Subgroups — C₂×#19136820 #25251882

Order 11008.

726 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583276	C₂	1	1	yes	maximal, characteristic
2	#19136820	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14582699	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₄₄	—	2	1	no	—
7	C₂²×D₃₄₄	C₂²	1	2	yes	—
8	C₂×#13063322	—	2	1	no	—
9	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
10	#14583276	—	2	2	no	—
11	#14583276	C₂²	1	4	yes	—
12	#14582699	C₂²	1	16	yes	complemented
13	C₄×C₄₃⋊C₁₆	C₂²	1	4	yes	complemented
14	C₂×C₄⋊Q₈×C₄₃	—	2	1	no	—
15	C₄₃×#500	—	2	2	no	—
16	C₄₃×#500	C₂²	1	4	yes	complemented
17	C₂×#107×C₄₃	—	2	1	no	—
18	C₂×#107×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×D₁₇₂	—	2	2	no	—
21	C₂²×D₁₇₂	—	4	1	no	—
22	C₂²×D₁₇₂	—	4	1	no	—
23	C₂×D₃₄₄	—	2	8	no	—
24	C₂×D₃₄₄	—	2	4	no	—
25	C₂×D₃₄₄	—	2	2	no	—
26	C₂×D₃₄₄	—	4	4	no	—
27	C₂×D₃₄₄	C₂³	1	4	yes	—
28	#13063322	—	2	4	no	—
29	#13063322	—	4	2	no	—
30	C₂×C₄₃⋊C₁₆	—	2	4	no	—
31	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
32	C₂²×Q₈×C₄₃	—	4	1	no	—
33	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₄₃	—	2	4	no	—
36	C₄⋊Q₈×C₄₃	—	4	2	no	—
37	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₃	—	4	1	no	—
39	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
40	#107×C₄₃	—	2	2	no	—
41	#107×C₄₃	—	2	4	no	—
42	#107×C₄₃	C₂³	1	4	yes	—
43	C₄×C₈×C₄₃	D₄	1	2	yes	—
44	C₄×C₈×C₄₃	C₂³	1	2	yes	—
45	C₂³×D₄₃	—	4	2	no	—
46	C₂×Q₈×C₄₃	—	4	4	no	—
47	C₂×Q₈×C₄₃	—	8	4	no	—
48	C₂²×C₄×C₄₃	—	4	2	no	—
49	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂×D₁₇₂	—	2	4	no	—
53	C₂×D₁₇₂	—	4	4	no	—
54	C₂×D₁₇₂	—	4	4	no	—
55	C₂×D₁₇₂	—	4	2	no	—
56	C₂×D₁₇₂	—	4	4	no	—
57	C₂×D₁₇₂	—	4	4	no	—
58	C₂×D₁₇₂	—	8	4	no	—
59	C₂×D₁₇₂	—	8	4	no	—
60	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
61	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₄₃	#41	1	4	yes	—
64	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₄₃	—	2	4	no	—
66	C₄⋊C₄×C₄₃	—	4	2	no	—
67	C₄⋊C₄×C₄₃	—	4	4	no	—
68	C₄²×C₄₃	D₈	1	2	yes	—
69	C₄²×C₄₃	C₂×D₄	1	2	yes	—
70	D₃₄₄	—	2	8	no	—
71	D₃₄₄	—	4	4	no	—
72	D₃₄₄	—	4	8	no	—
73	C₄₃⋊C₁₆	—	2	4	no	—
74	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
75	C₂²×D₄₃	—	4	4	no	—
76	C₂²×D₄₃	—	8	4	no	—
77	C₂²×D₄₃	—	8	4	no	—
78	C₂²×D₄₃	—	8	4	no	—
79	Q₈×C₄₃	—	8	8	no	—
80	C₂×C₄×C₄₃	—	4	4	no	—
81	C₂×C₄×C₄₃	—	8	4	no	—
82	C₂×C₄×C₄₃	D₁₆	1	4	yes	—
83	C₂×C₄×C₄₃	SD₃₂	1	4	yes	—
84	C₂×C₄×C₄₃	#124	1	2	yes	—
85	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
90	D₁₇₂	—	4	4	no	—
91	D₁₇₂	—	4	4	no	—
92	D₁₇₂	—	8	4	no	—
93	D₁₇₂	—	8	8	no	—
94	D₁₇₂	—	8	8	no	—
95	C₈×C₄₃	#124	1	2	yes	—
96	C₈×C₄₃	C₂×SD₁₆	1	4	yes	—
97	C₈×C₄₃	C₂×#41	1	2	yes	—
98	C₂×#2240	—	43	1	no	maximal, Sylow
99	C₂²×C₄₃	#486	1	2	yes	—
100	C₂²×C₄₃	C₂×D₁₆	1	2	yes	—
101	C₂²×C₄₃	C₂×SD₃₂	1	2	yes	—
102	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
103	C₂×D₄₃	—	8	4	no	—
104	C₂×D₄₃	—	8	4	no	—
105	C₂×D₄₃	—	8	4	no	—
106	C₂×D₄₃	—	16	8	no	—
107	C₄×C₄₃	—	8	4	no	—
108	C₄×C₄₃	#486	1	2	yes	—
109	C₄×C₄₃	C₂×D₁₆	1	4	yes	—
110	C₄×C₄₃	C₂×SD₃₂	1	4	yes	—
111	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
112	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
113	C₂×#500	D₄₃	1	1	yes	characteristic, complemented
114	C₂×#493	—	43	1	no	—
115	#2240	—	43	8	no	—
116	C₂×#357	—	43	4	no	—
117	C₂×C₄×C₁₆	—	43	1	no	—
118	C₂×C₄₃	#2240	1	4	yes	complemented
119	C₂×C₄₃	C₂×#486	1	1	yes	characteristic
120	C₂×C₄₃	C₂²×D₁₆	1	1	yes	characteristic
121	C₂×C₄₃	C₂²×SD₃₂	1	1	yes	characteristic
122	D₄₃	—	16	4	no	—
123	C₂²×D₈	—	43	2	no	—
124	C₂²×D₈	—	86	1	no	—
125	C₂×C₄⋊Q₈	—	2	1	no	—
126	C₂×#127	—	86	1	no	—
127	C₂²×C₁₆	—	43	2	no	—
128	#500	—	2	2	no	—
129	#500	C₂×D₄₃	1	4	yes	complemented
130	#493	—	43	4	no	—
131	#493	—	86	2	no	—
132	C₂×#107	—	2	1	no	—
133	C₂×#107	C₂×D₄₃	1	2	yes	—
134	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
135	#357	—	43	16	no	—
136	C₄×C₁₆	—	43	4	no	—
137	C₄₃	C₂×#2240	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	4	1	no	—
139	C₂²×D₄	—	86	2	no	—
140	C₂²×D₄	—	172	1	no	—
141	C₂²×D₄	—	172	1	no	—
142	C₂×D₈	—	43	4	no	—
143	C₂×D₈	—	86	8	no	—
144	C₂×D₈	—	86	2	no	—
145	C₂×D₈	—	86	4	no	—
146	C₂×D₈	—	172	4	no	—
147	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	4	no	—
150	C₄⋊Q₈	—	4	2	no	—
151	#127	—	86	4	no	—
152	#127	—	172	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	4	1	no	—
155	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
156	C₂×C₁₆	—	43	4	no	—
157	C₂×C₁₆	—	86	4	no	—
158	#107	—	2	4	no	—
159	#107	—	2	2	no	—
160	#107	C₂²×D₄₃	1	4	yes	—
161	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
162	C₄×C₈	C₂²×D₄₃	1	2	yes	—
163	C₂⁴	—	172	2	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	8	4	no	—
166	C₂×D₄	—	86	4	no	—
167	C₂×D₄	—	344	4	no	—
168	C₂×D₄	—	344	4	no	—
169	C₂×D₄	—	172	4	no	—
170	C₂×D₄	—	172	4	no	—
171	C₂×D₄	—	172	4	no	—
172	C₂×D₄	—	172	2	no	—
173	C₂×D₄	—	172	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	#10621882	1	1	yes	characteristic
176	C₂²×C₄	#10621882	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
178	D₈	—	86	8	no	—
179	D₈	—	172	4	no	—
180	D₈	—	172	8	no	—
181	C₂×C₈	#10621884	1	4	yes	—
182	C₂×C₈	#10621902	1	4	yes	—
183	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
184	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
185	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄²	#10621882	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
191	C₁₆	—	86	4	no	—
192	C₂³	—	344	4	no	—
193	C₂³	—	344	4	no	—
194	C₂³	—	344	4	no	—
195	C₂³	—	172	4	no	—
196	C₂³	C₂×#10621882	1	1	yes	characteristic
197	Q₈	—	8	8	no	—
198	D₄	—	344	8	no	—
199	D₄	—	344	8	no	—
200	D₄	—	344	4	no	—
201	D₄	—	172	4	no	—
202	D₄	—	172	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	#13063266	1	4	yes	—
206	C₂×C₄	#13063268	1	4	yes	—
207	C₂×C₄	#13063314	1	2	yes	—
208	C₂×C₄	C₂×#10621882	1	2	yes	—
209	C₂×C₄	C₂×#10621882	1	2	yes	—
210	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
211	C₂×C₄	C₂×#10621884	1	2	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
213	C₈	#13063314	1	2	yes	—
214	C₈	C₂×#10621884	1	4	yes	—
215	C₈	C₂×#10621902	1	2	yes	—
216	C₂²	—	344	4	no	—
217	C₂²	—	344	4	no	—
218	C₂²	—	344	4	no	—
219	C₂²	—	688	8	no	—
220	C₂²	#14583247	1	2	yes	—
221	C₂²	C₂×#13063266	1	2	yes	—
222	C₂²	C₂×#13063268	1	2	yes	—
223	C₂²	C₂²×#10621882	1	1	yes	characteristic
224	C₄	—	8	4	no	—
225	C₄	#14583247	1	2	yes	—
226	C₄	C₂×#13063266	1	4	yes	—
227	C₄	C₂×#13063268	1	4	yes	—
228	C₄	C₂×#13063314	1	1	yes	characteristic
229	C₄	C₂²×#10621884	1	1	yes	characteristic
230	C₂	—	688	4	no	—
231	C₂	#19136820	1	4	yes	direct product factor
232	C₂	C₂×#14583247	1	1	yes	characteristic
233	C₂	C₂²×#13063266	1	1	yes	characteristic
234	C₂	C₂²×#13063268	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#500	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂×#14583276	C₂
hypercenter	C₂×#500	D₄₃
hypocenter	C₄₃	C₂×#2240
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #19136820	C₂	direct (N × H)
3. C₂×#14582699	C₂	semidirect
4. C₂×C₄×C₄₃⋊C₁₆	C₂	semidirect
5. C₂×C₄₃×#500	C₂	semidirect
12. #14582699	C₂²	semidirect
13. C₄×C₄₃⋊C₁₆	C₂²	semidirect
16. C₄₃×#500	C₂²	semidirect
113. C₂×#500	D₄₃	semidirect
118. C₂×C₄₃	#2240	semidirect
129. #500	C₂×D₄₃	semidirect
137. C₄₃	C₂×#2240	semidirect
231. C₂	#19136820	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#14583276	2
#19136820	2
C₂×#14582699	2
C₂×C₄×C₄₃⋊C₁₆	2
C₂×C₄₃×#500	2
C₂×#2240	43

Subgroups — C2×#19136820 #25251882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136820 #25251882