Subgroups — #25097200

Order 11008.

842 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10621872	C₂	1	1	yes	maximal, characteristic
2	#19136988	C₂	1	2	yes	maximal
3	#19134759	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134725	C₂	1	2	yes	maximal, complemented
5	#19134629	C₂	1	2	yes	maximal
6	#19134625	C₂	1	2	yes	maximal
7	#19134495	C₂	1	2	yes	maximal, complemented
8	#19134494	C₂	1	2	yes	maximal, complemented
9	C₄₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10621872	C₂²	1	1	yes	characteristic
11	C₂²×#10621872	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₄₃	C₂²	1	1	yes	characteristic
13	C₂×#13063259	C₂²	1	2	yes	—
14	C₂×#13063257	C₂²	1	2	yes	—
15	C₄×#10621872	C₄	1	8	yes	complemented
16	C₂×#13063235	C₂²	1	1	yes	characteristic
17	C₂×#13063235	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
19	#14582766	—	2	2	no	—
20	#14582765	C₂²	1	2	yes	—
21	#14582761	—	2	2	no	—
22	#14582756	C₂²	1	4	yes	—
23	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
24	#14582746	—	2	2	no	—
25	#14582744	—	2	2	no	—
26	#14582661	C₂²	1	4	yes	—
27	#14582661	C₂²	1	4	yes	—
28	C₂×#107×C₄₃	C₂²	1	2	yes	—
29	C₂×#106×C₄₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₄₃×#382	C₂²	1	2	yes	—
32	C₂²×C₄×D₄₃	D₄	1	2	yes	—
33	C₂²×C₄×D₄₃	C₂³	1	1	yes	characteristic
34	C₂×#10621872	—	2	2	no	—
35	C₂×#10621872	—	2	2	no	—
36	C₂×#10621872	—	2	2	no	—
37	C₂×#10621872	—	2	2	no	—
38	C₂×#10621872	C₂×C₄	1	4	yes	—
39	C₂×#10621872	C₂×C₄	1	4	yes	—
40	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
43	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
44	C₂×C₄×Dic₄₃	—	2	4	no	—
45	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
46	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
47	C₄²×D₄₃	C₂×C₄	1	4	yes	—
48	C₄²×D₄₃	Q₈	1	4	yes	—
49	#13063259	—	2	4	no	—
50	#13063257	—	2	4	no	—
51	#13063235	C₂×C₄	1	4	yes	—
52	#13063235	C₂×C₄	1	4	yes	—
53	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
54	C₄₃⋊#95	—	2	2	no	—
55	C₄₃⋊#95	—	2	2	no	—
56	C₄₃⋊#95	—	2	4	no	—
57	C₄₃⋊#95	—	2	4	no	—
58	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
61	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	#107×C₄₃	—	2	4	no	—
63	#106×C₄₃	—	2	4	no	—
64	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
65	#95×C₄₃	—	2	2	no	—
66	#95×C₄₃	—	2	2	no	—
67	C₂³×D₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	—	2	4	no	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₃	—	2	4	no	—
72	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×D₄₃	—	2	1	no	—
75	C₂×C₄×D₄₃	—	2	1	no	—
76	C₂×C₄×D₄₃	—	2	1	no	—
77	C₂×C₄×D₄₃	—	2	1	no	—
78	C₂×C₄×D₄₃	—	4	4	no	—
79	C₂×C₄×D₄₃	C₄⋊C₄	1	8	yes	—
80	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₃	#41	1	2	yes	—
85	C₂×C₈×C₄₃	#41	1	2	yes	—
86	C₂×C₈×C₄₃	#41	1	2	yes	—
87	C₂×C₈×C₄₃	#41	1	2	yes	—
88	C₄⋊C₄×C₄₃	—	2	4	no	—
89	C₄⋊C₄×C₄₃	—	2	4	no	—
90	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
92	(C₂×D₄₃)⋊C₄	—	4	8	no	—
93	C₄⋊Dic₄₃	—	2	4	no	—
94	C₄⋊Dic₄₃	—	2	4	no	—
95	C₄×Dic₄₃	—	4	8	no	—
96	C₄×Dic₄₃	C₄⋊C₄	1	8	yes	—
97	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
98	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
99	C₂×C₄₃⋊C₈	—	2	1	no	—
100	C₂×C₄₃⋊C₈	—	2	1	no	—
101	C₂×C₄₃⋊C₈	—	2	1	no	—
102	C₂×C₄₃⋊C₈	—	2	1	no	—
103	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
105	#10621872	—	2	8	no	—
106	#10621872	—	2	8	no	—
107	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
108	C₂²×D₄₃	—	2	1	no	—
109	C₂²×D₄₃	—	2	1	no	—
110	C₂²×D₄₃	—	2	4	no	—
111	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₃	—	2	4	no	—
113	C₂×C₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	—	4	4	no	—
115	C₂×C₄×C₄₃	—	4	4	no	—
116	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	4	no	—
126	C₂×Dic₄₃	—	4	4	no	—
127	C₂×Dic₄₃	—	4	4	no	—
128	C₂×Dic₄₃	—	4	4	no	—
129	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
131	C₄×D₄₃	—	2	2	no	—
132	C₄×D₄₃	—	2	2	no	—
133	C₄×D₄₃	—	2	2	no	—
134	C₄×D₄₃	—	2	2	no	—
135	C₄×D₄₃	—	4	8	no	—
136	C₈×C₄₃	#117	1	4	yes	—
137	C₈×C₄₃	#117	1	4	yes	—
138	C₄₃⋊C₈	—	2	2	no	—
139	C₄₃⋊C₈	—	2	2	no	—
140	#20825	—	43	1	no	maximal, Sylow
141	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₄₃	#532	1	1	yes	characteristic
143	C₂²×C₄₃	#532	1	1	yes	characteristic
144	C₂²×C₄₃	#575	1	4	yes	—
145	C₂×D₄₃	—	2	4	no	—
146	C₂×D₄₃	—	2	2	no	—
147	C₂×D₄₃	—	4	2	no	—
148	C₂×D₄₃	—	4	2	no	—
149	C₄×C₄₃	—	2	4	no	—
150	C₄×C₄₃	—	8	4	no	—
151	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
152	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
153	C₄×C₄₃	#533	1	1	yes	characteristic
154	C₄×C₄₃	#533	1	1	yes	characteristic
155	Dic₄₃	—	2	4	no	—
156	Dic₄₃	—	2	2	no	—
157	Dic₄₃	—	8	4	no	—
158	C₂×C₄×M₁₆	—	43	1	no	—
159	#2283	—	43	2	no	—
160	#1846	—	43	1	no	—
161	#1846	D₄₃	1	1	yes	characteristic, complemented
162	#1835	—	43	2	no	—
163	#1809	—	43	2	no	—
164	#1809	—	43	2	no	—
165	#1776	—	43	2	no	—
166	#1776	—	43	2	no	—
167	C₂×C₄₃	#2285	1	1	yes	characteristic
168	C₂×C₄₃	#2911	1	2	yes	—
169	C₂×C₄₃	#3183	1	2	yes	—
170	C₂×C₄₃	#3183	1	2	yes	—
171	D₄₃	—	4	2	no	—
172	C₂²×M₁₆	—	43	1	no	—
173	C₂²×M₁₆	—	43	1	no	—
174	C₂²×C₄²	—	43	1	no	—
175	C₂×#107	—	43	2	no	—
176	C₂×#107	C₂×D₄₃	1	2	yes	—
177	C₂×#106	—	43	2	no	—
178	C₂×#106	C₂×D₄₃	1	2	yes	—
179	C₄×M₁₆	—	43	8	no	—
180	C₂×#97	—	43	1	no	—
181	C₂×#97	—	43	1	no	—
182	C₂×C₄×C₈	—	43	1	no	—
183	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
184	#383	—	86	2	no	—
185	#382	—	43	2	no	—
186	#382	C₂×D₄₃	1	2	yes	—
187	#381	—	86	2	no	—
188	#380	—	43	4	no	—
189	C₄×C₄⋊C₄	—	43	4	no	—
190	C₄×C₂²⋊C₄	—	86	2	no	—
191	#376	—	86	2	no	—
192	#340	—	43	4	no	—
193	#340	—	43	4	no	—
194	C₄₃	#20825	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	—	43	1	no	—
196	C₂³×C₄	—	43	2	no	—
197	C₂×M₁₆	—	43	4	no	—
198	C₂×M₁₆	—	43	4	no	—
199	C₂×M₁₆	—	86	2	no	—
200	C₂×M₁₆	—	86	2	no	—
201	C₂×M₁₆	—	86	2	no	—
202	C₂×M₁₆	—	86	2	no	—
203	C₂²×C₈	—	43	1	no	—
204	C₂²×C₈	—	43	1	no	—
205	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	43	4	no	—
208	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
209	C₂×C₂²⋊C₄	—	86	4	no	—
210	C₂×C₄²	—	43	2	no	—
211	C₂×C₄²	—	43	1	no	—
212	C₂×C₄²	—	43	4	no	—
213	C₂×C₄²	—	43	4	no	—
214	C₂×C₄²	—	86	4	no	—
215	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
216	#107	—	2	4	no	—
217	#107	—	86	4	no	—
218	#106	—	2	4	no	—
219	#106	—	86	4	no	—
220	#97	—	43	4	no	—
221	#97	—	43	4	no	—
222	C₄×C₈	—	43	4	no	—
223	C₄×C₈	C₄×D₄₃	1	4	yes	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	86	4	no	—
227	#95	—	86	4	no	—
228	#95	—	86	2	no	—
229	#95	—	86	2	no	—
230	C₂⁴	—	43	1	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	43	1	no	—
233	C₂²×C₄	—	43	2	no	—
234	C₂²×C₄	—	43	2	no	—
235	C₂²×C₄	—	43	2	no	—
236	C₂²×C₄	—	43	8	no	—
237	C₂²×C₄	—	86	1	no	—
238	C₂²×C₄	—	86	1	no	—
239	C₂²×C₄	—	86	1	no	—
240	C₂²×C₄	—	86	1	no	—
241	C₂²×C₄	—	86	4	no	—
242	C₂²×C₄	—	172	4	no	—
243	C₂²×C₄	D₄×D₄₃	1	2	yes	—
244	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
245	M₁₆	—	86	8	no	—
246	M₁₆	—	86	8	no	—
247	C₂×C₈	—	43	2	no	—
248	C₂×C₈	—	43	2	no	—
249	C₂×C₈	—	86	1	no	—
250	C₂×C₈	—	86	1	no	—
251	C₂×C₈	—	86	1	no	—
252	C₂×C₈	—	86	1	no	—
253	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
254	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
255	C₂×C₈	#10621900	1	2	yes	—
256	C₂×C₈	#10621900	1	2	yes	—
257	C₂×C₈	#10621902	1	2	yes	—
258	C₂×C₈	#10621902	1	2	yes	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	86	4	no	—
262	C₄⋊C₄	—	86	4	no	—
263	C₂²⋊C₄	—	172	8	no	—
264	C₄²	—	43	8	no	—
265	C₄²	—	43	2	no	—
266	C₄²	—	43	2	no	—
267	C₄²	—	172	8	no	—
268	C₄²	C₂×C₄×D₄₃	1	2	yes	—
269	C₄²	Q₈×D₄₃	1	2	yes	—
270	C₂³	—	43	2	no	—
271	C₂³	—	86	4	no	—
272	C₂³	—	86	1	no	—
273	C₂³	—	86	1	no	—
274	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	43	4	no	—
280	C₂×C₄	—	43	2	no	—
281	C₂×C₄	—	86	4	no	—
282	C₂×C₄	—	86	1	no	—
283	C₂×C₄	—	86	1	no	—
284	C₂×C₄	—	86	2	no	—
285	C₂×C₄	—	86	2	no	—
286	C₂×C₄	—	86	2	no	—
287	C₂×C₄	—	86	2	no	—
288	C₂×C₄	—	172	8	no	—
289	C₂×C₄	—	172	4	no	—
290	C₂×C₄	—	172	4	no	—
291	C₂×C₄	—	172	4	no	—
292	C₂×C₄	C₄⋊C₄×D₄₃	1	4	yes	—
293	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
295	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
296	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
297	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
298	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
299	C₈	—	86	2	no	—
300	C₈	—	86	2	no	—
301	C₈	#13063283	1	4	yes	—
302	C₈	#13063283	1	4	yes	—
303	C₂²	—	86	4	no	—
304	C₂²	—	86	2	no	—
305	C₂²	—	172	2	no	—
306	C₂²	—	172	2	no	—
307	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
308	C₂²	#14583447	1	1	yes	characteristic
309	C₂²	#14583450	1	1	yes	characteristic
310	C₂²	#14583696	1	4	yes	—
311	C₄	—	2	4	no	—
312	C₄	—	8	4	no	—
313	C₄	—	86	4	no	—
314	C₄	—	86	2	no	—
315	C₄	—	344	4	no	—
316	C₄	C₂×#13063283	1	1	yes	characteristic
317	C₄	C₂×#13063283	1	1	yes	characteristic
318	C₄	#14583454	1	1	yes	characteristic
319	C₄	#14583454	1	1	yes	characteristic
320	C₂	—	172	2	no	—
321	C₂	#19137000	1	1	yes	characteristic
322	C₂	#19143183	1	2	yes	—
323	C₂	#19145321	1	2	yes	—
324	C₂	#19145326	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1846	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂³×D₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1846	D₄₃
hypocenter	C₄₃	#20825
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
3. #19134759	C₂	semidirect
4. #19134725	C₂	semidirect
7. #19134495	C₂	semidirect
8. #19134494	C₂	semidirect
9. C₄₃×#1846	C₂	semidirect
15. C₄×#10621872	C₄	semidirect
161. #1846	D₄₃	semidirect
194. C₄₃	#20825	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10621872	2
#19136988	2
#19134759	2
#19134725	2
#19134629	2
#19134625	2
#19134495	2
#19134494	2
C₄₃×#1846	2
#20825	43