Subgroups — #25013091

Order 11008.

376 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147697	C₂	1	1	yes	maximal, characteristic
2	#19136360	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136351	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136222	C₂	1	1	yes	maximal, characteristic
5	#19134432	C₂	1	1	yes	maximal, characteristic
6	#19132682	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063331	C₂²	1	1	yes	characteristic
9	#14583317	—	2	2	no	—
10	#14583315	—	2	2	no	—
11	C₂×#13063265	C₂²	1	1	yes	characteristic
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	M₃₂×D₄₃	—	2	2	no	—
14	C₂×#13063253	C₄	1	1	yes	characteristic
15	#14582896	C₄	1	1	yes	characteristic
16	#14582854	C₂²	1	1	yes	characteristic
17	#14582719	—	2	2	no	—
18	C₄⋊(C₄₃⋊C₁₆)	—	2	2	no	—
19	#14582682	—	2	2	no	—
20	#14582664	C₄	1	1	yes	characteristic
21	#14582659	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
26	#13063331	—	2	1	no	—
27	#13063331	—	2	1	no	—
28	#13063331	D₄	1	2	yes	—
29	#13063331	Q₈	1	2	yes	—
30	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
31	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
32	C₂×#10621872	—	2	1	no	—
33	C₂×C₈×D₄₃	—	2	1	no	—
34	#13063284	C₂×C₄	1	1	yes	characteristic
35	#13063265	—	2	1	no	—
36	#13063265	—	2	1	no	—
37	#13063265	—	2	1	no	—
38	#13063265	—	2	1	no	—
39	#13063264	—	2	4	no	—
40	C₂×C₄₃⋊C₁₆	—	2	1	no	—
41	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
42	C₁₆×D₄₃	—	2	4	no	—
43	#13063253	C₈	1	2	yes	—
44	#13063253	C₈	1	2	yes	—
45	#13063250	—	2	1	no	—
46	#13063247	—	2	1	no	—
47	#13063235	—	2	1	no	—
48	C₈×Dic₄₃	—	2	1	no	—
49	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
52	M₃₂×C₄₃	—	2	1	no	—
53	M₃₂×C₄₃	—	2	1	no	—
54	M₃₂×C₄₃	—	2	1	no	—
55	M₃₂×C₄₃	D₄	1	2	yes	—
56	C₂×C₁₆×C₄₃	—	2	1	no	—
57	C₂×C₁₆×C₄₃	—	2	1	no	—
58	C₄×C₈×C₄₃	—	2	1	no	—
59	C₄×C₈×C₄₃	—	2	1	no	—
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₃	—	2	1	no	—
63	#10621898	—	2	1	no	—
64	#10621898	—	2	1	no	—
65	#10621898	C₄⋊C₄	1	2	yes	—
66	#10621898	C₂×C₈	1	2	yes	—
67	C₂×D₁₇₂	M₁₆	1	1	yes	characteristic
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×Dic₈₆	M₁₆	1	1	yes	characteristic
70	C₂×C₈×C₄₃	—	2	1	no	—
71	C₂×C₈×C₄₃	—	2	1	no	—
72	C₂×C₈×C₄₃	—	2	1	no	—
73	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
78	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
79	C₄²×C₄₃	—	2	1	no	—
80	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
82	C₂²⋊Dic₄₃	—	2	1	no	—
83	C₄⋊Dic₄₃	—	2	1	no	—
84	C₄×Dic₄₃	—	2	1	no	—
85	#10621877	—	2	1	no	—
86	#10621877	—	2	1	no	—
87	#10621877	C₄⋊C₄	1	2	yes	—
88	#10621877	C₂×C₈	1	2	yes	—
89	C₂×C₄₃⋊C₈	—	2	1	no	—
90	C₂×C₄₃⋊C₈	—	2	1	no	—
91	#10621872	—	2	2	no	—
92	#10621872	—	2	2	no	—
93	C₈×D₄₃	—	2	2	no	—
94	C₈×D₄₃	—	2	2	no	—
95	C₁₆×C₄₃	—	2	2	no	—
96	C₁₆×C₄₃	—	4	1	no	—
97	C₄₃⋊C₁₆	—	2	2	no	—
98	C₄₃⋊C₁₆	—	4	1	no	—
99	C₂³×C₄₃	#131	1	1	yes	characteristic
100	C₂²×D₄₃	—	2	1	no	—
101	C₂×C₄×C₄₃	—	2	1	no	—
102	C₂×C₄×C₄₃	—	2	1	no	—
103	C₂×C₄×C₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
110	C₂²⋊D₄₃	—	2	2	no	—
111	C₂²⋊D₄₃	—	2	2	no	—
112	C₂×Dic₄₃	—	2	1	no	—
113	C₂×Dic₄₃	—	4	1	no	—
114	D₁₇₂	—	2	1	no	—
115	D₁₇₂	C₄⋊C₈	1	2	yes	—
116	C₄×D₄₃	—	2	2	no	—
117	C₄×D₄₃	—	2	2	no	—
118	Dic₈₆	—	2	1	no	—
119	Dic₈₆	C₄⋊C₈	1	2	yes	—
120	C₈×C₄₃	—	4	1	no	—
121	C₈×C₄₃	#120	1	1	yes	characteristic
122	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₄₃	#123	1	1	yes	characteristic
125	C₄₃⋊C₈	—	2	2	no	—
126	C₄₃⋊C₈	—	2	2	no	—
127	#12918	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	4	1	no	—
129	C₂²×C₄₃	#424	1	1	yes	characteristic
130	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
131	C₂²×C₄₃	#435	1	1	yes	characteristic
132	C₂×D₄₃	—	2	2	no	—
133	C₂×D₄₃	—	8	1	no	—
134	C₄×C₄₃	—	2	2	no	—
135	C₄×C₄₃	—	4	1	no	—
136	C₄×C₄₃	#385	1	1	yes	characteristic
137	C₄×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
139	C₄×C₄₃	#435	1	1	yes	characteristic
140	Dic₄₃	—	2	2	no	—
141	Dic₄₃	—	8	1	no	—
142	#3407	—	43	1	no	—
143	#2153	—	43	1	no	—
144	#2150	—	43	1	no	—
145	#2117	—	43	1	no	—
146	#1764	—	43	1	no	—
147	#1378	—	43	1	no	—
148	#1376	D₄₃	1	1	yes	characteristic, complemented
149	C₂×C₄₃	—	2	1	no	—
150	C₂×C₄₃	—	2	1	no	—
151	C₂×C₄₃	—	2	1	no	—
152	C₂×C₄₃	#1850	1	1	yes	characteristic
153	D₄₃	—	4	2	no	—
154	C₂×#131	—	43	1	no	—
155	#504	—	86	2	no	—
156	#504	—	86	2	no	—
157	C₂×M₃₂	—	43	1	no	—
158	C₂×M₃₂	—	43	1	no	—
159	C₂×M₃₂	—	86	2	no	—
160	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
161	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
162	C₂×#104	—	43	1	no	—
163	#416	—	43	1	no	—
164	#405	—	43	1	no	—
165	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
166	#364	—	86	2	no	—
167	C₄⋊C₁₆	—	86	2	no	—
168	#350	—	86	2	no	—
169	#341	—	43	1	no	—
170	#339	—	43	1	no	—
171	C₄₃	#12918	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	43	1	no	—
173	#131	—	43	2	no	—
174	#131	—	43	2	no	—
175	#131	—	86	1	no	—
176	#131	—	86	1	no	—
177	C₂×M₁₆	—	43	1	no	—
178	C₂×M₁₆	—	43	1	no	—
179	C₂×M₁₆	—	86	1	no	—
180	C₂²×C₈	—	86	1	no	—
181	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
183	#117	—	43	1	no	—
184	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
185	M₃₂	—	2	1	no	—
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	86	4	no	—
189	M₃₂	—	86	1	no	—
190	M₃₂	—	86	1	no	—
191	M₃₂	—	86	1	no	—
192	M₃₂	—	86	1	no	—
193	M₃₂	D₁₇₂	1	2	yes	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	43	2	no	—
197	C₂×C₁₆	—	86	4	no	—
198	C₂×C₁₆	—	86	1	no	—
199	#104	—	43	2	no	—
200	#104	—	43	2	no	—
201	#103	—	86	1	no	—
202	#102	—	86	1	no	—
203	#97	—	86	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	86	1	no	—
207	#41	—	43	2	no	—
208	#41	—	43	2	no	—
209	#41	—	86	1	no	—
210	#41	—	86	1	no	—
211	C₂×Q₈	—	43	1	no	—
212	C₂×D₄	—	43	1	no	—
213	C₂×D₄	—	86	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	86	1	no	—
216	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
217	M₁₆	—	43	2	no	—
218	M₁₆	—	43	2	no	—
219	M₁₆	—	86	1	no	—
220	M₁₆	—	86	1	no	—
221	M₁₆	—	86	2	no	—
222	M₁₆	—	86	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	86	1	no	—
227	C₂×C₈	—	86	1	no	—
228	C₂×C₈	—	86	2	no	—
229	C₂×C₈	—	86	2	no	—
230	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
231	C₂×C₈	#10621898	1	1	yes	characteristic
232	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
233	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
234	C₂×C₈	#10621900	1	1	yes	characteristic
235	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
236	C₄⋊C₄	—	86	1	no	—
237	C₂²⋊C₄	—	86	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	86	1	no	—
240	C₄²	C₈×D₄₃	1	1	yes	characteristic
241	C₄²	C₈×D₄₃	1	1	yes	characteristic
242	C₁₆	—	2	2	no	—
243	C₁₆	—	4	1	no	—
244	C₁₆	—	86	2	no	—
245	C₁₆	—	172	1	no	—
246	C₂³	—	86	1	no	—
247	C₂³	#13063333	1	1	yes	characteristic
248	Q₈	—	43	2	no	—
249	Q₈	—	86	1	no	—
250	D₄	—	43	2	no	—
251	D₄	—	86	2	no	—
252	D₄	—	86	2	no	—
253	D₄	—	86	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	86	1	no	—
258	C₂×C₄	—	86	2	no	—
259	C₂×C₄	—	86	2	no	—
260	C₂×C₄	—	172	1	no	—
261	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
262	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
265	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
266	C₂×C₄	#13063331	1	1	yes	characteristic
267	C₈	—	4	1	no	—
268	C₈	—	86	2	no	—
269	C₈	—	86	2	no	—
270	C₈	#13063295	1	1	yes	characteristic
271	C₈	#13063303	1	1	yes	characteristic
272	C₈	#13063305	1	1	yes	characteristic
273	C₈	#13063307	1	1	yes	characteristic
274	C₂²	—	4	1	no	—
275	C₂²	—	86	2	no	—
276	C₂²	—	344	1	no	—
277	C₂²	#14582924	1	1	yes	characteristic
278	C₂²	(C₂²⋊D₄₃)⋊C₈	1	1	yes	characteristic
279	C₂²	#14582972	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	86	2	no	—
283	C₄	—	344	1	no	—
284	C₄	#14582773	1	1	yes	characteristic
285	C₄	C₄⋊C₈×D₄₃	1	1	yes	characteristic
286	C₄	C₈×D₁₇₂	1	1	yes	characteristic
287	C₄	#14582973	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	172	2	no	—
292	C₂	#19134774	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13063295
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1376	C₂
socle	C₂×C₄₃	#1850
prime radical	C₂×#10621898	C₂×C₄
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1376	D₄₃
hypocenter	C₄₃	#12918
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
2. #19136360	C₂	semidirect
3. #19136351	C₂	semidirect
6. #19132682	C₂	semidirect
7. C₄₃×#1376	C₂	semidirect
148. #1376	D₄₃	semidirect
171. C₄₃	#12918	semidirect

Maximal subgroups

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

subgroup	index
#19147697	2
#19136360	2
#19136351	2
#19136222	2
#19134432	2
#19132682	2
C₄₃×#1376	2
#12918	43