Subgroups — C₂×#16840565 #21460570

Order 7808.

1512 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14519823	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840565	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14519803	C₂	1	3	yes	maximal, complemented
4	C₂×#14519802	C₂	1	3	yes	maximal, complemented
5	C₂×#14519797	C₂	1	3	yes	maximal, complemented
6	C₂²×#11758199	C₂	1	3	yes	maximal, complemented
7	C₂×#14519784	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×H(5,𝔽₂)×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#14519823	—	2	4	no	—
10	#14519823	C₂²	1	8	yes	complemented
11	C₂×#11758226	—	2	1	no	—
12	C₂×#11758222	C₂²	1	3	yes	complemented
13	C₂²×D₂₄₄	C₂²	1	3	yes	complemented
14	C₂×D₄×D₆₁	—	2	3	no	—
15	#14519803	C₂²	1	24	yes	complemented
16	#14519802	C₂²	1	24	yes	complemented
17	#14519797	C₂²	1	24	yes	complemented
18	C₂×#11758202	C₂²	1	1	yes	characteristic, complemented
19	C₂×#11758200	C₂²	1	3	yes	complemented
20	C₂×#11758201	C₂²	1	3	yes	complemented
21	C₂×#11758199	C₂²	1	3	yes	complemented
22	C₂×#11758199	C₂²	1	6	yes	complemented
23	C₂×#11758199	C₂²	1	24	yes	complemented
24	#14519784	C₂²	1	8	yes	complemented
25	C₂×#11758194	C₂²	1	3	yes	complemented
26	C₂²×#92418	C₂²	1	3	yes	complemented
27	H(5,𝔽₂)×C₆₁	—	2	4	no	—
28	H(5,𝔽₂)×C₆₁	C₂²	1	8	yes	complemented
29	C₂×#41×C₆₁	—	2	1	no	—
30	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×#41×C₆₁	C₂²	1	3	yes	complemented
32	C₂²×D₄×C₆₁	—	2	3	no	—
33	C₂²×D₄×C₆₁	C₂²	1	3	yes	complemented
34	C₂⁴×C₆₁	—	2	3	no	—
35	C₂³×D₆₁	—	2	3	no	—
36	#41×C₆₁	—	2	8	no	—
37	#41×C₆₁	—	2	6	no	—
38	#41×C₆₁	D₄	1	4	yes	—
39	#41×C₆₁	C₂³	1	4	yes	—
40	#41×C₆₁	C₂³	1	12	yes	complemented
41	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
43	C₂×D₄×C₆₁	—	2	3	no	—
44	C₂×D₄×C₆₁	—	2	24	no	—
45	C₂×D₄×C₆₁	—	2	3	no	—
46	C₂×D₄×C₆₁	—	2	6	no	—
47	C₂×D₄×C₆₁	D₄	1	3	yes	—
48	C₂×D₄×C₆₁	C₂³	1	3	yes	—
49	C₂×D₄×C₆₁	C₂³	1	12	yes	complemented
50	C₂²×C₄×C₆₁	—	2	3	no	—
51	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
52	C₂×C₂²⋊D₆₁	—	2	3	no	—
53	#11758226	—	2	8	no	—
54	D₄×D₆₁	—	2	24	no	—
55	#11758222	—	2	6	no	—
56	#11758222	C₂³	1	12	yes	complemented
57	C₂×D₂₄₄	—	2	3	no	—
58	C₂×D₂₄₄	—	2	6	no	—
59	C₂×D₂₄₄	C₂³	1	3	yes	—
60	C₂×D₂₄₄	C₂³	1	12	yes	complemented
61	C₂×C₄×D₆₁	—	2	3	no	—
62	C₂×Dic₁₂₂	C₂³	1	1	yes	characteristic
63	#11758202	C₂³	1	4	yes	complemented
64	#11758201	C₂³	1	12	yes	complemented
65	#11758200	C₂³	1	12	yes	complemented
66	#11758199	C₂³	1	12	yes	complemented
67	#11758199	C₂³	1	24	yes	complemented
68	#11758194	C₂³	1	12	yes	complemented
69	C₂×#92418	C₂³	1	1	yes	characteristic
70	C₂×#92418	C₂³	1	3	yes	—
71	C₂×#92418	C₂³	1	12	yes	complemented
72	C₂³×C₆₁	—	2	3	no	—
73	C₂³×C₆₁	—	2	12	no	—
74	C₂³×C₆₁	—	8	6	no	—
75	C₂³×C₆₁	C₂×D₄	1	3	yes	—
76	C₂²×D₆₁	—	2	3	no	—
77	C₂²×D₆₁	—	2	12	no	—
78	C₂²×D₆₁	—	8	6	no	—
79	Q₈×C₆₁	—	2	1	no	—
80	Q₈×C₆₁	C₂×D₄	1	4	yes	—
81	Q₈×C₆₁	C₂⁴	1	2	yes	—
82	D₄×C₆₁	—	2	12	no	—
83	D₄×C₆₁	—	2	3	no	—
84	D₄×C₆₁	—	2	12	no	—
85	D₄×C₆₁	C₂×D₄	1	12	yes	—
86	D₄×C₆₁	C₂⁴	1	6	yes	—
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	—	2	12	no	—
89	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
90	C₂×C₄×C₆₁	C₂×D₄	1	6	yes	—
91	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂⁴	1	6	yes	—
93	C₂²⋊D₆₁	—	2	12	no	—
94	C₂×Dic₆₁	—	2	1	no	—
95	D₂₄₄	—	2	3	no	—
96	D₂₄₄	—	2	12	no	—
97	D₂₄₄	C₂⁴	1	6	yes	—
98	C₄×D₆₁	—	2	12	no	—
99	Dic₁₂₂	—	2	1	no	—
100	Dic₁₂₂	C₂⁴	1	2	yes	—
101	#92418	C₂⁴	1	2	yes	—
102	#92418	C₂⁴	1	6	yes	—
103	C₂²×C₆₁	—	2	6	no	—
104	C₂²×C₆₁	—	2	6	no	—
105	C₂²×C₆₁	—	4	6	no	—
106	C₂²×C₆₁	—	8	12	no	—
107	C₂²×C₆₁	C₂²×D₄	1	6	yes	—
108	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×D₆₁	—	2	6	no	—
110	C₂×D₆₁	—	4	6	no	—
111	C₂×D₆₁	—	8	12	no	—
112	C₄×C₆₁	—	2	2	no	—
113	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₆₁	C₂²×D₄	1	6	yes	—
115	C₄×C₆₁	C₂⁵	1	1	yes	characteristic
116	Dic₆₁	—	2	2	no	—
117	C₂×#576	—	61	1	no	maximal, Sylow
118	C₂×C₆₁	—	2	6	no	—
119	C₂×C₆₁	—	4	6	no	—
120	C₂×C₆₁	#576	1	2	yes	complemented
121	C₂×C₆₁	C₂³×D₄	1	1	yes	characteristic
122	D₆₁	—	4	6	no	—
123	C₂×H(5,𝔽₂)	—	61	1	no	—
124	C₂×H(5,𝔽₂)	D₆₁	1	1	yes	characteristic, complemented
125	#576	—	61	16	no	—
126	C₂×AGL(1,ℤ/8)	—	61	3	no	—
127	C₂×AGL(1,ℤ/8)	—	61	3	no	—
128	C₂×#135	—	61	3	no	—
129	C₂²×D₈	—	61	3	no	—
130	C₂×#131	—	61	1	no	—
131	C₆₁	C₂×#576	1	1	yes	characteristic, complemented, Sylow
132	H(5,𝔽₂)	—	2	4	no	—
133	H(5,𝔽₂)	—	61	8	no	—
134	H(5,𝔽₂)	—	122	4	no	—
135	H(5,𝔽₂)	C₂×D₆₁	1	8	yes	complemented
136	C₂×#41	—	2	1	no	—
137	C₂×#41	—	61	3	no	—
138	C₂×#41	—	122	1	no	—
139	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
140	C₂×#41	C₂×D₆₁	1	3	yes	complemented
141	C₂²×D₄	—	2	3	no	—
142	C₂²×D₄	—	61	3	no	—
143	C₂²×D₄	—	122	3	no	—
144	C₂²×D₄	C₂×D₆₁	1	3	yes	complemented
145	AGL(1,ℤ/8)	—	61	24	no	—
146	AGL(1,ℤ/8)	—	61	24	no	—
147	#135	—	61	24	no	—
148	C₂×Q₁₆	—	61	1	no	—
149	C₂×SD₁₆	—	61	3	no	—
150	C₂×SD₁₆	—	61	3	no	—
151	C₂×D₈	—	61	3	no	—
152	C₂×D₈	—	61	6	no	—
153	C₂×D₈	—	61	24	no	—
154	#131	—	61	8	no	—
155	C₂×M₁₆	—	61	3	no	—
156	C₂²×C₈	—	61	3	no	—
157	C₂⁴	—	2	3	no	—
158	C₂⁴	—	122	3	no	—
159	#41	—	2	8	no	—
160	#41	—	2	6	no	—
161	#41	—	61	12	no	—
162	#41	—	122	8	no	—
163	#41	—	122	6	no	—
164	#41	C₂²⋊D₆₁	1	4	yes	—
165	#41	C₂²×D₆₁	1	4	yes	—
166	#41	C₂²×D₆₁	1	12	yes	complemented
167	C₂×Q₈	—	61	1	no	—
168	C₂×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
169	C₂×Q₈	C₂²×D₆₁	1	1	yes	characteristic
170	C₂×D₄	—	2	3	no	—
171	C₂×D₄	—	2	24	no	—
172	C₂×D₄	—	2	3	no	—
173	C₂×D₄	—	2	6	no	—
174	C₂×D₄	—	61	3	no	—
175	C₂×D₄	—	61	12	no	—
176	C₂×D₄	—	122	3	no	—
177	C₂×D₄	—	122	24	no	—
178	C₂×D₄	—	122	3	no	—
179	C₂×D₄	—	122	6	no	—
180	C₂×D₄	C₂²⋊D₆₁	1	3	yes	—
181	C₂×D₄	C₂²×D₆₁	1	3	yes	—
182	C₂×D₄	C₂²×D₆₁	1	12	yes	complemented
183	C₂²×C₄	—	2	3	no	—
184	C₂²×C₄	—	122	3	no	—
185	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
186	Q₁₆	—	61	4	no	—
187	SD₁₆	—	61	12	no	—
188	SD₁₆	—	61	12	no	—
189	D₈	—	61	12	no	—
190	D₈	—	61	24	no	—
191	M₁₆	—	61	12	no	—
192	C₂×C₈	—	61	1	no	—
193	C₂×C₈	—	61	3	no	—
194	C₂×C₈	—	61	12	no	—
195	C₂³	—	2	3	no	—
196	C₂³	—	2	12	no	—
197	C₂³	—	8	6	no	—
198	C₂³	—	122	12	no	—
199	C₂³	—	122	3	no	—
200	C₂³	—	488	6	no	—
201	C₂³	C₂×C₂²⋊D₆₁	1	3	yes	—
202	Q₈	—	2	1	no	—
203	Q₈	—	61	2	no	—
204	Q₈	—	122	1	no	—
205	Q₈	C₂×C₂²⋊D₆₁	1	4	yes	—
206	Q₈	C₂³×D₆₁	1	2	yes	—
207	D₄	—	2	12	no	—
208	D₄	—	2	3	no	—
209	D₄	—	2	12	no	—
210	D₄	—	61	6	no	—
211	D₄	—	122	12	no	—
212	D₄	—	122	12	no	—
213	D₄	—	122	3	no	—
214	D₄	C₂×C₂²⋊D₆₁	1	12	yes	—
215	D₄	C₂³×D₆₁	1	6	yes	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	122	1	no	—
219	C₂×C₄	—	122	12	no	—
220	C₂×C₄	C₂×C₂²⋊D₆₁	1	6	yes	—
221	C₂×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
222	C₂×C₄	C₂³×D₆₁	1	6	yes	—
223	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
224	C₈	—	61	2	no	—
225	C₈	—	61	6	no	—
226	C₂²	—	2	6	no	—
227	C₂²	—	2	6	no	—
228	C₂²	—	4	6	no	—
229	C₂²	—	8	12	no	—
230	C₂²	—	122	6	no	—
231	C₂²	—	488	12	no	—
232	C₂²	—	244	6	no	—
233	C₂²	C₂²×C₂²⋊D₆₁	1	6	yes	—
234	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	122	2	no	—
237	C₄	C₂²×C₂²⋊D₆₁	1	6	yes	—
238	C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
239	C₄	C₂⁴×D₆₁	1	1	yes	characteristic
240	C₂	—	2	6	no	—
241	C₂	—	4	6	no	—
242	C₂	—	244	6	no	—
243	C₂	#16840565	1	2	yes	direct product factor
244	C₂	C₂³×C₂²⋊D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂⁵
Frattini	C₄	C₂⁴×D₆₁
Fitting	C₂×H(5,𝔽₂)×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₂×H(5,𝔽₂)	D₆₁
hypocenter	C₆₁	C₂×#576
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#14519823	C₂	semidirect
2. #16840565	C₂	direct (N × H)
3. C₂×#14519803	C₂	semidirect
4. C₂×#14519802	C₂	semidirect
5. C₂×#14519797	C₂	semidirect
6. C₂²×#11758199	C₂	semidirect
7. C₂×#14519784	C₂	semidirect
8. C₂×H(5,𝔽₂)×C₆₁	C₂	semidirect
10. #14519823	C₂²	semidirect
12. C₂×#11758222	C₂²	semidirect
13. C₂²×D₂₄₄	C₂²	semidirect
15. #14519803	C₂²	semidirect
16. #14519802	C₂²	semidirect
17. #14519797	C₂²	semidirect
18. C₂×#11758202	C₂²	semidirect
19. C₂×#11758200	C₂²	semidirect
20. C₂×#11758201	C₂²	semidirect
21. C₂×#11758199	C₂²	semidirect
22. C₂×#11758199	C₂²	semidirect
23. C₂×#11758199	C₂²	semidirect
24. #14519784	C₂²	semidirect
25. C₂×#11758194	C₂²	semidirect
26. C₂²×#92418	C₂²	semidirect
28. H(5,𝔽₂)×C₆₁	C₂²	semidirect
31. C₂×#41×C₆₁	C₂²	semidirect
33. C₂²×D₄×C₆₁	C₂²	semidirect
40. #41×C₆₁	C₂³	semidirect
49. C₂×D₄×C₆₁	C₂³	semidirect
56. #11758222	C₂³	semidirect
60. C₂×D₂₄₄	C₂³	semidirect
63. #11758202	C₂³	semidirect
64. #11758201	C₂³	semidirect
65. #11758200	C₂³	semidirect
66. #11758199	C₂³	semidirect
67. #11758199	C₂³	semidirect
68. #11758194	C₂³	semidirect
71. C₂×#92418	C₂³	semidirect
120. C₂×C₆₁	#576	semidirect
124. C₂×H(5,𝔽₂)	D₆₁	semidirect
131. C₆₁	C₂×#576	semidirect
135. H(5,𝔽₂)	C₂×D₆₁	semidirect
140. C₂×#41	C₂×D₆₁	semidirect
144. C₂²×D₄	C₂×D₆₁	semidirect
166. #41	C₂²×D₆₁	semidirect
182. C₂×D₄	C₂²×D₆₁	semidirect
243. C₂	#16840565	direct (N × H)

Subgroups — C2×#16840565 #21460570

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840565 #21460570