Subgroups — #25004797

Order 11008.

654 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134312	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582660	C₂	1	2	yes	maximal
3	#19134293	C₂	1	2	yes	maximal
4	C₂×#14582653	C₂	1	1	yes	maximal, characteristic
5	C₂²×#108×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×#13063284	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
9	#14582662	C₄	1	4	yes	complemented
10	#14582660	—	2	4	no	—
11	#14582653	—	2	4	no	—
12	#14582653	—	2	2	no	—
13	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
14	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×#108×C₄₃	—	2	4	no	—
16	C₂×#108×C₄₃	—	2	2	no	—
17	C₂³×Dic₄₃	—	2	1	no	—
18	C₂²×C₄₃⋊C₈	—	2	1	no	—
19	C₂²×C₄₃⋊C₈	—	2	2	no	—
20	C₂²×C₄₃⋊C₈	—	2	1	no	—
21	C₂²×C₄₃⋊C₈	—	2	2	no	—
22	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
23	#13063284	—	2	2	no	—
24	#13063284	—	2	2	no	—
25	#13063284	—	2	2	no	—
26	#13063284	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
29	C₂×C₄×Dic₄₃	—	2	2	no	—
30	C₂×C₄×Dic₄₃	—	2	2	no	—
31	C₂×C₄×Dic₄₃	—	2	1	no	—
32	C₂×C₄×Dic₄₃	—	2	2	no	—
33	C₂×C₄×Dic₄₃	Q₈	1	2	yes	—
34	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₄₃	—	2	2	no	—
36	C₂×M₁₆×C₄₃	—	2	2	no	—
37	C₂×M₁₆×C₄₃	—	2	2	no	—
38	C₂×M₁₆×C₄₃	—	2	2	no	—
39	C₂×M₁₆×C₄₃	—	2	2	no	—
40	C₂×M₁₆×C₄₃	D₄	1	4	yes	—
41	C₂²×C₈×C₄₃	—	2	2	no	—
42	C₂²×C₈×C₄₃	—	2	2	no	—
43	C₂²×C₈×C₄₃	—	2	2	no	—
44	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₄₃	D₄	1	2	yes	—
46	C₂²×C₈×C₄₃	Q₈	1	2	yes	—
47	#108×C₄₃	—	2	8	no	—
48	C₂⁴×C₄₃	#41	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	#41	1	2	yes	—
56	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	#41	1	2	yes	—
58	C₂²×C₄×C₄₃	#41	1	2	yes	—
59	C₂²×Dic₄₃	—	2	4	no	—
60	C₂²×Dic₄₃	—	2	4	no	—
61	C₂²×Dic₄₃	—	2	2	no	—
62	C₂²×Dic₄₃	—	4	2	no	—
63	C₂²×Dic₄₃	—	4	2	no	—
64	M₁₆×C₄₃	—	2	4	no	—
65	M₁₆×C₄₃	—	2	4	no	—
66	M₁₆×C₄₃	—	4	2	no	—
67	M₁₆×C₄₃	—	4	2	no	—
68	C₂×C₈×C₄₃	—	2	4	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	—	2	4	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	2	4	no	—
73	C₂×C₈×C₄₃	—	4	2	no	—
74	C₂×C₈×C₄₃	—	4	2	no	—
75	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
76	C₂²⋊Dic₄₃	—	4	2	no	—
77	C₂²⋊Dic₄₃	—	4	2	no	—
78	C₄⋊Dic₄₃	—	2	4	no	—
79	C₄⋊Dic₄₃	—	2	4	no	—
80	C₄×Dic₄₃	—	2	4	no	—
81	C₄×Dic₄₃	—	4	1	no	—
82	C₄×Dic₄₃	—	4	2	no	—
83	C₄×Dic₄₃	—	4	2	no	—
84	C₄×Dic₄₃	—	4	1	no	—
85	C₂×C₄₃⋊C₈	—	2	2	no	—
86	C₂×C₄₃⋊C₈	—	2	4	no	—
87	C₂×C₄₃⋊C₈	—	4	1	no	—
88	C₂×C₄₃⋊C₈	—	4	1	no	—
89	C₂×C₄₃⋊C₈	—	4	2	no	—
90	C₂³×C₄₃	—	2	2	no	—
91	C₂³×C₄₃	—	2	2	no	—
92	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂³×C₄₃	#124	1	2	yes	—
95	C₂³×C₄₃	#125	1	1	yes	characteristic
96	C₂³×C₄₃	#126	1	2	yes	—
97	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	#117	1	2	yes	—
99	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
102	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₄₃	#121	1	2	yes	—
104	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₄₃	#123	1	2	yes	—
107	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#123	1	2	yes	—
109	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#125	1	2	yes	—
112	C₂×C₄×C₄₃	#125	1	2	yes	—
113	C₂×C₄×C₄₃	#126	1	2	yes	—
114	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×Dic₄₃	—	2	4	no	—
116	C₂×Dic₄₃	—	4	8	no	—
117	C₂×Dic₄₃	—	4	2	no	—
118	C₂×Dic₄₃	—	4	2	no	—
119	C₂×Dic₄₃	—	4	4	no	—
120	C₂×Dic₄₃	—	4	4	no	—
121	C₂×Dic₄₃	—	4	4	no	—
122	C₈×C₄₃	—	2	4	no	—
123	C₈×C₄₃	—	4	4	no	—
124	C₄₃⋊C₈	—	4	2	no	—
125	#11258	—	43	1	no	maximal, Sylow
126	C₂²×C₄₃	—	2	4	no	—
127	C₂²×C₄₃	—	2	4	no	—
128	C₂²×C₄₃	—	2	1	no	—
129	C₂²×C₄₃	—	2	1	no	—
130	C₂²×C₄₃	—	2	1	no	—
131	C₂²×C₄₃	—	2	1	no	—
132	C₂²×C₄₃	#388	1	2	yes	—
133	C₂²×C₄₃	#389	1	1	yes	characteristic
134	C₂²×C₄₃	#393	1	1	yes	characteristic
135	C₂²×C₄₃	#400	1	1	yes	characteristic
136	C₂²×C₄₃	#400	1	2	yes	—
137	C₂²×C₄₃	#443	1	2	yes	—
138	C₂²×C₄₃	#444	1	2	yes	—
139	C₄×C₄₃	#382	1	1	yes	characteristic
140	C₄×C₄₃	#385	1	1	yes	characteristic
141	C₄×C₄₃	#387	1	2	yes	—
142	C₄×C₄₃	#394	1	1	yes	characteristic
143	C₄×C₄₃	#397	1	1	yes	characteristic
144	C₄×C₄₃	#398	1	2	yes	—
145	Dic₄₃	—	4	4	no	—
146	Dic₄₃	—	8	4	no	—
147	C₂²×#108	D₄₃	1	1	yes	characteristic
148	#1735	—	43	1	no	—
149	C₂×#339	—	43	2	no	—
150	#1729	—	43	2	no	—
151	C₂×#336	—	43	1	no	—
152	C₂×C₄₃	—	2	1	no	—
153	C₂×C₄₃	—	2	1	no	—
154	C₂×C₄₃	—	2	2	no	—
155	C₂×C₄₃	#1442	1	1	yes	characteristic
156	C₂×C₄₃	#1929	1	4	yes	—
157	C₂×C₄₃	#1945	1	1	yes	characteristic
158	C₂×C₄₃	#1946	1	1	yes	characteristic
159	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
160	C₂³×C₈	—	43	1	no	—
161	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
162	C₂×#117	—	43	2	no	—
163	C₂²×C₄²	—	43	1	no	—
164	C₂×#108	—	2	4	no	—
165	C₂×#108	—	2	2	no	—
166	#340	—	43	4	no	—
167	#339	—	86	4	no	—
168	#336	—	86	4	no	—
169	#336	—	86	2	no	—
170	C₄₃	#11258	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	—	86	1	no	—
172	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	C₂²⋊D₄₃	1	4	yes	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	43	2	no	—
183	C₂²×C₈	—	86	2	no	—
184	C₂²×C₈	—	86	1	no	—
185	C₂²×C₈	—	86	2	no	—
186	C₂²×C₈	—	86	1	no	—
187	C₂²×C₈	Dic₈₆	1	2	yes	—
188	C₂²×C₈	C₄×D₄₃	1	2	yes	—
189	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
190	#117	—	86	2	no	—
191	#117	—	86	2	no	—
192	#117	—	86	2	no	—
193	#117	—	86	2	no	—
194	C₂×C₄⋊C₄	—	43	4	no	—
195	C₂×C₂²⋊C₄	—	86	2	no	—
196	C₂×C₄²	—	43	2	no	—
197	C₂×C₄²	—	86	2	no	—
198	C₂×C₄²	—	86	2	no	—
199	C₂×C₄²	—	86	2	no	—
200	C₂×C₄²	—	86	1	no	—
201	#108	—	2	8	no	—
202	C₂⁴	#10621902	1	1	yes	characteristic
203	C₂²×C₄	—	86	4	no	—
204	C₂²×C₄	—	86	2	no	—
205	C₂²×C₄	—	86	4	no	—
206	C₂²×C₄	—	172	2	no	—
207	C₂²×C₄	—	172	2	no	—
208	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
210	C₂²×C₄	#10621898	1	2	yes	—
211	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
212	C₂²×C₄	#10621900	1	2	yes	—
213	C₂²×C₄	#10621900	1	1	yes	characteristic
214	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
215	C₂²×C₄	#10621902	1	2	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	86	2	no	—
230	C₂×C₈	—	86	4	no	—
231	C₂×C₈	—	172	1	no	—
232	C₂×C₈	—	172	1	no	—
233	C₂×C₈	—	172	2	no	—
234	C₂×C₈	Dic₄₃⋊C₄	1	4	yes	—
235	C₄⋊C₄	—	86	4	no	—
236	C₄⋊C₄	—	86	4	no	—
237	C₂²⋊C₄	—	172	2	no	—
238	C₂²⋊C₄	—	172	2	no	—
239	C₄²	—	86	4	no	—
240	C₄²	—	172	1	no	—
241	C₄²	—	172	2	no	—
242	C₄²	—	172	2	no	—
243	C₄²	—	172	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	#13063287	1	1	yes	characteristic
247	C₂³	#13063301	1	1	yes	characteristic
248	C₂³	#13063314	1	2	yes	—
249	C₂³	#13063316	1	1	yes	characteristic
250	C₂³	#13063320	1	2	yes	—
251	C₂×C₄	—	86	4	no	—
252	C₂×C₄	—	172	4	no	—
253	C₂×C₄	—	172	4	no	—
254	C₂×C₄	—	172	4	no	—
255	C₂×C₄	—	172	8	no	—
256	C₂×C₄	—	172	2	no	—
257	C₂×C₄	—	172	2	no	—
258	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
259	C₂×C₄	#13063283	1	2	yes	—
260	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
261	C₂×C₄	#13063290	1	1	yes	characteristic
262	C₂×C₄	#13063294	1	1	yes	characteristic
263	C₂×C₄	#13063296	1	1	yes	characteristic
264	C₂×C₄	#13063297	1	2	yes	—
265	C₂×C₄	#13063301	1	2	yes	—
266	C₂×C₄	#13063304	1	2	yes	—
267	C₂×C₄	#13063306	1	2	yes	—
268	C₂×C₄	#13063308	1	1	yes	characteristic
269	C₂×C₄	#13063310	1	2	yes	—
270	C₂×C₄	#13063312	1	1	yes	characteristic
271	C₂×C₄	#13063314	1	1	yes	characteristic
272	C₂×C₄	#13063315	1	2	yes	—
273	C₂×C₄	#13063316	1	2	yes	—
274	C₂×C₄	#13063320	1	2	yes	—
275	C₂×C₄	#13063324	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	4	4	no	—
278	C₈	—	172	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	#14582789	1	2	yes	—
286	C₂²	#14582795	1	1	yes	characteristic
287	C₂²	#14582810	1	1	yes	characteristic
288	C₂²	#14582837	1	1	yes	characteristic
289	C₂²	#14582838	1	2	yes	—
290	C₂²	#14583015	1	2	yes	—
291	C₂²	#14583020	1	2	yes	—
292	C₄	—	344	4	no	—
293	C₄	—	172	4	no	—
294	C₄	#14582764	1	1	yes	characteristic
295	C₄	#14582772	1	1	yes	characteristic
296	C₄	#14582780	1	2	yes	—
297	C₄	#14582813	1	1	yes	characteristic
298	C₄	#14582825	1	1	yes	characteristic
299	C₄	#14582832	1	2	yes	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	#19132854	1	1	yes	characteristic
304	C₂	#19135242	1	4	yes	—
305	C₂	#19135341	1	1	yes	characteristic
306	C₂	#19135346	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂²×#108×C₄₃	C₂
socle	C₂³×C₄₃	C₄×D₄
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂²×#108	D₄₃
hypocenter	C₄₃	#11258
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
9. #14582662	C₄	semidirect
170. C₄₃	#11258	semidirect

Maximal subgroups

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

subgroup	index
#19134312	2
C₂×#14582660	2
#19134293	2
C₂×#14582653	2
C₂²×#108×C₄₃	2
#11258	43