Subgroups — #25012692

Order 11008.

400 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583015	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582719	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136324	C₂	1	1	yes	maximal, characteristic
4	#19136322	C₂	1	1	yes	maximal, characteristic
5	#19134765	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132708	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₂²	1	1	yes	characteristic
9	C₂×#13063333	—	2	1	no	—
10	C₂×#13063265	C₂²	1	1	yes	characteristic
11	#14583015	—	2	1	no	—
12	#14583015	—	2	1	no	—
13	#14583015	C₄	1	2	yes	—
14	#14583015	C₄	1	2	yes	—
15	C₂×#13063261	C₂²	1	1	yes	characteristic
16	C₂×#13063260	—	2	1	no	—
17	#14582922	—	2	1	no	—
18	C₂×#13063251	—	2	1	no	—
19	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
20	#14582719	—	2	1	no	—
21	#14582719	—	2	1	no	—
22	#14582707	—	2	2	no	—
23	#14582705	—	2	2	no	—
24	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#10621898	—	2	1	no	—
28	#13063347	—	2	1	no	—
29	#13063347	—	2	1	no	—
30	#13063347	C₂×C₄	1	2	yes	—
31	#13063347	D₄	1	2	yes	—
32	C₂×Dic₁₇₂	Q₈	1	1	yes	characteristic
33	C₂×#10621873	—	2	1	no	—
34	C₂×D₃₄₄	Q₈	1	1	yes	characteristic
35	#13063333	—	2	2	no	—
36	#13063333	—	2	2	no	—
37	#13063333	—	4	1	no	—
38	#13063333	—	4	1	no	—
39	C₂×#10621877	—	2	1	no	—
40	C₂×#10621872	—	4	1	no	—
41	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
42	C₂×C₈×D₄₃	—	4	1	no	—
43	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
44	#13063265	—	2	1	no	—
45	#13063265	—	2	1	no	—
46	C₂×C₄₃⋊C₁₆	—	2	1	no	—
47	#13063261	—	2	1	no	—
48	#13063261	—	2	1	no	—
49	#13063260	—	4	1	no	—
50	#13063260	—	4	1	no	—
51	Dic₄₃⋊C₈	—	4	2	no	—
52	#13063251	—	2	2	no	—
53	#13063251	—	2	2	no	—
54	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₄₃	D₄	1	1	yes	characteristic
57	C₈×Dic₄₃	D₄	1	1	yes	characteristic
58	C₂×M₁₆×C₄₃	—	2	1	no	—
59	C₂×M₁₆×C₄₃	—	2	1	no	—
60	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	M₃₂×C₄₃	—	2	1	no	—
62	M₃₂×C₄₃	—	2	1	no	—
63	C₂×C₁₆×C₄₃	—	2	1	no	—
64	#108×C₄₃	—	2	1	no	—
65	#108×C₄₃	—	2	1	no	—
66	#108×C₄₃	—	2	1	no	—
67	#108×C₄₃	C₂×C₄	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₄₃	—	4	1	no	—
70	C₂²×Dic₄₃	—	2	1	no	—
71	#10621898	—	2	2	no	—
72	#10621898	—	2	2	no	—
73	#10621898	—	4	1	no	—
74	#10621898	—	4	1	no	—
75	C₂×D₁₇₂	—	2	1	no	—
76	C₂×C₄×D₄₃	—	4	1	no	—
77	C₂×Dic₈₆	—	2	1	no	—
78	M₁₆×C₄₃	—	2	2	no	—
79	M₁₆×C₄₃	—	4	1	no	—
80	M₁₆×C₄₃	—	4	1	no	—
81	M₁₆×C₄₃	—	4	1	no	—
82	C₂×C₈×C₄₃	—	4	1	no	—
83	C₂×C₈×C₄₃	—	4	1	no	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
90	C₄×Dic₄₃	—	2	1	no	—
91	C₄×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
93	#10621877	—	4	1	no	—
94	#10621877	—	4	1	no	—
95	C₂×C₄₃⋊C₈	—	4	1	no	—
96	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
99	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
100	C₂×C₄₃⋊C₈	SD₁₆	1	1	yes	characteristic
101	C₂×C₄₃⋊C₈	SD₁₆	1	1	yes	characteristic
102	Dic₁₇₂	—	2	1	no	—
103	Dic₁₇₂	C₄⋊C₄	1	2	yes	—
104	D₃₄₄	—	2	1	no	—
105	D₃₄₄	C₄⋊C₄	1	2	yes	—
106	#10621873	—	2	2	no	—
107	#10621873	—	2	2	no	—
108	#10621872	—	4	2	no	—
109	#10621872	—	4	2	no	—
110	C₈×D₄₃	—	4	2	no	—
111	C₈×D₄₃	—	4	2	no	—
112	C₁₆×C₄₃	—	4	1	no	—
113	C₄₃⋊C₁₆	—	4	1	no	—
114	C₂³×C₄₃	#137	1	1	yes	characteristic
115	C₂²×D₄₃	—	4	1	no	—
116	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂²⋊D₄₃	—	4	2	no	—
123	C₂²⋊D₄₃	—	4	2	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	—	4	2	no	—
127	C₂×Dic₄₃	—	4	1	no	—
128	D₁₇₂	—	2	2	no	—
129	D₁₇₂	—	4	1	no	—
130	C₄×D₄₃	—	4	2	no	—
131	C₄×D₄₃	—	4	2	no	—
132	Dic₈₆	—	2	2	no	—
133	Dic₈₆	—	4	1	no	—
134	C₈×C₄₃	—	4	2	no	—
135	C₈×C₄₃	—	8	1	no	—
136	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₈×C₄₃	#117	1	1	yes	characteristic
138	C₈×C₄₃	#121	1	1	yes	characteristic
139	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
140	C₄₃⋊C₈	—	8	1	no	—
141	C₄₃⋊C₈	#102	1	1	yes	characteristic
142	C₄₃⋊C₈	#102	1	1	yes	characteristic
143	C₄₃⋊C₈	#102	1	1	yes	characteristic
144	C₄₃⋊C₈	#102	1	1	yes	characteristic
145	#12822	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	#417	1	1	yes	characteristic
148	C₂²×C₄₃	#461	1	1	yes	characteristic
149	C₂²×C₄₃	#474	1	1	yes	characteristic
150	C₂×D₄₃	—	4	2	no	—
151	C₂×D₄₃	—	8	2	no	—
152	C₄×C₄₃	#380	1	1	yes	characteristic
153	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
154	C₄×C₄₃	#459	1	1	yes	characteristic
155	C₄×C₄₃	#476	1	1	yes	characteristic
156	Dic₄₃	—	2	2	no	—
157	Dic₄₃	—	4	1	no	—
158	Dic₄₃	—	4	2	no	—
159	C₂×#443	—	43	1	no	—
160	C₂×#364	—	43	1	no	—
161	#2143	—	43	1	no	—
162	#2143	—	43	1	no	—
163	#1848	—	43	1	no	—
164	#1388	—	43	1	no	—
165	#1388	D₄₃	1	1	yes	characteristic, complemented
166	C₂×C₄₃	—	2	1	no	—
167	C₂×C₄₃	—	2	1	no	—
168	C₂×C₄₃	—	2	1	no	—
169	C₂×C₄₃	#1802	1	1	yes	characteristic
170	D₄₃	—	8	2	no	—
171	C₂×#135	—	43	1	no	—
172	C₂×#131	—	86	1	no	—
173	C₂×M₃₂	—	43	1	no	—
174	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
175	#443	—	43	2	no	—
176	#443	—	43	2	no	—
177	#443	—	86	1	no	—
178	#443	—	86	1	no	—
179	C₂×#108	—	43	1	no	—
180	C₂×#108	—	86	1	no	—
181	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
182	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
183	#423	—	86	1	no	—
184	C₂×#104	—	86	1	no	—
185	C₂×C₄×C₈	—	43	1	no	—
186	#364	—	86	1	no	—
187	#364	—	86	1	no	—
188	#359	—	86	2	no	—
189	#359	—	86	2	no	—
190	C₄₃	#12822	1	1	yes	characteristic, complemented, Sylow
191	C₂×#41	—	86	1	no	—
192	#135	—	43	2	no	—
193	#135	—	43	2	no	—
194	#135	—	86	1	no	—
195	#135	—	86	1	no	—
196	C₂×Q₁₆	—	43	1	no	—
197	C₂×SD₁₆	—	86	1	no	—
198	C₂×D₈	—	43	1	no	—
199	#131	—	86	2	no	—
200	#131	—	86	2	no	—
201	#131	—	172	1	no	—
202	#131	—	172	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	86	1	no	—
206	C₂×M₁₆	—	172	1	no	—
207	C₂²×C₈	—	43	1	no	—
208	C₂²×C₈	—	172	1	no	—
209	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
210	C₂×C₄²	—	43	1	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	2	1	no	—
213	M₃₂	—	86	1	no	—
214	M₃₂	—	86	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	86	1	no	—
217	#108	—	2	1	no	—
218	#108	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	—	86	1	no	—
221	#108	—	86	1	no	—
222	#108	—	172	1	no	—
223	#108	—	172	1	no	—
224	#108	C₄×D₄₃	1	2	yes	—
225	C₄⋊C₈	—	172	2	no	—
226	#104	—	86	2	no	—
227	#104	—	86	2	no	—
228	C₄×C₈	—	43	1	no	—
229	C₄×C₈	—	43	1	no	—
230	C₄×C₈	—	43	1	no	—
231	C₄×C₈	—	43	1	no	—
232	#41	—	86	2	no	—
233	#41	—	86	2	no	—
234	#41	—	172	1	no	—
235	#41	—	172	1	no	—
236	C₂×Q₈	—	86	1	no	—
237	C₂×D₄	—	86	1	no	—
238	C₂×D₄	—	172	1	no	—
239	C₂²×C₄	—	86	1	no	—
240	C₂²×C₄	—	172	1	no	—
241	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
242	Q₁₆	—	43	2	no	—
243	Q₁₆	—	86	1	no	—
244	SD₁₆	—	86	2	no	—
245	SD₁₆	—	86	2	no	—
246	D₈	—	43	2	no	—
247	D₈	—	86	1	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	172	2	no	—
253	M₁₆	—	172	2	no	—
254	M₁₆	—	172	1	no	—
255	M₁₆	—	172	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	43	1	no	—
259	C₂×C₈	—	43	1	no	—
260	C₂×C₈	—	43	1	no	—
261	C₂×C₈	—	43	1	no	—
262	C₂×C₈	—	43	1	no	—
263	C₂×C₈	—	43	1	no	—
264	C₂×C₈	—	172	2	no	—
265	C₂×C₈	—	172	2	no	—
266	C₂×C₈	—	172	1	no	—
267	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
268	C₂×C₈	#10621898	1	1	yes	characteristic
269	C₂×C₈	#10621898	1	1	yes	characteristic
270	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
271	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
272	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
273	C₄²	—	43	1	no	—
274	C₄²	—	43	1	no	—
275	C₄²	—	86	1	no	—
276	C₁₆	—	4	1	no	—
277	C₁₆	—	172	1	no	—
278	C₂³	—	172	1	no	—
279	C₂³	#13063357	1	1	yes	characteristic
280	Q₈	—	86	2	no	—
281	Q₈	—	172	1	no	—
282	D₄	—	86	2	no	—
283	D₄	—	172	2	no	—
284	D₄	—	172	2	no	—
285	D₄	—	172	1	no	—
286	C₂×C₄	—	86	1	no	—
287	C₂×C₄	—	86	1	no	—
288	C₂×C₄	—	172	2	no	—
289	C₂×C₄	—	172	1	no	—
290	C₂×C₄	—	172	2	no	—
291	C₂×C₄	—	172	2	no	—
292	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
293	C₂×C₄	#13063298	1	1	yes	characteristic
294	C₂×C₄	#13063303	1	1	yes	characteristic
295	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
296	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
297	C₂×C₄	#13063352	1	1	yes	characteristic
298	C₈	—	4	2	no	—
299	C₈	—	8	1	no	—
300	C₈	—	43	1	no	—
301	C₈	—	43	1	no	—
302	C₈	—	43	1	no	—
303	C₈	—	43	1	no	—
304	C₈	—	344	1	no	—
305	C₈	C₄⋊C₄×D₄₃	1	1	yes	characteristic
306	C₈	#13063282	1	1	yes	characteristic
307	C₈	#13063298	1	1	yes	characteristic
308	C₈	#13063303	1	1	yes	characteristic
309	C₂²	—	4	1	no	—
310	C₂²	—	344	2	no	—
311	C₂²	—	172	2	no	—
312	C₂²	#14582899	1	1	yes	characteristic
313	C₂²	#14583103	1	1	yes	characteristic
314	C₂²	#14583178	1	1	yes	characteristic
315	C₄	—	86	2	no	—
316	C₄	—	172	1	no	—
317	C₄	—	172	2	no	—
318	C₄	#14582758	1	1	yes	characteristic
319	C₄	#102×D₄₃	1	1	yes	characteristic
320	C₄	#14583093	1	1	yes	characteristic
321	C₄	#14583188	1	1	yes	characteristic
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	344	2	no	—
326	C₂	#19134594	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14583093
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1388	C₂
socle	C₂×C₄₃	#1802
prime radical	C₂×#13063347	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1388	D₄₃
hypocenter	C₄₃	#12822
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1388

Complements

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

N	H ≅ G/N	type
2. C₂×#14582719	C₂	semidirect
5. #19134765	C₂	semidirect
6. #19132708	C₂	semidirect
7. C₄₃×#1388	C₂	semidirect
165. #1388	D₄₃	semidirect
190. C₄₃	#12822	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583015	2
C₂×#14582719	2
#19136324	2
#19136322	2
#19134765	2
#19132708	2
C₄₃×#1388	2
#12822	43