Subgroups — #25015100

Order 11008.

482 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583283	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582720	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582714	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582711	C₂	1	1	yes	maximal, characteristic
5	#19136314	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136310	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
9	C₂²×#10621873	—	2	1	no	—
10	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
11	C₂×#13063313	—	2	1	no	—
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	#14583283	—	2	2	no	—
15	#14583283	C₄	1	4	yes	complemented
16	#14582720	—	2	2	no	—
17	#14582714	—	2	2	no	—
18	#14582711	—	2	2	no	—
19	#14582702	—	2	2	no	—
20	#14582699	—	2	2	no	—
21	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#427	C₂²	1	1	yes	characteristic
23	C₄₃×#405	—	2	1	no	—
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#339	—	2	1	no	—
26	C₂²×Dic₈₆	—	2	1	no	—
27	C₂²×D₁₇₂	—	2	1	no	—
28	C₂×Dic₁₇₂	—	2	1	no	—
29	C₂×Dic₁₇₂	—	2	1	no	—
30	C₂×Dic₁₇₂	—	2	1	no	—
31	C₂×Dic₁₇₂	—	2	1	no	—
32	C₂×Dic₁₇₂	—	2	1	no	—
33	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
34	C₂×#10621873	—	2	4	no	—
35	C₂×#10621873	—	4	1	no	—
36	C₂×#10621873	—	4	1	no	—
37	C₂×#10621873	—	4	2	no	—
38	C₂×D₃₄₄	—	2	1	no	—
39	C₂×D₃₄₄	—	2	1	no	—
40	C₂×D₃₄₄	—	2	1	no	—
41	C₂×D₃₄₄	—	2	1	no	—
42	C₂×D₃₄₄	—	2	1	no	—
43	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
44	#13063313	—	2	4	no	—
45	#13063313	—	4	2	no	—
46	C₂×(C₂×D₄₃)⋊C₄	—	4	2	no	—
47	#13063265	—	2	1	no	—
48	#13063265	—	2	1	no	—
49	#13063265	—	2	1	no	—
50	#13063265	—	2	1	no	—
51	C₂×C₄₃⋊C₁₆	—	2	1	no	—
52	C₂×C₄₃⋊C₁₆	—	2	1	no	—
53	C₂×M₁₆×C₄₃	—	2	1	no	—
54	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	#117×C₄₃	—	2	1	no	—
57	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
58	#108×C₄₃	—	2	1	no	—
59	#108×C₄₃	D₄	1	2	yes	—
60	#107×C₄₃	D₄	1	2	yes	—
61	#106×C₄₃	—	2	1	no	—
62	#97×C₄₃	—	4	1	no	—
63	C₄×C₈×C₄₃	—	4	1	no	—
64	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₄₃	D₄	1	2	yes	—
66	C₂³×D₄₃	—	4	1	no	—
67	C₂²×C₄×C₄₃	—	2	1	no	—
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₃	—	4	1	no	—
70	C₂×D₁₇₂	—	2	2	no	—
71	C₂×D₁₇₂	—	4	1	no	—
72	C₂×D₁₇₂	—	4	1	no	—
73	C₂×D₁₇₂	—	4	1	no	—
74	C₂×D₁₇₂	—	4	1	no	—
75	C₂×D₁₇₂	—	4	1	no	—
76	C₂×Dic₈₆	—	2	2	no	—
77	C₂×Dic₈₆	—	4	1	no	—
78	C₂×Dic₈₆	—	4	1	no	—
79	C₂×Dic₈₆	—	4	1	no	—
80	C₂×Dic₈₆	—	4	1	no	—
81	C₂×Dic₈₆	—	4	1	no	—
82	M₁₆×C₄₃	—	2	2	no	—
83	M₁₆×C₄₃	—	4	1	no	—
84	C₂×C₈×C₄₃	—	4	1	no	—
85	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
87	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
88	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
89	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
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	#41	1	1	yes	characteristic
94	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
95	C₄⋊C₄×C₄₃	—	2	2	no	—
96	C₂²⋊C₄×C₄₃	—	4	1	no	—
97	C₄²×C₄₃	—	2	1	no	—
98	C₄²×C₄₃	—	4	1	no	—
99	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
100	(C₂×D₄₃)⋊C₄	—	4	8	no	—
101	Dic₁₇₂	—	2	2	no	—
102	Dic₁₇₂	—	2	2	no	—
103	Dic₁₇₂	—	4	1	no	—
104	Dic₁₇₂	—	4	1	no	—
105	D₃₄₄	—	2	2	no	—
106	D₃₄₄	—	2	2	no	—
107	D₃₄₄	—	4	1	no	—
108	D₃₄₄	—	4	1	no	—
109	#10621873	—	4	4	no	—
110	#10621873	—	4	4	no	—
111	C₄₃⋊C₁₆	—	4	1	no	—
112	C₄₃⋊C₁₆	—	4	1	no	—
113	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂²×D₄₃	—	4	2	no	—
115	C₂²×D₄₃	—	8	4	no	—
116	C₂²×D₄₃	—	8	1	no	—
117	C₂²×D₄₃	—	8	1	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	4	2	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×Dic₄₃	—	4	2	no	—
128	C₂×Dic₄₃	—	8	1	no	—
129	C₂×Dic₄₃	—	8	1	no	—
130	D₁₇₂	—	4	2	no	—
131	D₁₇₂	—	4	2	no	—
132	D₁₇₂	—	8	1	no	—
133	D₁₇₂	—	8	1	no	—
134	Dic₈₆	—	4	2	no	—
135	Dic₈₆	—	4	2	no	—
136	Dic₈₆	—	8	1	no	—
137	Dic₈₆	—	8	1	no	—
138	C₈×C₄₃	—	4	2	no	—
139	C₈×C₄₃	#102	1	2	yes	—
140	C₈×C₄₃	#102	1	2	yes	—
141	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
142	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
143	C₈×C₄₃	#124	1	1	yes	characteristic
144	C₈×C₄₃	#127	1	1	yes	characteristic
145	#13260	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	—	2	2	no	—
147	C₂²×C₄₃	#418	1	1	yes	characteristic
148	C₂²×C₄₃	#459	1	1	yes	characteristic
149	C₂²×C₄₃	#460	1	1	yes	characteristic
150	C₂×D₄₃	—	8	4	no	—
151	C₂×D₄₃	—	8	2	no	—
152	C₂×D₄₃	—	16	2	no	—
153	C₂×D₄₃	—	16	2	no	—
154	C₄×C₄₃	—	4	2	no	—
155	C₄×C₄₃	—	8	2	no	—
156	C₄×C₄₃	#390	1	1	yes	characteristic
157	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
158	C₄×C₄₃	#459	1	1	yes	characteristic
159	C₄×C₄₃	#460	1	1	yes	characteristic
160	Dic₄₃	—	8	2	no	—
161	C₂×#495	—	43	1	no	—
162	C₂×#364	—	43	1	no	—
163	C₂×#362	—	43	1	no	—
164	C₂×#361	—	43	1	no	—
165	#2141	—	43	1	no	—
166	#2140	—	43	1	no	—
167	#1772	D₄₃	1	1	yes	characteristic, complemented
168	C₂×C₄₃	—	2	1	no	—
169	C₂×C₄₃	—	2	1	no	—
170	C₂×C₄₃	#1982	1	1	yes	characteristic
171	C₂×C₄₃	#2212	1	2	yes	—
172	D₄₃	—	16	2	no	—
173	C₂²×Q₁₆	—	43	1	no	—
174	C₂²×SD₁₆	—	86	1	no	—
175	C₂²×D₈	—	43	1	no	—
176	C₂×#124	—	86	1	no	—
177	C₂×M₃₂	—	43	1	no	—
178	C₂×M₃₂	—	43	1	no	—
179	#495	—	43	4	no	—
180	#495	—	86	2	no	—
181	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
182	#427	C₂×D₄₃	1	1	yes	characteristic
183	#405	—	2	1	no	—
184	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
185	#364	—	86	2	no	—
186	#362	—	86	2	no	—
187	#361	—	86	2	no	—
188	#358	—	86	2	no	—
189	#357	—	86	2	no	—
190	#339	—	2	1	no	—
191	C₄₃	#13260	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	86	1	no	—
193	C₂²×D₄	—	86	1	no	—
194	C₂×Q₁₆	—	43	2	no	—
195	C₂×Q₁₆	—	86	1	no	—
196	C₂×Q₁₆	—	86	1	no	—
197	C₂×Q₁₆	—	86	1	no	—
198	C₂×Q₁₆	—	86	1	no	—
199	C₂×Q₁₆	—	86	1	no	—
200	C₂×SD₁₆	—	86	4	no	—
201	C₂×SD₁₆	—	172	1	no	—
202	C₂×SD₁₆	—	172	1	no	—
203	C₂×SD₁₆	—	172	2	no	—
204	C₂×D₈	—	43	2	no	—
205	C₂×D₈	—	86	1	no	—
206	C₂×D₈	—	86	1	no	—
207	C₂×D₈	—	86	1	no	—
208	C₂×D₈	—	86	1	no	—
209	C₂×D₈	—	86	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
213	#124	—	86	4	no	—
214	#124	—	172	2	no	—
215	#117	—	2	1	no	—
216	C₂×C₂²⋊C₄	—	172	2	no	—
217	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
218	M₃₂	—	86	1	no	—
219	M₃₂	—	86	1	no	—
220	M₃₂	—	86	1	no	—
221	M₃₂	—	86	1	no	—
222	C₂×C₁₆	—	86	1	no	—
223	C₂×C₁₆	—	86	1	no	—
224	#108	—	2	1	no	—
225	#108	D₁₇₂	1	2	yes	—
226	#107	D₁₇₂	1	2	yes	—
227	#106	—	2	1	no	—
228	#97	—	4	1	no	—
229	C₄×C₈	—	4	1	no	—
230	C₄×C₈	C₄×D₄₃	1	2	yes	—
231	C₄×C₈	D₁₇₂	1	2	yes	—
232	C₂⁴	—	172	1	no	—
233	C₂×Q₈	—	86	2	no	—
234	C₂×Q₈	—	172	1	no	—
235	C₂×Q₈	—	172	1	no	—
236	C₂×Q₈	—	172	1	no	—
237	C₂×Q₈	—	172	1	no	—
238	C₂×Q₈	—	172	1	no	—
239	C₂×D₄	—	86	2	no	—
240	C₂×D₄	—	172	1	no	—
241	C₂×D₄	—	172	1	no	—
242	C₂×D₄	—	172	1	no	—
243	C₂×D₄	—	172	1	no	—
244	C₂×D₄	—	172	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	172	1	no	—
247	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
248	Q₁₆	—	86	2	no	—
249	Q₁₆	—	86	2	no	—
250	Q₁₆	—	172	1	no	—
251	Q₁₆	—	172	1	no	—
252	SD₁₆	—	172	4	no	—
253	SD₁₆	—	172	4	no	—
254	D₈	—	86	2	no	—
255	D₈	—	86	2	no	—
256	D₈	—	172	1	no	—
257	D₈	—	172	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
262	C₂×C₈	#10621882	1	1	yes	characteristic
263	C₂×C₈	#10621882	1	1	yes	characteristic
264	C₂×C₈	#10621884	1	2	yes	—
265	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
266	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
267	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
268	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
269	C₂×C₈	#10621898	1	1	yes	characteristic
270	C₂×C₈	#10621898	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	172	8	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
277	C₁₆	—	172	1	no	—
278	C₁₆	—	172	1	no	—
279	C₂³	—	344	4	no	—
280	C₂³	—	344	1	no	—
281	C₂³	—	344	1	no	—
282	C₂³	—	172	2	no	—
283	C₂³	#13063350	1	1	yes	characteristic
284	Q₈	—	344	1	no	—
285	Q₈	—	344	1	no	—
286	Q₈	—	172	2	no	—
287	Q₈	—	172	2	no	—
288	D₄	—	344	1	no	—
289	D₄	—	344	1	no	—
290	D₄	—	172	2	no	—
291	D₄	—	172	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	344	1	no	—
296	C₂×C₄	—	344	1	no	—
297	C₂×C₄	—	172	2	no	—
298	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
299	C₂×C₄	#13063299	1	1	yes	characteristic
300	C₂×C₄	#13063299	1	1	yes	characteristic
301	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
302	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
303	C₂×C₄	#13063350	1	1	yes	characteristic
304	C₈	—	4	2	no	—
305	C₈	#13063245	1	2	yes	—
306	C₈	#13063245	1	2	yes	—
307	C₈	C₄×D₁₇₂	1	1	yes	characteristic
308	C₈	C₄×D₁₇₂	1	1	yes	characteristic
309	C₈	#13063313	1	1	yes	characteristic
310	C₈	#13063322	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	344	4	no	—
313	C₂²	—	344	2	no	—
314	C₂²	—	688	2	no	—
315	C₂²	—	688	2	no	—
316	C₂²	#14582903	1	1	yes	characteristic
317	C₂²	#14583094	1	1	yes	characteristic
318	C₂²	#14583099	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	344	2	no	—
322	C₄	#14582801	1	1	yes	characteristic
323	C₄	C₂×#13063245	1	1	yes	characteristic
324	C₄	#14583094	1	1	yes	characteristic
325	C₄	#14583099	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	688	2	no	—
329	C₂	#19135522	1	1	yes	characteristic
330	C₂	#19136664	1	2	yes	—

	subgroup	G/H
center	C₂²	#14583094
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1772	C₂
socle	C₂²×C₄₃	#459
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1772	D₄₃
hypocenter	C₄₃	#13260
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
2. C₂×#14582720	C₂	semidirect
3. C₂×#14582714	C₂	semidirect
5. #19136314	C₂	semidirect
7. C₄₃×#1772	C₂	semidirect
15. #14583283	C₄	semidirect
167. #1772	D₄₃	semidirect
191. C₄₃	#13260	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583283	2
C₂×#14582720	2
C₂×#14582714	2
C₂×#14582711	2
#19136314	2
#19136310	2
C₄₃×#1772	2
#13260	43