Subgroups — #25013468

Order 11008.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 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, complemented
2	#19143499	C₂	1	1	yes	maximal, characteristic
3	#19136222	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582684	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136210	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132684	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583699	C₄	1	1	yes	characteristic
9	C₂×#13063331	C₂²	1	1	yes	characteristic
10	C₂×#13063331	C₂²	1	1	yes	characteristic
11	#14583317	—	2	2	no	—
12	#14583315	—	2	2	no	—
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	M₃₂×D₄₃	—	2	2	no	—
15	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14583022	—	2	2	no	—
17	#14583017	—	2	2	no	—
18	#14582948	C₄	1	1	yes	characteristic
19	#14582917	C₄	1	1	yes	characteristic
20	#14582917	C₄	1	1	yes	characteristic
21	#14582684	—	2	2	no	—
22	#14582682	—	2	2	no	—
23	(C₂×D₄₃)⋊C₁₆	—	2	2	no	—
24	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#405	C₂²	1	1	yes	characteristic
27	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
28	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
29	#13063358	—	2	1	no	—
30	#13063358	C₈	1	2	yes	—
31	#13063353	—	2	1	no	—
32	#13063353	C₈	1	2	yes	—
33	#13063347	—	2	2	no	—
34	#13063331	—	2	1	no	—
35	#13063331	—	2	1	no	—
36	#13063331	—	2	1	no	—
37	#13063331	—	2	1	no	—
38	#13063331	—	2	1	no	—
39	#13063331	—	2	1	no	—
40	#13063331	D₄	1	2	yes	—
41	#13063331	D₄	1	2	yes	—
42	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
43	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
44	C₂×#10621872	—	2	1	no	—
45	C₂×#10621872	—	2	1	no	—
46	C₂×C₈×D₄₃	—	2	1	no	—
47	C₂×C₈×D₄₃	—	2	1	no	—
48	#13063265	—	2	1	no	—
49	#13063265	D₄	1	2	yes	—
50	#13063264	—	2	4	no	—
51	C₂×C₄₃⋊C₁₆	—	2	1	no	—
52	C₂×C₄₃⋊C₁₆	—	2	1	no	—
53	C₂×C₄₃⋊C₁₆	—	2	1	no	—
54	C₂×C₄₃⋊C₁₆	—	2	1	no	—
55	C₁₆×D₄₃	—	2	4	no	—
56	#13063261	—	2	2	no	—
57	#13063253	—	2	2	no	—
58	#13063253	—	2	2	no	—
59	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	#117×C₄₃	C₂×C₄	1	1	yes	characteristic
62	M₃₂×C₄₃	—	2	1	no	—
63	M₃₂×C₄₃	—	2	1	no	—
64	M₃₂×C₄₃	—	2	1	no	—
65	M₃₂×C₄₃	—	2	1	no	—
66	C₂×C₁₆×C₄₃	—	2	1	no	—
67	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
68	#97×C₄₃	—	2	1	no	—
69	C₄×C₈×C₄₃	—	2	1	no	—
70	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊D₄₃	—	2	1	no	—
72	C₂×C₂²⋊D₄₃	—	2	1	no	—
73	#10621898	—	2	1	no	—
74	#10621898	—	2	1	no	—
75	#10621898	—	2	1	no	—
76	#10621898	—	2	1	no	—
77	#10621898	—	2	1	no	—
78	#10621898	—	2	1	no	—
79	#10621898	C₂²⋊C₄	1	2	yes	—
80	#10621898	C₂×C₈	1	2	yes	—
81	C₂×D₁₇₂	C₂×C₈	1	1	yes	characteristic
82	C₂×D₁₇₂	M₁₆	1	1	yes	characteristic
83	C₂×C₄×D₄₃	—	2	1	no	—
84	C₂×C₄×D₄₃	—	2	1	no	—
85	C₂×Dic₈₆	C₂×C₈	1	1	yes	characteristic
86	C₂×Dic₈₆	M₁₆	1	1	yes	characteristic
87	M₁₆×C₄₃	—	2	1	no	—
88	M₁₆×C₄₃	C₂×C₈	1	2	yes	—
89	C₂×C₈×C₄₃	—	2	1	no	—
90	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
96	C₄⋊C₄×C₄₃	—	2	1	no	—
97	C₂²⋊C₄×C₄₃	—	2	1	no	—
98	C₄²×C₄₃	—	2	1	no	—
99	#10621877	—	2	1	no	—
100	#10621877	—	2	1	no	—
101	#10621877	—	2	1	no	—
102	#10621877	C₂²⋊C₄	1	2	yes	—
103	C₂×C₄₃⋊C₈	—	2	1	no	—
104	C₂×C₄₃⋊C₈	—	2	1	no	—
105	Dic₁₇₂	—	2	2	no	—
106	D₃₄₄	—	2	2	no	—
107	#10621873	—	2	2	no	—
108	#10621873	—	2	2	no	—
109	#10621872	—	2	2	no	—
110	#10621872	—	2	2	no	—
111	#10621872	—	4	1	no	—
112	#10621872	—	4	1	no	—
113	C₈×D₄₃	—	2	2	no	—
114	C₈×D₄₃	—	2	2	no	—
115	C₈×D₄₃	—	4	1	no	—
116	C₈×D₄₃	—	4	1	no	—
117	C₁₆×C₄₃	—	2	2	no	—
118	C₁₆×C₄₃	—	4	1	no	—
119	C₄₃⋊C₁₆	—	2	2	no	—
120	C₄₃⋊C₁₆	—	4	1	no	—
121	C₂³×C₄₃	#131	1	1	yes	characteristic
122	C₂²×D₄₃	—	2	1	no	—
123	C₂²×D₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	4	1	no	—
125	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
131	C₂²⋊D₄₃	—	2	2	no	—
132	C₂²⋊D₄₃	—	2	2	no	—
133	C₂²⋊D₄₃	—	4	1	no	—
134	C₂²⋊D₄₃	—	4	1	no	—
135	C₂×Dic₄₃	—	2	1	no	—
136	C₂×Dic₄₃	—	2	1	no	—
137	D₁₇₂	—	2	1	no	—
138	D₁₇₂	—	2	1	no	—
139	D₁₇₂	—	2	1	no	—
140	D₁₇₂	C₂²⋊C₈	1	2	yes	—
141	C₄×D₄₃	—	2	2	no	—
142	C₄×D₄₃	—	2	2	no	—
143	C₄×D₄₃	—	4	1	no	—
144	C₄×D₄₃	—	4	1	no	—
145	Dic₈₆	—	2	1	no	—
146	Dic₈₆	—	2	1	no	—
147	Dic₈₆	—	2	1	no	—
148	Dic₈₆	C₂²⋊C₈	1	2	yes	—
149	C₈×C₄₃	—	2	2	no	—
150	C₈×C₄₃	#120	1	1	yes	characteristic
151	C₈×C₄₃	#121	1	1	yes	characteristic
152	C₈×C₄₃	#121	1	1	yes	characteristic
153	C₈×C₄₃	#123	1	1	yes	characteristic
154	C₄₃⋊C₈	—	2	2	no	—
155	C₄₃⋊C₈	—	4	1	no	—
156	#12979	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	—	4	1	no	—
158	C₂²×C₄₃	#408	1	1	yes	characteristic
159	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
160	C₂²×C₄₃	#435	1	1	yes	characteristic
161	C₂×D₄₃	—	2	2	no	—
162	C₂×D₄₃	—	4	2	no	—
163	C₂×D₄₃	—	4	1	no	—
164	C₂×D₄₃	—	8	1	no	—
165	C₄×C₄₃	—	8	1	no	—
166	C₄×C₄₃	#386	1	1	yes	characteristic
167	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
169	C₄×C₄₃	#435	1	1	yes	characteristic
170	Dic₄₃	—	2	2	no	—
171	Dic₄₃	—	4	1	no	—
172	#3407	—	43	1	no	—
173	#2955	—	43	1	no	—
174	#2117	—	43	1	no	—
175	C₂×#350	—	43	1	no	—
176	#2113	—	43	1	no	—
177	#1379	—	43	1	no	—
178	#1378	D₄₃	1	1	yes	characteristic, complemented
179	C₂×C₄₃	—	2	1	no	—
180	C₂×C₄₃	—	2	1	no	—
181	C₂×C₄₃	—	2	1	no	—
182	C₂×C₄₃	#1869	1	1	yes	characteristic
183	D₄₃	—	4	2	no	—
184	D₄₃	—	8	1	no	—
185	#575	—	43	1	no	—
186	C₂×#131	—	43	1	no	—
187	C₂×#131	—	43	1	no	—
188	#504	—	86	2	no	—
189	#504	—	86	2	no	—
190	C₂×M₃₂	—	43	1	no	—
191	C₂×M₃₂	—	86	2	no	—
192	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
193	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
194	C₂²×C₁₆	—	43	1	no	—
195	#444	—	86	2	no	—
196	#443	—	86	2	no	—
197	#430	—	43	1	no	—
198	#421	—	43	1	no	—
199	#421	—	43	1	no	—
200	#405	C₂×D₄₃	1	1	yes	characteristic
201	#350	—	86	2	no	—
202	#350	—	86	2	no	—
203	C₂²⋊C₁₆	—	86	2	no	—
204	C₄₃	#12979	1	1	yes	characteristic, complemented, Sylow
205	C₂×#41	—	43	1	no	—
206	C₂×#41	—	43	1	no	—
207	#137	—	43	2	no	—
208	#137	—	86	1	no	—
209	AGL(1,ℤ/8)	—	43	2	no	—
210	AGL(1,ℤ/8)	—	86	1	no	—
211	#135	—	86	2	no	—
212	#131	—	43	2	no	—
213	#131	—	43	2	no	—
214	#131	—	86	1	no	—
215	#131	—	86	1	no	—
216	#131	—	86	1	no	—
217	#131	—	86	1	no	—
218	#131	—	86	1	no	—
219	#131	—	86	1	no	—
220	C₂×M₁₆	—	43	1	no	—
221	C₂×M₁₆	—	43	1	no	—
222	C₂×M₁₆	—	86	1	no	—
223	C₂×M₁₆	—	86	1	no	—
224	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
225	C₂²×C₈	—	86	1	no	—
226	C₂²×C₈	—	86	1	no	—
227	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
228	#117	C₄×D₄₃	1	1	yes	characteristic
229	M₃₂	—	2	1	no	—
230	M₃₂	—	2	1	no	—
231	M₃₂	—	2	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	43	2	no	—
234	M₃₂	—	86	4	no	—
235	M₃₂	—	86	1	no	—
236	C₂×C₁₆	—	2	1	no	—
237	C₂×C₁₆	—	86	4	no	—
238	C₂×C₁₆	—	86	1	no	—
239	C₂×C₁₆	—	86	1	no	—
240	C₂×C₁₆	—	86	1	no	—
241	C₂×C₁₆	—	86	1	no	—
242	C₂×C₁₆	D₁₇₂	1	2	yes	—
243	#108	—	86	2	no	—
244	#104	—	86	2	no	—
245	#104	—	86	2	no	—
246	#97	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	#41	—	43	2	no	—
249	#41	—	43	2	no	—
250	#41	—	86	1	no	—
251	#41	—	86	1	no	—
252	#41	—	86	1	no	—
253	#41	—	86	1	no	—
254	#41	—	86	1	no	—
255	#41	—	86	1	no	—
256	C₂×Q₈	—	43	1	no	—
257	C₂×Q₈	—	43	1	no	—
258	C₂×D₄	—	43	1	no	—
259	C₂×D₄	—	43	1	no	—
260	C₂×D₄	—	86	1	no	—
261	C₂×D₄	—	86	1	no	—
262	C₂²×C₄	—	86	1	no	—
263	C₂²×C₄	—	86	1	no	—
264	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
265	Q₁₆	—	86	2	no	—
266	SD₁₆	—	86	2	no	—
267	SD₁₆	—	86	2	no	—
268	D₈	—	86	2	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	43	2	no	—
271	M₁₆	—	86	1	no	—
272	M₁₆	—	86	2	no	—
273	M₁₆	—	86	2	no	—
274	M₁₆	—	86	1	no	—
275	M₁₆	—	86	1	no	—
276	M₁₆	—	172	1	no	—
277	M₁₆	—	172	1	no	—
278	M₁₆	C₈×D₄₃	1	2	yes	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	86	1	no	—
281	C₂×C₈	—	86	2	no	—
282	C₂×C₈	—	86	2	no	—
283	C₂×C₈	—	86	1	no	—
284	C₂×C₈	—	172	1	no	—
285	C₂×C₈	—	172	1	no	—
286	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
287	C₂×C₈	#10621898	1	1	yes	characteristic
288	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
289	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
290	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
291	C₂×C₈	#10621902	1	1	yes	characteristic
292	C₄⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	4	1	no	—
297	C₁₆	—	86	2	no	—
298	C₁₆	—	172	1	no	—
299	C₂³	—	86	1	no	—
300	C₂³	—	86	1	no	—
301	C₂³	#13063331	1	1	yes	characteristic
302	Q₈	—	43	2	no	—
303	Q₈	—	86	1	no	—
304	Q₈	—	86	1	no	—
305	Q₈	—	86	1	no	—
306	D₄	—	43	2	no	—
307	D₄	—	86	2	no	—
308	D₄	—	86	2	no	—
309	D₄	—	86	1	no	—
310	D₄	—	86	1	no	—
311	D₄	—	86	1	no	—
312	D₄	—	172	1	no	—
313	D₄	—	172	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	86	1	no	—
316	C₂×C₄	—	86	2	no	—
317	C₂×C₄	—	86	2	no	—
318	C₂×C₄	—	86	1	no	—
319	C₂×C₄	—	172	1	no	—
320	C₂×C₄	—	172	1	no	—
321	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
322	C₂×C₄	#13063287	1	1	yes	characteristic
323	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
324	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
325	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
326	C₂×C₄	#13063333	1	1	yes	characteristic
327	C₈	—	2	2	no	—
328	C₈	—	86	2	no	—
329	C₈	—	172	1	no	—
330	C₈	#13063295	1	1	yes	characteristic
331	C₈	#13063298	1	1	yes	characteristic
332	C₈	#13063300	1	1	yes	characteristic
333	C₈	#13063310	1	1	yes	characteristic
334	C₂²	—	4	1	no	—
335	C₂²	—	86	2	no	—
336	C₂²	—	344	1	no	—
337	C₂²	—	172	2	no	—
338	C₂²	—	172	1	no	—
339	C₂²	#14582864	1	1	yes	characteristic
340	C₂²	C₈×D₁₇₂	1	1	yes	characteristic
341	C₂²	#14582969	1	1	yes	characteristic
342	C₄	—	8	1	no	—
343	C₄	—	86	2	no	—
344	C₄	—	172	1	no	—
345	C₄	#14582777	1	1	yes	characteristic
346	C₄	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
347	C₄	#14582964	1	1	yes	characteristic
348	C₄	#14582972	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	344	1	no	—
353	C₂	—	172	2	no	—
354	C₂	#19134886	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13063295
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1378	C₂
socle	C₂×C₄₃	#1869
prime radical	#14583699	C₄
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1378	D₄₃
hypocenter	C₄₃	#12979
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1378

Complements

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

N	H ≅ G/N	type
1. #19147697	C₂	semidirect
3. #19136222	C₂	semidirect
4. C₂×#14582684	C₂	semidirect
5. #19136210	C₂	semidirect
6. #19132684	C₂	semidirect
7. C₄₃×#1378	C₂	semidirect
178. #1378	D₄₃	semidirect
204. C₄₃	#12979	semidirect

Maximal subgroups

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

subgroup	index
#19147697	2
#19143499	2
#19136222	2
C₂×#14582684	2
#19136210	2
#19132684	2
C₄₃×#1378	2
#12979	43