×C₇×D₄	C₂³	1	84	yes	complemented
30	C₂³×C₄×C₇	C₂³	1	3	yes	—
31	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₇	C₂³	1	84	yes	complemented
33	C₂²×C₂²⋊D₇	C₂³	1	168	yes	complemented
34	C₂³×Dic₇	C₂³	1	2	yes	—
35	#6553	C₂³	1	4096	yes	direct product factor
36	C₂×#1161	C₂³	1	896	yes	complemented
37	C₂×D₄×D₇	C₂³	1	2688	yes	complemented
38	C₂×#1157	C₂³	1	1344	yes	complemented
39	C₂²×D₂₈	C₂³	1	84	yes	complemented
40	C₂²×D₂₈	C₂³	1	168	yes	complemented
41	C₂²×D₂₈	C₂³	1	1344	yes	complemented
42	C₂²×C₄×D₇	C₂³	1	168	yes	complemented
43	C₂²×Dic₁₄	C₂³	1	28	yes	complemented
44	C₂²×H(5,𝔽₂)	—	7	112	no	—
45	C₂³×#41	—	7	2	no	—
46	C₂³×#41	—	7	3	no	—
47	C₂³×#41	D₇	1	1	yes	characteristic, complemented
48	C₂⁴×D₄	—	7	6	no	—
49	C₂⁴×D₄	—	7	3	no	—
50	C₂⁴×C₇	—	2	24	no	—
51	C₂⁴×C₇	C₂⁴	1	42	yes	—
52	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
53	C₂³×D₇	—	2	48	no	—
54	C₂³×D₇	—	4	672	no	—
55	C₂³×D₇	C₂⁴	1	84	yes	—
56	C₂³×D₇	C₂⁴	1	672	yes	complemented
57	C₇×#41	C₂⁴	1	512	yes	complemented
58	C₂×C₇×Q₈	C₂⁴	1	112	yes	complemented
59	C₂×C₇×D₄	C₂⁴	1	336	yes	complemented
60	C₂²×C₄×C₇	C₂⁴	1	14	yes	—
61	C₂²×C₄×C₇	C₂⁴	1	42	yes	—
62	C₂²×C₄×C₇	C₂⁴	1	336	yes	complemented
63	C₂×C₂²⋊D₇	C₂⁴	1	672	yes	complemented
64	C₂²×Dic₇	C₂⁴	1	28	yes	—
65	#1161	C₂⁴	1	1024	yes	complemented
66	D₄×D₇	C₂⁴	1	3072	yes	complemented
67	#1157	C₂⁴	1	1536	yes	complemented
68	C₂×D₂₈	C₂⁴	1	336	yes	complemented
69	C₂×D₂₈	C₂⁴	1	672	yes	complemented
70	C₂×D₂₈	C₂⁴	1	1536	yes	complemented
71	C₂×C₄×D₇	C₂⁴	1	672	yes	complemented
72	C₂×Dic₁₄	C₂⁴	1	112	yes	complemented
73	C₂⁶	—	7	6	no	—
74	C₂×H(5,𝔽₂)	—	7	1792	no	—
75	C₂²×#41	—	7	112	no	—
76	C₂²×#41	—	7	168	no	—
77	C₂²×#41	C₂×D₇	1	56	yes	complemented
78	C₂³×Q₈	—	7	1	no	—
79	C₂³×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
80	C₂³×D₄	—	7	6	no	—
81	C₂³×D₄	—	7	336	no	—
82	C₂³×D₄	—	7	6	no	—
83	C₂³×D₄	—	7	3	no	—
84	C₂³×D₄	—	7	168	no	—
85	C₂³×D₄	C₂×D₇	1	3	yes	complemented
86	C₂⁴×C₄	—	7	6	no	—
87	C₂⁴×C₄	C₂×D₇	1	3	yes	complemented
88	C₂³×C₇	—	2	168	no	—
89	C₂³×C₇	H(5,𝔽₂)	1	8	yes	complemented
90	C₂³×C₇	C₂⁵	1	84	yes	—
91	C₂³×C₇	C₂⁵	1	7	yes	—
92	C₂²×D₇	—	2	336	no	—
93	C₂²×D₇	—	4	1344	no	—
94	C₂²×D₇	C₂⁵	1	168	yes	—
95	C₂²×D₇	C₂⁵	1	384	yes	complemented
96	C₇×Q₈	C₂⁵	1	64	yes	complemented
97	C₇×D₄	C₂⁵	1	192	yes	complemented
98	C₂×C₄×C₇	C₂⁵	1	28	yes	—
99	C₂×C₄×C₇	C₂⁵	1	84	yes	—
100	C₂×C₄×C₇	C₂⁵	1	192	yes	complemented
101	C₂²⋊D₇	C₂⁵	1	384	yes	complemented
102	C₂×Dic₇	C₂⁵	1	56	yes	—
103	D₂₈	C₂⁵	1	192	yes	complemented
104	D₂₈	C₂⁵	1	384	yes	complemented
105	C₄×D₇	C₂⁵	1	384	yes	complemented
106	Dic₁₄	C₂⁵	1	64	yes	complemented
107	C₂⁵	—	7	6	no	—
108	C₂⁵	—	7	168	no	—
109	C₂⁵	—	28	48	no	—
110	C₂⁵	C₂²×D₇	1	3	yes	—
111	H(5,𝔽₂)	—	7	4096	no	—
112	C₂×#41	—	7	896	no	—
113	C₂×#41	—	7	1344	no	—
114	C₂×#41	C₂²×D₇	1	448	yes	complemented
115	C₂²×Q₈	—	7	28	no	—
116	C₂²×Q₈	C₂²×D₇	1	28	yes	complemented
117	C₂²×D₄	—	7	168	no	—
118	C₂²×D₄	—	7	2688	no	—
119	C₂²×D₄	—	7	168	no	—
120	C₂²×D₄	—	7	84	no	—
121	C₂²×D₄	—	7	1344	no	—
122	C₂²×D₄	C₂²×D₇	1	84	yes	complemented
123	C₂³×C₄	—	7	2	no	—
124	C₂³×C₄	—	7	168	no	—
125	C₂³×C₄	C₂²×D₇	1	3	yes	—
126	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
127	C₂³×C₄	C₂²×D₇	1	84	yes	complemented
128	C₂²×C₇	—	2	168	no	—
129	C₂²×C₇	C₂×H(5,𝔽₂)	1	28	yes	complemented
130	C₂²×C₇	C₂⁶	1	24	yes	—
131	C₂²×C₇	C₂⁶	1	7	yes	—
132	C₂×D₇	—	2	336	no	—
133	C₂×D₇	—	4	384	no	—
134	C₂×D₇	C₂⁶	1	48	yes	—
135	C₄×C₇	C₂⁶	1	8	yes	—
136	C₄×C₇	C₂⁶	1	24	yes	—
137	Dic₇	C₂⁶	1	16	yes	—
138	C₂⁴	—	2	24	no	—
139	C₂⁴	—	7	84	no	—
140	C₂⁴	—	7	672	no	—
141	C₂⁴	—	14	48	no	—
142	C₂⁴	—	28	672	no	—
143	C₂⁴	C₂³×D₇	1	42	yes	—
144	C₂⁴	C₂³×D₇	1	1	yes	characteristic
145	#41	—	7	1024	no	—
146	#41	—	7	1536	no	—
147	#41	C₂³×D₇	1	512	yes	complemented
148	C₂×Q₈	—	7	112	no	—
149	C₂×Q₈	C₂³×D₇	1	112	yes	complemented
150	C₂×D₄	—	7	672	no	—
151	C₂×D₄	—	7	3072	no	—
152	C₂×D₄	—	7	672	no	—
153	C₂×D₄	—	7	336	no	—
154	C₂×D₄	—	7	1536	no	—
155	C₂×D₄	C₂³×D₇	1	336	yes	complemented
156	C₂²×C₄	—	7	28	no	—
157	C₂²×C₄	—	7	672	no	—
158	C₂²×C₄	C₂³×D₇	1	42	yes	—
159	C₂²×C₄	C₂³×D₇	1	14	yes	—
160	C₂²×C₄	C₂³×D₇	1	336	yes	complemented
161	C₂×C₇	—	2	24	no	—
162	C₂×C₇	C₂²×H(5,𝔽₂)	1	14	yes	complemented
163	C₂×C₇	C₂⁷	1	1	yes	characteristic
164	D₇	—	2	48	no	—
165	C₂³	—	2	168	no	—
166	C₂³	—	7	168	no	—
167	C₂³	—	7	384	no	—
168	C₂³	—	14	336	no	—
169	C₂³	—	28	1344	no	—
170	C₂³	#6553	1	8	yes	direct product factor
171	C₂³	C₂⁴×D₇	1	84	yes	—
172	C₂³	C₂⁴×D₇	1	7	yes	—
173	Q₈	—	7	64	no	—
174	Q₈	C₂⁴×D₇	1	64	yes	complemented
175	D₄	—	7	384	no	—
176	D₄	—	7	384	no	—
177	D₄	—	7	192	no	—
178	D₄	C₂⁴×D₇	1	192	yes	complemented
179	C₂×C₄	—	7	56	no	—
180	C₂×C₄	—	7	384	no	—
181	C₂×C₄	C₂⁴×D₇	1	84	yes	—
182	C₂×C₄	C₂⁴×D₇	1	28	yes	—
183	C₂×C₄	C₂⁴×D₇	1	192	yes	complemented
184	C₇	C₂³×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
185	C₂²	—	2	168	no	—
186	C₂²	—	7	48	no	—
187	C₂²	—	14	336	no	—
188	C₂²	—	28	384	no	—
189	C₂²	C₂×#6553	1	28	yes	direct product factor
190	C₂²	C₂⁵×D₇	1	24	yes	—
191	C₂²	C₂⁵×D₇	1	7	yes	—
192	C₄	—	7	16	no	—
193	C₄	C₂⁵×D₇	1	24	yes	—
194	C₄	C₂⁵×D₇	1	8	yes	—
195	C₂	—	2	24	no	—
196	C₂	—	14	48	no	—
197	C₂	C₂²×#6553	1	14	yes	direct product factor
198	C₂	C₂⁶×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₇
commutator, cosocle	C₂×C₇	C₂⁷
Frattini	C₂	C₂⁶×D₇
Fitting	C₂³×C₇×#41	C₂
socle	C₂⁴×C₇	C₂⁴
hypercenter	C₂³×#41	D₇
hypocenter	C₇	C₂³×H(5,𝔽₂)
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×C₇×#41	C₂	semidirect
2. C₂²×#6553	C₂	direct (N × H)
3. C₂³×#1161	C₂	semidirect
4. C₂³×D₄×D₇	C₂	semidirect
5. C₂³×#1157	C₂	semidirect
6. C₂⁴×D₂₈	C₂	semidirect
7. C₂⁵×D₇	C₂²	semidirect
8. C₂²×C₇×#41	C₂²	semidirect
9. C₂³×C₇×Q₈	C₂²	semidirect
10. C₂³×C₇×D₄	C₂²	semidirect
11. C₂⁴×C₄×C₇	C₂²	semidirect
12. C₂³×C₂²⋊D₇	C₂²	semidirect
13. C₂×#6553	C₂²	direct (N × H)
14. C₂²×#1161	C₂²	semidirect
15. C₂²×D₄×D₇	C₂²	semidirect
16. C₂²×#1157	C₂²	semidirect
17. C₂³×D₂₈	C₂²	semidirect
18. C₂³×D₂₈	C₂²	semidirect
19. C₂³×D₂₈	C₂²	semidirect
20. C₂³×C₄×D₇	C₂²	semidirect
21. C₂³×Dic₁₄	C₂²	semidirect
26. C₂⁴×D₇	C₂³	semidirect
27. C₂×C₇×#41	C₂³	semidirect
28. C₂²×C₇×Q₈	C₂³	semidirect
29. C₂²×C₇×D₄	C₂³	semidirect
32. C₂³×C₄×C₇	C₂³	semidirect
33. C₂²×C₂²⋊D₇	C₂³	semidirect
35. #6553	C₂³	direct (N × H)
36. C₂×#1161	C₂³	semidirect
37. C₂×D₄×D₇	C₂³	semidirect
38. C₂×#1157	C₂³	semidirect
39. C₂²×D₂₈	C₂³	semidirect
40. C₂²×D₂₈	C₂³	semidirect
41. C₂²×D₂₈	C₂³	semidirect
42. C₂²×C₄×D₇	C₂³	semidirect
43. C₂²×Dic₁₄	C₂³	semidirect
47. C₂³×#41	D₇	semidirect
56. C₂³×D₇	C₂⁴	semidirect
57. C₇×#41	C₂⁴	semidirect
58. C₂×C₇×Q₈	C₂⁴	semidirect
59. C₂×C₇×D₄	C₂⁴	semidirect
62. C₂²×C₄×C₇	C₂⁴	semidirect
63. C₂×C₂²⋊D₇	C₂⁴	semidirect
65. #1161	C₂⁴	semidirect
66. D₄×D₇	C₂⁴	semidirect
67. #1157	C₂⁴	semidirect
68. C₂×D₂₈	C₂⁴	semidirect
69. C₂×D₂₈	C₂⁴	semidirect
70. C₂×D₂₈	C₂⁴	semidirect
71. C₂×C₄×D₇	C₂⁴	semidirect
72. C₂×Dic₁₄	C₂⁴	semidirect
77. C₂²×#41	C₂×D₇	semidirect
79. C₂³×Q₈	C₂×D₇	semidirect
85. C₂³×D₄	C₂×D₇	semidirect
87. C₂⁴×C₄	C₂×D₇	semidirect
89. C₂³×C₇	H(5,𝔽₂)	semidirect
95. C₂²×D₇	C₂⁵	semidirect
96. C₇×Q₈	C₂⁵	semidirect
97. C₇×D₄	C₂⁵	semidirect
100. C₂×C₄×C₇	C₂⁵	semidirect
101. C₂²⋊D₇	C₂⁵	semidirect
103. D₂₈	C₂⁵	semidirect
104. D₂₈	C₂⁵	semidirect
105. C₄×D₇	C₂⁵	semidirect
106. Dic₁₄	C₂⁵	semidirect
114. C₂×#41	C₂²×D₇	semidirect
116. C₂²×Q₈	C₂²×D₇	semidirect
122. C₂²×D₄	C₂²×D₇	semidirect
127. C₂³×C₄	C₂²×D₇	semidirect
129. C₂²×C₇	C₂×H(5,𝔽₂)	semidirect
147. #41	C₂³×D₇	semidirect
149. C₂×Q₈	C₂³×D₇	semidirect
155. C₂×D₄	C₂³×D₇	semidirect
160. C₂²×C₄	C₂³×D₇	semidirect
162. C₂×C₇	C₂²×H(5,𝔽₂)	semidirect
170. C₂³	#6553	direct (N × H)
174. Q₈	C₂⁴×D₇	semidirect
178. D₄	C₂⁴×D₇	semidirect
183. C₂×C₄	C₂⁴×D₇	semidirect
184. C₇	C₂³×H(5,𝔽₂)	semidirect
189. C₂²	C₂×#6553	direct (N × H)
197. C₂	C₂²×#6553	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×#41	2
C₂²×#6553	2
C₂³×#1161	2
C₂³×D₄×D₇	2
C₂³×#1157	2
C₂⁴×D₂₈	2
C₂³×H(5,𝔽₂)	7

Subgroups — C23×#6553 #14267683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6553 #14267683