Subgroups — #25011862

Order 11008.

428 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	C₂×#14583017	C₂	1	1	yes	maximal, characteristic
2	#19136345	C₂	1	1	yes	maximal, characteristic
3	#19136332	C₂	1	1	yes	maximal, characteristic
4	#19134474	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132710	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132707	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₂²	1	1	yes	characteristic
9	C₂×#13063331	—	2	1	no	—
10	#14583017	—	2	1	no	—
11	#14583017	—	2	1	no	—
12	#14583017	C₄	1	2	yes	complemented
13	#14583017	C₄	1	2	yes	complemented
14	C₂×#13063261	C₂²	1	1	yes	characteristic
15	C₂×#13063261	C₂²	1	1	yes	characteristic
16	#14582937	C₂²	1	1	yes	characteristic
17	C₂×#13063253	—	2	1	no	—
18	#14582854	—	2	1	no	—
19	#14582716	C₄	1	2	yes	complemented
20	#14582713	C₄	1	2	yes	complemented
21	#14582710	C₄	1	4	yes	complemented
22	#14582659	—	2	1	no	—
23	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#346	—	2	1	no	—
27	C₄×C₁₆×C₄₃	—	2	1	no	—
28	C₂×#10621898	—	2	1	no	—
29	#13063347	—	2	1	no	—
30	#13063347	—	2	1	no	—
31	#13063347	C₂×C₄	1	2	yes	—
32	#13063347	C₂×C₄	1	2	yes	—
33	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
34	C₂×#10621873	—	2	1	no	—
35	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
36	#13063331	—	2	2	no	—
37	#13063331	—	2	2	no	—
38	#13063331	—	4	1	no	—
39	#13063331	—	4	1	no	—
40	C₂×#10621877	—	2	1	no	—
41	C₂×#10621877	—	2	1	no	—
42	C₂×#10621872	—	4	1	no	—
43	C₂×C₈×D₄₃	—	4	1	no	—
44	#13063284	—	2	1	no	—
45	#13063261	—	2	1	no	—
46	#13063261	—	2	1	no	—
47	#13063261	C₂×C₄	1	2	yes	—
48	#13063261	C₂×C₄	1	2	yes	—
49	#13063259	—	2	1	no	—
50	#13063257	C₂×C₄	1	2	yes	—
51	#13063253	—	2	2	no	—
52	#13063253	—	2	2	no	—
53	#13063235	—	4	1	no	—
54	C₈×Dic₄₃	—	4	1	no	—
55	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
58	M₃₂×C₄₃	—	2	1	no	—
59	M₃₂×C₄₃	—	2	1	no	—
60	M₃₂×C₄₃	C₂×C₄	1	2	yes	—
61	M₃₂×C₄₃	C₂×C₄	1	2	yes	—
62	C₂×C₁₆×C₄₃	—	2	1	no	—
63	C₂×C₁₆×C₄₃	—	2	1	no	—
64	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	—	2	1	no	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₄₃	—	4	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	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
81	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
82	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
83	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
84	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
90	C₄²×C₄₃	—	2	1	no	—
91	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂²⋊Dic₄₃	—	4	1	no	—
94	C₄⋊Dic₄₃	—	2	2	no	—
95	C₄×Dic₄₃	—	4	1	no	—
96	#10621877	—	2	2	no	—
97	#10621877	—	2	2	no	—
98	#10621877	—	4	1	no	—
99	#10621877	—	4	1	no	—
100	C₂×C₄₃⋊C₈	—	4	1	no	—
101	C₂×C₄₃⋊C₈	—	4	1	no	—
102	Dic₁₇₂	—	2	1	no	—
103	Dic₁₇₂	C₄²	1	2	yes	—
104	D₃₄₄	—	2	1	no	—
105	D₃₄₄	C₄²	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	—	2	2	no	—
113	C₁₆×C₄₃	—	2	2	no	—
114	C₂³×C₄₃	#135	1	1	yes	characteristic
115	C₂²×D₄₃	—	4	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	4	2	no	—
119	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
125	C₂²⋊D₄₃	—	4	2	no	—
126	C₂²⋊D₄₃	—	4	2	no	—
127	C₂×Dic₄₃	—	4	1	no	—
128	C₂×Dic₄₃	—	4	2	no	—
129	D₁₇₂	—	2	2	no	—
130	D₁₇₂	—	4	1	no	—
131	C₄×D₄₃	—	4	2	no	—
132	C₄×D₄₃	—	4	2	no	—
133	Dic₈₆	—	2	2	no	—
134	Dic₈₆	—	4	1	no	—
135	C₈×C₄₃	#102	1	1	yes	characteristic
136	C₈×C₄₃	#102	1	1	yes	characteristic
137	C₈×C₄₃	#102	1	1	yes	characteristic
138	C₈×C₄₃	#102	1	1	yes	characteristic
139	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
140	C₈×C₄₃	#117	1	1	yes	characteristic
141	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
142	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
143	C₄₃⋊C₈	—	4	2	no	—
144	C₄₃⋊C₈	—	4	2	no	—
145	#12646	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	#416	1	1	yes	characteristic
148	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
149	C₂²×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
150	C₂×D₄₃	—	4	2	no	—
151	C₂×D₄₃	—	8	2	no	—
152	C₄×C₄₃	—	2	2	no	—
153	C₄×C₄₃	—	4	1	no	—
154	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
156	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
157	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
158	Dic₄₃	—	4	2	no	—
159	Dic₄₃	—	8	2	no	—
160	C₂×#443	—	43	1	no	—
161	#2148	—	43	1	no	—
162	#2145	—	43	1	no	—
163	C₄×M₃₂	D₄₃	1	1	yes	characteristic, complemented
164	#1772	—	43	1	no	—
165	#1389	—	43	1	no	—
166	#1388	—	43	1	no	—
167	C₂×C₄₃	—	2	1	no	—
168	C₂×C₄₃	—	2	1	no	—
169	C₂×C₄₃	—	2	1	no	—
170	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
171	D₄₃	—	8	2	no	—
172	C₂×#135	—	43	1	no	—
173	C₂×#131	—	86	1	no	—
174	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
175	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
176	#443	—	43	2	no	—
177	#443	—	43	2	no	—
178	#443	—	86	1	no	—
179	#443	—	86	1	no	—
180	C₂×#108	—	43	1	no	—
181	C₂×#108	—	43	1	no	—
182	#427	—	43	1	no	—
183	C₂×#104	—	86	1	no	—
184	#405	—	86	1	no	—
185	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
186	#362	—	43	2	no	—
187	#361	—	43	2	no	—
188	#360	—	43	4	no	—
189	#346	—	2	1	no	—
190	C₄×C₁₆	—	2	1	no	—
191	#339	—	86	1	no	—
192	C₄₃	#12646	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	86	1	no	—
194	#135	—	43	2	no	—
195	#135	—	43	2	no	—
196	#135	—	86	1	no	—
197	#135	—	86	1	no	—
198	C₂×Q₁₆	—	43	1	no	—
199	C₂×SD₁₆	—	86	1	no	—
200	C₂×D₈	—	43	1	no	—
201	#131	—	86	2	no	—
202	#131	—	86	2	no	—
203	#131	—	172	1	no	—
204	#131	—	172	1	no	—
205	C₂×M₁₆	—	86	1	no	—
206	C₂×M₁₆	—	86	1	no	—
207	C₂×M₁₆	—	172	1	no	—
208	C₂²×C₈	—	172	1	no	—
209	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
211	#117	—	86	1	no	—
212	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
213	M₃₂	—	2	1	no	—
214	M₃₂	—	2	1	no	—
215	M₃₂	C₄×D₄₃	1	2	yes	—
216	M₃₂	C₄×D₄₃	1	2	yes	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	2	1	no	—
219	#108	—	43	2	no	—
220	#108	—	43	2	no	—
221	#108	—	86	1	no	—
222	#108	—	86	1	no	—
223	#107	—	86	1	no	—
224	#106	—	43	2	no	—
225	#104	—	86	2	no	—
226	#104	—	86	2	no	—
227	#97	—	172	1	no	—
228	C₄×C₈	—	172	1	no	—
229	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
230	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
231	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
232	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
233	#41	—	86	2	no	—
234	#41	—	86	2	no	—
235	#41	—	172	1	no	—
236	#41	—	172	1	no	—
237	C₂×Q₈	—	86	1	no	—
238	C₂×D₄	—	86	1	no	—
239	C₂×D₄	—	172	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	172	1	no	—
242	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
243	Q₁₆	—	43	2	no	—
244	Q₁₆	—	86	1	no	—
245	SD₁₆	—	86	2	no	—
246	SD₁₆	—	86	2	no	—
247	D₈	—	43	2	no	—
248	D₈	—	86	1	no	—
249	M₁₆	—	86	2	no	—
250	M₁₆	—	86	2	no	—
251	M₁₆	—	172	1	no	—
252	M₁₆	—	172	1	no	—
253	M₁₆	—	172	2	no	—
254	M₁₆	—	172	2	no	—
255	C₂×C₈	—	172	1	no	—
256	C₂×C₈	—	172	1	no	—
257	C₂×C₈	—	172	2	no	—
258	C₂×C₈	—	172	2	no	—
259	C₂×C₈	#10621873	1	1	yes	characteristic
260	C₂×C₈	#10621873	1	1	yes	characteristic
261	C₂×C₈	D₃₄₄	1	1	yes	characteristic
262	C₂×C₈	D₃₄₄	1	1	yes	characteristic
263	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
264	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
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	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
271	C₄⋊C₄	—	86	2	no	—
272	C₂²⋊C₄	—	172	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	172	1	no	—
275	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
276	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
277	C₁₆	—	2	2	no	—
278	C₁₆	—	2	2	no	—
279	C₂³	—	172	1	no	—
280	C₂³	#13063347	1	1	yes	characteristic
281	Q₈	—	86	2	no	—
282	Q₈	—	172	1	no	—
283	D₄	—	86	2	no	—
284	D₄	—	172	2	no	—
285	D₄	—	172	2	no	—
286	D₄	—	172	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	172	2	no	—
291	C₂×C₄	—	172	1	no	—
292	C₂×C₄	—	172	2	no	—
293	C₂×C₄	—	172	2	no	—
294	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
295	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
296	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
297	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
298	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
299	C₂×C₄	#13063347	1	1	yes	characteristic
300	C₈	—	172	2	no	—
301	C₈	—	172	2	no	—
302	C₈	#13063247	1	1	yes	characteristic
303	C₈	#13063247	1	1	yes	characteristic
304	C₈	#13063247	1	1	yes	characteristic
305	C₈	#13063247	1	1	yes	characteristic
306	C₈	C₄²×D₄₃	1	1	yes	characteristic
307	C₈	#13063282	1	1	yes	characteristic
308	C₈	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
309	C₈	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
310	C₂²	—	4	1	no	—
311	C₂²	—	344	2	no	—
312	C₂²	—	172	2	no	—
313	C₂²	#14582896	1	1	yes	characteristic
314	C₂²	C₄×D₃₄₄	1	1	yes	characteristic
315	C₂²	C₄×#10621873	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	344	2	no	—
319	C₄	—	172	2	no	—
320	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
321	C₄	C₂×#13063247	1	1	yes	characteristic
322	C₄	C₄×D₃₄₄	1	1	yes	characteristic
323	C₄	C₄×#10621873	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	344	2	no	—
328	C₂	C₄×#13063247	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13063247
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄×M₃₂×C₄₃	C₂
socle	C₂×C₄₃	C₄×#102
prime radical	C₂×#13063347	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₄×M₃₂	D₄₃
hypocenter	C₄₃	#12646
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
4. #19134474	C₂	semidirect
5. #19132710	C₂	semidirect
6. #19132707	C₂	semidirect
7. C₄×M₃₂×C₄₃	C₂	semidirect
12. #14583017	C₄	semidirect
13. #14583017	C₄	semidirect
19. #14582716	C₄	semidirect
20. #14582713	C₄	semidirect
21. #14582710	C₄	semidirect
163. C₄×M₃₂	D₄₃	semidirect
192. C₄₃	#12646	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583017	2
#19136345	2
#19136332	2
#19134474	2
#19132710	2
#19132707	2
C₄×M₃₂×C₄₃	2
#12646	43