Subgroups — C₂×#24909340 #35557708

Order 21248.

798 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	C₂²×D₁₃₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₆₆₄	C₂	1	2	yes	maximal, complemented
3	#24909340	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19127529	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19127503	C₂	1	2	yes	maximal, complemented
6	C₂×#346×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₆₆₄	C₂²	1	2	yes	—
8	C₂×C₄×D₃₃₂	—	2	2	no	—
9	C₂×D₁₃₂₈	C₄	1	8	yes	complemented
10	C₂×D₁₃₂₈	C₂²	1	4	yes	—
11	C₄×D₆₆₄	—	2	4	no	—
12	C₄×D₆₆₄	C₂²	1	8	yes	complemented
13	C₂×#14581726	C₂²	1	2	yes	—
14	C₂×#14581714	—	2	2	no	—
15	#19127529	C₂²	1	4	yes	complemented
16	#19127503	C₂²	1	8	yes	complemented
17	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	#346×C₈₃	C₂²	1	4	yes	complemented
21	C₂²×D₃₃₂	—	2	2	no	—
22	C₂²×C₄×D₈₃	—	4	2	no	—
23	C₂×D₆₆₄	—	2	2	no	—
24	C₂×D₆₆₄	—	2	4	no	—
25	C₂×D₆₆₄	C₂×C₄	1	8	yes	complemented
26	C₂×D₆₆₄	C₂³	1	4	yes	—
27	C₄×D₃₃₂	—	2	8	no	—
28	C₄×D₃₃₂	—	4	4	no	—
29	C₂×C₄⋊Dic₈₃	—	2	2	no	—
30	C₂×(C₂×D₈₃)⋊C₄	—	4	2	no	—
31	D₁₃₂₈	C₂×C₄	1	16	yes	complemented
32	#14581726	—	2	2	no	—
33	#14581726	C₂³	1	4	yes	—
34	#14581714	—	2	8	no	—
35	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₈₃	—	2	2	no	—
39	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	complemented
40	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
41	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
42	C₄×C₈×C₈₃	D₄	1	2	yes	—
43	C₄×C₈×C₈₃	C₂³	1	2	yes	—
44	C₂³×D₈₃	—	4	2	no	—
45	C₂×D₃₃₂	—	2	8	no	—
46	C₂×D₃₃₂	—	2	4	no	—
47	C₂×D₃₃₂	—	4	2	no	—
48	C₂×D₃₃₂	—	4	4	no	—
49	C₂²×Dic₈₃	—	4	2	no	—
50	C₂×C₄×D₈₃	—	4	8	no	—
51	C₂×C₄×D₈₃	—	8	8	no	—
52	D₆₆₄	—	2	8	no	—
53	D₆₆₄	—	2	4	no	—
54	D₆₆₄	C₂²×C₄	1	8	yes	complemented
55	C₄⋊Dic₈₃	—	2	4	no	—
56	C₄⋊Dic₈₃	—	4	2	no	—
57	(C₂×D₈₃)⋊C₄	—	4	8	no	—
58	C₂²×C₄×C₈₃	D₈	1	2	yes	—
59	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	—	2	2	no	—
61	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₃	#41	1	2	yes	—
65	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
66	C₄²×C₈₃	D₈	1	2	yes	—
67	C₄²×C₈₃	C₂×D₄	1	2	yes	—
68	C₁₆×C₈₃	—	2	2	no	—
69	C₁₆×C₈₃	C₂²×C₄	1	4	yes	complemented
70	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
71	C₂²×D₈₃	—	4	4	no	—
72	C₂²×D₈₃	—	4	8	no	—
73	C₂²×D₈₃	—	8	4	no	—
74	C₂²×D₈₃	—	8	4	no	—
75	C₂×C₄×C₈₃	—	2	4	no	—
76	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₈₃	C₂×D₈	1	4	yes	—
78	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₈₃	#135	1	2	yes	—
80	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₈₃	—	4	4	no	—
82	C₂×Dic₈₃	—	8	4	no	—
83	D₃₃₂	—	2	8	no	—
84	D₃₃₂	—	4	4	no	—
85	D₃₃₂	—	4	8	no	—
86	C₄×D₈₃	—	8	16	no	—
87	C₈×C₈₃	—	2	2	no	—
88	C₈×C₈₃	C₄×D₄	1	2	yes	—
89	C₈×C₈₃	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
91	C₂²×C₈₃	C₄×D₈	1	2	yes	—
92	C₂²×C₈₃	#509	1	4	yes	—
93	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
94	C₂×D₈₃	—	4	8	no	—
95	C₂×D₈₃	—	8	4	no	—
96	C₂×D₈₃	—	8	4	no	—
97	C₂×D₈₃	—	8	16	no	—
98	C₄×C₈₃	—	2	4	no	—
99	C₄×C₈₃	C₄×D₈	1	2	yes	—
100	C₄×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
101	C₄×C₈₃	C₂×#135	1	1	yes	characteristic
102	Dic₈₃	—	8	4	no	—
103	C₂×#2177	—	83	1	no	maximal, Sylow
104	C₂×C₈₃	#2177	1	4	yes	complemented
105	C₂×C₈₃	C₂×C₄×D₈	1	1	yes	characteristic
106	C₂×C₈₃	C₂×#509	1	2	yes	—
107	D₈₃	—	8	8	no	—
108	C₂²×D₁₆	—	83	1	no	—
109	C₂×C₄×D₈	—	83	2	no	—
110	#2177	—	83	8	no	—
111	C₂×#367	—	83	1	no	—
112	C₂×#357	—	83	2	no	—
113	C₂×#346	D₈₃	1	1	yes	characteristic, complemented
114	C₈₃	C₂×#2177	1	1	yes	characteristic, complemented, Sylow
115	C₂²×D₈	—	83	2	no	—
116	C₂×C₄×D₄	—	166	2	no	—
117	C₂×D₁₆	—	83	8	no	—
118	C₂×D₁₆	—	83	4	no	—
119	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
120	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
121	C₄×D₈	—	83	8	no	—
122	C₄×D₈	—	166	4	no	—
123	C₂×#107	—	83	2	no	—
124	C₂×#102	—	166	2	no	—
125	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
126	#367	—	83	4	no	—
127	#357	—	83	8	no	—
128	#346	C₂×D₈₃	1	4	yes	complemented
129	C₂²×D₄	—	166	2	no	—
130	C₂³×C₄	—	332	2	no	—
131	C₂×D₈	—	83	8	no	—
132	C₂×D₈	—	83	4	no	—
133	C₂×D₈	—	166	2	no	—
134	C₂×D₈	—	166	4	no	—
135	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
137	C₄×D₄	—	332	4	no	—
138	C₄×D₄	—	166	8	no	—
139	C₂×C₄⋊C₄	—	166	2	no	—
140	C₂×C₂²⋊C₄	—	332	2	no	—
141	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
142	D₁₆	—	83	16	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	C₄×D₈₃	1	4	yes	complemented
145	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
146	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
147	#107	—	83	4	no	—
148	#107	—	166	2	no	—
149	#102	—	166	8	no	—
150	C₄×C₈	D₃₃₂	1	2	yes	—
151	C₄×C₈	C₂²×D₈₃	1	2	yes	—
152	C₂⁴	—	332	2	no	—
153	C₂×D₄	—	332	2	no	—
154	C₂×D₄	—	332	4	no	—
155	C₂×D₄	—	166	4	no	—
156	C₂×D₄	—	166	8	no	—
157	C₂²×C₄	—	332	2	no	—
158	C₂²×C₄	—	332	8	no	—
159	C₂²×C₄	—	664	8	no	—
160	C₂²×C₄	D₆₆₄	1	2	yes	—
161	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
162	D₈	—	83	8	no	—
163	D₈	—	166	4	no	—
164	D₈	—	166	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
167	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
168	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
169	C₂×C₈	#13050590	1	2	yes	—
170	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
171	C₄⋊C₄	—	332	2	no	—
172	C₄⋊C₄	—	166	4	no	—
173	C₂²⋊C₄	—	332	8	no	—
174	C₄²	D₆₆₄	1	2	yes	—
175	C₄²	C₂×D₃₃₂	1	2	yes	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂×C₄×D₈₃	1	4	yes	complemented
178	C₂³	—	332	4	no	—
179	C₂³	—	332	8	no	—
180	C₂³	—	664	4	no	—
181	C₂³	—	664	4	no	—
182	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
183	D₄	—	332	4	no	—
184	D₄	—	332	8	no	—
185	D₄	—	166	8	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	332	4	no	—
188	C₂×C₄	—	664	16	no	—
189	C₂×C₄	—	664	4	no	—
190	C₂×C₄	C₄×D₃₃₂	1	2	yes	—
191	C₂×C₄	C₂×D₆₆₄	1	4	yes	—
192	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
193	C₂×C₄	#14581814	1	2	yes	—
194	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	C₄×D₃₃₂	1	2	yes	—
197	C₈	C₂²×C₄×D₈₃	1	1	yes	characteristic
198	C₈	C₂×#13050590	1	1	yes	characteristic
199	C₂²	—	332	8	no	—
200	C₂²	—	664	4	no	—
201	C₂²	—	664	4	no	—
202	C₂²	—	664	16	no	—
203	C₂²	C₄×D₆₆₄	1	2	yes	—
204	C₂²	#19128139	1	4	yes	—
205	C₂²	C₂²×D₆₆₄	1	1	yes	characteristic
206	C₄	—	2	4	no	—
207	C₄	—	664	4	no	—
208	C₄	C₄×D₆₆₄	1	2	yes	—
209	C₄	C₂×C₄×D₃₃₂	1	1	yes	characteristic
210	C₄	C₂×#14581814	1	1	yes	characteristic
211	C₂	—	664	8	no	—
212	C₂	#24909340	1	4	yes	direct product factor
213	C₂	C₂×C₄×D₆₆₄	1	1	yes	characteristic
214	C₂	C₂×#19128139	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₆₆₄
commutator	C₈×C₈₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#346×C₈₃	C₂
socle	C₂³×C₈₃	C₂×D₈
prime radical	C₂²×D₁₃₂₈	C₂
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₂×#346	D₈₃
hypocenter	C₈₃	C₂×#2177
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₆₆₄	C₂	semidirect
3. #24909340	C₂	direct (N × H)
4. C₂×#19127529	C₂	semidirect
5. C₂×#19127503	C₂	semidirect
6. C₂×#346×C₈₃	C₂	semidirect
9. C₂×D₁₃₂₈	C₄	semidirect
12. C₄×D₆₆₄	C₂²	semidirect
15. #19127529	C₂²	semidirect
16. #19127503	C₂²	semidirect
20. #346×C₈₃	C₂²	semidirect
25. C₂×D₆₆₄	C₂×C₄	semidirect
31. D₁₃₂₈	C₂×C₄	semidirect
39. C₂×C₁₆×C₈₃	C₂×C₄	semidirect
54. D₆₆₄	C₂²×C₄	semidirect
69. C₁₆×C₈₃	C₂²×C₄	semidirect
104. C₂×C₈₃	#2177	semidirect
113. C₂×#346	D₈₃	semidirect
114. C₈₃	C₂×#2177	semidirect
128. #346	C₂×D₈₃	semidirect
144. C₂×C₁₆	C₄×D₈₃	semidirect
177. C₁₆	C₂×C₄×D₈₃	semidirect
212. C₂	#24909340	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₃₂₈	2
C₂×C₄×D₆₆₄	2
#24909340	2
C₂×#19127529	2
C₂×#19127503	2
C₂×#346×C₈₃	2
C₂×#2177	83

Subgroups — C2×#24909340 #35557708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909340 #35557708