Subgroups — #25005276

Order 11008.

694 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582948	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582926	C₂	1	2	yes	maximal
3	#19134305	C₂	1	2	yes	maximal
4	#19134293	C₂	1	1	yes	maximal, characteristic
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	#14582948	—	2	4	no	—
10	C₂×#13063261	—	2	2	no	—
11	#14582926	—	2	4	no	—
12	#14582926	—	2	4	no	—
13	C₂×C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
14	#14582659	—	2	4	no	—
15	#14582653	—	2	2	no	—
16	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#117×C₄₃	C₂²	1	2	yes	—
18	C₄₃×H(3,ℤ/4)	C₄	1	4	yes	complemented
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	D₄	1	4	yes	—
25	C₂²×C₄₃⋊C₈	—	2	1	no	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	C₂²×C₄₃⋊C₈	—	2	2	no	—
28	C₂²×C₄₃⋊C₈	—	2	2	no	—
29	C₂²×C₄₃⋊C₈	—	2	1	no	—
30	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
31	#13063284	—	2	2	no	—
32	#13063284	—	2	2	no	—
33	C₂×C₄⋊Dic₄₃	Q₈	1	2	yes	—
34	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
35	C₂×C₄×Dic₄₃	—	2	1	no	—
36	#13063261	—	2	8	no	—
37	C₄⋊(C₄₃⋊C₈)	—	2	16	no	—
38	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₄₃	—	2	2	no	—
40	C₂×M₁₆×C₄₃	—	2	2	no	—
41	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₄₃	Q₈	1	2	yes	—
43	C₂²×C₈×C₄₃	D₄	1	2	yes	—
44	#117×C₄₃	—	2	2	no	—
45	#117×C₄₃	—	2	2	no	—
46	#117×C₄₃	C₂×C₄	1	4	yes	—
47	#117×C₄₃	Q₈	1	4	yes	—
48	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
50	C₂×C₄²×C₄₃	D₄	1	4	yes	—
51	C₂⁴×C₄₃	#41	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	—	2	4	no	—
53	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	#41	1	2	yes	—
62	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
63	C₂²×Dic₄₃	—	2	2	no	—
64	M₁₆×C₄₃	—	2	4	no	—
65	M₁₆×C₄₃	—	2	4	no	—
66	C₂×C₈×C₄₃	—	2	2	no	—
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
69	C₄⋊C₄×C₄₃	—	2	4	no	—
70	C₄⋊C₄×C₄₃	—	2	4	no	—
71	C₂²⋊C₄×C₄₃	—	2	4	no	—
72	C₂²⋊C₄×C₄₃	—	2	4	no	—
73	C₄²×C₄₃	—	2	4	no	—
74	C₄²×C₄₃	C₄⋊C₄	1	8	yes	—
75	C₂²⋊Dic₄₃	—	4	2	no	—
76	C₄⋊Dic₄₃	—	2	4	no	—
77	C₄×Dic₄₃	—	4	1	no	—
78	C₄×Dic₄₃	—	4	1	no	—
79	#10621877	—	2	4	no	—
80	#10621877	—	2	4	no	—
81	#10621877	—	4	2	no	—
82	#10621877	—	4	2	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂×C₄₃⋊C₈	—	2	2	no	—
86	C₂×C₄₃⋊C₈	—	4	1	no	—
87	C₂×C₄₃⋊C₈	—	4	2	no	—
88	C₂×C₄₃⋊C₈	—	4	2	no	—
89	C₂×C₄₃⋊C₈	—	4	1	no	—
90	C₂×C₄₃⋊C₈	—	4	2	no	—
91	C₂³×C₄₃	—	4	2	no	—
92	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₄₃	C₂²⋊Q₈	1	2	yes	—
94	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂³×C₄₃	#123	1	2	yes	—
96	C₂³×C₄₃	#125	1	1	yes	characteristic
97	C₂×C₄×C₄₃	—	2	8	no	—
98	C₂×C₄×C₄₃	—	4	8	no	—
99	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
104	C₂×C₄×C₄₃	#120	1	2	yes	—
105	C₂×C₄×C₄₃	#121	1	2	yes	—
106	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#125	1	2	yes	—
112	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×Dic₄₃	—	4	4	no	—
118	C₂×Dic₄₃	—	4	2	no	—
119	C₈×C₄₃	—	2	4	no	—
120	C₄₃⋊C₈	—	4	2	no	—
121	C₄₃⋊C₈	—	4	4	no	—
122	#11336	—	43	1	no	maximal, Sylow
123	C₂²×C₄₃	—	2	2	no	—
124	C₂²×C₄₃	—	2	4	no	—
125	C₂²×C₄₃	—	4	1	no	—
126	C₂²×C₄₃	—	4	2	no	—
127	C₂²×C₄₃	—	4	1	no	—
128	C₂²×C₄₃	#385	1	2	yes	—
129	C₂²×C₄₃	#389	1	1	yes	characteristic
130	C₂²×C₄₃	#395	1	1	yes	characteristic
131	C₂²×C₄₃	#398	1	2	yes	—
132	C₂²×C₄₃	#399	1	1	yes	characteristic
133	C₄×C₄₃	—	4	8	no	—
134	C₄×C₄₃	#384	1	1	yes	characteristic
135	C₄×C₄₃	#385	1	1	yes	characteristic
136	C₄×C₄₃	#389	1	2	yes	—
137	C₄×C₄₃	#392	1	1	yes	characteristic
138	C₄×C₄₃	#395	1	2	yes	—
139	C₄×C₄₃	#398	1	1	yes	characteristic
140	Dic₄₃	—	8	2	no	—
141	C₂×#430	—	43	1	no	—
142	C₂×#424	—	43	2	no	—
143	#1733	—	43	2	no	—
144	C₂×H(3,ℤ/4)	D₄₃	1	1	yes	characteristic
145	#1729	—	43	1	no	—
146	C₂×C₄₃	—	2	2	no	—
147	C₂×C₄₃	—	2	1	no	—
148	C₂×C₄₃	—	2	1	no	—
149	C₂×C₄₃	—	2	2	no	—
150	C₂×C₄₃	#1444	1	1	yes	characteristic
151	C₂×C₄₃	#2098	1	2	yes	—
152	C₂²×M₁₆	—	43	2	no	—
153	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
154	C₂³×C₈	—	43	1	no	—
155	C₂×#117	—	43	1	no	—
156	C₂×#117	C₂×D₄₃	1	2	yes	—
157	#430	—	86	4	no	—
158	C₂×#108	—	86	2	no	—
159	#424	—	86	4	no	—
160	#424	—	86	4	no	—
161	C₂×C₄⋊C₈	—	86	4	no	—
162	#339	—	86	4	no	—
163	H(3,ℤ/4)	Dic₄₃	1	4	yes	complemented
164	#336	—	86	2	no	—
165	C₄₃	#11336	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	43	4	no	—
170	C₂×M₁₆	—	86	2	no	—
171	C₂×M₁₆	—	86	2	no	—
172	C₂×M₁₆	—	86	2	no	—
173	C₂×M₁₆	—	86	2	no	—
174	C₂×M₁₆	—	86	2	no	—
175	C₂×M₁₆	Dic₈₆	1	2	yes	—
176	C₂×M₁₆	C₂×Dic₄₃	1	2	yes	—
177	C₂²×C₈	—	43	2	no	—
178	C₂²×C₈	—	86	1	no	—
179	C₂²×C₈	—	86	2	no	—
180	C₂²×C₈	—	86	2	no	—
181	C₂²×C₈	—	86	2	no	—
182	C₂²×C₈	—	86	1	no	—
183	C₂²×C₈	D₁₇₂	1	2	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	86	2	no	—
187	#117	—	86	2	no	—
188	#117	Dic₈₆	1	4	yes	—
189	#117	C₂×Dic₄₃	1	4	yes	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	43	2	no	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₂²⋊C₄	—	86	1	no	—
194	C₂×C₄²	—	86	1	no	—
195	C₂×C₄²	D₁₇₂	1	4	yes	—
196	#108	—	86	8	no	—
197	C₄⋊C₈	—	86	16	no	—
198	C₂⁴	#10621900	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	86	2	no	—
201	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
202	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
203	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
204	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₄₃	1	2	yes	—
206	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
207	C₂²×C₄	#10621900	1	2	yes	—
208	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
209	C₂²×C₄	#10621902	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	86	4	no	—
214	M₁₆	—	86	4	no	—
215	M₁₆	—	172	2	no	—
216	M₁₆	—	172	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	86	4	no	—
220	C₂×C₈	—	86	4	no	—
221	C₂×C₈	—	86	2	no	—
222	C₂×C₈	—	172	1	no	—
223	C₂×C₈	—	172	2	no	—
224	C₂×C₈	—	172	2	no	—
225	C₂×C₈	—	172	1	no	—
226	C₂×C₈	—	172	2	no	—
227	C₂×C₈	C₄⋊Dic₄₃	1	4	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	86	4	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	172	2	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	172	1	no	—
236	C₄²	—	172	1	no	—
237	C₄²	C₄⋊Dic₄₃	1	8	yes	—
238	C₂³	—	4	2	no	—
239	C₂³	D₄×Dic₄₃	1	1	yes	characteristic
240	C₂³	#13063302	1	2	yes	—
241	C₂³	#13063305	1	1	yes	characteristic
242	C₂³	#13063309	1	2	yes	—
243	C₂³	#13063316	1	1	yes	characteristic
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	—	172	4	no	—
247	C₂×C₄	—	172	2	no	—
248	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
249	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
250	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
251	C₂×C₄	Q₈×Dic₄₃	1	1	yes	characteristic
252	C₂×C₄	#13063295	1	2	yes	—
253	C₂×C₄	#13063295	1	1	yes	characteristic
254	C₂×C₄	#13063300	1	2	yes	—
255	C₂×C₄	#13063300	1	1	yes	characteristic
256	C₂×C₄	#13063302	1	2	yes	—
257	C₂×C₄	#13063302	1	2	yes	—
258	C₂×C₄	#13063305	1	2	yes	—
259	C₂×C₄	#13063309	1	1	yes	characteristic
260	C₂×C₄	#13063316	1	2	yes	—
261	C₂×C₄	#13063322	1	1	yes	characteristic
262	C₂×C₄	#13063324	1	2	yes	—
263	C₂×C₄	#13063324	1	1	yes	characteristic
264	C₂×C₄	#13063325	1	2	yes	—
265	C₂×C₄	#13063325	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	172	2	no	—
268	C₈	—	172	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	1	no	—
274	C₂²	#14582774	1	2	yes	—
275	C₂²	#14582798	1	1	yes	characteristic
276	C₂²	#14582820	1	1	yes	characteristic
277	C₂²	#14582831	1	2	yes	—
278	C₂²	#14582836	1	1	yes	characteristic
279	C₄	—	4	8	no	—
280	C₄	—	344	2	no	—
281	C₄	#14582769	1	1	yes	characteristic
282	C₄	#14582774	1	1	yes	characteristic
283	C₄	#14582798	1	2	yes	—
284	C₄	#14582809	1	1	yes	characteristic
285	C₄	#14582820	1	2	yes	—
286	C₄	#14582831	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	#19132860	1	1	yes	characteristic
292	C₂	#19136149	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063322
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×H(3,ℤ/4)	C₂
socle	C₂²×C₄₃	#399
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₄₃
hypocenter	C₄₃	#11336
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
18. C₄₃×H(3,ℤ/4)	C₄	semidirect
163. H(3,ℤ/4)	Dic₄₃	semidirect
165. C₄₃	#11336	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582948	2
C₂×#14582926	2
#19134305	2
#19134293	2
C₂×C₄₃×H(3,ℤ/4)	2
#11336	43