Subgroups — #25011887

Order 11008.

438 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583020	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582708	C₂	1	1	yes	maximal, characteristic
3	#19136308	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582676	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134495	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132705	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1383	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	#14583020	—	2	2	no	—
12	#14583020	C₄	1	4	yes	complemented
13	C₂×#13063259	C₂²	1	1	yes	characteristic
14	C₂×#13063251	—	2	1	no	—
15	C₂×#13063235	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×Dic₄₃	—	2	1	no	—
17	#14582708	C₄	1	4	yes	complemented
18	#14582704	C₄	1	2	yes	complemented
19	#14582701	C₄	1	2	yes	complemented
20	#14582676	—	2	2	no	—
21	#14582661	—	2	1	no	—
22	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#10621898	—	2	1	no	—
26	#13063347	—	2	1	no	—
27	#13063347	—	2	1	no	—
28	#13063347	C₂×C₄	1	2	yes	—
29	#13063347	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
31	C₂×#10621873	—	2	1	no	—
32	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
33	#13063333	—	2	4	no	—
34	#13063333	—	4	2	no	—
35	C₂×#10621872	—	4	1	no	—
36	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
37	C₂×C₈×D₄₃	—	4	1	no	—
38	C₂×C₄⋊Dic₄₃	—	2	1	no	—
39	C₂×C₄×Dic₄₃	—	4	1	no	—
40	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
41	#13063265	—	2	1	no	—
42	#13063265	C₂×C₄	1	2	yes	—
43	C₂×C₄₃⋊C₁₆	—	2	1	no	—
44	#13063259	—	2	1	no	—
45	#13063259	C₂×C₄	1	2	yes	—
46	#13063251	—	2	4	no	—
47	#13063235	C₂×C₄	1	2	yes	—
48	#13063235	D₄	1	2	yes	—
49	C₄₃⋊#95	—	4	1	no	—
50	C₂×M₁₆×C₄₃	—	2	1	no	—
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
53	M₃₂×C₄₃	—	2	1	no	—
54	M₃₂×C₄₃	—	2	1	no	—
55	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
56	#108×C₄₃	—	2	1	no	—
57	#108×C₄₃	C₂×C₄	1	2	yes	—
58	#106×C₄₃	—	2	1	no	—
59	#106×C₄₃	C₂×C₄	1	2	yes	—
60	C₂²×C₄×C₄₃	—	4	1	no	—
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₃	—	4	1	no	—
63	C₂²×Dic₄₃	—	2	1	no	—
64	C₂²×Dic₄₃	—	4	1	no	—
65	#10621898	—	2	2	no	—
66	#10621898	—	2	2	no	—
67	#10621898	—	4	1	no	—
68	#10621898	—	4	1	no	—
69	C₂×D₁₇₂	—	2	1	no	—
70	C₂×C₄×D₄₃	—	4	1	no	—
71	C₂×Dic₈₆	—	2	1	no	—
72	M₁₆×C₄₃	—	2	2	no	—
73	M₁₆×C₄₃	—	4	1	no	—
74	C₂×C₈×C₄₃	—	4	1	no	—
75	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
80	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
81	C₄⋊C₄×C₄₃	—	2	2	no	—
82	C₄⋊C₄×C₄₃	—	4	1	no	—
83	C₄⋊Dic₄₃	—	2	2	no	—
84	C₄⋊Dic₄₃	—	4	1	no	—
85	C₄×Dic₄₃	—	2	1	no	—
86	C₄×Dic₄₃	—	8	2	no	—
87	C₄×Dic₄₃	C₂²⋊C₄	1	2	yes	—
88	C₂×C₄₃⋊C₈	—	2	1	no	—
89	C₂×C₄₃⋊C₈	—	2	1	no	—
90	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
91	Dic₁₇₂	—	2	1	no	—
92	Dic₁₇₂	C₄²	1	2	yes	complemented
93	D₃₄₄	—	2	1	no	—
94	D₃₄₄	C₄²	1	2	yes	complemented
95	#10621873	—	2	2	no	—
96	#10621873	—	2	2	no	—
97	#10621872	—	4	4	no	—
98	C₈×D₄₃	—	4	4	no	—
99	C₁₆×C₄₃	—	2	2	no	—
100	C₄₃⋊C₁₆	—	2	2	no	—
101	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂²×D₄₃	—	4	1	no	—
103	C₂×C₄×C₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	—	8	1	no	—
105	C₂×C₄×C₄₃	—	8	1	no	—
106	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
112	C₂²⋊D₄₃	—	4	2	no	—
113	C₂²⋊D₄₃	—	4	2	no	—
114	C₂×Dic₄₃	—	2	2	no	—
115	C₂×Dic₄₃	—	4	1	no	—
116	C₂×Dic₄₃	—	4	1	no	—
117	C₂×Dic₄₃	—	4	1	no	—
118	C₂×Dic₄₃	—	4	2	no	—
119	C₂×Dic₄₃	—	8	1	no	—
120	C₂×Dic₄₃	—	8	1	no	—
121	D₁₇₂	—	2	2	no	—
122	D₁₇₂	—	4	1	no	—
123	C₄×D₄₃	—	4	2	no	—
124	C₄×D₄₃	—	4	2	no	—
125	Dic₈₆	—	2	2	no	—
126	Dic₈₆	—	4	1	no	—
127	C₈×C₄₃	—	4	2	no	—
128	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₄₃	#117	1	1	yes	characteristic
130	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
131	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
132	C₄₃⋊C₈	—	2	2	no	—
133	#12650	—	43	1	no	maximal, Sylow
134	C₂²×C₄₃	—	2	1	no	—
135	C₂²×C₄₃	—	2	1	no	—
136	C₂²×C₄₃	#417	1	1	yes	characteristic
137	C₂²×C₄₃	#442	1	1	yes	characteristic
138	C₂²×C₄₃	#442	1	1	yes	characteristic
139	C₂×D₄₃	—	4	2	no	—
140	C₂×D₄₃	—	8	1	no	—
141	C₂×D₄₃	—	8	1	no	—
142	C₄×C₄₃	—	8	2	no	—
143	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₄₃	#418	1	1	yes	characteristic
145	C₄×C₄₃	#440	1	1	yes	characteristic
146	C₄×C₄₃	#440	1	1	yes	characteristic
147	Dic₄₃	—	4	2	no	—
148	Dic₄₃	—	4	2	no	—
149	Dic₄₃	—	8	2	no	—
150	C₂×#444	—	43	1	no	—
151	C₂×#360	—	43	1	no	—
152	#2139	—	43	1	no	—
153	C₂×#347	—	43	1	no	—
154	#1776	—	43	1	no	—
155	#1387	—	43	1	no	—
156	#1383	D₄₃	1	1	yes	characteristic, complemented
157	C₂×C₄₃	—	2	1	no	—
158	C₂×C₄₃	—	2	1	no	—
159	C₂×C₄₃	#1762	1	1	yes	characteristic
160	C₂×C₄₃	#2180	1	1	yes	characteristic
161	C₂×C₄₃	AGL(1,ℤ/16)	1	1	yes	characteristic
162	D₄₃	—	8	2	no	—
163	C₂×#135	—	43	1	no	—
164	C₂×#131	—	86	1	no	—
165	C₂×M₃₂	—	43	1	no	—
166	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
167	#444	—	43	4	no	—
168	#444	—	86	2	no	—
169	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
170	C₂×#107	—	43	1	no	—
171	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
172	C₂×#104	—	86	1	no	—
173	C₂×#97	—	43	1	no	—
174	C₄×C₄⋊C₄	—	86	1	no	—
175	#360	—	43	4	no	—
176	#358	—	43	2	no	—
177	#357	—	43	2	no	—
178	#347	—	86	2	no	—
179	#340	—	86	1	no	—
180	C₄₃	#12650	1	1	yes	characteristic, complemented, Sylow
181	C₂×#41	—	86	1	no	—
182	#135	—	43	2	no	—
183	#135	—	43	2	no	—
184	#135	—	86	1	no	—
185	#135	—	86	1	no	—
186	C₂×Q₁₆	—	43	1	no	—
187	C₂×SD₁₆	—	86	1	no	—
188	C₂×D₈	—	43	1	no	—
189	#131	—	86	4	no	—
190	#131	—	172	2	no	—
191	C₂×M₁₆	—	2	1	no	—
192	C₂×M₁₆	—	172	1	no	—
193	C₂²×C₈	—	43	1	no	—
194	C₂²×C₈	—	172	1	no	—
195	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	86	1	no	—
198	C₂×C₄²	—	43	1	no	—
199	C₂×C₄²	—	172	1	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	—	43	2	no	—
203	M₃₂	—	86	1	no	—
204	C₂×C₁₆	—	86	1	no	—
205	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
206	#108	—	2	1	no	—
207	#108	C₄×D₄₃	1	2	yes	—
208	#107	—	43	2	no	—
209	#107	—	86	1	no	—
210	#106	—	2	1	no	—
211	#106	C₄×D₄₃	1	2	yes	—
212	#104	—	86	4	no	—
213	#97	—	43	2	no	—
214	#97	—	43	2	no	—
215	#95	—	172	1	no	—
216	#41	—	86	2	no	—
217	#41	—	86	2	no	—
218	#41	—	172	1	no	—
219	#41	—	172	1	no	—
220	C₂×Q₈	—	86	1	no	—
221	C₂×D₄	—	86	1	no	—
222	C₂×D₄	—	172	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	86	1	no	—
225	C₂²×C₄	—	172	1	no	—
226	C₂²×C₄	—	172	1	no	—
227	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
228	Q₁₆	—	43	2	no	—
229	Q₁₆	—	86	1	no	—
230	SD₁₆	—	86	2	no	—
231	SD₁₆	—	86	2	no	—
232	D₈	—	43	2	no	—
233	D₈	—	86	1	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	4	1	no	—
236	M₁₆	—	172	4	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	43	2	no	—
239	C₂×C₈	—	86	1	no	—
240	C₂×C₈	—	86	1	no	—
241	C₂×C₈	—	172	4	no	—
242	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
245	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
246	C₂×C₈	#10621900	1	1	yes	characteristic
247	C₂×C₈	#10621902	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	86	2	no	—
251	C₄⋊C₄	—	172	1	no	—
252	C₄²	—	43	2	no	—
253	C₄²	—	86	1	no	—
254	C₄²	—	344	2	no	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	86	2	no	—
257	C₂³	—	172	1	no	—
258	C₂³	#13063352	1	1	yes	characteristic
259	Q₈	—	86	2	no	—
260	Q₈	—	172	1	no	—
261	D₄	—	86	2	no	—
262	D₄	—	172	2	no	—
263	D₄	—	172	2	no	—
264	D₄	—	172	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	1	no	—
267	C₂×C₄	—	8	1	no	—
268	C₂×C₄	—	86	2	no	—
269	C₂×C₄	—	344	1	no	—
270	C₂×C₄	—	344	1	no	—
271	C₂×C₄	—	172	2	no	—
272	C₂×C₄	—	172	1	no	—
273	C₂×C₄	—	172	1	no	—
274	C₂×C₄	—	172	1	no	—
275	C₂×C₄	—	172	2	no	—
276	C₂×C₄	—	172	2	no	—
277	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
278	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
279	C₂×C₄	#13063287	1	1	yes	characteristic
280	C₂×C₄	#13063349	1	1	yes	characteristic
281	C₂×C₄	#13063352	1	1	yes	characteristic
282	C₂×C₄	#13063357	1	1	yes	characteristic
283	C₈	—	4	2	no	—
284	C₈	—	86	2	no	—
285	C₈	C₄²×D₄₃	1	1	yes	characteristic
286	C₈	#13063283	1	1	yes	characteristic
287	C₈	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
288	C₈	#13063287	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	344	1	no	—
292	C₂²	—	344	1	no	—
293	C₂²	—	172	2	no	—
294	C₂²	#14582899	1	1	yes	characteristic
295	C₂²	#14583008	1	1	yes	characteristic
296	C₂²	#14583010	1	1	yes	characteristic
297	C₄	—	8	2	no	—
298	C₄	—	344	2	no	—
299	C₄	—	172	2	no	—
300	C₄	—	172	2	no	—
301	C₄	#14582746	1	1	yes	characteristic
302	C₄	#14582902	1	1	yes	characteristic
303	C₄	#14582998	1	1	yes	characteristic
304	C₄	#14583002	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	344	2	no	—
308	C₂	#19134419	1	1	yes	characteristic
309	C₂	#19136485	1	1	yes	characteristic
310	C₂	#19136492	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14583010
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1383	C₂
socle	C₂²×C₄₃	#442
prime radical	C₂×#13063347	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1383	D₄₃
hypocenter	C₄₃	#12650
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1383

Complements

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

N	H ≅ G/N	type
4. C₂×#14582676	C₂	semidirect
5. #19134495	C₂	semidirect
6. #19132705	C₂	semidirect
7. C₄₃×#1383	C₂	semidirect
12. #14583020	C₄	semidirect
17. #14582708	C₄	semidirect
18. #14582704	C₄	semidirect
19. #14582701	C₄	semidirect
92. Dic₁₇₂	C₄²	semidirect
94. D₃₄₄	C₄²	semidirect
156. #1383	D₄₃	semidirect
180. C₄₃	#12650	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583020	2
C₂×#14582708	2
#19136308	2
C₂×#14582676	2
#19134495	2
#19132705	2
C₄₃×#1383	2
#12650	43