Subgroups — #25116397

Order 11008.

922 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137544	C₂	1	2	yes	maximal, complemented
2	#19135472	C₂	1	2	yes	maximal
3	#19135224	C₂	1	4	yes	maximal, complemented
4	#19134752	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134595	C₂	1	2	yes	maximal, complemented
6	#19134494	C₂	1	2	yes	maximal, complemented
7	#19134462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063284	—	2	2	no	—
10	C₂×#13063257	C₂²	1	2	yes	—
11	C₂×#13063257	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#13063246	C₂²	1	4	yes	—
14	C₂×#13063235	C₂²	1	2	yes	—
15	#14582802	C₂²	1	4	yes	—
16	#14582792	—	2	4	no	—
17	#14582769	C₂²	1	2	yes	—
18	#14582759	—	2	2	no	—
19	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
20	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
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	#13063284	—	4	8	no	—
31	C₂×C₄⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄⋊Dic₄₃	—	2	2	no	—
33	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
34	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
36	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
37	C₂×C₄×Dic₄₃	—	2	4	no	—
38	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
39	#13063257	—	2	4	no	—
40	#13063257	—	4	2	no	—
41	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
42	#13063246	—	4	4	no	—
43	#13063235	—	2	4	no	—
44	C₄₃⋊#95	—	2	4	no	—
45	C₄₃⋊#95	D₄	1	2	yes	—
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	—	4	4	no	—
71	C₂²×C₄×C₄₃	—	4	4	no	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₄₃	—	2	4	no	—
76	C₂²×Dic₄₃	C₂×D₄	1	4	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	—	4	4	no	—
95	C₄⋊Dic₄₃	—	8	2	no	—
96	C₄×Dic₄₃	—	2	2	no	—
97	C₄×Dic₄₃	—	2	2	no	—
98	C₄×Dic₄₃	—	4	8	no	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	C₂×C₄₃⋊C₈	—	2	2	no	—
102	C₂³×C₄₃	—	2	4	no	—
103	C₂³×C₄₃	—	4	2	no	—
104	C₂³×C₄₃	—	4	2	no	—
105	C₂³×C₄₃	—	4	2	no	—
106	C₂³×C₄₃	—	4	2	no	—
107	C₂³×C₄₃	—	4	2	no	—
108	C₂³×C₄₃	—	4	2	no	—
109	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
110	D₄×C₄₃	—	2	4	no	—
111	D₄×C₄₃	—	2	4	no	—
112	D₄×C₄₃	—	4	2	no	—
113	D₄×C₄₃	—	4	2	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	4	no	—
116	C₂×C₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	4	16	no	—
119	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×Dic₄₃	—	2	4	no	—
129	C₂×Dic₄₃	—	2	4	no	—
130	C₂×Dic₄₃	—	2	4	no	—
131	C₂×Dic₄₃	—	4	4	no	—
132	C₂×Dic₄₃	—	4	4	no	—
133	C₂×Dic₄₃	—	4	4	no	—
134	C₈×C₄₃	—	2	2	no	—
135	C₈×C₄₃	#117	1	4	yes	complemented
136	C₄₃⋊C₈	—	4	2	no	—
137	#22813	—	43	1	no	maximal, Sylow
138	C₂²×C₄₃	—	4	4	no	—
139	C₂²×C₄₃	—	4	4	no	—
140	C₂²×C₄₃	—	4	4	no	—
141	C₂²×C₄₃	—	8	2	no	—
142	C₂²×C₄₃	—	8	2	no	—
143	C₂²×C₄₃	—	8	2	no	—
144	C₂²×C₄₃	—	8	2	no	—
145	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂²×C₄₃	#547	1	1	yes	characteristic
147	C₂²×C₄₃	#550	1	1	yes	characteristic
148	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
149	C₂²×C₄₃	#577	1	2	yes	—
150	C₄×C₄₃	—	2	2	no	—
151	C₄×C₄₃	—	4	4	no	—
152	C₄×C₄₃	#442	1	4	yes	—
153	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
154	C₄×C₄₃	#520	1	1	yes	characteristic
155	C₄×C₄₃	#555	1	1	yes	characteristic
156	C₄×C₄₃	#556	1	1	yes	characteristic
157	Dic₄₃	—	4	4	no	—
158	Dic₄₃	—	8	4	no	—
159	C₂×C₄×D₈	D₄₃	1	1	yes	characteristic
160	#2354	—	43	2	no	—
161	#1972	—	43	2	no	—
162	#1926	—	43	4	no	—
163	#1844	—	43	1	no	—
164	#1802	—	43	2	no	—
165	#1776	—	43	2	no	—
166	#1770	—	43	1	no	—
167	C₂×C₄₃	—	8	4	no	—
168	C₂×C₄₃	#2354	1	1	yes	characteristic
169	C₂×C₄₃	C₂×#442	1	1	yes	characteristic
170	C₂×C₄₃	#2952	1	1	yes	characteristic
171	C₂×C₄₃	#3354	1	2	yes	—
172	C₂×C₄₃	#3397	1	2	yes	—
173	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
174	C₂×C₄×D₄	C₂×D₄₃	1	2	yes	—
175	C₂×#117	—	86	2	no	—
176	C₄×D₈	Dic₄₃	1	8	yes	complemented
177	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
178	C₂×#106	—	43	2	no	—
179	C₂×#106	—	43	2	no	—
180	C₂×C₄⋊C₈	—	43	1	no	—
181	C₂×#102	—	43	4	no	—
182	C₂×#102	C₂×D₄₃	1	2	yes	—
183	C₂×#97	—	43	2	no	—
184	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
185	#390	—	43	4	no	—
186	#388	—	86	4	no	—
187	#384	—	43	2	no	—
188	#380	—	86	2	no	—
189	C₄×C₄⋊C₄	—	43	1	no	—
190	C₄×C₄⋊C₄	—	43	4	no	—
191	#340	—	43	4	no	—
192	#336	—	43	2	no	—
193	C₄₃	#22813	1	1	yes	characteristic, complemented, Sylow
194	C₂²×D₄	C₂²×D₄₃	1	2	yes	—
195	C₂³×C₄	—	2	2	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	—	2	2	no	—
198	C₂×D₈	C₂×Dic₄₃	1	4	yes	—
199	C₂²×C₈	—	43	2	no	—
200	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
202	C₄×D₄	—	2	4	no	—
203	C₄×D₄	C₂×Dic₄₃	1	8	yes	—
204	#117	—	172	8	no	—
205	C₂×C₄⋊C₄	—	43	2	no	—
206	C₂×C₄⋊C₄	—	43	4	no	—
207	C₂×C₄⋊C₄	—	86	2	no	—
208	C₂×C₄⋊C₄	—	86	2	no	—
209	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	—	86	4	no	—
212	C₂×C₂²⋊C₄	—	86	4	no	—
213	C₂×C₄²	—	43	2	no	—
214	C₂×C₄²	—	86	4	no	—
215	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
216	#107	C₂×Dic₄₃	1	4	yes	—
217	#106	—	86	4	no	—
218	#106	—	172	2	no	—
219	C₄⋊C₈	—	86	2	no	—
220	#102	—	172	4	no	—
221	#102	C₂×Dic₄₃	1	8	yes	—
222	#97	—	86	4	no	—
223	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
224	#95	—	43	2	no	—
225	#95	—	86	4	no	—
226	C₂⁴	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	C₂²×Dic₄₃	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	8	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	43	4	no	—
238	C₂²×C₄	—	86	4	no	—
239	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
240	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
241	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
242	D₈	—	2	8	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	86	2	no	—
246	C₂×C₈	—	86	2	no	—
247	C₂×C₈	—	86	2	no	—
248	C₂×C₈	#10621898	1	2	yes	—
249	C₂×C₈	#10621898	1	2	yes	—
250	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
251	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	86	4	no	—
254	C₄⋊C₄	—	86	4	no	—
255	C₄⋊C₄	—	86	4	no	—
256	C₄⋊C₄	—	344	2	no	—
257	C₄⋊C₄	—	172	4	no	—
258	C₄⋊C₄	C₂²×Dic₄₃	1	4	yes	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₂²⋊C₄	—	344	4	no	—
261	C₂²⋊C₄	—	172	8	no	—
262	C₄²	—	86	2	no	—
263	C₄²	—	86	2	no	—
264	C₄²	—	172	8	no	—
265	C₄²	#10621900	1	2	yes	—
266	C₄²	C₂²×Dic₄₃	1	2	yes	—
267	C₂³	—	2	4	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
275	D₄	—	2	4	no	—
276	D₄	—	2	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	16	no	—
284	C₂×C₄	—	86	4	no	—
285	C₂×C₄	—	86	4	no	—
286	C₂×C₄	—	86	4	no	—
287	C₂×C₄	—	172	4	no	—
288	C₂×C₄	—	172	4	no	—
289	C₂×C₄	—	172	4	no	—
290	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
291	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
292	C₂×C₄	#13063349	1	4	yes	—
293	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
294	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
295	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
296	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
297	C₂×C₄	#13063371	1	1	yes	characteristic
298	C₂×C₄	#13063372	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	172	2	no	—
301	C₈	#13063284	1	4	yes	complemented
302	C₂²	—	4	4	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
310	C₂²	#14583554	1	1	yes	characteristic
311	C₂²	#14583570	1	1	yes	characteristic
312	C₂²	C₂×#13063349	1	2	yes	—
313	C₂²	#14583706	1	2	yes	—
314	C₄	—	2	2	no	—
315	C₄	—	4	4	no	—
316	C₄	—	344	4	no	—
317	C₄	—	172	4	no	—
318	C₄	#14583011	1	4	yes	—
319	C₄	C₂×#13063284	1	1	yes	characteristic
320	C₄	#14583382	1	1	yes	characteristic
321	C₄	#14583614	1	1	yes	characteristic
322	C₄	#14583619	1	1	yes	characteristic
323	C₂	—	8	4	no	—
324	C₂	#19137544	1	1	yes	characteristic
325	C₂	C₂×#14583011	1	1	yes	characteristic
326	C₂	#19143479	1	1	yes	characteristic
327	C₂	#19147192	1	2	yes	—
328	C₂	#19147613	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	#22813
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. #19137544	C₂	semidirect
3. #19135224	C₂	semidirect
4. #19134752	C₂	semidirect
5. #19134595	C₂	semidirect
6. #19134494	C₂	semidirect
7. #19134462	C₂	semidirect
25. C₄×D₈×C₄₃	C₄	semidirect
135. C₈×C₄₃	#117	semidirect
176. C₄×D₈	Dic₄₃	semidirect
193. C₄₃	#22813	semidirect
301. C₈	#13063284	semidirect

Maximal subgroups

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

subgroup	index
#19137544	2
#19135472	2
#19135224	2
#19134752	2
#19134595	2
#19134494	2
#19134462	2
C₂×C₄×D₈×C₄₃	2
#22813	43