Subgroups — C₂×#41×#6923 #21462795

Order 7808.

1728 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#6923	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	6	yes	—
12	C₂×C₂²⋊#6923	C₂²	1	6	yes	—
13	C₂×C₄×#6923	C₂²	1	1	yes	characteristic
14	C₂×C₄×#6923	C₂²	1	3	yes	—
15	C₂×C₄×#6923	C₂²	1	6	yes	—
16	C₂×C₄×#6923	C₂²	1	24	yes	complemented
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	—	2	24	no	—
28	C₂²×#6923	C₂³	1	6	yes	—
29	C₂²×#6923	C₂³	1	24	yes	—
30	C₂²×#6923	C₂³	1	2	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	24	yes	—
54	C₄×#6923	C₂³	1	4	yes	—
55	C₄×#6923	C₂³	1	12	yes	—
56	C₄×#6923	C₂³	1	24	yes	—
57	C₂³×C₆₁	C₂²×C₄	1	3	yes	—
58	C₂²×D₆₁	—	2	12	no	—
59	C₂²×D₆₁	C₂²×C₄	1	3	yes	—
60	C₂²×D₆₁	C₂²×C₄	1	6	yes	—
61	C₂²×D₆₁	C₂⁴	1	6	yes	—
62	C₂²×D₆₁	C₂⁴	1	1	yes	characteristic
63	C₂×#6923	—	2	12	no	—
64	C₂×#6923	—	2	48	no	—
65	C₂×#6923	#41	1	8	yes	direct product factor
66	C₂×#6923	C₂⁴	1	12	yes	—
67	C₂×#6923	C₂⁴	1	4	yes	—
68	Q₈×C₆₁	C₂²×C₄	1	4	yes	complemented
69	D₄×C₆₁	C₂²×C₄	1	12	yes	complemented
70	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
71	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂²×C₄	1	12	yes	complemented
73	C₂²⋊D₆₁	C₂²×C₄	1	24	yes	complemented
74	C₂×Dic₆₁	C₂²×C₄	1	3	yes	—
75	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₆₁	C₂²×C₄	1	12	yes	complemented
77	D₂₄₄	C₂²×C₄	1	12	yes	complemented
78	C₄×D₆₁	C₂²×C₄	1	2	yes	—
79	C₄×D₆₁	C₂²×C₄	1	6	yes	—
80	C₄×D₆₁	C₂²×C₄	1	12	yes	complemented
81	C₄×D₆₁	C₂²×C₄	1	12	yes	complemented
82	C₄×D₆₁	C₂⁴	1	2	yes	—
83	C₄×D₆₁	C₂⁴	1	6	yes	—
84	Dic₁₂₂	C₂²×C₄	1	12	yes	complemented
85	C₂²×C₆₁	—	2	6	no	—
86	C₂²×C₆₁	C₂³×C₄	1	6	yes	—
87	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
88	C₂×D₆₁	—	2	6	no	—
89	C₂×D₆₁	—	2	12	no	—
90	C₂×D₆₁	—	2	12	no	—
91	C₂×D₆₁	C₂³×C₄	1	6	yes	—
92	C₂×D₆₁	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₆₁	C₂×#41	1	4	yes	—
94	C₂×D₆₁	C₂⁵	1	1	yes	characteristic
95	#6923	—	2	12	no	—
96	#6923	C₂×#41	1	8	yes	direct product factor
97	C₄×C₆₁	C₂³×C₄	1	6	yes	—
98	C₄×C₆₁	C₂³×C₄	1	2	yes	—
99	Dic₆₁	C₂³×C₄	1	6	yes	—
100	Dic₆₁	C₂³×C₄	1	2	yes	—
101	C₂×C₄×#41	—	61	1	no	maximal, Sylow
102	C₂×C₆₁	—	2	6	no	—
103	C₂×C₆₁	C₄×#41	1	2	yes	complemented
104	C₂×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
105	D₆₁	—	2	6	no	—
106	D₆₁	C₄×#41	1	2	yes	complemented
107	D₆₁	C₂²×#41	1	2	yes	—
108	C₂²×#41	—	61	1	no	—
109	C₄×#41	—	61	16	no	—
110	C₂×C₄×Q₈	—	61	2	no	—
111	C₂×C₄×D₄	—	61	6	no	—
112	C₂×#117	—	61	3	no	—
113	C₂²×C₄²	—	61	3	no	—
114	C₆₁	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
115	C₂×#41	—	61	8	no	—
116	C₂×#41	—	61	8	no	—
117	C₂×#41	—	61	1	no	—
118	C₂×#41	—	61	3	no	—
119	C₂×#41	—	61	3	no	—
120	C₂×#41	#6923	1	1	yes	characteristic, direct product factor
121	C₂²×Q₈	—	61	1	no	—
122	C₂²×D₄	—	61	3	no	—
123	C₂³×C₄	—	61	3	no	—
124	C₂³×C₄	—	61	6	no	—
125	C₄×Q₈	—	61	16	no	—
126	C₄×D₄	—	61	48	no	—
127	#117	—	61	24	no	—
128	C₂×C₄⋊C₄	—	61	6	no	—
129	C₂×C₂²⋊C₄	—	61	6	no	—
130	C₂×C₄²	—	61	3	no	—
131	C₂×C₄²	—	61	6	no	—
132	C₂×C₄²	—	61	24	no	—
133	C₂×C₄²	—	61	1	no	—
134	C₂⁴	—	61	3	no	—
135	#41	—	61	8	no	—
136	#41	—	61	24	no	—
137	#41	—	61	24	no	—
138	#41	C₂×#6923	1	8	yes	direct product factor
139	C₂×Q₈	—	61	4	no	—
140	C₂×Q₈	—	61	4	no	—
141	C₂×Q₈	—	61	3	no	—
142	C₂×Q₈	C₂×#6923	1	1	yes	characteristic, complemented
143	C₂×D₄	—	61	6	no	—
144	C₂×D₄	—	61	12	no	—
145	C₂×D₄	—	61	12	no	—
146	C₂×D₄	—	61	3	no	—
147	C₂×D₄	C₂×#6923	1	3	yes	complemented
148	C₂²×C₄	—	61	2	no	—
149	C₂²×C₄	—	61	3	no	—
150	C₂²×C₄	—	61	3	no	—
151	C₂²×C₄	—	61	3	no	—
152	C₂²×C₄	—	61	1	no	—
153	C₂²×C₄	—	61	3	no	—
154	C₂²×C₄	—	61	12	no	—
155	C₂²×C₄	—	61	12	no	—
156	C₂²×C₄	—	61	6	no	—
157	C₂²×C₄	—	61	24	no	—
158	C₂²×C₄	—	122	24	no	—
159	C₂²×C₄	C₂×#6923	1	3	yes	complemented
160	C₄⋊C₄	—	61	24	no	—
161	C₂²⋊C₄	—	61	24	no	—
162	C₄²	—	61	4	no	—
163	C₄²	—	61	12	no	—
164	C₄²	—	61	24	no	—
165	C₂³	—	61	1	no	—
166	C₂³	—	61	6	no	—
167	C₂³	—	61	3	no	—
168	C₂³	—	61	6	no	—
169	C₂³	—	122	12	no	—
170	C₂³	C₂²×#6923	1	3	yes	—
171	Q₈	—	61	12	no	—
172	Q₈	C₂²×#6923	1	4	yes	complemented
173	D₄	—	61	24	no	—
174	D₄	—	61	12	no	—
175	D₄	C₂²×#6923	1	12	yes	complemented
176	C₂×C₄	—	61	8	no	—
177	C₂×C₄	—	61	4	no	—
178	C₂×C₄	—	61	1	no	—
179	C₂×C₄	—	61	3	no	—
180	C₂×C₄	—	61	12	no	—
181	C₂×C₄	—	61	2	no	—
182	C₂×C₄	—	61	6	no	—
183	C₂×C₄	—	61	12	no	—
184	C₂×C₄	—	61	12	no	—
185	C₂×C₄	—	61	2	no	—
186	C₂×C₄	—	61	6	no	—
187	C₂×C₄	—	61	12	no	—
188	C₂×C₄	—	122	48	no	—
189	C₂×C₄	—	122	12	no	—
190	C₂×C₄	C₂²×#6923	1	3	yes	—
191	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
192	C₂×C₄	C₂²×#6923	1	12	yes	complemented
193	C₂²	—	2	6	no	—
194	C₂²	—	61	4	no	—
195	C₂²	—	61	1	no	—
196	C₂²	—	61	1	no	—
197	C₂²	—	61	6	no	—
198	C₂²	—	122	12	no	—
199	C₂²	—	122	12	no	—
200	C₂²	—	122	6	no	—
201	C₂²	C₂³×#6923	1	6	yes	—
202	C₂²	C₂³×#6923	1	1	yes	characteristic
203	C₄	—	61	8	no	—
204	C₄	—	61	2	no	—
205	C₄	—	61	6	no	—
206	C₄	—	122	12	no	—
207	C₄	C₂³×#6923	1	2	yes	—
208	C₄	C₂³×#6923	1	6	yes	—
209	C₂	—	2	6	no	—
210	C₂	—	61	2	no	—
211	C₂	—	61	2	no	—
212	C₂	—	122	6	no	—
213	C₂	#41×#6923	1	2	yes	direct product factor
214	C₂	C₂⁴×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#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×C₄×#41
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #41×#6923	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
16. 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
65. C₂×#6923	#41	direct (N × H)
68. Q₈×C₆₁	C₂²×C₄	semidirect
69. D₄×C₆₁	C₂²×C₄	semidirect
72. C₂×C₄×C₆₁	C₂²×C₄	semidirect
73. C₂²⋊D₆₁	C₂²×C₄	semidirect
76. C₂×Dic₆₁	C₂²×C₄	semidirect
77. D₂₄₄	C₂²×C₄	semidirect
80. C₄×D₆₁	C₂²×C₄	semidirect
81. C₄×D₆₁	C₂²×C₄	semidirect
84. Dic₁₂₂	C₂²×C₄	semidirect
96. #6923	C₂×#41	direct (N × H)
103. C₂×C₆₁	C₄×#41	semidirect
106. D₆₁	C₄×#41	semidirect
114. C₆₁	C₂×C₄×#41	semidirect
120. C₂×#41	#6923	direct (N × H)
138. #41	C₂×#6923	direct (N × H)
142. C₂×Q₈	C₂×#6923	semidirect
147. C₂×D₄	C₂×#6923	semidirect
159. C₂²×C₄	C₂×#6923	semidirect
172. Q₈	C₂²×#6923	semidirect
175. D₄	C₂²×#6923	semidirect
192. C₂×C₄	C₂²×#6923	semidirect
213. C₂	#41×#6923	direct (N × H)

Subgroups — C2×#41×#6923 #21462795

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#41×#6923 #21462795