Subgroups — C₂×#16840831 #21462798

Order 7808.

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	#16840831	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×#6923	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×#6923	C₂	1	6	yes	maximal, complemented
4	C₂×#14519856	C₂	1	3	yes	maximal, complemented
5	C₂²×C₄⋊#6923	C₂	1	3	yes	maximal, complemented
6	C₂×#41×D₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×#6923	C₂²	1	6	yes	—
8	Q₈×#6923	C₂²	1	16	yes	complemented
9	D₄×#6923	C₂²	1	48	yes	complemented
10	#14519856	C₂²	1	24	yes	complemented
11	C₂×C₄⋊#6923	C₂²	1	3	yes	—
12	C₂×C₄⋊#6923	C₂²	1	6	yes	—
13	C₂×C₄⋊#6923	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#6923	C₂²	1	24	yes	complemented
15	C₂×C₂²⋊#6923	C₂²	1	6	yes	—
16	C₂×C₄×#6923	C₂²	1	6	yes	—
17	C₂×#11758226	C₄	1	1	yes	characteristic, complemented
18	C₂×#11758222	C₄	1	3	yes	complemented
19	C₂×#11758224	C₄	1	3	yes	complemented
20	#41×D₆₁	C₄	1	8	yes	complemented
21	#41×D₆₁	C₂²	1	8	yes	—
22	C₂×Q₈×D₆₁	C₂²	1	1	yes	characteristic
23	C₂×D₄×D₆₁	C₂²	1	3	yes	—
24	C₂²×C₄×D₆₁	C₂²	1	3	yes	—
25	C₂×#41×C₆₁	C₄	1	1	yes	characteristic, complemented
26	C₂³×D₆₁	C₂³	1	3	yes	—
27	C₂²×#6923	—	4	12	no	—
28	C₂²×#6923	C₂³	1	2	yes	—
29	C₂²×#6923	C₂³	1	6	yes	—
30	C₂²×#6923	C₂³	1	24	yes	—
31	#41×C₆₁	C₂×C₄	1	8	yes	complemented
32	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
33	C₂×D₄×C₆₁	C₂×C₄	1	3	yes	complemented
34	C₂²×C₄×C₆₁	C₂×C₄	1	3	yes	complemented
35	C₂×C₂²⋊D₆₁	C₂×C₄	1	6	yes	complemented
36	C₂²×Dic₆₁	C₂×C₄	1	3	yes	complemented
37	#11758226	C₂×C₄	1	8	yes	complemented
38	Q₈×D₆₁	C₂×C₄	1	4	yes	complemented
39	Q₈×D₆₁	C₂³	1	4	yes	—
40	#11758224	C₂×C₄	1	24	yes	complemented
41	D₄×D₆₁	C₂×C₄	1	12	yes	complemented
42	D₄×D₆₁	C₂³	1	12	yes	—
43	#11758222	C₂×C₄	1	24	yes	complemented
44	C₂×D₂₄₄	C₂×C₄	1	3	yes	complemented
45	C₂×C₄×D₆₁	C₂×C₄	1	12	yes	complemented
46	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	complemented
47	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	complemented
48	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₆₁	C₂³	1	3	yes	—
50	C₂×C₄×D₆₁	C₂³	1	12	yes	—
51	C₂×Dic₁₂₂	C₂×C₄	1	3	yes	complemented
52	C₂²⋊#6923	C₂³	1	24	yes	—
53	C₄⋊#6923	C₂³	1	12	yes	—
54	C₄⋊#6923	C₂³	1	24	yes	—
55	C₄⋊#6923	C₂³	1	4	yes	—
56	C₄×#6923	C₂³	1	24	yes	—
57	C₂³×C₆₁	C₂²×C₄	1	3	yes	—
58	C₂²×D₆₁	—	2	6	no	—
59	C₂²×D₆₁	—	2	6	no	—
60	C₂²×D₆₁	C₂²×C₄	1	3	yes	—
61	C₂²×D₆₁	C₂²×C₄	1	6	yes	—
62	C₂²×D₆₁	C₂⁴	1	6	yes	—
63	C₂²×D₆₁	C₂⁴	1	1	yes	characteristic
64	C₂×#6923	—	2	12	no	—
65	C₂×#6923	—	2	4	no	—
66	C₂×#6923	—	4	24	no	—
67	C₂×#6923	C₂⁴	1	4	yes	—
68	C₂×#6923	C₂⁴	1	12	yes	—
69	Q₈×C₆₁	C₂²×C₄	1	4	yes	complemented
70	D₄×C₆₁	C₂²×C₄	1	12	yes	complemented
71	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
72	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂²×C₄	1	12	yes	complemented
74	C₂²⋊D₆₁	C₂²×C₄	1	24	yes	complemented
75	C₂×Dic₆₁	C₂²×C₄	1	3	yes	—
76	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₆₁	C₂²×C₄	1	12	yes	complemented
78	D₂₄₄	C₂²×C₄	1	12	yes	complemented
79	C₄×D₆₁	C₂²×C₄	1	2	yes	—
80	C₄×D₆₁	C₂²×C₄	1	6	yes	—
81	C₄×D₆₁	C₂²×C₄	1	12	yes	complemented
82	C₄×D₆₁	C₂²×C₄	1	12	yes	complemented
83	C₄×D₆₁	C₂⁴	1	6	yes	—
84	C₄×D₆₁	C₂⁴	1	2	yes	—
85	Dic₁₂₂	C₂²×C₄	1	12	yes	complemented
86	C₂²×C₆₁	—	2	6	no	—
87	C₂²×C₆₁	C₂³×C₄	1	6	yes	—
88	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
89	C₂×D₆₁	—	2	6	no	—
90	C₂×D₆₁	—	2	12	no	—
91	C₂×D₆₁	—	2	6	no	—
92	C₂×D₆₁	—	2	6	no	—
93	C₂×D₆₁	C₂³×C₄	1	6	yes	—
94	C₂×D₆₁	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₆₁	H(5,𝔽₂)	1	2	yes	—
96	C₂×D₆₁	2₋¹⁺⁴	1	2	yes	—
97	C₂×D₆₁	C₂⁵	1	1	yes	characteristic
98	#6923	—	2	12	no	—
99	#6923	—	2	4	no	—
100	C₄×C₆₁	C₂³×C₄	1	6	yes	—
101	C₄×C₆₁	C₂³×C₄	1	2	yes	—
102	Dic₆₁	C₂³×C₄	1	6	yes	—
103	Dic₆₁	C₂³×C₄	1	2	yes	—
104	C₂×#520	—	61	1	no	maximal, Sylow
105	C₂×C₆₁	—	2	6	no	—
106	C₂×C₆₁	#520	1	2	yes	complemented
107	C₂×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
108	D₆₁	—	2	6	no	—
109	D₆₁	#520	1	2	yes	complemented
110	D₆₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
111	D₆₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
112	C₂²×#41	—	61	1	no	—
113	#520	—	61	16	no	—
114	C₂×C₄×Q₈	—	61	2	no	—
115	C₂×C₄×D₄	—	61	6	no	—
116	C₂×#117	—	61	3	no	—
117	C₂²×C₄⋊C₄	—	61	3	no	—
118	C₆₁	C₂×#520	1	1	yes	characteristic, complemented, Sylow
119	C₂×#41	—	61	8	no	—
120	C₂×#41	—	61	8	no	—
121	C₂×#41	—	61	1	no	—
122	C₂×#41	—	61	3	no	—
123	C₂×#41	—	61	3	no	—
124	C₂×#41	#6923	1	1	yes	characteristic, complemented
125	C₂²×Q₈	—	61	1	no	—
126	C₂²×D₄	—	61	3	no	—
127	C₂³×C₄	—	61	3	no	—
128	C₂³×C₄	—	61	6	no	—
129	C₄×Q₈	—	61	16	no	—
130	C₄×D₄	—	61	48	no	—
131	#117	—	61	24	no	—
132	C₂×C₄⋊C₄	—	61	3	no	—
133	C₂×C₄⋊C₄	—	61	6	no	—
134	C₂×C₄⋊C₄	—	61	24	no	—
135	C₂×C₄⋊C₄	—	61	1	no	—
136	C₂×C₂²⋊C₄	—	61	6	no	—
137	C₂×C₄²	—	61	6	no	—
138	C₂⁴	—	61	3	no	—
139	#41	—	61	8	no	—
140	#41	—	61	24	no	—
141	#41	—	61	24	no	—
142	#41	C₂×#6923	1	8	yes	complemented
143	C₂×Q₈	—	61	4	no	—
144	C₂×Q₈	—	61	4	no	—
145	C₂×Q₈	—	61	3	no	—
146	C₂×Q₈	C₂×#6923	1	1	yes	characteristic, complemented
147	C₂×D₄	—	61	6	no	—
148	C₂×D₄	—	61	12	no	—
149	C₂×D₄	—	61	12	no	—
150	C₂×D₄	—	61	3	no	—
151	C₂×D₄	C₂×#6923	1	3	yes	complemented
152	C₂²×C₄	—	61	3	no	—
153	C₂²×C₄	—	61	3	no	—
154	C₂²×C₄	—	61	1	no	—
155	C₂²×C₄	—	61	3	no	—
156	C₂²×C₄	—	61	3	no	—
157	C₂²×C₄	—	61	12	no	—
158	C₂²×C₄	—	61	12	no	—
159	C₂²×C₄	—	61	2	no	—
160	C₂²×C₄	—	61	6	no	—
161	C₂²×C₄	—	61	24	no	—
162	C₂²×C₄	—	244	12	no	—
163	C₂²×C₄	C₂×#6923	1	3	yes	complemented
164	C₄⋊C₄	—	61	12	no	—
165	C₄⋊C₄	—	61	24	no	—
166	C₄⋊C₄	—	61	4	no	—
167	C₂²⋊C₄	—	61	24	no	—
168	C₄²	—	61	24	no	—
169	C₂³	—	61	1	no	—
170	C₂³	—	61	6	no	—
171	C₂³	—	61	6	no	—
172	C₂³	—	61	3	no	—
173	C₂³	—	122	6	no	—
174	C₂³	—	122	6	no	—
175	C₂³	C₂²×#6923	1	3	yes	—
176	Q₈	—	61	12	no	—
177	Q₈	C₂²×#6923	1	4	yes	complemented
178	D₄	—	61	24	no	—
179	D₄	—	61	12	no	—
180	D₄	C₂²×#6923	1	12	yes	complemented
181	C₂×C₄	—	61	12	no	—
182	C₂×C₄	—	61	1	no	—
183	C₂×C₄	—	61	3	no	—
184	C₂×C₄	—	61	2	no	—
185	C₂×C₄	—	61	6	no	—
186	C₂×C₄	—	61	12	no	—
187	C₂×C₄	—	61	12	no	—
188	C₂×C₄	—	61	6	no	—
189	C₂×C₄	—	61	2	no	—
190	C₂×C₄	—	61	4	no	—
191	C₂×C₄	—	61	12	no	—
192	C₂×C₄	—	122	4	no	—
193	C₂×C₄	—	122	12	no	—
194	C₂×C₄	—	244	24	no	—
195	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
196	C₂×C₄	C₂²×#6923	1	3	yes	—
197	C₂×C₄	C₂²×#6923	1	12	yes	complemented
198	C₂²	—	2	6	no	—
199	C₂²	—	61	2	no	—
200	C₂²	—	61	2	no	—
201	C₂²	—	61	1	no	—
202	C₂²	—	61	1	no	—
203	C₂²	—	61	6	no	—
204	C₂²	—	122	6	no	—
205	C₂²	—	122	6	no	—
206	C₂²	—	122	12	no	—
207	C₂²	—	122	6	no	—
208	C₂²	C₂³×#6923	1	6	yes	—
209	C₂²	C₂³×#6923	1	1	yes	characteristic
210	C₄	—	61	2	no	—
211	C₄	—	61	6	no	—
212	C₄	—	122	12	no	—
213	C₄	—	122	4	no	—
214	C₄	C₂³×#6923	1	2	yes	—
215	C₄	C₂³×#6923	1	6	yes	—
216	C₂	—	2	6	no	—
217	C₂	—	61	1	no	—
218	C₂	—	61	1	no	—
219	C₂	—	61	2	no	—
220	C₂	—	122	6	no	—
221	C₂	#16840831	1	2	yes	direct product factor
222	C₂	C₂⁴×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×#6923
commutator	C₂×C₆₁	C₂⁴×C₄
Frattini	C₂	C₂⁴×#6923
Fitting	C₂×#41×C₆₁	C₄
socle	C₂²×C₆₁	C₂³×C₄
hypercenter	C₂×#41	#6923
hypocenter	C₆₁	C₂×#520
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #16840831	C₂	direct (N × H)
2. C₂×Q₈×#6923	C₂	semidirect
3. C₂×D₄×#6923	C₂	semidirect
4. C₂×#14519856	C₂	semidirect
5. C₂²×C₄⋊#6923	C₂	semidirect
8. Q₈×#6923	C₂²	semidirect
9. D₄×#6923	C₂²	semidirect
10. #14519856	C₂²	semidirect
14. C₂×C₄⋊#6923	C₂²	semidirect
17. C₂×#11758226	C₄	semidirect
18. C₂×#11758222	C₄	semidirect
19. C₂×#11758224	C₄	semidirect
20. #41×D₆₁	C₄	semidirect
25. C₂×#41×C₆₁	C₄	semidirect
31. #41×C₆₁	C₂×C₄	semidirect
32. C₂×Q₈×C₆₁	C₂×C₄	semidirect
33. C₂×D₄×C₆₁	C₂×C₄	semidirect
34. C₂²×C₄×C₆₁	C₂×C₄	semidirect
35. C₂×C₂²⋊D₆₁	C₂×C₄	semidirect
36. C₂²×Dic₆₁	C₂×C₄	semidirect
37. #11758226	C₂×C₄	semidirect
38. Q₈×D₆₁	C₂×C₄	semidirect
40. #11758224	C₂×C₄	semidirect
41. D₄×D₆₁	C₂×C₄	semidirect
43. #11758222	C₂×C₄	semidirect
44. C₂×D₂₄₄	C₂×C₄	semidirect
45. C₂×C₄×D₆₁	C₂×C₄	semidirect
46. C₂×C₄×D₆₁	C₂×C₄	semidirect
47. C₂×C₄×D₆₁	C₂×C₄	semidirect
51. C₂×Dic₁₂₂	C₂×C₄	semidirect
69. Q₈×C₆₁	C₂²×C₄	semidirect
70. D₄×C₆₁	C₂²×C₄	semidirect
73. C₂×C₄×C₆₁	C₂²×C₄	semidirect
74. C₂²⋊D₆₁	C₂²×C₄	semidirect
77. C₂×Dic₆₁	C₂²×C₄	semidirect
78. D₂₄₄	C₂²×C₄	semidirect
81. C₄×D₆₁	C₂²×C₄	semidirect
82. C₄×D₆₁	C₂²×C₄	semidirect
85. Dic₁₂₂	C₂²×C₄	semidirect
106. C₂×C₆₁	#520	semidirect
109. D₆₁	#520	semidirect
118. C₆₁	C₂×#520	semidirect
124. C₂×#41	#6923	semidirect
142. #41	C₂×#6923	semidirect
146. C₂×Q₈	C₂×#6923	semidirect
151. C₂×D₄	C₂×#6923	semidirect
163. C₂²×C₄	C₂×#6923	semidirect
177. Q₈	C₂²×#6923	semidirect
180. D₄	C₂²×#6923	semidirect
197. C₂×C₄	C₂²×#6923	semidirect
221. C₂	#16840831	direct (N × H)

Subgroups — C2×#16840831 #21462798

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840831 #21462798