Subgroups — #25116367

Order 11008.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137568	C₂	1	2	yes	maximal, complemented
2	#19135462	C₂	1	2	yes	maximal
3	#19135224	C₂	1	4	yes	maximal, complemented
4	#19134759	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134605	C₂	1	2	yes	maximal, complemented
6	#107×Dic₄₃	C₂	1	2	yes	maximal, complemented
7	C₈×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063259	C₂²	1	2	yes	—
10	C₂×#13063257	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#13063246	C₂²	1	4	yes	—
13	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
14	#14582802	C₂²	1	4	yes	—
15	#14582792	—	2	4	no	—
16	#14582765	C₂²	1	2	yes	—
17	#14582762	—	2	2	no	—
18	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊Dic₄₃	—	2	2	no	—
21	#14582661	C₂²	1	4	yes	—
22	#14582653	C₂²	1	2	yes	—
23	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₄₃	C₂²	1	2	yes	—
25	C₄×D₈×C₄₃	C₄	1	8	yes	complemented
26	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#102×C₄₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
31	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
32	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
33	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
34	C₂×C₄×Dic₄₃	—	2	4	no	—
35	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
36	#13063259	—	2	4	no	—
37	#13063257	—	4	2	no	—
38	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
39	#13063246	—	4	4	no	—
40	C₈×Dic₄₃	—	2	4	no	—
41	C₄₃⋊#95	—	2	2	no	—
42	C₄₃⋊#95	—	2	2	no	—
43	C₄₃⋊#95	—	2	4	no	—
44	C₄₃⋊#95	D₄	1	1	yes	characteristic
45	C₄₃⋊#95	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₄₃	C₂³	1	2	yes	—
47	C₂³×C₄×C₄₃	—	2	2	no	—
48	C₂×D₈×C₄₃	—	2	2	no	—
49	C₂×D₈×C₄₃	—	2	2	no	—
50	C₂×D₈×C₄₃	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₄×D₄×C₄₃	—	2	4	no	—
54	C₄×D₄×C₄₃	C₂×C₄	1	8	yes	—
55	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	#107×C₄₃	C₂×C₄	1	4	yes	—
59	#102×C₄₃	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
61	C₂⁴×C₄₃	—	2	2	no	—
62	C₂×D₄×C₄₃	—	2	2	no	—
63	C₂×D₄×C₄₃	—	2	2	no	—
64	C₂×D₄×C₄₃	—	2	2	no	—
65	C₂×D₄×C₄₃	—	2	2	no	—
66	C₂×D₄×C₄₃	—	2	2	no	—
67	C₂×D₄×C₄₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₃	—	2	8	no	—
69	C₂²×C₄×C₄₃	—	2	2	no	—
70	C₂²×C₄×C₄₃	—	8	4	no	—
71	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₃	—	2	4	no	—
75	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
76	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
77	D₈×C₄₃	—	2	8	no	—
78	C₂×C₈×C₄₃	—	2	1	no	—
79	C₂×C₈×C₄₃	—	2	1	no	—
80	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₃	#41	1	2	yes	—
83	C₂×C₈×C₄₃	#41	1	2	yes	—
84	C₄⋊C₄×C₄₃	—	2	2	no	—
85	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
86	C₂²⋊C₄×C₄₃	—	2	8	no	—
87	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₃	#41	1	2	yes	—
89	C₂²⋊Dic₄₃	—	4	8	no	—
90	C₂²⋊Dic₄₃	—	8	4	no	—
91	C₄⋊Dic₄₃	—	2	4	no	—
92	C₄⋊Dic₄₃	—	2	4	no	—
93	C₄⋊Dic₄₃	—	2	4	no	—
94	C₄×Dic₄₃	—	2	2	no	—
95	C₄×Dic₄₃	—	2	2	no	—
96	C₄×Dic₄₃	—	8	4	no	—
97	C₂×C₄₃⋊C₈	—	2	2	no	—
98	C₂×C₄₃⋊C₈	—	2	2	no	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂³×C₄₃	—	2	4	no	—
101	C₂³×C₄₃	—	4	2	no	—
102	C₂³×C₄₃	—	4	2	no	—
103	C₂³×C₄₃	—	4	4	no	—
104	C₂³×C₄₃	—	4	4	no	—
105	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
106	D₄×C₄₃	—	2	4	no	—
107	D₄×C₄₃	—	2	4	no	—
108	D₄×C₄₃	—	4	2	no	—
109	D₄×C₄₃	—	4	2	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	4	no	—
113	C₂×C₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	—	8	8	no	—
115	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×Dic₄₃	—	2	2	no	—
124	C₂×Dic₄₃	—	2	2	no	—
125	C₂×Dic₄₃	—	2	2	no	—
126	C₂×Dic₄₃	—	2	2	no	—
127	C₂×Dic₄₃	—	2	2	no	—
128	C₂×Dic₄₃	—	4	4	no	—
129	C₂×Dic₄₃	—	4	4	no	—
130	C₂×Dic₄₃	—	4	4	no	—
131	C₂×Dic₄₃	#135	1	4	yes	—
132	C₈×C₄₃	—	2	2	no	—
133	C₈×C₄₃	#117	1	4	yes	—
134	C₄₃⋊C₈	—	4	2	no	—
135	#22810	—	43	1	no	maximal, Sylow
136	C₂²×C₄₃	—	4	4	no	—
137	C₂²×C₄₃	—	4	4	no	—
138	C₂²×C₄₃	—	4	4	no	—
139	C₂²×C₄₃	—	8	4	no	—
140	C₂²×C₄₃	—	8	4	no	—
141	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₄₃	#547	1	1	yes	characteristic
143	C₂²×C₄₃	#550	1	1	yes	characteristic
144	C₂²×C₄₃	C₂×#135	1	2	yes	—
145	C₂²×C₄₃	#576	1	2	yes	—
146	C₄×C₄₃	—	2	2	no	—
147	C₄×C₄₃	—	2	2	no	—
148	C₄×C₄₃	—	4	4	no	—
149	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
150	C₄×C₄₃	#520	1	1	yes	characteristic
151	C₄×C₄₃	#555	1	1	yes	characteristic
152	C₄×C₄₃	#556	1	1	yes	characteristic
153	Dic₄₃	—	2	4	no	—
154	Dic₄₃	—	4	2	no	—
155	Dic₄₃	—	8	4	no	—
156	C₂×C₄×D₈	D₄₃	1	1	yes	characteristic
157	#2357	—	43	2	no	—
158	#1970	—	43	2	no	—
159	#1926	—	43	4	no	—
160	#1846	—	43	1	no	—
161	#1804	—	43	2	no	—
162	C₄×#107	—	43	2	no	—
163	C₈×C₄⋊C₄	—	43	1	no	—
164	C₂×C₄₃	—	8	4	no	—
165	C₂×C₄₃	#2354	1	1	yes	characteristic
166	C₂×C₄₃	C₄×#135	1	1	yes	characteristic
167	C₂×C₄₃	#2948	1	1	yes	characteristic
168	C₂×C₄₃	#3339	1	2	yes	—
169	C₂×C₄₃	#3396	1	2	yes	—
170	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
171	C₂×C₄×D₄	C₂×D₄₃	1	2	yes	—
172	C₄×D₈	Dic₄₃	1	8	yes	complemented
173	C₂×#107	—	43	2	no	—
174	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
175	C₂×#106	—	43	2	no	—
176	C₂×C₄⋊C₈	—	43	1	no	—
177	C₂×#102	—	43	4	no	—
178	C₂×#102	C₂×D₄₃	1	2	yes	—
179	C₂×C₄×C₈	—	43	2	no	—
180	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
181	#390	—	43	4	no	—
182	#388	—	86	4	no	—
183	#382	—	43	2	no	—
184	#381	—	86	2	no	—
185	C₄×C₄⋊C₄	—	43	1	no	—
186	C₄×C₄⋊C₄	—	43	4	no	—
187	C₄×C₂²⋊C₄	—	86	2	no	—
188	#340	—	43	4	no	—
189	#336	—	43	2	no	—
190	C₄₃	#22810	1	1	yes	characteristic, complemented, Sylow
191	C₂²×D₄	C₂²×D₄₃	1	2	yes	—
192	C₂³×C₄	—	2	2	no	—
193	C₂×D₈	—	2	2	no	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	C₂×Dic₄₃	1	4	yes	—
196	C₂²×C₈	—	43	2	no	—
197	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
199	C₄×D₄	—	2	4	no	—
200	C₄×D₄	C₂×Dic₄₃	1	8	yes	—
201	C₂×C₄⋊C₄	—	43	2	no	—
202	C₂×C₄⋊C₄	—	43	4	no	—
203	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	86	4	no	—
206	C₂×C₂²⋊C₄	—	86	4	no	—
207	C₂×C₄²	—	43	2	no	—
208	C₂×C₄²	—	86	4	no	—
209	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
210	#107	—	86	4	no	—
211	#107	C₂×Dic₄₃	1	4	yes	—
212	#106	—	172	2	no	—
213	C₄⋊C₈	—	86	2	no	—
214	#102	—	172	4	no	—
215	#102	C₂×Dic₄₃	1	8	yes	—
216	C₄×C₈	—	86	4	no	—
217	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
218	#95	—	43	1	no	—
219	#95	—	43	1	no	—
220	#95	—	86	4	no	—
221	#95	—	86	2	no	—
222	#95	—	86	2	no	—
223	C₂⁴	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×Dic₄₃	1	4	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	8	4	no	—
233	C₂²×C₄	—	43	2	no	—
234	C₂²×C₄	—	43	2	no	—
235	C₂²×C₄	—	86	4	no	—
236	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
238	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
239	D₈	—	2	8	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	86	2	no	—
243	C₂×C₈	—	86	2	no	—
244	C₂×C₈	—	86	2	no	—
245	C₂×C₈	#10621898	1	2	yes	—
246	C₂×C₈	#10621898	1	2	yes	—
247	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
248	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	86	4	no	—
251	C₄⋊C₄	—	86	4	no	—
252	C₄⋊C₄	—	86	4	no	—
253	C₄⋊C₄	C₂²×Dic₄₃	1	4	yes	—
254	C₂²⋊C₄	—	2	8	no	—
255	C₂²⋊C₄	—	344	4	no	—
256	C₂²⋊C₄	—	172	8	no	—
257	C₄²	—	86	2	no	—
258	C₄²	—	86	2	no	—
259	C₄²	—	344	4	no	—
260	C₄²	#10621900	1	2	yes	—
261	C₄²	C₂²×Dic₄₃	1	2	yes	—
262	C₂³	—	2	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	4	4	no	—
267	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
268	D₄	—	2	4	no	—
269	D₄	—	2	4	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	8	no	—
277	C₂×C₄	—	43	4	no	—
278	C₂×C₄	—	86	2	no	—
279	C₂×C₄	—	86	2	no	—
280	C₂×C₄	—	86	2	no	—
281	C₂×C₄	—	86	2	no	—
282	C₂×C₄	—	86	2	no	—
283	C₂×C₄	—	172	4	no	—
284	C₂×C₄	—	172	4	no	—
285	C₂×C₄	—	172	4	no	—
286	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
287	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
288	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
289	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
290	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
291	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
292	C₂×C₄	#13063371	1	1	yes	characteristic
293	C₂×C₄	#13063372	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	172	2	no	—
296	C₈	#13063284	1	4	yes	—
297	C₂²	—	4	4	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
303	C₂²	#14583554	1	1	yes	characteristic
304	C₂²	#14583570	1	1	yes	characteristic
305	C₂²	#135×D₄₃	1	2	yes	—
306	C₂²	#14583700	1	2	yes	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	4	no	—
310	C₄	—	86	4	no	—
311	C₄	—	344	4	no	—
312	C₄	—	172	2	no	—
313	C₄	C₂×#13063284	1	1	yes	characteristic
314	C₄	#14583382	1	1	yes	characteristic
315	C₄	#14583614	1	1	yes	characteristic
316	C₄	#14583619	1	1	yes	characteristic
317	C₂	—	8	4	no	—
318	C₂	#19137544	1	1	yes	characteristic
319	C₂	#135×Dic₄₃	1	1	yes	characteristic
320	C₂	#19143447	1	1	yes	characteristic
321	C₂	#19146999	1	2	yes	—
322	C₂	#19147607	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄×D₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₈×C₄₃	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₃
hypocenter	C₄₃	#22810
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #19137568	C₂	semidirect
3. #19135224	C₂	semidirect
4. #19134759	C₂	semidirect
5. #19134605	C₂	semidirect
6. #107×Dic₄₃	C₂	semidirect
7. C₈×C₄⋊Dic₄₃	C₂	semidirect
25. C₄×D₈×C₄₃	C₄	semidirect
172. C₄×D₈	Dic₄₃	semidirect
190. C₄₃	#22810	semidirect

Maximal subgroups

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

subgroup	index
#19137568	2
#19135462	2
#19135224	2
#19134759	2
#19134605	2
#107×Dic₄₃	2
C₈×C₄⋊Dic₄₃	2
C₂×C₄×D₈×C₄₃	2
#22810	43