Subgroups — #25008308

Order 11008.

244 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	#19136427	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#19132665	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132512	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063265	—	2	1	no	—
9	C₂²×C₄₃⋊C₁₆	—	2	1	no	—
10	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
11	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
12	C₂²⋊(C₄₃⋊C₁₆)	—	2	1	no	—
13	C₂²⋊(C₄₃⋊C₁₆)	—	2	1	no	—
14	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₄×D₈×C₄₃	C₄	1	1	yes	characteristic
17	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
18	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#335	C₄	1	1	yes	characteristic
20	C₄₃×D₄⋊C₈	C₄	1	1	yes	characteristic
21	C₄₃×D₄⋊C₈	C₄	1	1	yes	characteristic
22	C₈²×C₄₃	C₂²	1	1	yes	characteristic
23	#13063265	—	4	2	no	—
24	C₂×C₄₃⋊C₁₆	—	2	1	no	—
25	C₂×C₄₃⋊C₁₆	—	2	1	no	—
26	C₂×C₄₃⋊C₁₆	—	2	1	no	—
27	C₂×C₄₃⋊C₁₆	—	4	2	no	—
28	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
30	C₂×D₈×C₄₃	C₈	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	—	2	1	no	—
32	C₂²×C₈×C₄₃	—	2	1	no	—
33	C₄×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₄×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
35	#107×C₄₃	C₈	1	1	yes	characteristic
36	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
38	#102×C₄₃	C₈	1	1	yes	characteristic
39	#102×C₄₃	C₈	1	1	yes	characteristic
40	C₂²⋊C₈×C₄₃	—	2	1	no	—
41	C₂²⋊C₈×C₄₃	—	2	1	no	—
42	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
46	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	—	2	1	no	—
48	C₂²×C₄×C₄₃	—	2	1	no	—
49	D₈×C₄₃	—	2	2	no	—
50	C₂×C₈×C₄₃	—	2	1	no	—
51	C₂×C₈×C₄₃	—	2	1	no	—
52	C₂×C₈×C₄₃	—	2	1	no	—
53	C₂×C₈×C₄₃	—	4	1	no	—
54	C₂×C₈×C₄₃	—	4	1	no	—
55	C₂×C₈×C₄₃	—	4	1	no	—
56	C₂×C₈×C₄₃	—	4	1	no	—
57	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
58	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
59	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
62	C₄⋊C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
63	C₂²⋊C₄×C₄₃	—	2	1	no	—
64	C₂²⋊C₄×C₄₃	—	2	1	no	—
65	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
66	C₄₃⋊C₁₆	—	2	2	no	—
67	C₄₃⋊C₁₆	—	2	1	no	—
68	C₄₃⋊C₁₆	—	4	1	no	—
69	C₄₃⋊C₁₆	—	4	1	no	—
70	C₄₃⋊C₁₆	D₈	1	2	yes	complemented
71	C₂³×C₄₃	—	2	1	no	—
72	C₂³×C₄₃	—	2	1	no	—
73	D₄×C₄₃	—	2	1	no	—
74	D₄×C₄₃	—	2	1	no	—
75	D₄×C₄₃	—	2	1	no	—
76	D₄×C₄₃	M₃₂	1	2	yes	complemented
77	C₂×C₄×C₄₃	—	2	1	no	—
78	C₂×C₄×C₄₃	—	2	1	no	—
79	C₂×C₄×C₄₃	—	4	1	no	—
80	C₂×C₄×C₄₃	—	4	1	no	—
81	C₂×C₄×C₄₃	—	4	1	no	—
82	C₂×C₄×C₄₃	—	4	1	no	—
83	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
86	C₈×C₄₃	—	2	1	no	—
87	C₈×C₄₃	—	2	1	no	—
88	C₈×C₄₃	—	2	1	no	—
89	C₈×C₄₃	—	4	1	no	—
90	C₈×C₄₃	—	4	1	no	—
91	C₈×C₄₃	—	4	1	no	—
92	C₈×C₄₃	C₂×D₈	1	1	yes	characteristic
93	C₈×C₄₃	#135	1	1	yes	characteristic
94	#12038	—	43	1	no	maximal, Sylow
95	C₂²×C₄₃	—	2	2	no	—
96	C₂²×C₄₃	—	4	1	no	—
97	C₂²×C₄₃	—	4	1	no	—
98	C₂²×C₄₃	—	4	1	no	—
99	C₂²×C₄₃	—	4	1	no	—
100	C₂²×C₄₃	—	4	1	no	—
101	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
102	C₄×C₄₃	—	2	1	no	—
103	C₄×C₄₃	—	4	1	no	—
104	C₄×C₄₃	—	4	1	no	—
105	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
106	C₄×C₄₃	#443	1	1	yes	characteristic
107	C₄×C₄₃	C₂×M₃₂	1	1	yes	characteristic
108	C₄×C₄₃	#504	1	1	yes	characteristic
109	#2169	—	43	1	no	—
110	D₄×C₁₆	—	43	1	no	—
111	C₈×D₈	D₄₃	1	1	yes	characteristic
112	#1370	—	43	1	no	—
113	D₄⋊C₁₆	—	43	1	no	—
114	D₄⋊C₁₆	—	43	1	no	—
115	#1311	—	43	1	no	—
116	C₂×C₄₃	—	4	2	no	—
117	C₂×C₄₃	—	8	1	no	—
118	C₂×C₄₃	C₈×D₈	1	1	yes	characteristic
119	C₂×C₄₃	#2168	1	1	yes	characteristic
120	C₂×C₄₃	#2171	1	1	yes	characteristic
121	C₂×M₃₂	—	86	1	no	—
122	C₂²×C₁₆	—	86	1	no	—
123	C₄×D₈	Dic₄₃	1	1	yes	characteristic
124	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
125	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
126	C₄⋊C₁₆	—	43	1	no	—
127	C₄⋊C₁₆	—	43	1	no	—
128	C₂²⋊C₁₆	—	86	1	no	—
129	C₂²⋊C₁₆	—	86	1	no	—
130	C₄×C₁₆	—	43	1	no	—
131	C₄×C₁₆	—	43	1	no	—
132	#335	Dic₄₃	1	1	yes	characteristic
133	D₄⋊C₈	Dic₄₃	1	1	yes	characteristic
134	D₄⋊C₈	Dic₄₃	1	1	yes	characteristic
135	C₈²	C₂×D₄₃	1	1	yes	characteristic
136	C₄₃	#12038	1	1	yes	characteristic, complemented, Sylow
137	C₂×D₈	C₄₃⋊C₈	1	1	yes	characteristic
138	C₂²×C₈	—	2	1	no	—
139	C₂²×C₈	—	2	1	no	—
140	C₄×D₄	C₂×Dic₄₃	1	1	yes	characteristic
141	C₄×D₄	C₂×Dic₄₃	1	1	yes	characteristic
142	M₃₂	—	172	2	no	—
143	C₂×C₁₆	—	43	1	no	—
144	C₂×C₁₆	—	43	1	no	—
145	C₂×C₁₆	—	86	1	no	—
146	C₂×C₁₆	—	86	1	no	—
147	C₂×C₁₆	—	86	1	no	—
148	C₂×C₁₆	—	172	2	no	—
149	#107	C₄₃⋊C₈	1	1	yes	characteristic
150	C₄⋊C₈	C₂×Dic₄₃	1	1	yes	characteristic
151	C₄⋊C₈	C₂×Dic₄₃	1	1	yes	characteristic
152	#102	C₄₃⋊C₈	1	1	yes	characteristic
153	#102	C₄₃⋊C₈	1	1	yes	characteristic
154	C₂²⋊C₈	—	2	1	no	—
155	C₂²⋊C₈	—	2	1	no	—
156	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
157	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
159	C₂×D₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
160	C₂×D₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	D₈	—	2	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	C₂×C₄₃⋊C₈	1	1	yes	characteristic
172	C₂×C₈	C₂×C₄₃⋊C₈	1	1	yes	characteristic
173	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
174	C₂×C₈	#10621900	1	1	yes	characteristic
175	C₄⋊C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
176	C₄⋊C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
177	C₂²⋊C₄	—	2	1	no	—
178	C₂²⋊C₄	—	2	1	no	—
179	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
180	C₁₆	—	43	2	no	—
181	C₁₆	—	86	2	no	—
182	C₁₆	—	86	1	no	—
183	C₁₆	—	172	1	no	—
184	C₁₆	—	172	1	no	—
185	C₂³	—	2	1	no	—
186	C₂³	—	2	1	no	—
187	D₄	—	2	1	no	—
188	D₄	—	2	1	no	—
189	D₄	—	2	1	no	—
190	D₄	#13063265	1	2	yes	complemented
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
198	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
199	C₂×C₄	#13063332	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	4	1	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	1	no	—
206	C₈	D₈×D₄₃	1	1	yes	characteristic
207	C₈	#13063344	1	1	yes	characteristic
208	C₂²	—	2	2	no	—
209	C₂²	—	4	1	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	D₄×C₄₃⋊C₈	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	D₈×Dic₄₃	1	1	yes	characteristic
219	C₄	#14583016	1	1	yes	characteristic
220	C₄	C₂×#13063265	1	1	yes	characteristic
221	C₄	#14583316	1	1	yes	characteristic
222	C₂	—	4	2	no	—
223	C₂	—	8	1	no	—
224	C₂	D₈×C₄₃⋊C₈	1	1	yes	characteristic
225	C₂	#19136421	1	1	yes	characteristic
226	C₂	#19136437	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₈×D₈×C₄₃	C₂
socle	C₂²×C₄₃	C₈×D₄
prime radical	C₂×D₈×C₄₃	C₈
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	C₈×D₈	D₄₃
hypocenter	C₄₃	#12038
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
1. #19136427	C₂	semidirect
2. D₄×C₄₃⋊C₁₆	C₂	semidirect
3. #19132665	C₂	semidirect
4. D₄⋊(C₄₃⋊C₁₆)	C₂	semidirect
5. D₄⋊(C₄₃⋊C₁₆)	C₂	semidirect
6. #19132512	C₂	semidirect
70. C₄₃⋊C₁₆	D₈	semidirect
76. D₄×C₄₃	M₃₂	semidirect
136. C₄₃	#12038	semidirect
190. D₄	#13063265	semidirect

Maximal subgroups

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

subgroup	index
#19136427	2
D₄×C₄₃⋊C₁₆	2
#19132665	2
D₄⋊(C₄₃⋊C₁₆)	2
#19132512	2
C₈×D₈×C₄₃	2
#12038	43