Subgroups — C₂×#19136441 #25249857

Order 11008.

806 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₆₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582981	C₂	1	2	yes	maximal, complemented
3	#19136441	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582699	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₄₄	C₂²	1	2	yes	—
8	C₂×#13063287	—	2	2	no	—
9	C₂×D₆₈₈	C₄	1	8	yes	complemented
10	C₂×D₆₈₈	C₂²	1	4	yes	—
11	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	#14582981	—	2	4	no	—
13	#14582981	C₂²	1	8	yes	complemented
14	C₂×#13063245	—	2	2	no	—
15	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
16	#14582699	C₂²	1	8	yes	complemented
17	C₁₆×Dic₄₃	C₂²	1	4	yes	complemented
18	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×#107×C₄₃	C₂²	1	2	yes	—
20	C₄₃×#366	C₂²	1	4	yes	complemented
21	C₂²×D₁₇₂	—	2	2	no	—
22	C₂²×C₄×D₄₃	—	4	2	no	—
23	C₂×D₃₄₄	—	2	2	no	—
24	C₂×D₃₄₄	—	2	4	no	—
25	C₂×D₃₄₄	C₂×C₄	1	8	yes	complemented
26	C₂×D₃₄₄	C₂³	1	4	yes	—
27	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
28	#13063287	—	2	8	no	—
29	#13063287	—	4	4	no	—
30	C₂×(C₂×D₄₃)⋊C₄	—	4	2	no	—
31	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
32	D₆₈₈	C₂×C₄	1	16	yes	complemented
33	C₂×C₄₃⋊C₁₆	—	2	2	no	—
34	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
35	#13063245	—	2	8	no	—
36	C₈×Dic₄₃	D₄	1	2	yes	—
37	C₈×Dic₄₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
40	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
41	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
42	#107×C₄₃	—	2	2	no	—
43	#107×C₄₃	C₂³	1	4	yes	—
44	C₂³×D₄₃	—	4	2	no	—
45	C₂²×C₄×C₄₃	—	4	2	no	—
46	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
48	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
49	C₂×D₁₇₂	—	2	8	no	—
50	C₂×D₁₇₂	—	2	4	no	—
51	C₂×D₁₇₂	—	4	2	no	—
52	C₂×D₁₇₂	—	4	4	no	—
53	C₂×C₄×D₄₃	—	4	8	no	—
54	C₂×C₄×D₄₃	—	8	8	no	—
55	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₄₃	#41	1	2	yes	—
58	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₄₃	—	2	4	no	—
60	C₄⋊C₄×C₄₃	—	4	2	no	—
61	(C₂×D₄₃)⋊C₄	—	4	8	no	—
62	C₄×Dic₄₃	D₈	1	2	yes	—
63	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
64	C₂×C₄₃⋊C₈	—	2	2	no	—
65	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
66	D₃₄₄	—	2	8	no	—
67	D₃₄₄	—	2	4	no	—
68	D₃₄₄	C₂²×C₄	1	8	yes	complemented
69	C₁₆×C₄₃	C₂²×C₄	1	4	yes	complemented
70	C₄₃⋊C₁₆	—	2	2	no	—
71	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
72	C₂²×D₄₃	—	4	4	no	—
73	C₂²×D₄₃	—	4	8	no	—
74	C₂²×D₄₃	—	8	4	no	—
75	C₂²×D₄₃	—	8	4	no	—
76	C₂×C₄×C₄₃	—	4	4	no	—
77	C₂×C₄×C₄₃	—	8	4	no	—
78	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#135	1	2	yes	—
81	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×Dic₄₃	—	2	2	no	—
83	C₂×Dic₄₃	D₁₆	1	4	yes	complemented
84	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
85	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
86	D₁₇₂	—	2	8	no	—
87	D₁₇₂	—	4	4	no	—
88	D₁₇₂	—	4	8	no	—
89	C₄×D₄₃	—	8	16	no	—
90	C₈×C₄₃	C₄×D₄	1	2	yes	—
91	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
93	C₄₃⋊C₈	—	2	2	no	—
94	C₂×C₄×D₁₆	—	43	1	no	maximal, Sylow
95	C₂²×C₄₃	C₄×D₈	1	2	yes	—
96	C₂²×C₄₃	C₂×D₁₆	1	2	yes	—
97	C₂²×C₄₃	#508	1	2	yes	—
98	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
99	C₂×D₄₃	—	4	8	no	—
100	C₂×D₄₃	—	8	4	no	—
101	C₂×D₄₃	—	8	4	no	—
102	C₂×D₄₃	—	8	16	no	—
103	C₄×C₄₃	—	8	4	no	—
104	C₄×C₄₃	C₄×D₈	1	2	yes	—
105	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
106	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
107	Dic₄₃	—	2	2	no	—
108	Dic₄₃	C₂×D₁₆	1	4	yes	complemented
109	C₂²×D₁₆	—	43	1	no	—
110	C₂×C₄×D₈	—	43	2	no	—
111	C₄×D₁₆	—	43	8	no	—
112	C₂×#366	D₄₃	1	1	yes	characteristic, complemented
113	C₂×#357	—	43	2	no	—
114	C₂×C₄×C₁₆	—	43	1	no	—
115	C₂×C₄₃	C₄×D₁₆	1	4	yes	complemented
116	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
117	C₂×C₄₃	C₂²×D₁₆	1	1	yes	characteristic
118	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
119	D₄₃	—	8	8	no	—
120	C₂²×D₈	—	43	2	no	—
121	C₂×C₄×D₄	—	86	2	no	—
122	C₂×D₁₆	—	43	4	no	—
123	C₂×D₁₆	—	43	8	no	—
124	C₂²×C₁₆	—	43	1	no	—
125	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
126	C₄×D₈	—	43	8	no	—
127	C₄×D₈	—	86	4	no	—
128	C₂×#107	C₂×D₄₃	1	2	yes	—
129	C₂×#102	—	86	2	no	—
130	C₂×C₄×C₈	—	43	1	no	—
131	#366	C₂×D₄₃	1	4	yes	complemented
132	#357	—	43	8	no	—
133	C₄×C₁₆	—	43	4	no	—
134	C₄₃	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
135	C₂²×D₄	—	86	2	no	—
136	C₂³×C₄	—	172	2	no	—
137	C₂×D₈	—	43	8	no	—
138	C₂×D₈	—	43	4	no	—
139	C₂×D₈	—	86	2	no	—
140	C₂×D₈	—	86	4	no	—
141	C₂²×C₈	—	43	1	no	—
142	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
143	C₄×D₄	—	86	8	no	—
144	C₄×D₄	—	172	4	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₂²⋊C₄	—	172	2	no	—
147	C₂×C₄²	—	43	1	no	—
148	D₁₆	—	43	16	no	—
149	C₂×C₁₆	—	43	2	no	—
150	C₂×C₁₆	—	86	2	no	—
151	C₂×C₁₆	C₄×D₄₃	1	4	yes	complemented
152	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
153	#107	—	2	2	no	—
154	#107	C₂²×D₄₃	1	4	yes	—
155	#102	—	86	8	no	—
156	C₄×C₈	—	43	2	no	—
157	C₄×C₈	—	43	2	no	—
158	C₂⁴	—	172	2	no	—
159	C₂×D₄	—	86	8	no	—
160	C₂×D₄	—	86	4	no	—
161	C₂×D₄	—	172	2	no	—
162	C₂×D₄	—	172	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	43	1	no	—
165	C₂²×C₄	—	43	1	no	—
166	C₂²×C₄	—	344	8	no	—
167	C₂²×C₄	—	172	8	no	—
168	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
169	D₈	—	43	8	no	—
170	D₈	—	86	8	no	—
171	D₈	—	86	4	no	—
172	C₂×C₈	—	43	2	no	—
173	C₂×C₈	—	86	2	no	—
174	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
175	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
176	C₂×C₈	#10621902	1	2	yes	—
177	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	172	8	no	—
181	C₄²	—	43	2	no	—
182	C₄²	—	43	2	no	—
183	C₁₆	—	86	2	no	—
184	C₁₆	C₂×C₄×D₄₃	1	4	yes	complemented
185	C₂³	—	344	4	no	—
186	C₂³	—	344	4	no	—
187	C₂³	—	172	4	no	—
188	C₂³	—	172	8	no	—
189	C₂³	D₈×D₄₃	1	1	yes	characteristic
190	D₄	—	86	8	no	—
191	D₄	—	172	4	no	—
192	D₄	—	172	8	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	8	4	no	—
195	C₂×C₄	—	43	4	no	—
196	C₂×C₄	—	43	2	no	—
197	C₂×C₄	—	43	2	no	—
198	C₂×C₄	—	86	2	no	—
199	C₂×C₄	—	344	16	no	—
200	C₂×C₄	#13063287	1	2	yes	—
201	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
202	C₂×C₄	#13063348	1	2	yes	—
203	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
204	C₈	—	86	2	no	—
205	C₈	#13063287	1	2	yes	—
206	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
207	C₈	C₂×#10621902	1	1	yes	characteristic
208	C₂²	—	344	4	no	—
209	C₂²	—	344	16	no	—
210	C₂²	—	344	4	no	—
211	C₂²	—	172	8	no	—
212	C₂²	#14582981	1	2	yes	—
213	C₂²	D₁₆×D₄₃	1	2	yes	—
214	C₂²	#14583332	1	2	yes	—
215	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
216	C₄	—	8	4	no	—
217	C₄	—	43	4	no	—
218	C₄	—	86	2	no	—
219	C₄	#14582981	1	2	yes	—
220	C₄	C₂×#13063287	1	1	yes	characteristic
221	C₄	C₂×#13063348	1	1	yes	characteristic
222	C₂	—	344	8	no	—
223	C₂	#19136441	1	4	yes	direct product factor
224	C₂	C₂×#14582981	1	1	yes	characteristic
225	C₂	C₂×D₁₆×D₄₃	1	1	yes	characteristic
226	C₂	C₂×#14583332	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#366	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₆₈₈	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#366	D₄₃
hypocenter	C₄₃	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. C₂×#14582981	C₂	semidirect
3. #19136441	C₂	direct (N × H)
4. C₂×#14582699	C₂	semidirect
5. C₂×C₁₆×Dic₄₃	C₂	semidirect
6. C₂×C₄₃×#366	C₂	semidirect
9. C₂×D₆₈₈	C₄	semidirect
13. #14582981	C₂²	semidirect
16. #14582699	C₂²	semidirect
17. C₁₆×Dic₄₃	C₂²	semidirect
20. C₄₃×#366	C₂²	semidirect
25. C₂×D₃₄₄	C₂×C₄	semidirect
32. D₆₈₈	C₂×C₄	semidirect
40. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
68. D₃₄₄	C₂²×C₄	semidirect
69. C₁₆×C₄₃	C₂²×C₄	semidirect
83. C₂×Dic₄₃	D₁₆	semidirect
108. Dic₄₃	C₂×D₁₆	semidirect
112. C₂×#366	D₄₃	semidirect
115. C₂×C₄₃	C₄×D₁₆	semidirect
131. #366	C₂×D₄₃	semidirect
134. C₄₃	C₂×C₄×D₁₆	semidirect
151. C₂×C₁₆	C₄×D₄₃	semidirect
184. C₁₆	C₂×C₄×D₄₃	semidirect
223. C₂	#19136441	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×D₆₈₈	2
C₂×#14582981	2
#19136441	2
C₂×#14582699	2
C₂×C₁₆×Dic₄₃	2
C₂×C₄₃×#366	2
C₂×C₄×D₁₆	43

Subgroups — C2×#19136441 #25249857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136441 #25249857