Subgroups — #25002305

Order 11008.

722 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	#19134334	C₂	1	1	yes	maximal, characteristic
2	#19134328	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582659	C₂	1	2	yes	maximal
4	C₂×#14582655	C₂	1	2	yes	maximal
5	C₄×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂×#13063284	C₂²	1	2	yes	—
8	#14582668	—	2	2	no	—
9	#14582666	—	2	2	no	—
10	#14582659	—	2	4	no	—
11	#14582655	C₄	1	8	yes	complemented
12	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
16	C₄×C₂²⋊C₄×C₄₃	—	2	2	no	—
17	C₄₃×#376	C₄	1	2	yes	complemented
18	C₄³×C₄₃	C₄	1	2	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	C₂×C₄	1	4	yes	—
25	C₂²×C₄₃⋊C₈	—	2	2	no	—
26	#13063284	—	2	2	no	—
27	#13063284	—	2	2	no	—
28	#13063284	—	2	2	no	—
29	#13063284	C₂×C₄	1	4	yes	—
30	C₂×C₄⋊Dic₄₃	—	2	2	no	—
31	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄×Dic₄₃	—	2	2	no	—
33	C₂³×C₄×C₄₃	—	2	1	no	—
34	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
35	#117×C₄₃	—	2	2	no	—
36	#117×C₄₃	—	2	2	no	—
37	#117×C₄₃	—	2	2	no	—
38	#117×C₄₃	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
42	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
43	C₂×C₄²×C₄₃	—	2	1	no	—
44	C₂×C₄²×C₄₃	—	2	1	no	—
45	C₂×C₄²×C₄₃	—	2	1	no	—
46	C₂×C₄²×C₄₃	—	2	4	no	—
47	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	D₄	1	2	yes	—
52	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
53	#95×C₄₃	—	2	4	no	—
54	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	—	2	1	no	—
56	C₂²×C₄×C₄₃	—	2	1	no	—
57	C₂²×C₄×C₄₃	—	2	2	no	—
58	C₂²×C₄×C₄₃	—	2	2	no	—
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	4	4	no	—
61	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	#41	1	2	yes	—
68	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	#41	1	2	yes	—
71	C₂²×C₄×C₄₃	#41	1	2	yes	—
72	C₂²×Dic₄₃	—	4	2	no	—
73	C₄⋊C₄×C₄₃	—	2	4	no	—
74	C₄⋊C₄×C₄₃	—	2	4	no	—
75	C₂²⋊C₄×C₄₃	—	2	4	no	—
76	C₂²⋊C₄×C₄₃	—	2	4	no	—
77	C₄²×C₄₃	—	2	1	no	—
78	C₄²×C₄₃	—	2	1	no	—
79	C₄²×C₄₃	—	2	2	no	—
80	C₄²×C₄₃	—	2	2	no	—
81	C₄²×C₄₃	—	2	2	no	—
82	C₄²×C₄₃	—	2	2	no	—
83	C₄²×C₄₃	—	2	2	no	—
84	C₄²×C₄₃	—	4	8	no	—
85	C₄²×C₄₃	C₄²	1	4	yes	—
86	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
87	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
88	C₂²⋊Dic₄₃	—	2	4	no	—
89	C₂²⋊Dic₄₃	—	4	2	no	—
90	C₄⋊Dic₄₃	—	2	4	no	—
91	C₄⋊Dic₄₃	—	4	2	no	—
92	C₄×Dic₄₃	—	2	4	no	—
93	C₄×Dic₄₃	—	4	2	no	—
94	#10621877	—	2	4	no	—
95	#10621877	—	2	4	no	—
96	#10621877	—	4	2	no	—
97	#10621877	—	4	2	no	—
98	C₂×C₄₃⋊C₈	—	2	4	no	—
99	C₂×C₄₃⋊C₈	—	4	2	no	—
100	C₂×C₄₃⋊C₈	—	4	2	no	—
101	C₂³×C₄₃	—	4	2	no	—
102	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₄₃	C₄×D₄	1	2	yes	—
104	C₂³×C₄₃	#121	1	1	yes	characteristic
105	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₄₃	#123	1	2	yes	—
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	4	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	—	4	4	no	—
113	C₂×C₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	—	4	4	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	4	4	no	—
117	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	#117	1	2	yes	—
120	C₂×C₄×C₄₃	#117	1	2	yes	—
121	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#123	1	2	yes	—
131	C₂×C₄×C₄₃	#124	1	2	yes	—
132	C₂×C₄×C₄₃	#125	1	2	yes	—
133	C₂×C₄×C₄₃	#126	1	2	yes	—
134	C₂×C₄×C₄₃	#126	1	2	yes	—
135	C₂×Dic₄₃	—	4	4	no	—
136	C₂×Dic₄₃	—	8	4	no	—
137	C₄₃⋊C₈	—	4	4	no	—
138	#10816	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	—	2	2	no	—
140	C₂²×C₄₃	—	2	4	no	—
141	C₂²×C₄₃	—	4	2	no	—
142	C₂²×C₄₃	—	4	1	no	—
143	C₂²×C₄₃	—	4	1	no	—
144	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄₃	#380	1	1	yes	characteristic
146	C₂²×C₄₃	#381	1	1	yes	characteristic
147	C₂²×C₄₃	#387	1	2	yes	—
148	C₂²×C₄₃	#388	1	2	yes	—
149	C₄×C₄₃	—	2	4	no	—
150	C₄×C₄₃	—	4	4	no	—
151	C₄×C₄₃	—	4	2	no	—
152	C₄×C₄₃	—	4	4	no	—
153	C₄×C₄₃	#376	1	1	yes	characteristic
154	C₄×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₄₃	#382	1	1	yes	characteristic
156	C₄×C₄₃	#383	1	1	yes	characteristic
157	C₄×C₄₃	#387	1	2	yes	—
158	C₄×C₄₃	#388	1	2	yes	—
159	Dic₄₃	—	8	4	no	—
160	C₄×#117	D₄₃	1	1	yes	characteristic
161	#1742	—	43	1	no	—
162	#1740	—	43	1	no	—
163	C₂×#339	—	43	2	no	—
164	C₂×H(3,ℤ/4)	—	43	2	no	—
165	C₂×C₄₃	—	2	2	no	—
166	C₂×C₄₃	—	2	1	no	—
167	C₂×C₄₃	—	2	1	no	—
168	C₂×C₄₃	—	2	2	no	—
169	C₂×C₄₃	#1434	1	1	yes	characteristic
170	C₂×C₄₃	#1827	1	2	yes	—
171	C₂²×M₁₆	—	43	2	no	—
172	C₂×#117	—	43	2	no	—
173	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
174	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
175	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
176	C₄×C₄⋊C₄	—	2	2	no	—
177	C₄×C₂²⋊C₄	—	2	2	no	—
178	#376	Dic₄₃	1	2	yes	complemented
179	C₄³	Dic₄₃	1	2	yes	complemented
180	#343	—	86	2	no	—
181	#342	—	86	2	no	—
182	#339	—	86	4	no	—
183	H(3,ℤ/4)	—	43	8	no	—
184	C₄₃	#10816	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	—	2	1	no	—
186	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
187	C₂×M₁₆	—	43	4	no	—
188	C₂×M₁₆	—	86	2	no	—
189	C₂×M₁₆	—	86	2	no	—
190	C₂×M₁₆	—	86	2	no	—
191	C₂×M₁₆	—	86	2	no	—
192	C₂×M₁₆	—	86	2	no	—
193	C₂²×C₈	—	86	2	no	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	43	4	no	—
198	#117	—	86	2	no	—
199	#117	—	86	2	no	—
200	#117	—	86	2	no	—
201	#117	C₄×D₄₃	1	4	yes	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	86	2	no	—
204	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	2	yes	—
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₂²⋊C₄	—	86	2	no	—
207	C₂×C₂²⋊C₄	C₂²⋊D₄₃	1	2	yes	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	—	2	4	no	—
212	C₂×C₄²	—	86	2	no	—
213	C₂×C₄²	Dic₈₆	1	2	yes	—
214	C₂×C₄²	D₁₇₂	1	2	yes	—
215	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
216	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
217	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
218	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
219	#95	—	2	4	no	—
220	C₂⁴	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	172	2	no	—
228	C₂²×C₄	C₂²⋊Dic₄₃	1	4	yes	—
229	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
231	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
232	C₂²×C₄	#10621898	1	1	yes	characteristic
233	C₂²×C₄	#10621898	1	1	yes	characteristic
234	C₂²×C₄	#10621898	1	2	yes	—
235	C₂²×C₄	#10621898	1	2	yes	—
236	C₂²×C₄	#10621898	1	2	yes	—
237	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
239	M₁₆	—	86	4	no	—
240	M₁₆	—	86	4	no	—
241	M₁₆	—	172	2	no	—
242	M₁₆	—	172	2	no	—
243	C₂×C₈	—	86	4	no	—
244	C₂×C₈	—	172	2	no	—
245	C₂×C₈	—	172	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	86	4	no	—
249	C₄⋊C₄	—	172	2	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	86	4	no	—
253	C₂²⋊C₄	—	172	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	4	8	no	—
262	C₄²	—	86	4	no	—
263	C₄²	—	172	2	no	—
264	C₄²	C₄×Dic₄₃	1	4	yes	—
265	C₄²	C₄⋊Dic₄₃	1	2	yes	—
266	C₄²	C₄⋊Dic₄₃	1	2	yes	—
267	C₂³	—	4	2	no	—
268	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
269	C₂³	C₄×C₂²⋊D₄₃	1	2	yes	—
270	C₂³	#13063299	1	1	yes	characteristic
271	C₂³	#13063304	1	1	yes	characteristic
272	C₂³	#13063308	1	2	yes	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	344	4	no	—
284	C₂×C₄	—	172	4	no	—
285	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
287	C₂×C₄	#13063282	1	2	yes	—
288	C₂×C₄	#13063282	1	2	yes	—
289	C₂×C₄	#13063284	1	1	yes	characteristic
290	C₂×C₄	#13063284	1	1	yes	characteristic
291	C₂×C₄	#13063284	1	1	yes	characteristic
292	C₂×C₄	#13063284	1	1	yes	characteristic
293	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
294	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
295	C₂×C₄	C₄×Dic₈₆	1	2	yes	—
296	C₂×C₄	#13063299	1	1	yes	characteristic
297	C₂×C₄	#13063304	1	1	yes	characteristic
298	C₂×C₄	#13063308	1	2	yes	—
299	C₂×C₄	#13063313	1	2	yes	—
300	C₂×C₄	#13063317	1	2	yes	—
301	C₂×C₄	#13063319	1	2	yes	—
302	C₂×C₄	#13063319	1	2	yes	—
303	C₈	—	172	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₄×C₂²⋊Dic₄₃	1	1	yes	characteristic
310	C₂²	#14582759	1	1	yes	characteristic
311	C₂²	#14582762	1	1	yes	characteristic
312	C₂²	#14582783	1	2	yes	—
313	C₂²	#14582790	1	2	yes	—
314	C₄	—	2	4	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	344	4	no	—
319	C₄	#14582745	1	1	yes	characteristic
320	C₄	C₄×C₄⋊Dic₄₃	1	1	yes	characteristic
321	C₄	#14582765	1	1	yes	characteristic
322	C₄	#14582767	1	1	yes	characteristic
323	C₄	#14582783	1	2	yes	—
324	C₄	#14582790	1	2	yes	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	2	2	no	—
328	C₂	—	2	2	no	—
329	C₂	#19132822	1	1	yes	characteristic
330	C₂	#19134699	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063284
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄×#117×C₄₃	C₂
socle	C₂²×C₄₃	#381
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₄×#117	D₄₃
hypocenter	C₄₃	#10816
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
11. #14582655	C₄	semidirect
17. C₄₃×#376	C₄	semidirect
18. C₄³×C₄₃	C₄	semidirect
178. #376	Dic₄₃	semidirect
179. C₄³	Dic₄₃	semidirect
184. C₄₃	#10816	semidirect

Maximal subgroups

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

subgroup	index
#19134334	2
#19134328	2
C₂×#14582659	2
C₂×#14582655	2
C₄×#117×C₄₃	2
#10816	43