Subgroups — C₂×#19136778 #25251380

Order 11008.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063259	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136778	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14582726	C₂	1	2	yes	maximal, complemented
4	C₂×#14582700	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₃×#466	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₄₃	—	2	1	no	—
8	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
9	C₂×#13063259	—	2	1	no	—
10	C₂×#13063259	—	2	2	no	—
11	C₂×#13063259	C₂²	1	2	yes	—
12	C₂×#13063259	C₂²	1	4	yes	complemented
13	#14582726	C₂²	1	8	yes	complemented
14	#14582700	C₂²	1	8	yes	complemented
15	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	complemented
16	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂×#121×C₄₃	—	2	1	no	—
19	C₄₃×#466	—	2	2	no	—
20	C₄₃×#466	C₂²	1	4	yes	—
21	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×#102×C₄₃	—	2	1	no	—
23	C₂³×Dic₄₃	—	4	1	no	—
24	C₂×C₄⋊Dic₄₃	—	2	2	no	—
25	C₂×C₄⋊Dic₄₃	—	2	4	no	—
26	C₂×C₄⋊Dic₄₃	—	4	1	no	—
27	C₂×C₄⋊Dic₄₃	—	4	2	no	—
28	C₂×C₄₃⋊C₁₆	—	2	4	no	—
29	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
30	#13063259	—	2	2	no	—
31	#13063259	—	2	4	no	—
32	#13063259	C₂³	1	4	yes	—
33	C₂²×D₄×C₄₃	—	2	1	no	—
34	C₂²×D₄×C₄₃	—	4	1	no	—
35	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
36	C₂×D₈×C₄₃	—	2	1	no	—
37	C₂×D₈×C₄₃	—	2	4	no	—
38	C₂×D₈×C₄₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₄₃	—	2	4	no	—
40	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₃	D₄	1	2	yes	—
42	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
44	#121×C₄₃	—	2	4	no	—
45	#121×C₄₃	—	4	2	no	—
46	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₄₃	—	4	1	no	—
48	#107×C₄₃	—	2	1	no	—
49	#107×C₄₃	C₂³	1	2	yes	—
50	#102×C₄₃	—	2	4	no	—
51	C₂⁴×C₄₃	—	4	1	no	—
52	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
53	C₂×D₄×C₄₃	—	2	2	no	—
54	C₂×D₄×C₄₃	—	4	1	no	—
55	C₂×D₄×C₄₃	—	4	4	no	—
56	C₂×D₄×C₄₃	—	4	2	no	—
57	C₂×D₄×C₄₃	—	4	2	no	—
58	C₂×D₄×C₄₃	—	8	4	no	—
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	2	2	no	—
61	C₂²×C₄×C₄₃	—	4	1	no	—
62	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	D₈	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₄₃	—	4	4	no	—
67	C₂²×Dic₄₃	—	4	2	no	—
68	C₂²×Dic₄₃	—	8	4	no	—
69	D₈×C₄₃	—	2	4	no	—
70	D₈×C₄₃	—	4	2	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	2	4	no	—
73	C₂×C₈×C₄₃	—	2	4	no	—
74	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₄₃	#41	1	4	yes	—
77	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₄₃	—	2	2	no	—
79	C₄⋊C₄×C₄₃	—	4	1	no	—
80	C₂²⋊C₄×C₄₃	—	4	4	no	—
81	C₄⋊Dic₄₃	—	2	4	no	—
82	C₄⋊Dic₄₃	—	4	4	no	—
83	C₄⋊Dic₄₃	—	4	2	no	—
84	C₄₃⋊C₁₆	—	2	4	no	—
85	C₂³×C₄₃	—	2	2	no	—
86	C₂³×C₄₃	—	2	2	no	—
87	C₂³×C₄₃	—	4	2	no	—
88	C₂³×C₄₃	—	8	2	no	—
89	C₂³×C₄₃	—	8	2	no	—
90	C₂³×C₄₃	—	8	2	no	—
91	C₂³×C₄₃	SD₃₂	1	4	yes	—
92	C₂³×C₄₃	C₂×D₈	1	2	yes	—
93	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
94	D₄×C₄₃	—	4	2	no	—
95	D₄×C₄₃	—	4	2	no	—
96	D₄×C₄₃	—	8	8	no	—
97	D₄×C₄₃	—	8	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	—	4	2	no	—
103	C₂×C₄×C₄₃	—	8	2	no	—
104	C₂×C₄×C₄₃	#123	1	2	yes	—
105	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₄₃	#137	1	2	yes	—
108	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×Dic₄₃	—	4	4	no	—
110	C₂×Dic₄₃	—	8	8	no	—
111	C₂×Dic₄₃	—	8	4	no	—
112	C₈×C₄₃	—	2	2	no	—
113	C₈×C₄₃	#123	1	2	yes	—
114	C₈×C₄₃	C₂×#41	1	2	yes	—
115	C₂×#2233	—	43	1	no	maximal, Sylow
116	C₂²×C₄₃	—	2	2	no	—
117	C₂²×C₄₃	—	2	2	no	—
118	C₂²×C₄₃	—	2	8	no	—
119	C₂²×C₄₃	—	8	2	no	—
120	C₂²×C₄₃	—	8	2	no	—
121	C₂²×C₄₃	—	8	2	no	—
122	C₂²×C₄₃	—	16	4	no	—
123	C₂²×C₄₃	#480	1	2	yes	—
124	C₂²×C₄₃	C₂×SD₃₂	1	4	yes	—
125	C₂²×C₄₃	C₂×SD₃₂	1	2	yes	—
126	C₂²×C₄₃	#509	1	2	yes	—
127	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
128	C₄×C₄₃	—	2	2	no	—
129	C₄×C₄₃	—	8	2	no	—
130	C₄×C₄₃	#480	1	2	yes	—
131	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
132	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
133	Dic₄₃	—	8	4	no	—
134	C₂×#466	D₄₃	1	1	yes	characteristic
135	C₂²×#107	—	43	1	no	—
136	#2233	—	43	8	no	—
137	C₂×#367	—	43	2	no	—
138	C₂×#357	—	43	2	no	—
139	C₂×C₂²⋊C₁₆	—	43	1	no	—
140	C₂×C₄₃	—	2	4	no	—
141	C₂×C₄₃	—	16	2	no	—
142	C₂×C₄₃	#2233	1	4	yes	complemented
143	C₂×C₄₃	C₂×#480	1	1	yes	characteristic
144	C₂×C₄₃	C₂²×SD₃₂	1	1	yes	characteristic
145	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
146	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
147	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
148	C₂×#121	—	2	1	no	—
149	C₂²×C₄⋊C₄	—	86	1	no	—
150	C₂²×C₁₆	—	43	2	no	—
151	#466	—	2	2	no	—
152	#466	C₂×D₄₃	1	4	yes	—
153	C₂×#107	—	43	2	no	—
154	C₂×#107	—	43	4	no	—
155	C₂×#107	—	86	1	no	—
156	C₂×#107	—	86	2	no	—
157	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
158	C₂×#102	—	2	1	no	—
159	#367	—	43	8	no	—
160	#357	—	43	8	no	—
161	C₂²⋊C₁₆	—	43	4	no	—
162	C₄₃	C₂×#2233	1	1	yes	characteristic, complemented, Sylow
163	C₂²×D₄	—	2	1	no	—
164	C₂²×D₄	—	4	1	no	—
165	C₂³×C₄	—	172	1	no	—
166	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
167	C₂×D₈	—	2	4	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×D₈	C₂²×D₄₃	1	2	yes	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
173	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
174	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
175	#121	—	2	4	no	—
176	#121	—	4	2	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	86	4	no	—
179	C₂×C₄⋊C₄	—	86	2	no	—
180	C₂×C₄⋊C₄	—	172	2	no	—
181	C₂×C₄⋊C₄	—	172	1	no	—
182	C₂×C₂²⋊C₄	—	4	1	no	—
183	C₂×C₁₆	—	43	4	no	—
184	C₂×C₁₆	—	86	4	no	—
185	#107	—	2	1	no	—
186	#107	—	43	4	no	—
187	#107	—	86	4	no	—
188	#107	—	86	2	no	—
189	#107	C₂²×D₄₃	1	2	yes	—
190	#102	—	2	4	no	—
191	C₂⁴	—	4	1	no	—
192	C₂⁴	#10621882	1	1	yes	characteristic
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	8	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	344	4	no	—
203	C₂²×C₄	—	172	4	no	—
204	C₂²×C₄	—	172	2	no	—
205	C₂²×C₄	#10621882	1	2	yes	—
206	C₂²×C₄	#10621882	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
209	D₈	—	2	4	no	—
210	D₈	—	4	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	#10621900	1	4	yes	—
215	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
216	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
217	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	86	4	no	—
221	C₄⋊C₄	—	172	4	no	—
222	C₄⋊C₄	—	172	2	no	—
223	C₂²⋊C₄	—	4	4	no	—
224	C₁₆	—	86	4	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	—	8	2	no	—
231	C₂³	#13063269	1	4	yes	—
232	C₂³	C₂×#10621882	1	2	yes	—
233	C₂³	C₂×#10621882	1	1	yes	characteristic
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	8	8	no	—
237	D₄	—	8	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	—	344	8	no	—
245	C₂×C₄	—	344	4	no	—
246	C₂×C₄	—	172	4	no	—
247	C₂×C₄	#13063306	1	2	yes	—
248	C₂×C₄	C₂×#10621882	1	2	yes	—
249	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
250	C₂×C₄	#13063355	1	2	yes	—
251	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	#13063306	1	2	yes	—
254	C₈	C₂×#10621900	1	2	yes	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	16	4	no	—
262	C₂²	#14583210	1	2	yes	—
263	C₂²	C₂×#13063269	1	4	yes	—
264	C₂²	C₂×#13063269	1	2	yes	—
265	C₂²	#14583334	1	2	yes	—
266	C₂²	C₂²×#10621882	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	344	4	no	—
270	C₄	#14583210	1	2	yes	—
271	C₄	C₂×#13063306	1	1	yes	characteristic
272	C₄	C₂×#13063355	1	1	yes	characteristic
273	C₂	—	2	4	no	—
274	C₂	—	16	2	no	—
275	C₂	#19136778	1	4	yes	direct product factor
276	C₂	C₂×#14583210	1	1	yes	characteristic
277	C₂	C₂²×#13063269	1	1	yes	characteristic
278	C₂	C₂×#14583334	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting, prime radical	C₂×C₄₃×#466	C₂
socle	C₂³×C₄₃	C₂×D₈
hypercenter	C₂×#466	D₄₃
hypocenter	C₄₃	C₂×#2233
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063259	C₂	semidirect
2. #19136778	C₂	direct (N × H)
3. C₂×#14582726	C₂	semidirect
4. C₂×#14582700	C₂	semidirect
5. C₂×C₂²⋊(C₄₃⋊C₁₆)	C₂	semidirect
12. C₂×#13063259	C₂²	semidirect
13. #14582726	C₂²	semidirect
14. #14582700	C₂²	semidirect
15. C₂²⋊(C₄₃⋊C₁₆)	C₂²	semidirect
142. C₂×C₄₃	#2233	semidirect
162. C₄₃	C₂×#2233	semidirect
275. C₂	#19136778	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×#13063259	2
#19136778	2
C₂×#14582726	2
C₂×#14582700	2
C₂×C₂²⋊(C₄₃⋊C₁₆)	2
C₂×C₄₃×#466	2
C₂×#2233	43

Subgroups — C2×#19136778 #25251380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136778 #25251380