Subgroups — #25078966

Order 11008.

606 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	#19145109	C₂	1	1	yes	maximal, characteristic
2	#19144634	C₂	1	2	yes	maximal
3	#19133999	C₂	1	2	yes	maximal, complemented
4	#19133989	C₂	1	2	yes	maximal, complemented
5	#19133977	C₂	1	4	yes	maximal, complemented
6	#19133343	C₂	1	1	yes	maximal, characteristic
7	#19133253	C₂	1	2	yes	maximal
8	C₂×C₈×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
9	#14583456	C₂²	1	2	yes	—
10	#14583304	C₂²	1	1	yes	characteristic
11	#14583304	C₂²	1	1	yes	characteristic
12	#14583301	—	2	1	no	—
13	#14583301	—	2	1	no	—
14	#14583297	C₂²	1	2	yes	—
15	#14583234	—	2	2	no	—
16	#14583194	C₂²	1	4	yes	—
17	#14583109	C₂²	1	4	yes	—
18	#14583055	—	2	2	no	—
19	C₄×#10621885	—	2	4	no	—
20	#14582952	C₂²	1	2	yes	—
21	#14582937	—	2	1	no	—
22	#14582937	—	2	1	no	—
23	#14582652	C₂²	1	2	yes	—
24	#14582650	C₂²	1	2	yes	—
25	#14582647	C₂²	1	4	yes	—
26	#14582644	C₂²	1	4	yes	—
27	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₄₃	—	2	1	no	—
29	C₈×Q₈×C₄₃	—	2	1	no	—
30	C₈×Q₈×C₄₃	C₂²	1	2	yes	—
31	C₈×Q₈×C₄₃	C₂²	1	2	yes	—
32	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
33	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₄₃	—	2	1	no	—
35	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#10621885	—	2	4	no	—
37	#13063325	C₂³	1	2	yes	—
38	#13063325	C₂³	1	2	yes	—
39	#13063317	—	2	2	no	—
40	#13063304	—	2	2	no	—
41	#13063304	—	2	2	no	—
42	C₄×Dic₈₆	—	2	2	no	—
43	C₄×Dic₈₆	—	2	2	no	—
44	#13063284	—	2	2	no	—
45	#13063259	—	2	1	no	—
46	#13063259	—	2	1	no	—
47	#13063259	—	2	1	no	—
48	#13063259	—	2	1	no	—
49	#13063258	—	2	4	no	—
50	#13063257	—	2	2	no	—
51	#13063257	—	2	2	no	—
52	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
53	#13063249	—	2	4	no	—
54	#13063248	—	2	4	no	—
55	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
56	C₄×C₄₃⋊C₈	—	2	2	no	—
57	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₃	—	2	1	no	—
59	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
61	C₄×Q₈×C₄₃	—	2	1	no	—
62	C₄×Q₈×C₄₃	—	2	1	no	—
63	C₄×Q₈×C₄₃	D₄	1	2	yes	—
64	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₄₃	—	2	1	no	—
68	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₄₃	—	2	2	no	—
70	C₄⋊C₈×C₄₃	—	2	2	no	—
71	C₄⋊C₈×C₄₃	—	2	1	no	—
72	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
73	C₄×C₈×C₄₃	—	2	2	no	—
74	C₄×C₈×C₄₃	—	2	2	no	—
75	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₄₃	—	2	4	no	—
80	C₂×Q₈×C₄₃	—	2	1	no	—
81	C₂×Q₈×C₄₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₃	—	2	1	no	—
83	C₂²×C₄×C₄₃	—	2	1	no	—
84	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
87	C₂×Dic₈₆	—	2	2	no	—
88	C₂×Dic₈₆	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	2	2	no	—
91	C₂×C₈×C₄₃	—	4	2	no	—
92	C₂×C₈×C₄₃	Q₁₆	1	4	yes	complemented
93	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₄₃	—	2	2	no	—
98	C₄⋊C₄×C₄₃	—	2	1	no	—
99	C₄⋊C₄×C₄₃	—	2	2	no	—
100	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
101	C₄²×C₄₃	—	2	2	no	—
102	C₄²×C₄₃	—	2	2	no	—
103	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₃	#41	1	1	yes	characteristic
106	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₄₃	—	4	2	no	—
108	#10621885	—	4	8	no	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	—	2	2	no	—
113	Dic₄₃⋊C₄	—	4	2	no	—
114	Dic₄₃⋊C₄	—	4	2	no	—
115	C₄×Dic₄₃	—	4	2	no	—
116	C₂×C₄₃⋊C₈	—	2	4	no	—
117	C₂³×C₄₃	#121	1	1	yes	characteristic
118	Q₈×C₄₃	—	2	4	no	—
119	Q₈×C₄₃	—	2	2	no	—
120	Q₈×C₄₃	#135	1	4	yes	—
121	C₂×C₄×C₄₃	—	2	4	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	4	1	no	—
126	C₂×C₄×C₄₃	—	4	2	no	—
127	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
131	C₂×C₄×C₄₃	#137	1	2	yes	—
132	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₄₃	—	4	2	no	—
136	C₂×Dic₄₃	—	4	2	no	—
137	Dic₈₆	—	4	4	no	—
138	Dic₈₆	—	8	2	no	—
139	C₈×C₄₃	—	2	1	no	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	—	4	2	no	—
142	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
143	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
144	C₄₃⋊C₈	—	4	4	no	—
145	#19118	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	#462	1	2	yes	—
149	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
150	C₄×C₄₃	—	2	4	no	—
151	C₄×C₄₃	—	2	1	no	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	4	2	no	—
155	C₄×C₄₃	#462	1	2	yes	—
156	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
157	C₄×C₄₃	C₂×#135	1	2	yes	—
158	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
159	Dic₄₃	—	8	2	no	—
160	Dic₄₃	—	8	2	no	—
161	#3160	—	43	1	no	—
162	#3104	—	43	2	no	—
163	C₂×C₈×Q₈	D₄₃	1	1	yes	characteristic
164	#1676	—	43	2	no	—
165	#1674	—	43	2	no	—
166	#1672	—	43	4	no	—
167	#1564	—	43	1	no	—
168	#1542	—	43	2	no	—
169	C₂×C₄₃	—	2	2	no	—
170	C₂×C₄₃	C₂×#462	1	1	yes	characteristic
171	C₂×C₄₃	#3050	1	1	yes	characteristic
172	C₂×C₄₃	#3067	1	1	yes	characteristic
173	#533	—	43	2	no	—
174	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
175	#500	—	43	1	no	—
176	#500	—	43	1	no	—
177	#499	—	86	1	no	—
178	#499	—	86	1	no	—
179	#498	—	43	2	no	—
180	#484	—	86	2	no	—
181	#477	—	43	4	no	—
182	#462	—	43	4	no	—
183	#451	—	86	2	no	—
184	C₈×Q₈	—	2	1	no	—
185	C₈×Q₈	—	2	1	no	—
186	C₈×Q₈	C₂×D₄₃	1	2	yes	—
187	C₈×Q₈	C₂×D₄₃	1	2	yes	—
188	C₄×Q₁₆	—	86	4	no	—
189	#431	—	43	2	no	—
190	#427	—	86	1	no	—
191	#427	—	86	1	no	—
192	C₂×C₄⋊C₈	—	2	1	no	—
193	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
194	C₂×C₄×C₈	—	2	1	no	—
195	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
196	#335	—	43	2	no	—
197	#334	—	43	2	no	—
198	#333	—	43	4	no	—
199	#332	—	43	4	no	—
200	C₄₃	#19118	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₈	D₁₇₂	1	1	yes	characteristic
202	C₂×Q₁₆	—	86	4	no	—
203	C₂²×C₈	—	2	1	no	—
204	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
206	C₄⋊Q₈	—	43	2	no	—
207	C₄⋊Q₈	—	43	2	no	—
208	#125	—	86	2	no	—
209	C₂²⋊Q₈	—	86	2	no	—
210	C₂²⋊Q₈	—	86	2	no	—
211	C₄×Q₈	—	2	1	no	—
212	C₄×Q₈	—	2	1	no	—
213	C₄×Q₈	—	86	2	no	—
214	C₄×Q₈	—	86	2	no	—
215	C₄×Q₈	D₁₇₂	1	2	yes	—
216	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
217	#117	—	86	2	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
222	#107	—	86	1	no	—
223	#107	—	86	1	no	—
224	#107	—	86	1	no	—
225	#107	—	86	1	no	—
226	#107	—	86	4	no	—
227	#106	—	86	2	no	—
228	#106	—	86	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	43	4	no	—
233	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
234	#103	—	86	4	no	—
235	#103	—	86	4	no	—
236	C₂²⋊C₈	—	86	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	—	2	2	no	—
239	C₄×C₈	—	86	2	no	—
240	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
241	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	86	2	no	—
247	C₂×Q₈	—	86	2	no	—
248	C₂×Q₈	C₂×D₁₇₂	1	2	yes	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
252	C₂²×C₄	#10621898	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
254	Q₁₆	—	172	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	86	4	no	—
259	C₂×C₈	#10621885	1	4	yes	complemented
260	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	86	2	no	—
268	C₄⋊C₄	—	86	2	no	—
269	C₄⋊C₄	—	86	2	no	—
270	C₄⋊C₄	—	86	2	no	—
271	C₄⋊C₄	—	172	2	no	—
272	C₄⋊C₄	—	172	2	no	—
273	C₄⋊C₄	C₂×D₁₇₂	1	2	yes	—
274	C₂²⋊C₄	—	172	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	172	2	no	—
278	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
279	C₄²	#10621898	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
281	C₄²	C₂³×D₄₃	1	1	yes	characteristic
282	C₂³	#13063299	1	1	yes	characteristic
283	Q₈	—	2	4	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	344	2	no	—
286	Q₈	—	172	4	no	—
287	Q₈	#13063347	1	4	yes	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	—	172	2	no	—
296	C₂×C₄	#13063299	1	1	yes	characteristic
297	C₂×C₄	#13063299	1	1	yes	characteristic
298	C₂×C₄	#13063299	1	1	yes	characteristic
299	C₂×C₄	C₂×#10621885	1	2	yes	—
300	C₂×C₄	#13063355	1	2	yes	—
301	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
303	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	4	2	no	—
307	C₈	—	172	4	no	—
308	C₈	C₂×#10621885	1	2	yes	—
309	C₈	C₂×#10621885	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#14583109	1	2	yes	—
313	C₂²	C₂×#13063299	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	4	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	344	2	no	—
320	C₄	—	344	2	no	—
321	C₄	#14583109	1	2	yes	—
322	C₄	C₂²×#10621885	1	1	yes	characteristic
323	C₄	C₂×#13063347	1	2	yes	—
324	C₄	C₂×#13063355	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	C₂×#14583109	1	1	yes	characteristic
327	C₂	#19144176	1	1	yes	characteristic
328	C₂	#19144315	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063299
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₈×Q₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂³×C₄₃	#121
hypercenter	C₂×C₈×Q₈	D₄₃
hypocenter	C₄₃	#19118
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #19133999	C₂	semidirect
4. #19133989	C₂	semidirect
5. #19133977	C₂	semidirect
92. C₂×C₈×C₄₃	Q₁₆	semidirect
200. C₄₃	#19118	semidirect
259. C₂×C₈	#10621885	semidirect

Maximal subgroups

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

subgroup	index
#19145109	2
#19144634	2
#19133999	2
#19133989	2
#19133977	2
#19133343	2
#19133253	2
C₂×C₈×Q₈×C₄₃	2
#19118	43