Subgroups — #24987241

Order 11008.

486 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137073	C₂	1	1	yes	maximal, characteristic
2	#19134450	C₂	1	1	yes	maximal, characteristic
3	#19134340	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#19132435	C₂	1	1	yes	maximal, characteristic
6	#19132434	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×#13063235	—	2	1	no	—
10	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
11	#14582806	C₄	1	2	yes	—
12	#14582798	C₄	1	2	yes	—
13	#14582785	—	2	2	no	—
14	C₄×C₄⋊Dic₄₃	—	2	1	no	—
15	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
16	#14582745	—	2	1	no	—
17	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
18	#14582653	—	2	2	no	—
19	#14582653	—	2	1	no	—
20	Q₈⋊(C₄₃⋊C₈)	C₄	1	4	yes	complemented
21	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×Q₈⋊C₈	C₄	1	4	yes	complemented
25	C₂²×C₄₃⋊C₈	—	2	1	no	—
26	C₂²×C₄₃⋊C₈	—	2	1	no	—
27	C₂×C₄⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₄₃	—	2	1	no	—
30	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
32	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
34	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
35	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
36	#13063235	—	4	2	no	—
37	C₄×C₄₃⋊C₈	—	2	1	no	—
38	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₄₃⋊#95	—	2	1	no	—
40	C₄₃⋊#95	—	2	2	no	—
41	C₄₃⋊#95	—	2	2	no	—
42	C₄₃⋊#95	—	2	2	no	—
43	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₃	—	2	1	no	—
45	C₂²×C₈×C₄₃	—	2	1	no	—
46	C₄×Q₈×C₄₃	—	2	1	no	—
47	C₄×Q₈×C₄₃	—	2	1	no	—
48	C₄×Q₈×C₄₃	C₂×C₄	1	2	yes	—
49	C₄×Q₈×C₄₃	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
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	C₂×Q₈×C₄₃	—	2	1	no	—
59	C₂×Q₈×C₄₃	—	2	2	no	—
60	C₂×Q₈×C₄₃	—	2	2	no	—
61	C₂×Q₈×C₄₃	C₄²	1	2	yes	—
62	C₂²×C₄×C₄₃	—	2	1	no	—
63	C₂²×C₄×C₄₃	—	2	1	no	—
64	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₄₃	—	2	2	no	—
68	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₄₃	—	2	2	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	4	2	no	—
73	C₂×C₈×C₄₃	—	4	1	no	—
74	C₂×C₈×C₄₃	—	4	1	no	—
75	C₄⋊C₄×C₄₃	—	2	1	no	—
76	C₄⋊C₄×C₄₃	—	2	2	no	—
77	C₄⋊C₄×C₄₃	—	2	2	no	—
78	C₄⋊C₄×C₄₃	C₄²	1	2	yes	—
79	C₄²×C₄₃	—	2	2	no	—
80	C₄²×C₄₃	—	2	2	no	—
81	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₃	#41	1	1	yes	characteristic
84	C₄²×C₄₃	#41	1	1	yes	characteristic
85	C₄⋊Dic₄₃	—	2	2	no	—
86	C₄⋊Dic₄₃	—	2	2	no	—
87	C₄⋊Dic₄₃	—	4	4	no	—
88	C₄×Dic₄₃	—	2	2	no	—
89	C₄×Dic₄₃	—	2	2	no	—
90	C₄×Dic₄₃	—	2	1	no	—
91	C₄×Dic₄₃	—	2	1	no	—
92	C₄×Dic₄₃	—	2	1	no	—
93	C₄×Dic₄₃	—	2	1	no	—
94	C₄×Dic₄₃	—	4	2	no	—
95	C₄×Dic₄₃	—	4	1	no	—
96	C₄×Dic₄₃	—	4	1	no	—
97	C₂×C₄₃⋊C₈	—	2	2	no	—
98	C₂×C₄₃⋊C₈	—	2	2	no	—
99	C₂×C₄₃⋊C₈	—	4	2	no	—
100	C₂×C₄₃⋊C₈	—	4	1	no	—
101	C₂×C₄₃⋊C₈	—	4	1	no	—
102	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	Q₈×C₄₃	—	2	2	no	—
104	Q₈×C₄₃	—	2	2	no	—
105	Q₈×C₄₃	—	4	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	4	2	no	—
110	C₂×C₄×C₄₃	—	4	2	no	—
111	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#131	1	2	yes	—
118	C₂×C₄×C₄₃	#131	1	2	yes	—
119	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#137	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	—	2	2	no	—
129	C₂×Dic₄₃	—	4	2	no	—
130	C₂×Dic₄₃	—	4	2	no	—
131	C₂×Dic₄₃	—	4	2	no	—
132	C₈×C₄₃	—	4	2	no	—
133	C₈×C₄₃	—	4	2	no	—
134	C₄₃⋊C₈	—	4	2	no	—
135	C₄₃⋊C₈	—	4	2	no	—
136	#8178	—	43	1	no	maximal, Sylow
137	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₄₃	#408	1	2	yes	—
139	C₂²×C₄₃	#417	1	1	yes	characteristic
140	C₂²×C₄₃	#419	1	1	yes	characteristic
141	C₂²×C₄₃	#421	1	2	yes	—
142	C₄×C₄₃	—	2	1	no	—
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	4	2	no	—
145	C₄×C₄₃	—	4	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	—	4	2	no	—
154	Dic₄₃	—	4	2	no	—
155	Dic₄₃	—	8	2	no	—
156	#2295	—	43	1	no	—
157	#1767	—	43	1	no	—
158	#1744	—	43	1	no	—
159	C₂×Q₈⋊C₈	—	43	1	no	—
160	C₂×Q₈⋊C₈	D₄₃	1	1	yes	characteristic
161	#1279	—	43	1	no	—
162	#1279	—	43	1	no	—
163	C₂×C₄₃	#1495	1	2	yes	—
164	C₂×C₄₃	#1497	1	2	yes	—
165	C₂×C₄₃	#1750	1	1	yes	characteristic
166	C₂×C₄₃	#1759	1	1	yes	characteristic
167	C₂×C₄₃	#1763	1	1	yes	characteristic
168	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
169	C₂×C₄⋊C₈	—	43	1	no	—
170	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
171	C₂×#97	—	86	1	no	—
172	C₂×C₄×C₈	—	43	1	no	—
173	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
174	#391	—	43	2	no	—
175	#389	—	43	2	no	—
176	#387	—	86	2	no	—
177	C₄×C₄⋊C₄	—	43	1	no	—
178	C₄×C₄⋊C₄	—	86	1	no	—
179	#376	—	86	1	no	—
180	C₄³	—	43	1	no	—
181	#336	—	86	2	no	—
182	#336	—	86	1	no	—
183	Q₈⋊C₈	—	43	4	no	—
184	Q₈⋊C₈	Dic₄₃	1	4	yes	complemented
185	C₄₃	#8178	1	1	yes	characteristic, complemented, Sylow
186	C₂²×Q₈	C₄×D₄₃	1	1	yes	characteristic
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	86	1	no	—
190	C₂²×C₈	—	86	1	no	—
191	C₄×Q₈	—	2	1	no	—
192	C₄×Q₈	—	2	1	no	—
193	C₄×Q₈	C₄×D₄₃	1	2	yes	—
194	C₄×Q₈	C₂×Dic₄₃	1	2	yes	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	43	2	no	—
197	C₂×C₄⋊C₄	—	86	2	no	—
198	C₂×C₄⋊C₄	C₄×D₄₃	1	1	yes	characteristic
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	43	2	no	—
201	C₂×C₄²	—	43	1	no	—
202	C₂×C₄²	—	43	1	no	—
203	C₂×C₄²	—	43	2	no	—
204	C₂×C₄²	—	86	1	no	—
205	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	43	2	no	—
208	C₄⋊C₈	—	86	1	no	—
209	C₄⋊C₈	C₂×Dic₄₃	1	2	yes	—
210	#97	—	172	2	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	43	2	no	—
213	C₄×C₈	—	86	1	no	—
214	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
215	#95	—	86	2	no	—
216	#95	—	86	2	no	—
217	#95	—	86	2	no	—
218	#95	—	86	1	no	—
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	C₄×Dic₄₃	1	2	yes	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	43	2	no	—
226	C₂²×C₄	—	43	2	no	—
227	C₂²×C₄	—	86	2	no	—
228	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
229	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
230	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	86	2	no	—
237	C₂×C₈	—	86	2	no	—
238	C₂×C₈	—	172	2	no	—
239	C₂×C₈	—	172	1	no	—
240	C₂×C₈	—	172	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	86	2	no	—
245	C₄⋊C₄	—	86	2	no	—
246	C₄⋊C₄	—	172	4	no	—
247	C₄⋊C₄	C₄×Dic₄₃	1	2	yes	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	86	2	no	—
251	C₄²	—	86	2	no	—
252	C₄²	—	86	1	no	—
253	C₄²	—	86	1	no	—
254	C₄²	—	86	1	no	—
255	C₄²	—	86	1	no	—
256	C₄²	—	172	2	no	—
257	C₄²	—	172	1	no	—
258	C₄²	—	172	1	no	—
259	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
260	C₄²	#10621900	1	1	yes	characteristic
261	C₄²	#10621900	1	1	yes	characteristic
262	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
263	C₂³	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
264	Q₈	—	2	2	no	—
265	Q₈	—	2	2	no	—
266	Q₈	—	4	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	86	2	no	—
273	C₂×C₄	—	86	2	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	—	172	2	no	—
279	C₂×C₄	—	172	2	no	—
280	C₂×C₄	—	172	2	no	—
281	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
282	C₂×C₄	#13063281	1	1	yes	characteristic
283	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
285	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
286	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
287	C₂×C₄	#13063331	1	2	yes	—
288	C₂×C₄	#13063333	1	2	yes	—
289	C₂×C₄	#13063352	1	1	yes	characteristic
290	C₂×C₄	#13063354	1	1	yes	characteristic
291	C₂×C₄	#13063357	1	1	yes	characteristic
292	C₂×C₄	#13063357	1	1	yes	characteristic
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₈	—	172	2	no	—
296	C₈	—	172	2	no	—
297	C₂²	C₂²⋊C₄×Dic₄₃	1	1	yes	characteristic
298	C₂²	#14582864	1	2	yes	—
299	C₂²	#14582899	1	1	yes	characteristic
300	C₂²	#14582906	1	1	yes	characteristic
301	C₂²	#14582914	1	2	yes	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	344	2	no	—
307	C₄	—	172	2	no	—
308	C₄	—	172	2	no	—
309	C₄	#14582854	1	1	yes	characteristic
310	C₄	#14582857	1	1	yes	characteristic
311	C₄	#14582957	1	2	yes	—
312	C₄	#14583000	1	1	yes	characteristic
313	C₄	#14583002	1	1	yes	characteristic
314	C₄	#14583003	1	1	yes	characteristic
315	C₄	#14583006	1	1	yes	characteristic
316	C₂	#19133059	1	2	yes	—
317	C₂	#19133066	1	2	yes	—
318	C₂	#19134363	1	1	yes	characteristic
319	C₂	#19134407	1	1	yes	characteristic
320	C₂	#19134427	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×Q₈⋊C₈	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×Q₈⋊C₈	D₄₃
hypocenter	C₄₃	#8178
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
20. Q₈⋊(C₄₃⋊C₈)	C₄	semidirect
24. C₄₃×Q₈⋊C₈	C₄	semidirect
184. Q₈⋊C₈	Dic₄₃	semidirect
185. C₄₃	#8178	semidirect

Maximal subgroups

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

subgroup	index
#19137073	2
#19134450	2
#19134340	2
C₂×Q₈⋊(C₄₃⋊C₈)	2
#19132435	2
#19132434	2
C₂×C₄₃×Q₈⋊C₈	2
#8178	43