Subgroups — C₂×#16811692 #20291694

Order 6784.

1640 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	#16811692	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×#6157	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×#6157	C₂	1	6	yes	maximal, complemented
4	C₂×#13133296	C₂	1	3	yes	maximal, complemented
5	C₂²×C₄⋊#6157	C₂	1	3	yes	maximal, complemented
6	C₂×#41×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#6157	C₂²	1	6	yes	—
8	Q₈×#6157	C₂²	1	16	yes	complemented
9	D₄×#6157	C₂²	1	48	yes	complemented
10	#13133296	C₂²	1	24	yes	complemented
11	C₂×C₄⋊#6157	C₂²	1	3	yes	—
12	C₂×C₄⋊#6157	C₂²	1	6	yes	—
13	C₂×C₄⋊#6157	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#6157	C₂²	1	24	yes	complemented
15	C₂×C₂²⋊#6157	C₂²	1	6	yes	—
16	C₂×C₄×#6157	C₂²	1	6	yes	—
17	C₂×#11719949	C₄	1	1	yes	characteristic, complemented
18	C₂×#11719945	C₄	1	3	yes	complemented
19	C₂×#11719947	C₄	1	3	yes	complemented
20	#41×D₅₃	C₄	1	8	yes	complemented
21	#41×D₅₃	C₂²	1	8	yes	—
22	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic
23	C₂×D₄×D₅₃	C₂²	1	3	yes	—
24	C₂²×C₄×D₅₃	C₂²	1	3	yes	—
25	C₂×#41×C₅₃	C₄	1	1	yes	characteristic, complemented
26	C₂³×D₅₃	C₂³	1	3	yes	—
27	C₂²×#6157	—	4	12	no	—
28	C₂²×#6157	C₂³	1	2	yes	—
29	C₂²×#6157	C₂³	1	6	yes	—
30	C₂²×#6157	C₂³	1	24	yes	—
31	#41×C₅₃	C₂×C₄	1	8	yes	complemented
32	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
33	C₂×D₄×C₅₃	C₂×C₄	1	3	yes	complemented
34	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	complemented
35	C₂×C₂²⋊D₅₃	C₂×C₄	1	6	yes	complemented
36	C₂²×Dic₅₃	C₂×C₄	1	3	yes	complemented
37	#11719949	C₂×C₄	1	8	yes	complemented
38	Q₈×D₅₃	C₂×C₄	1	4	yes	complemented
39	Q₈×D₅₃	C₂³	1	4	yes	—
40	#11719947	C₂×C₄	1	24	yes	complemented
41	D₄×D₅₃	C₂×C₄	1	12	yes	complemented
42	D₄×D₅₃	C₂³	1	12	yes	—
43	#11719945	C₂×C₄	1	24	yes	complemented
44	C₂×D₂₁₂	C₂×C₄	1	3	yes	complemented
45	C₂×C₄×D₅₃	C₂×C₄	1	12	yes	complemented
46	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	complemented
47	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	complemented
48	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₅₃	C₂³	1	3	yes	—
50	C₂×C₄×D₅₃	C₂³	1	12	yes	—
51	C₂×Dic₁₀₆	C₂×C₄	1	3	yes	complemented
52	C₂²⋊#6157	C₂³	1	24	yes	—
53	C₄⋊#6157	C₂³	1	12	yes	—
54	C₄⋊#6157	C₂³	1	24	yes	—
55	C₄⋊#6157	C₂³	1	4	yes	—
56	C₄×#6157	C₂³	1	24	yes	—
57	C₂³×C₅₃	C₂²×C₄	1	3	yes	—
58	C₂²×D₅₃	—	2	6	no	—
59	C₂²×D₅₃	—	2	6	no	—
60	C₂²×D₅₃	C₂²×C₄	1	3	yes	—
61	C₂²×D₅₃	C₂²×C₄	1	6	yes	—
62	C₂²×D₅₃	C₂⁴	1	6	yes	—
63	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
64	C₂×#6157	—	2	12	no	—
65	C₂×#6157	—	2	4	no	—
66	C₂×#6157	—	4	24	no	—
67	C₂×#6157	C₂⁴	1	4	yes	—
68	C₂×#6157	C₂⁴	1	12	yes	—
69	Q₈×C₅₃	C₂²×C₄	1	4	yes	complemented
70	D₄×C₅₃	C₂²×C₄	1	12	yes	complemented
71	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
72	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₄	1	12	yes	complemented
74	C₂²⋊D₅₃	C₂²×C₄	1	24	yes	complemented
75	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
76	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₅₃	C₂²×C₄	1	12	yes	complemented
78	D₂₁₂	C₂²×C₄	1	12	yes	complemented
79	C₄×D₅₃	C₂²×C₄	1	2	yes	—
80	C₄×D₅₃	C₂²×C₄	1	6	yes	—
81	C₄×D₅₃	C₂²×C₄	1	12	yes	complemented
82	C₄×D₅₃	C₂²×C₄	1	12	yes	complemented
83	C₄×D₅₃	C₂⁴	1	6	yes	—
84	C₄×D₅₃	C₂⁴	1	2	yes	—
85	Dic₁₀₆	C₂²×C₄	1	12	yes	complemented
86	C₂²×C₅₃	—	2	6	no	—
87	C₂²×C₅₃	C₂³×C₄	1	6	yes	—
88	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×D₅₃	—	2	6	no	—
90	C₂×D₅₃	—	2	12	no	—
91	C₂×D₅₃	—	2	6	no	—
92	C₂×D₅₃	—	2	6	no	—
93	C₂×D₅₃	C₂³×C₄	1	6	yes	—
94	C₂×D₅₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₅₃	H(5,𝔽₂)	1	2	yes	—
96	C₂×D₅₃	2₋¹⁺⁴	1	2	yes	—
97	C₂×D₅₃	C₂⁵	1	1	yes	characteristic
98	#6157	—	2	12	no	—
99	#6157	—	2	4	no	—
100	C₄×C₅₃	C₂³×C₄	1	6	yes	—
101	C₄×C₅₃	C₂³×C₄	1	2	yes	—
102	Dic₅₃	C₂³×C₄	1	6	yes	—
103	Dic₅₃	C₂³×C₄	1	2	yes	—
104	C₂×#520	—	53	1	no	maximal, Sylow
105	C₂×C₅₃	—	2	6	no	—
106	C₂×C₅₃	#520	1	2	yes	complemented
107	C₂×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
108	D₅₃	—	2	6	no	—
109	D₅₃	#520	1	2	yes	complemented
110	D₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
111	D₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
112	C₂²×#41	—	53	1	no	—
113	#520	—	53	16	no	—
114	C₂×C₄×Q₈	—	53	2	no	—
115	C₂×C₄×D₄	—	53	6	no	—
116	C₂×#117	—	53	3	no	—
117	C₂²×C₄⋊C₄	—	53	3	no	—
118	C₅₃	C₂×#520	1	1	yes	characteristic, complemented, Sylow
119	C₂×#41	—	53	8	no	—
120	C₂×#41	—	53	8	no	—
121	C₂×#41	—	53	1	no	—
122	C₂×#41	—	53	3	no	—
123	C₂×#41	—	53	3	no	—
124	C₂×#41	#6157	1	1	yes	characteristic, complemented
125	C₂²×Q₈	—	53	1	no	—
126	C₂²×D₄	—	53	3	no	—
127	C₂³×C₄	—	53	3	no	—
128	C₂³×C₄	—	53	6	no	—
129	C₄×Q₈	—	53	16	no	—
130	C₄×D₄	—	53	48	no	—
131	#117	—	53	24	no	—
132	C₂×C₄⋊C₄	—	53	3	no	—
133	C₂×C₄⋊C₄	—	53	6	no	—
134	C₂×C₄⋊C₄	—	53	24	no	—
135	C₂×C₄⋊C₄	—	53	1	no	—
136	C₂×C₂²⋊C₄	—	53	6	no	—
137	C₂×C₄²	—	53	6	no	—
138	C₂⁴	—	53	3	no	—
139	#41	—	53	8	no	—
140	#41	—	53	24	no	—
141	#41	—	53	24	no	—
142	#41	C₂×#6157	1	8	yes	complemented
143	C₂×Q₈	—	53	4	no	—
144	C₂×Q₈	—	53	4	no	—
145	C₂×Q₈	—	53	3	no	—
146	C₂×Q₈	C₂×#6157	1	1	yes	characteristic, complemented
147	C₂×D₄	—	53	6	no	—
148	C₂×D₄	—	53	12	no	—
149	C₂×D₄	—	53	12	no	—
150	C₂×D₄	—	53	3	no	—
151	C₂×D₄	C₂×#6157	1	3	yes	complemented
152	C₂²×C₄	—	53	3	no	—
153	C₂²×C₄	—	53	3	no	—
154	C₂²×C₄	—	53	3	no	—
155	C₂²×C₄	—	53	3	no	—
156	C₂²×C₄	—	53	1	no	—
157	C₂²×C₄	—	53	12	no	—
158	C₂²×C₄	—	53	12	no	—
159	C₂²×C₄	—	53	2	no	—
160	C₂²×C₄	—	53	6	no	—
161	C₂²×C₄	—	53	24	no	—
162	C₂²×C₄	—	212	12	no	—
163	C₂²×C₄	C₂×#6157	1	3	yes	complemented
164	C₄⋊C₄	—	53	12	no	—
165	C₄⋊C₄	—	53	24	no	—
166	C₄⋊C₄	—	53	4	no	—
167	C₂²⋊C₄	—	53	24	no	—
168	C₄²	—	53	24	no	—
169	C₂³	—	53	1	no	—
170	C₂³	—	53	6	no	—
171	C₂³	—	53	6	no	—
172	C₂³	—	53	3	no	—
173	C₂³	—	106	6	no	—
174	C₂³	—	106	6	no	—
175	C₂³	C₂²×#6157	1	3	yes	—
176	Q₈	—	53	12	no	—
177	Q₈	C₂²×#6157	1	4	yes	complemented
178	D₄	—	53	24	no	—
179	D₄	—	53	12	no	—
180	D₄	C₂²×#6157	1	12	yes	complemented
181	C₂×C₄	—	53	12	no	—
182	C₂×C₄	—	53	1	no	—
183	C₂×C₄	—	53	3	no	—
184	C₂×C₄	—	53	2	no	—
185	C₂×C₄	—	53	6	no	—
186	C₂×C₄	—	53	12	no	—
187	C₂×C₄	—	53	12	no	—
188	C₂×C₄	—	53	6	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	53	4	no	—
191	C₂×C₄	—	53	12	no	—
192	C₂×C₄	—	106	4	no	—
193	C₂×C₄	—	106	12	no	—
194	C₂×C₄	—	212	24	no	—
195	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
196	C₂×C₄	C₂²×#6157	1	3	yes	—
197	C₂×C₄	C₂²×#6157	1	12	yes	complemented
198	C₂²	—	2	6	no	—
199	C₂²	—	53	2	no	—
200	C₂²	—	53	2	no	—
201	C₂²	—	53	1	no	—
202	C₂²	—	53	1	no	—
203	C₂²	—	53	6	no	—
204	C₂²	—	106	6	no	—
205	C₂²	—	106	6	no	—
206	C₂²	—	106	12	no	—
207	C₂²	—	106	6	no	—
208	C₂²	C₂³×#6157	1	6	yes	—
209	C₂²	C₂³×#6157	1	1	yes	characteristic
210	C₄	—	53	2	no	—
211	C₄	—	53	6	no	—
212	C₄	—	106	12	no	—
213	C₄	—	106	4	no	—
214	C₄	C₂³×#6157	1	2	yes	—
215	C₄	C₂³×#6157	1	6	yes	—
216	C₂	—	2	6	no	—
217	C₂	—	53	1	no	—
218	C₂	—	53	1	no	—
219	C₂	—	53	2	no	—
220	C₂	—	106	6	no	—
221	C₂	#16811692	1	2	yes	direct product factor
222	C₂	C₂⁴×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×#6157
commutator	C₂×C₅₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#6157
Fitting	C₂×#41×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	C₂×#41	#6157
hypocenter	C₅₃	C₂×#520
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #16811692	C₂	direct (N × H)
2. C₂×Q₈×#6157	C₂	semidirect
3. C₂×D₄×#6157	C₂	semidirect
4. C₂×#13133296	C₂	semidirect
5. C₂²×C₄⋊#6157	C₂	semidirect
8. Q₈×#6157	C₂²	semidirect
9. D₄×#6157	C₂²	semidirect
10. #13133296	C₂²	semidirect
14. C₂×C₄⋊#6157	C₂²	semidirect
17. C₂×#11719949	C₄	semidirect
18. C₂×#11719945	C₄	semidirect
19. C₂×#11719947	C₄	semidirect
20. #41×D₅₃	C₄	semidirect
25. C₂×#41×C₅₃	C₄	semidirect
31. #41×C₅₃	C₂×C₄	semidirect
32. C₂×Q₈×C₅₃	C₂×C₄	semidirect
33. C₂×D₄×C₅₃	C₂×C₄	semidirect
34. C₂²×C₄×C₅₃	C₂×C₄	semidirect
35. C₂×C₂²⋊D₅₃	C₂×C₄	semidirect
36. C₂²×Dic₅₃	C₂×C₄	semidirect
37. #11719949	C₂×C₄	semidirect
38. Q₈×D₅₃	C₂×C₄	semidirect
40. #11719947	C₂×C₄	semidirect
41. D₄×D₅₃	C₂×C₄	semidirect
43. #11719945	C₂×C₄	semidirect
44. C₂×D₂₁₂	C₂×C₄	semidirect
45. C₂×C₄×D₅₃	C₂×C₄	semidirect
46. C₂×C₄×D₅₃	C₂×C₄	semidirect
47. C₂×C₄×D₅₃	C₂×C₄	semidirect
51. C₂×Dic₁₀₆	C₂×C₄	semidirect
69. Q₈×C₅₃	C₂²×C₄	semidirect
70. D₄×C₅₃	C₂²×C₄	semidirect
73. C₂×C₄×C₅₃	C₂²×C₄	semidirect
74. C₂²⋊D₅₃	C₂²×C₄	semidirect
77. C₂×Dic₅₃	C₂²×C₄	semidirect
78. D₂₁₂	C₂²×C₄	semidirect
81. C₄×D₅₃	C₂²×C₄	semidirect
82. C₄×D₅₃	C₂²×C₄	semidirect
85. Dic₁₀₆	C₂²×C₄	semidirect
106. C₂×C₅₃	#520	semidirect
109. D₅₃	#520	semidirect
118. C₅₃	C₂×#520	semidirect
124. C₂×#41	#6157	semidirect
142. #41	C₂×#6157	semidirect
146. C₂×Q₈	C₂×#6157	semidirect
151. C₂×D₄	C₂×#6157	semidirect
163. C₂²×C₄	C₂×#6157	semidirect
177. Q₈	C₂²×#6157	semidirect
180. D₄	C₂²×#6157	semidirect
197. C₂×C₄	C₂²×#6157	semidirect
221. C₂	#16811692	direct (N × H)

Subgroups — C2×#16811692 #20291694

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811692 #20291694