Subgroups — #24995372

Order 11008.

774 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13063322	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582626	C₂	1	4	yes	maximal, complemented
4	C₄₃×#1842	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13063322	C₄	1	1	yes	characteristic
6	C₂×C₄×D₁₇₂	—	2	2	no	—
7	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
8	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
9	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
10	#14582801	—	2	2	no	—
11	#14582801	C₄	1	2	yes	—
12	#14582769	C₄	1	1	yes	characteristic
13	#14582626	—	2	8	no	—
14	C₂×C₄⋊C₈×C₄₃	—	2	2	no	—
15	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
16	C₄³×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×D₁₇₂	—	2	2	no	—
18	C₂²×D₁₇₂	C₂×C₄	1	2	yes	—
19	C₂²×C₄×D₄₃	—	2	2	no	—
20	C₂²×C₄₃⋊C₈	—	2	2	no	—
21	#13063322	—	2	2	no	—
22	#13063322	C₈	1	4	yes	complemented
23	C₄×D₁₇₂	—	2	4	no	—
24	C₄×D₁₇₂	—	2	4	no	—
25	C₄×D₁₇₂	—	4	4	no	—
26	C₄×D₁₇₂	—	4	4	no	—
27	C₂×C₄⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
29	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
30	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
31	C₄×C₄₃⋊C₈	—	2	8	no	—
32	C₄×C₄₃⋊C₈	—	2	2	no	—
33	C₄×C₄₃⋊C₈	D₄	1	4	yes	complemented
34	C₂²×C₈×C₄₃	—	2	2	no	—
35	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₃	D₄	1	2	yes	—
38	C₂×C₄²×C₄₃	D₄	1	2	yes	—
39	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₄₃	—	2	4	no	—
41	C₄⋊C₈×C₄₃	—	4	4	no	—
42	C₂³×D₄₃	—	2	2	no	—
43	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₄₃	—	2	2	no	—
49	C₂×D₁₇₂	—	2	8	no	—
50	C₂×D₁₇₂	—	2	2	no	—
51	C₂×D₁₇₂	—	2	4	no	—
52	C₂×D₁₇₂	—	2	4	no	—
53	C₂×D₁₇₂	—	4	4	no	—
54	C₂×D₁₇₂	—	4	4	no	—
55	C₂×D₁₇₂	C₂×C₈	1	4	yes	—
56	C₂×C₄×D₄₃	—	4	2	no	—
57	C₂×C₄×D₄₃	—	4	2	no	—
58	C₂×C₄×D₄₃	—	4	4	no	—
59	C₂×C₄×D₄₃	—	4	4	no	—
60	C₂×C₈×C₄₃	—	4	2	no	—
61	C₂×C₈×C₄₃	—	4	4	no	—
62	C₄²×C₄₃	—	2	4	no	—
63	C₄²×C₄₃	—	2	2	no	—
64	C₄²×C₄₃	C₂×C₈	1	2	yes	—
65	C₄²×C₄₃	M₁₆	1	2	yes	—
66	C₄²×C₄₃	D₈	1	4	yes	—
67	C₄²×C₄₃	SD₁₆	1	4	yes	—
68	C₄²×C₄₃	C₂×D₄	1	2	yes	—
69	C₄²×C₄₃	#41	1	2	yes	—
70	(C₂×D₄₃)⋊C₄	—	4	8	no	—
71	(C₂×D₄₃)⋊C₄	—	4	4	no	—
72	C₄⋊Dic₄₃	—	2	4	no	—
73	C₄⋊Dic₄₃	—	4	4	no	—
74	C₂×C₄₃⋊C₈	—	2	8	no	—
75	C₂×C₄₃⋊C₈	—	2	4	no	—
76	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×D₄₃	—	2	4	no	—
78	C₂²×D₄₃	—	4	2	no	—
79	C₂²×D₄₃	—	4	2	no	—
80	C₂²×D₄₃	—	4	4	no	—
81	C₂²×D₄₃	—	4	4	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	C₂²⋊C₈	1	4	yes	—
86	C₂×C₄×C₄₃	#102	1	4	yes	—
87	C₂×C₄×C₄₃	#102	1	4	yes	—
88	C₂×C₄×C₄₃	#104	1	4	yes	—
89	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₄₃	#121	1	2	yes	—
91	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
92	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₄₃	#131	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
98	C₂×Dic₄₃	—	4	2	no	—
99	C₂×Dic₄₃	—	4	4	no	—
100	D₁₇₂	—	2	8	no	—
101	D₁₇₂	—	4	16	no	—
102	D₁₇₂	—	4	4	no	—
103	C₄×D₄₃	—	8	4	no	—
104	C₄×D₄₃	—	8	2	no	—
105	C₄×D₄₃	—	8	2	no	—
106	C₈×C₄₃	—	8	2	no	—
107	C₄₃⋊C₈	—	2	8	no	—
108	#9638	—	43	1	no	maximal, Sylow
109	C₂²×C₄₃	#390	1	1	yes	characteristic
110	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
111	C₂²×C₄₃	#408	1	1	yes	characteristic
112	C₂²×C₄₃	C₂×#102	1	2	yes	—
113	C₂²×C₄₃	C₂×#104	1	2	yes	—
114	C₂×D₄₃	—	4	4	no	—
115	C₂×D₄₃	—	4	8	no	—
116	C₂×D₄₃	—	8	2	no	—
117	C₂×D₄₃	—	8	2	no	—
118	C₂×D₄₃	—	8	4	no	—
119	C₄×C₄₃	—	2	2	no	—
120	C₄×C₄₃	—	2	4	no	—
121	C₄×C₄₃	D₄⋊C₈	1	8	yes	complemented
122	C₄×C₄₃	C₈×D₄	1	2	yes	—
123	C₄×C₄₃	#436	1	2	yes	—
124	C₄×C₄₃	#486	1	2	yes	—
125	C₄×C₄₃	#492	1	1	yes	characteristic
126	C₄×C₄₃	#493	1	1	yes	characteristic
127	Dic₄₃	—	8	2	no	—
128	C₄×#127	—	43	1	no	—
129	#1842	D₄₃	1	1	yes	characteristic, complemented
130	C₄²×C₈	—	43	1	no	—
131	C₂×D₄⋊C₈	—	43	4	no	—
132	C₂×C₄₃	C₂×D₄⋊C₈	1	2	yes	—
133	C₂×C₄₃	#1492	1	2	yes	—
134	C₂×C₄₃	#1954	1	1	yes	characteristic
135	C₂×C₄₃	#1957	1	1	yes	characteristic
136	C₂×C₄₃	#1962	1	1	yes	characteristic
137	D₄₃	—	8	4	no	—
138	C₂×#127	—	43	1	no	—
139	C₂×C₄×D₄	—	43	2	no	—
140	C₂×C₄×D₄	—	86	2	no	—
141	C₂×C₄⋊C₈	—	2	2	no	—
142	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
143	C₂×C₄×C₈	—	43	2	no	—
144	C₂×C₄×C₈	—	86	2	no	—
145	#390	—	43	2	no	—
146	#390	—	86	2	no	—
147	#384	—	43	1	no	—
148	C₄³	C₂×D₄₃	1	1	yes	characteristic
149	D₄⋊C₈	—	86	8	no	—
150	C₄₃	#9638	1	1	yes	characteristic, complemented, Sylow
151	C₂²×D₄	—	43	2	no	—
152	C₂²×D₄	—	86	2	no	—
153	C₂³×C₄	—	86	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	86	2	no	—
156	#127	—	43	4	no	—
157	#127	—	86	2	no	—
158	C₄×D₄	—	86	4	no	—
159	C₄×D₄	—	86	4	no	—
160	C₄×D₄	—	172	4	no	—
161	C₄×D₄	—	172	4	no	—
162	C₂×C₄⋊C₄	—	43	2	no	—
163	C₂×C₄⋊C₄	—	86	2	no	—
164	C₂×C₂²⋊C₄	—	86	4	no	—
165	C₂×C₂²⋊C₄	—	86	2	no	—
166	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
167	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
168	C₂×C₄²	D₁₇₂	1	2	yes	—
169	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
170	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	—	4	4	no	—
173	C₄×C₈	—	43	4	no	—
174	C₄×C₈	—	86	8	no	—
175	C₄×C₈	—	86	2	no	—
176	C₂⁴	—	86	2	no	—
177	C₂×D₄	—	43	4	no	—
178	C₂×D₄	—	86	8	no	—
179	C₂×D₄	—	86	4	no	—
180	C₂×D₄	—	86	2	no	—
181	C₂×D₄	—	86	4	no	—
182	C₂×D₄	—	172	4	no	—
183	C₂×D₄	—	172	4	no	—
184	C₂²×C₄	—	86	2	no	—
185	C₂²×C₄	—	172	2	no	—
186	C₂²×C₄	—	172	2	no	—
187	C₂²×C₄	—	172	4	no	—
188	C₂²×C₄	—	172	4	no	—
189	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
190	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
191	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	86	8	no	—
197	C₂×C₈	—	86	4	no	—
198	C₄⋊C₄	—	86	4	no	—
199	C₄⋊C₄	—	172	4	no	—
200	C₂²⋊C₄	—	172	8	no	—
201	C₂²⋊C₄	—	172	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	C₈×D₄₃	1	2	yes	—
205	C₄²	#10621872	1	2	yes	—
206	C₄²	#10621882	1	4	yes	—
207	C₄²	#10621884	1	4	yes	—
208	C₄²	D₄×D₄₃	1	2	yes	—
209	C₄²	#10621902	1	2	yes	—
210	C₂³	—	86	4	no	—
211	C₂³	—	172	2	no	—
212	C₂³	—	172	2	no	—
213	C₂³	—	172	4	no	—
214	C₂³	—	172	4	no	—
215	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
216	D₄	—	86	8	no	—
217	D₄	—	172	16	no	—
218	D₄	—	172	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	344	4	no	—
223	C₂×C₄	—	344	2	no	—
224	C₂×C₄	—	344	2	no	—
225	C₂×C₄	—	172	2	no	—
226	C₂×C₄	—	172	4	no	—
227	C₂×C₄	(C₂×D₄₃)⋊C₈	1	4	yes	—
228	C₂×C₄	#13063245	1	4	yes	—
229	C₂×C₄	#13063245	1	4	yes	—
230	C₂×C₄	#13063251	1	4	yes	—
231	C₂×C₄	#13063287	1	2	yes	—
232	C₂×C₄	#13063300	1	2	yes	—
233	C₂×C₄	#13063314	1	1	yes	characteristic
234	C₂×C₄	#13063321	1	1	yes	characteristic
235	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
236	C₂×C₄	C₂×#10621872	1	1	yes	characteristic
237	C₂×C₄	#13063333	1	2	yes	—
238	C₂×C₄	C₂×#10621882	1	2	yes	—
239	C₂×C₄	C₂×#10621884	1	2	yes	—
240	C₈	—	8	2	no	—
241	C₈	—	86	8	no	—
242	C₂²	—	344	4	no	—
243	C₂²	—	344	2	no	—
244	C₂²	—	344	2	no	—
245	C₂²	—	172	8	no	—
246	C₂²	—	172	4	no	—
247	C₂²	#14582800	1	1	yes	characteristic
248	C₂²	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
249	C₂²	#14582867	1	1	yes	characteristic
250	C₂²	C₂×#13063245	1	2	yes	—
251	C₂²	C₂×#13063251	1	2	yes	—
252	C₄	—	2	4	no	—
253	C₄	—	2	2	no	—
254	C₄	—	344	2	no	—
255	C₄	#14582626	1	8	yes	complemented
256	C₄	#14582964	1	2	yes	—
257	C₄	#14582976	1	2	yes	—
258	C₄	#14583247	1	2	yes	—
259	C₄	#14583273	1	1	yes	characteristic
260	C₄	#14583275	1	1	yes	characteristic
261	C₂	—	344	4	no	—
262	C₂	C₂×#14582626	1	2	yes	—
263	C₂	#19133045	1	2	yes	—
264	C₂	#19135388	1	1	yes	characteristic
265	C₂	#19135400	1	1	yes	characteristic
266	C₂	#19135422	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1842	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂×#13063322	C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1842	D₄₃
hypocenter	C₄₃	#9638
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
2. C₄²×C₄₃⋊C₈	C₂	semidirect
3. C₂×#14582626	C₂	semidirect
4. C₄₃×#1842	C₂	semidirect
22. #13063322	C₈	semidirect
33. C₄×C₄₃⋊C₈	D₄	semidirect
121. C₄×C₄₃	D₄⋊C₈	semidirect
129. #1842	D₄₃	semidirect
150. C₄₃	#9638	semidirect
255. C₄	#14582626	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13063322	2
C₄²×C₄₃⋊C₈	2
C₂×#14582626	2
C₄₃×#1842	2
#9638	43