Subgroups — #25002222

Order 11008.

650 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×#14582857	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃⋊#338	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582655	C₂	1	2	yes	maximal
4	#19134292	C₂	1	2	yes	maximal
5	C₂×C₄₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#13063284	C₂²	1	1	yes	characteristic
9	#14582857	—	2	2	no	—
10	#14582857	—	2	2	no	—
11	C₂×#13063235	C₄	1	2	yes	complemented
12	C₂×C₈×Dic₄₃	C₄	1	2	yes	complemented
13	C₄₃⋊#338	—	2	2	no	—
14	#14582655	—	2	4	no	—
15	#14582654	C₄	1	8	yes	complemented
16	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#117×C₄₃	C₂²	1	2	yes	—
18	C₄₃×H(3,ℤ/4)	—	2	2	no	—
19	C₂×#10621877	—	2	2	no	—
20	C₂×#10621877	—	2	2	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	2	no	—
25	C₂²×C₄₃⋊C₈	—	2	1	no	—
26	C₂²×C₄₃⋊C₈	—	2	1	no	—
27	C₂²×C₄₃⋊C₈	—	2	2	no	—
28	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
32	#13063284	—	2	1	no	—
33	#13063284	—	2	1	no	—
34	#13063284	D₄	1	2	yes	—
35	#13063284	Q₈	1	2	yes	—
36	C₂×C₄⋊Dic₄₃	—	2	1	no	—
37	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
38	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
39	#13063235	—	2	4	no	—
40	C₈×Dic₄₃	—	2	4	no	—
41	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₄₃	—	2	2	no	—
43	C₂×M₁₆×C₄₃	—	2	2	no	—
44	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
45	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
46	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
47	#117×C₄₃	—	2	2	no	—
48	#117×C₄₃	—	2	2	no	—
49	#117×C₄₃	—	2	2	no	—
50	#117×C₄₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
53	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
54	C₂⁴×C₄₃	#41	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	—	2	4	no	—
56	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	#41	1	2	yes	—
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	#41	1	2	yes	—
65	C₂²×C₄×C₄₃	#41	1	2	yes	—
66	C₂²×Dic₄₃	—	2	2	no	—
67	M₁₆×C₄₃	—	2	4	no	—
68	M₁₆×C₄₃	—	2	4	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	—	2	2	no	—
71	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
72	C₄⋊C₄×C₄₃	—	4	2	no	—
73	C₄⋊C₄×C₄₃	—	4	2	no	—
74	C₂²⋊C₄×C₄₃	—	4	2	no	—
75	C₂²⋊C₄×C₄₃	—	4	2	no	—
76	C₄²×C₄₃	—	2	4	no	—
77	C₄²×C₄₃	—	2	4	no	—
78	C₂²⋊Dic₄₃	—	2	2	no	—
79	C₂²⋊Dic₄₃	—	2	2	no	—
80	C₄⋊Dic₄₃	—	2	2	no	—
81	C₄⋊Dic₄₃	—	2	2	no	—
82	C₄×Dic₄₃	—	2	2	no	—
83	C₄×Dic₄₃	C₄⋊C₄	1	4	yes	—
84	#10621877	—	4	2	no	—
85	#10621877	—	4	2	no	—
86	#10621877	—	4	2	no	—
87	#10621877	—	4	2	no	—
88	C₂×C₄₃⋊C₈	—	2	1	no	—
89	C₂×C₄₃⋊C₈	—	2	1	no	—
90	C₂×C₄₃⋊C₈	—	2	1	no	—
91	C₂×C₄₃⋊C₈	—	2	1	no	—
92	C₂×C₄₃⋊C₈	—	2	4	no	—
93	C₂×C₄₃⋊C₈	—	2	4	no	—
94	C₂×C₄₃⋊C₈	—	2	4	no	—
95	C₂×C₄₃⋊C₈	—	2	4	no	—
96	C₂×C₄₃⋊C₈	—	2	4	no	—
97	C₂×C₄₃⋊C₈	C₄²	1	2	yes	—
98	C₂×C₄₃⋊C₈	C₄²	1	2	yes	—
99	C₂³×C₄₃	—	4	2	no	—
100	C₂³×C₄₃	#117	1	2	yes	—
101	C₂³×C₄₃	#117	1	1	yes	characteristic
102	C₂³×C₄₃	#121	1	1	yes	characteristic
103	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₄₃	#126	1	2	yes	—
105	C₂×C₄×C₄₃	—	4	8	no	—
106	C₂×C₄×C₄₃	—	4	4	no	—
107	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#117	1	2	yes	—
114	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#123	1	2	yes	—
121	C₂×C₄×C₄₃	#123	1	2	yes	—
122	C₂×C₄×C₄₃	#124	1	2	yes	—
123	C₂×C₄×C₄₃	#125	1	2	yes	—
124	C₂×C₄×C₄₃	#126	1	2	yes	—
125	C₂×Dic₄₃	—	2	4	no	—
126	C₂×Dic₄₃	—	4	4	no	—
127	C₈×C₄₃	—	2	4	no	—
128	C₄₃⋊C₈	—	2	2	no	—
129	C₄₃⋊C₈	—	2	2	no	—
130	C₄₃⋊C₈	—	4	4	no	—
131	#10800	—	43	1	no	maximal, Sylow
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	—	2	4	no	—
134	C₂²×C₄₃	—	4	1	no	—
135	C₂²×C₄₃	—	4	1	no	—
136	C₂²×C₄₃	—	4	2	no	—
137	C₂²×C₄₃	#376	1	1	yes	characteristic
138	C₂²×C₄₃	#380	1	1	yes	characteristic
139	C₂²×C₄₃	#383	1	1	yes	characteristic
140	C₂²×C₄₃	#387	1	2	yes	—
141	C₂²×C₄₃	#388	1	2	yes	—
142	C₄×C₄₃	—	8	4	no	—
143	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₄₃	#381	1	1	yes	characteristic
146	C₄×C₄₃	#382	1	1	yes	characteristic
147	C₄×C₄₃	#387	1	2	yes	—
148	C₄×C₄₃	#388	1	2	yes	—
149	Dic₄₃	—	4	4	no	—
150	C₂×#405	—	43	1	no	—
151	C₂×#338	—	43	1	no	—
152	C₂×H(3,ℤ/4)	—	43	2	no	—
153	C₂×H(3,ℤ/4)	D₄₃	1	1	yes	characteristic
154	#1729	—	43	2	no	—
155	C₂×C₄₃	—	2	1	no	—
156	C₂×C₄₃	—	2	2	no	—
157	C₂×C₄₃	—	2	1	no	—
158	C₂×C₄₃	—	2	2	no	—
159	C₂×C₄₃	#1434	1	1	yes	characteristic
160	C₂×C₄₃	#1747	1	2	yes	—
161	C₂²×M₁₆	—	43	2	no	—
162	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
163	C₂³×C₈	—	43	1	no	—
164	C₂×#117	—	43	1	no	—
165	C₂×#117	C₂×D₄₃	1	2	yes	—
166	#405	—	86	2	no	—
167	#405	—	86	2	no	—
168	C₂×#97	—	43	2	no	—
169	C₂×C₄×C₈	—	43	2	no	—
170	#338	—	86	2	no	—
171	H(3,ℤ/4)	—	2	2	no	—
172	H(3,ℤ/4)	—	86	4	no	—
173	#336	—	43	8	no	—
174	C₄₃	#10800	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	86	2	no	—
179	C₂×M₁₆	—	86	2	no	—
180	C₂×M₁₆	—	86	2	no	—
181	C₂×M₁₆	—	86	2	no	—
182	C₂×M₁₆	—	86	2	no	—
183	C₂×M₁₆	—	86	2	no	—
184	C₂×M₁₆	D₁₇₂	1	2	yes	—
185	C₂×M₁₆	C₂²⋊D₄₃	1	2	yes	—
186	C₂²×C₈	—	43	4	no	—
187	C₂²×C₈	—	43	1	no	—
188	C₂²×C₈	—	43	1	no	—
189	C₂²×C₈	—	43	4	no	—
190	C₂²×C₈	—	86	1	no	—
191	C₂²×C₈	—	86	2	no	—
192	C₂²×C₈	—	86	1	no	—
193	C₂²×C₈	C₄×D₄₃	1	2	yes	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	43	2	no	—
199	#117	—	43	2	no	—
200	#117	—	86	1	no	—
201	#117	—	86	1	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	86	1	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	86	1	no	—
206	C₂×C₄²	—	43	2	no	—
207	C₂×C₄²	C₄×D₄₃	1	4	yes	—
208	#97	—	86	4	no	—
209	C₄×C₈	—	86	4	no	—
210	C₂⁴	#10621900	1	1	yes	characteristic
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	86	2	no	—
213	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
215	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
216	C₂²×C₄	#10621898	1	2	yes	—
217	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
218	C₂²×C₄	#10621900	1	2	yes	—
219	C₂²×C₄	#10621900	1	2	yes	—
220	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
221	C₂²×C₄	#10621902	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	172	2	no	—
226	M₁₆	—	172	2	no	—
227	M₁₆	—	172	2	no	—
228	M₁₆	—	172	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	43	2	no	—
232	C₂×C₈	—	43	2	no	—
233	C₂×C₈	—	86	1	no	—
234	C₂×C₈	—	86	1	no	—
235	C₂×C₈	—	86	1	no	—
236	C₂×C₈	—	86	1	no	—
237	C₂×C₈	—	86	4	no	—
238	C₂×C₈	—	86	4	no	—
239	C₂×C₈	—	86	4	no	—
240	C₂×C₈	—	86	4	no	—
241	C₂×C₈	—	86	4	no	—
242	C₂×C₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	86	2	no	—
246	C₄⋊C₄	—	86	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	86	2	no	—
250	C₂²⋊C₄	—	86	2	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	43	4	no	—
254	C₄²	—	86	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	#13063281	1	2	yes	—
257	C₂³	#13063283	1	1	yes	characteristic
258	C₂³	#13063296	1	1	yes	characteristic
259	C₂³	#13063305	1	1	yes	characteristic
260	C₂³	#13063318	1	2	yes	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	86	4	no	—
264	C₂×C₄	—	172	4	no	—
265	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
266	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
268	C₂×C₄	#13063281	1	1	yes	characteristic
269	C₂×C₄	#13063281	1	1	yes	characteristic
270	C₂×C₄	#13063282	1	1	yes	characteristic
271	C₂×C₄	#13063283	1	2	yes	—
272	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
273	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
274	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
275	C₂×C₄	#13063290	1	2	yes	—
276	C₂×C₄	#13063300	1	1	yes	characteristic
277	C₂×C₄	#13063301	1	1	yes	characteristic
278	C₂×C₄	#13063306	1	2	yes	—
279	C₂×C₄	#13063309	1	2	yes	—
280	C₂×C₄	#13063311	1	2	yes	—
281	C₂×C₄	#13063315	1	2	yes	—
282	C₂×C₄	#13063320	1	2	yes	—
283	C₈	—	2	4	no	—
284	C₈	—	86	2	no	—
285	C₈	—	86	2	no	—
286	C₈	—	172	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	#14582744	1	1	yes	characteristic
293	C₂²	#14582756	1	1	yes	characteristic
294	C₂²	#14582766	1	1	yes	characteristic
295	C₂²	#14582784	1	2	yes	—
296	C₂²	#14582787	1	2	yes	—
297	C₄	—	8	4	no	—
298	C₄	—	172	4	no	—
299	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
300	C₄	#14582750	1	1	yes	characteristic
301	C₄	#14582760	1	1	yes	characteristic
302	C₄	#14582763	1	1	yes	characteristic
303	C₄	#14582780	1	2	yes	—
304	C₄	#14582791	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	1	no	—
309	C₂	#19132819	1	1	yes	characteristic
310	C₂	#19134349	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×H(3,ℤ/4)	C₂
socle	C₂²×C₄₃	#376
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₄₃
hypocenter	C₄₃	#10800
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
11. C₂×#13063235	C₄	semidirect
12. C₂×C₈×Dic₄₃	C₄	semidirect
15. #14582654	C₄	semidirect
174. C₄₃	#10800	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582857	2
C₂×C₄₃⋊#338	2
C₂×#14582655	2
#19134292	2
C₂×C₄₃×H(3,ℤ/4)	2
#10800	43