Subgroups — #24987248

Order 11008.

530 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	#19137069	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#13063235	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582631	C₂	1	1	yes	maximal, characteristic
5	C₂×#14582629	C₂	1	1	yes	maximal, characteristic
6	#19132436	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#13063235	—	2	2	no	—
11	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₄₃	—	2	1	no	—
13	#14582804	C₄	1	2	yes	—
14	#14582795	C₄	1	2	yes	—
15	#14582781	—	2	2	no	—
16	C₄×Dic₄₃⋊C₄	—	2	1	no	—
17	#14582745	—	2	1	no	—
18	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
19	#14582653	—	2	1	no	—
20	#14582631	C₄	1	4	yes	complemented
21	#14582629	C₄	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄₃⋊C₈	—	2	1	no	—
27	C₂²×C₄₃⋊C₈	—	2	1	no	—
28	C₄×Dic₈₆	—	2	1	no	—
29	C₄×Dic₈₆	—	2	1	no	—
30	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
31	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₄₃⋊C₄	—	2	1	no	—
34	C₂×Dic₄₃⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₄₃	—	2	1	no	—
36	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
39	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
40	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
41	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
42	#13063235	—	2	8	no	—
43	C₄×C₄₃⋊C₈	—	2	1	no	—
44	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
45	C₈×Dic₄₃	—	4	2	no	—
46	C₄₃⋊#95	—	2	2	no	—
47	C₄₃⋊#95	—	2	2	no	—
48	C₄₃⋊#95	—	2	2	no	—
49	C₂²×C₈×C₄₃	—	2	1	no	—
50	C₂²×C₈×C₄₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₃	—	2	1	no	—
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₃	—	2	1	no	—
55	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₄₃	—	2	1	no	—
57	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
58	#95×C₄₃	—	2	1	no	—
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₄₃	—	2	1	no	—
64	C₂²×Dic₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
67	C₂×Dic₈₆	—	2	2	no	—
68	C₂×Dic₈₆	—	2	2	no	—
69	C₂×Dic₈₆	—	2	1	no	—
70	C₂×Dic₈₆	C₄²	1	2	yes	—
71	C₂×C₈×C₄₃	—	2	4	no	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	—	2	2	no	—
74	C₂×C₈×C₄₃	—	4	1	no	—
75	C₂×C₈×C₄₃	—	4	1	no	—
76	C₄⋊C₄×C₄₃	—	2	2	no	—
77	C₄⋊C₄×C₄₃	—	2	2	no	—
78	C₄²×C₄₃	—	4	1	no	—
79	C₄²×C₄₃	—	4	1	no	—
80	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₃	#41	1	1	yes	characteristic
83	C₄²×C₄₃	#41	1	1	yes	characteristic
84	C₄⋊Dic₄₃	—	2	1	no	—
85	C₄⋊Dic₄₃	C₄²	1	2	yes	—
86	Dic₄₃⋊C₄	—	2	2	no	—
87	Dic₄₃⋊C₄	—	2	2	no	—
88	Dic₄₃⋊C₄	—	4	4	no	—
89	C₄×Dic₄₃	—	2	2	no	—
90	C₄×Dic₄₃	—	2	2	no	—
91	C₄×Dic₄₃	—	2	2	no	—
92	C₄×Dic₄₃	—	2	2	no	—
93	C₄×Dic₄₃	—	2	1	no	—
94	C₄×Dic₄₃	—	2	1	no	—
95	C₄×Dic₄₃	—	2	1	no	—
96	C₄×Dic₄₃	—	2	1	no	—
97	C₄×Dic₄₃	—	2	4	no	—
98	C₂×C₄₃⋊C₈	—	2	4	no	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	C₂×C₄₃⋊C₈	—	4	1	no	—
102	C₂×C₄₃⋊C₈	—	4	1	no	—
103	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	4	2	no	—
106	C₂×C₄×C₄₃	—	4	2	no	—
107	C₂×C₄×C₄₃	—	4	2	no	—
108	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#131	1	2	yes	—
115	C₂×C₄×C₄₃	#131	1	2	yes	—
116	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
119	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
120	C₂×Dic₄₃	—	2	2	no	—
121	C₂×Dic₄₃	—	2	2	no	—
122	C₂×Dic₄₃	—	2	2	no	—
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	—	4	2	no	—
128	C₂×Dic₄₃	—	4	2	no	—
129	C₂×Dic₄₃	#104	1	4	yes	—
130	Dic₈₆	—	2	2	no	—
131	Dic₈₆	—	2	2	no	—
132	Dic₈₆	—	4	2	no	—
133	C₈×C₄₃	—	2	4	no	—
134	C₈×C₄₃	—	4	2	no	—
135	C₄₃⋊C₈	—	2	4	no	—
136	C₄₃⋊C₈	—	4	2	no	—
137	#8180	—	43	1	no	maximal, Sylow
138	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₃	#408	1	2	yes	—
140	C₂²×C₄₃	#417	1	1	yes	characteristic
141	C₂²×C₄₃	#419	1	1	yes	characteristic
142	C₂²×C₄₃	C₂×#104	1	2	yes	—
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	2	1	no	—
145	C₄×C₄₃	—	8	2	no	—
146	C₄×C₄₃	#405	1	1	yes	characteristic
147	C₄×C₄₃	#405	1	1	yes	characteristic
148	C₄×C₄₃	#433	1	2	yes	—
149	C₄×C₄₃	#440	1	1	yes	characteristic
150	C₄×C₄₃	#440	1	1	yes	characteristic
151	C₄×C₄₃	#441	1	1	yes	characteristic
152	C₄×C₄₃	#441	1	1	yes	characteristic
153	Dic₄₃	—	2	4	no	—
154	Dic₄₃	—	4	2	no	—
155	Dic₄₃	—	4	2	no	—
156	Dic₄₃	—	4	2	no	—
157	#2295	—	43	1	no	—
158	C₄×C₄⋊C₈	—	43	1	no	—
159	C₄×#97	—	43	1	no	—
160	C₂×Q₈⋊C₈	—	43	1	no	—
161	C₂×Q₈⋊C₈	—	43	1	no	—
162	#1279	—	43	1	no	—
163	#1279	D₄₃	1	1	yes	characteristic
164	C₂×C₄₃	#1483	1	2	yes	—
165	C₂×C₄₃	#1485	1	2	yes	—
166	C₂×C₄₃	#1750	1	1	yes	characteristic
167	C₂×C₄₃	C₄×#104	1	1	yes	characteristic
168	C₂×C₄₃	#1763	1	1	yes	characteristic
169	C₂×C₄×Q₈	—	43	1	no	—
170	C₂×C₄⋊C₈	—	43	1	no	—
171	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
172	C₂×#97	—	86	2	no	—
173	C₂×C₄×C₈	—	43	1	no	—
174	C₂×C₄×C₈	—	86	1	no	—
175	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
176	#391	—	43	2	no	—
177	#389	—	43	2	no	—
178	#387	—	86	2	no	—
179	C₄×C₄⋊C₄	—	86	1	no	—
180	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
181	#376	—	86	1	no	—
182	C₄³	—	43	1	no	—
183	#336	—	86	1	no	—
184	Q₈⋊C₈	—	43	4	no	—
185	Q₈⋊C₈	—	43	4	no	—
186	C₄₃	#8180	1	1	yes	characteristic, complemented, Sylow
187	C₂²×Q₈	—	43	1	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	86	1	no	—
191	C₂²×C₈	—	86	1	no	—
192	C₄×Q₈	—	43	2	no	—
193	C₄×Q₈	—	43	2	no	—
194	C₄×Q₈	—	86	1	no	—
195	C₄×Q₈	—	86	1	no	—
196	C₂×C₄⋊C₄	—	43	1	no	—
197	C₂×C₄⋊C₄	—	86	1	no	—
198	C₂×C₄⋊C₄	—	86	2	no	—
199	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	43	2	no	—
202	C₂×C₄²	—	43	2	no	—
203	C₂×C₄²	—	43	1	no	—
204	C₂×C₄²	—	43	1	no	—
205	C₂×C₄²	—	86	1	no	—
206	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	43	2	no	—
209	C₄⋊C₈	—	86	1	no	—
210	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
211	#97	—	86	8	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	43	2	no	—
214	C₄×C₈	—	86	1	no	—
215	C₄×C₈	—	172	2	no	—
216	C₄×C₈	C₄×D₄₃	1	2	yes	—
217	#95	—	2	1	no	—
218	#95	—	86	2	no	—
219	#95	—	86	2	no	—
220	#95	—	86	2	no	—
221	C₂×Q₈	—	43	2	no	—
222	C₂×Q₈	—	86	1	no	—
223	C₂×Q₈	—	86	2	no	—
224	C₂×Q₈	—	86	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	43	2	no	—
227	C₂²×C₄	—	43	2	no	—
228	C₂²×C₄	—	86	1	no	—
229	C₂²×C₄	—	86	1	no	—
230	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	86	4	no	—
239	C₂×C₈	—	86	2	no	—
240	C₂×C₈	—	86	2	no	—
241	C₂×C₈	—	172	1	no	—
242	C₂×C₈	—	172	1	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	43	2	no	—
246	C₄⋊C₄	—	86	1	no	—
247	C₄⋊C₄	—	86	2	no	—
248	C₄⋊C₄	—	86	2	no	—
249	C₄⋊C₄	—	172	4	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	86	4	no	—
253	C₄²	—	86	2	no	—
254	C₄²	—	86	2	no	—
255	C₄²	—	86	2	no	—
256	C₄²	—	86	2	no	—
257	C₄²	—	86	1	no	—
258	C₄²	—	86	1	no	—
259	C₄²	—	86	1	no	—
260	C₄²	—	86	1	no	—
261	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
262	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
263	C₄²	#10621900	1	1	yes	characteristic
264	C₄²	#10621902	1	1	yes	characteristic
265	C₂³	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
266	Q₈	—	86	2	no	—
267	Q₈	—	86	2	no	—
268	Q₈	—	172	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	43	4	no	—
274	C₂×C₄	—	86	2	no	—
275	C₂×C₄	—	86	2	no	—
276	C₂×C₄	—	86	2	no	—
277	C₂×C₄	—	86	2	no	—
278	C₂×C₄	—	86	2	no	—
279	C₂×C₄	—	86	2	no	—
280	C₂×C₄	—	86	2	no	—
281	C₂×C₄	—	172	2	no	—
282	C₂×C₄	—	172	2	no	—
283	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
284	C₂×C₄	#13063283	1	1	yes	characteristic
285	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
287	C₂×C₄	#13063287	1	1	yes	characteristic
288	C₂×C₄	#13063287	1	1	yes	characteristic
289	C₂×C₄	#13063331	1	2	yes	—
290	C₂×C₄	#13063333	1	2	yes	—
291	C₂×C₄	#13063349	1	1	yes	characteristic
292	C₂×C₄	#13063354	1	1	yes	characteristic
293	C₂×C₄	#13063354	1	1	yes	characteristic
294	C₂×C₄	#13063357	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	4	2	no	—
297	C₈	—	86	4	no	—
298	C₈	—	172	2	no	—
299	C₂²	#14582746	1	1	yes	characteristic
300	C₂²	#14582864	1	2	yes	—
301	C₂²	#14582897	1	1	yes	characteristic
302	C₂²	#14582906	1	1	yes	characteristic
303	C₂²	#104×D₄₃	1	2	yes	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	8	2	no	—
307	C₄	—	86	4	no	—
308	C₄	—	172	2	no	—
309	C₄	—	172	2	no	—
310	C₄	—	172	2	no	—
311	C₄	#14582853	1	1	yes	characteristic
312	C₄	#14582856	1	1	yes	characteristic
313	C₄	#14582959	1	2	yes	—
314	C₄	#14582996	1	1	yes	characteristic
315	C₄	#14582998	1	1	yes	characteristic
316	C₄	#14583003	1	1	yes	characteristic
317	C₄	#14583005	1	1	yes	characteristic
318	C₂	#19133005	1	2	yes	—
319	C₂	#19133014	1	2	yes	—
320	C₂	#19134360	1	1	yes	characteristic
321	C₂	#19134399	1	1	yes	characteristic
322	C₂	#19134424	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1279	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1279	D₄₃
hypocenter	C₄₃	#8180
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
20. #14582631	C₄	semidirect
21. #14582629	C₄	semidirect
186. C₄₃	#8180	semidirect

Maximal subgroups

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

subgroup	index
#19137069	2
C₄⋊C₈×Dic₄₃	2
C₄×#13063235	2
C₂×#14582631	2
C₂×#14582629	2
#19132436	2
C₄₃×#1279	2
#8180	43