Subgroups — C₂×#19136742 #25251206

Order 11008.

638 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583304	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136742	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14582726	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582700	C₂	1	2	yes	maximal
6	C₂×C₄×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13063325	—	2	1	no	—
8	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
9	#14583304	—	2	2	no	—
10	#14583304	C₂²	1	4	yes	complemented
11	C₂×#13063259	—	2	1	no	—
12	C₂×#13063259	C₂²	1	2	yes	—
13	#14582726	C₂²	1	8	yes	complemented
14	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	complemented
15	#14582700	C₂²	1	8	yes	—
16	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₄₃	—	2	1	no	—
18	C₄×D₈×C₄₃	—	2	2	no	—
19	C₄×D₈×C₄₃	C₂²	1	4	yes	—
20	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×#102×C₄₃	—	2	1	no	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₈₆	—	4	1	no	—
24	#13063325	—	2	4	no	—
25	#13063325	—	4	2	no	—
26	C₂×C₄⋊Dic₄₃	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	—	4	1	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	C₂³×C₄×C₄₃	—	4	1	no	—
35	C₂×D₈×C₄₃	—	2	1	no	—
36	C₂×D₈×C₄₃	—	2	2	no	—
37	C₂×D₈×C₄₃	Q₈	1	4	yes	complemented
38	C₂×D₈×C₄₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
41	C₄×D₄×C₄₃	—	2	4	no	—
42	C₄×D₄×C₄₃	—	4	2	no	—
43	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₄₃	—	4	1	no	—
45	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
46	#107×C₄₃	—	2	1	no	—
47	#107×C₄₃	C₂³	1	2	yes	—
48	#102×C₄₃	—	2	4	no	—
49	C₄×C₈×C₄₃	D₄	1	2	yes	—
50	C₄×C₈×C₄₃	C₂³	1	2	yes	—
51	C₂⁴×C₄₃	—	4	1	no	—
52	C₂×D₄×C₄₃	—	2	4	no	—
53	C₂×D₄×C₄₃	—	2	2	no	—
54	C₂×D₄×C₄₃	—	4	1	no	—
55	C₂×D₄×C₄₃	—	4	2	no	—
56	C₂²×C₄×C₄₃	—	4	4	no	—
57	C₂²×C₄×C₄₃	—	4	1	no	—
58	C₂²×C₄×C₄₃	—	8	4	no	—
59	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₄₃	—	4	2	no	—
63	C₂×Dic₈₆	—	4	4	no	—
64	C₂×Dic₈₆	—	8	4	no	—
65	D₈×C₄₃	—	2	4	no	—
66	D₈×C₄₃	—	2	2	no	—
67	D₈×C₄₃	C₂×Q₈	1	4	yes	complemented
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₄₃	#41	1	2	yes	—
73	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₄₃	—	2	2	no	—
75	C₄⋊C₄×C₄₃	—	4	1	no	—
76	C₂²⋊C₄×C₄₃	—	4	4	no	—
77	C₄²×C₄₃	D₈	1	2	yes	—
78	C₄²×C₄₃	C₂×D₄	1	2	yes	—
79	C₄⋊Dic₄₃	—	2	4	no	—
80	C₄⋊Dic₄₃	—	4	2	no	—
81	C₄⋊Dic₄₃	—	4	4	no	—
82	C₄₃⋊C₁₆	—	2	4	no	—
83	C₂³×C₄₃	—	4	2	no	—
84	C₂³×C₄₃	—	4	4	no	—
85	C₂³×C₄₃	—	8	2	no	—
86	C₂³×C₄₃	—	8	2	no	—
87	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
88	D₄×C₄₃	—	2	4	no	—
89	D₄×C₄₃	—	4	2	no	—
90	D₄×C₄₃	—	4	4	no	—
91	C₂×C₄×C₄₃	—	2	2	no	—
92	C₂×C₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	—	8	2	no	—
94	C₂×C₄×C₄₃	—	8	8	no	—
95	C₂×C₄×C₄₃	SD₃₂	1	4	yes	—
96	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₄₃	#137	1	2	yes	—
101	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×Dic₄₃	—	4	4	no	—
103	C₂×Dic₄₃	—	8	4	no	—
104	Dic₈₆	—	8	8	no	—
105	C₈×C₄₃	—	2	2	no	—
106	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
107	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
108	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
109	C₂×#2226	—	43	1	no	maximal, Sylow
110	C₂²×C₄₃	—	4	4	no	—
111	C₂²×C₄₃	—	8	2	no	—
112	C₂²×C₄₃	—	8	2	no	—
113	C₂²×C₄₃	—	8	8	no	—
114	C₂²×C₄₃	#474	1	2	yes	—
115	C₂²×C₄₃	C₂×SD₃₂	1	2	yes	—
116	C₂²×C₄₃	#509	1	2	yes	—
117	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
118	C₄×C₄₃	—	2	2	no	—
119	C₄×C₄₃	—	8	2	no	—
120	C₄×C₄₃	#474	1	2	yes	—
121	C₄×C₄₃	C₂×SD₃₂	1	4	yes	—
122	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
123	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
124	Dic₄₃	—	8	4	no	—
125	C₂×#500	—	43	1	no	—
126	C₂×C₄×D₈	D₄₃	1	1	yes	characteristic
127	#2226	—	43	8	no	—
128	C₂×#367	—	43	2	no	—
129	C₂×C₄⋊C₁₆	—	43	1	no	—
130	C₂×#357	—	43	2	no	—
131	C₂×C₄₃	—	8	4	no	—
132	C₂×C₄₃	#2226	1	4	yes	complemented
133	C₂×C₄₃	C₂×#474	1	1	yes	characteristic
134	C₂×C₄₃	C₂²×SD₃₂	1	1	yes	characteristic
135	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
136	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
137	C₂×C₄⋊Q₈	—	86	1	no	—
138	C₂×C₄×D₄	—	2	1	no	—
139	C₂²×C₁₆	—	43	2	no	—
140	#500	—	43	4	no	—
141	#500	—	86	2	no	—
142	C₄×D₈	—	2	2	no	—
143	C₄×D₈	C₂×D₄₃	1	4	yes	—
144	C₂×#107	—	43	2	no	—
145	C₂×#107	—	86	1	no	—
146	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
147	C₂×#102	—	2	1	no	—
148	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
149	#367	—	43	8	no	—
150	C₄⋊C₁₆	—	43	4	no	—
151	#357	—	43	8	no	—
152	C₄₃	C₂×#2226	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	172	1	no	—
154	C₂²×D₄	—	2	1	no	—
155	C₂³×C₄	—	4	1	no	—
156	C₂×D₈	—	2	2	no	—
157	C₂×D₈	—	2	1	no	—
158	C₂×D₈	Dic₈₆	1	4	yes	complemented
159	C₂×D₈	C₂²×D₄₃	1	2	yes	—
160	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
162	C₄⋊Q₈	—	86	4	no	—
163	C₄⋊Q₈	—	172	2	no	—
164	C₄×D₄	—	2	4	no	—
165	C₄×D₄	—	4	2	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	86	2	no	—
168	C₂×C₄⋊C₄	—	172	1	no	—
169	C₂×C₂²⋊C₄	—	4	1	no	—
170	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
171	C₂×C₁₆	—	43	4	no	—
172	C₂×C₁₆	—	86	4	no	—
173	#107	—	2	1	no	—
174	#107	—	43	4	no	—
175	#107	—	86	2	no	—
176	#107	—	86	4	no	—
177	#107	C₂²×D₄₃	1	2	yes	—
178	#102	—	2	4	no	—
179	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
180	C₄×C₈	C₂²×D₄₃	1	2	yes	—
181	C₂⁴	—	4	1	no	—
182	C₂×Q₈	—	344	4	no	—
183	C₂×Q₈	—	172	4	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	—	8	4	no	—
191	C₂²×C₄	—	172	2	no	—
192	C₂²×C₄	#10621882	1	1	yes	characteristic
193	C₂²×C₄	#10621882	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
195	D₈	—	2	4	no	—
196	D₈	—	2	2	no	—
197	D₈	C₂×Dic₈₆	1	4	yes	complemented
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
200	C₂×C₈	#10621898	1	2	yes	—
201	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
203	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	86	4	no	—
207	C₄⋊C₄	—	172	2	no	—
208	C₄⋊C₄	—	172	4	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₄²	#10621882	1	2	yes	—
211	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
212	C₁₆	—	86	4	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	8	2	no	—
216	C₂³	—	8	2	no	—
217	C₂³	C₂×#10621882	1	1	yes	characteristic
218	Q₈	—	344	8	no	—
219	D₄	—	2	4	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	8	no	—
226	C₂×C₄	—	344	4	no	—
227	C₂×C₄	—	172	4	no	—
228	C₂×C₄	#13063269	1	4	yes	—
229	C₂×C₄	#13063304	1	2	yes	—
230	C₂×C₄	C₂×#10621882	1	2	yes	—
231	C₂×C₄	C₂×#10621882	1	2	yes	—
232	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
233	C₂×C₄	#13063355	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	#13063304	1	2	yes	—
237	C₈	C₂²×Dic₈₆	1	1	yes	characteristic
238	C₈	C₂×#10621898	1	1	yes	characteristic
239	C₂²	—	4	4	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	8	no	—
243	C₂²	#14583179	1	2	yes	—
244	C₂²	C₂×#13063269	1	2	yes	—
245	C₂²	#14583334	1	2	yes	—
246	C₂²	C₂²×#10621882	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	8	2	no	—
249	C₄	—	344	4	no	—
250	C₄	#14583179	1	2	yes	—
251	C₄	C₂×#13063269	1	4	yes	—
252	C₄	C₂×#13063304	1	1	yes	characteristic
253	C₄	C₂×#13063355	1	1	yes	characteristic
254	C₂	—	8	4	no	—
255	C₂	#19136742	1	4	yes	direct product factor
256	C₂	C₂×#14583179	1	1	yes	characteristic
257	C₂	C₂²×#13063269	1	1	yes	characteristic
258	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	C₂×C₄×D₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₈×C₄₃	C₂²
hypercenter	C₂×C₄×D₈	D₄₃
hypocenter	C₄₃	C₂×#2226
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14583304	C₂	semidirect
2. #19136742	C₂	direct (N × H)
3. C₂×#14582726	C₂	semidirect
4. C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	semidirect
10. #14583304	C₂²	semidirect
13. #14582726	C₂²	semidirect
14. C₄⋊(C₄₃⋊C₁₆)	C₂²	semidirect
37. C₂×D₈×C₄₃	Q₈	semidirect
67. D₈×C₄₃	C₂×Q₈	semidirect
132. C₂×C₄₃	#2226	semidirect
152. C₄₃	C₂×#2226	semidirect
158. C₂×D₈	Dic₈₆	semidirect
197. D₈	C₂×Dic₈₆	semidirect
255. C₂	#19136742	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×#14583304	2
#19136742	2
C₂×#14582726	2
C₂×C₄⋊(C₄₃⋊C₁₆)	2
C₂×#14582700	2
C₂×C₄×D₈×C₄₃	2
C₂×#2226	43

Subgroups — C2×#19136742 #25251206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136742 #25251206