Subgroups — #25094095

Order 11008.

1042 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×#14582855	C₂	1	2	yes	maximal
2	#19134347	C₂	1	4	yes	maximal, complemented
3	#19134344	C₂	1	2	yes	maximal, complemented
4	#19134341	C₂	1	2	yes	maximal, complemented
5	#19134292	C₂	1	2	yes	maximal
6	#19134289	C₂	1	1	yes	maximal, characteristic
7	C₂×#14582654	C₂	1	1	yes	maximal, characteristic
8	C₄×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10621877	C₂²	1	2	yes	—
10	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
11	#14582855	C₄	1	16	yes	complemented
12	C₂×#13063236	C₂²	1	4	yes	—
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	4	yes	—
14	#14582654	C₂²	1	1	yes	characteristic
15	#14582654	C₂²	1	1	yes	characteristic
16	#14582654	C₂²	1	2	yes	—
17	#14582654	C₂²	1	2	yes	—
18	#14582654	C₂²	1	2	yes	—
19	#14582654	C₂²	1	4	yes	—
20	#14582654	C₂²	1	4	yes	—
21	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
26	C₄×C₂²⋊C₄×C₄₃	C₄	1	4	yes	—
27	C₄₃×#376	C₂²	1	2	yes	—
28	C₄³×C₄₃	C₂²	1	2	yes	—
29	C₂×#10621877	—	2	4	no	—
30	C₂×#10621877	—	2	4	no	—
31	C₂×#10621877	C₂×C₄	1	8	yes	—
32	C₂²×C₄₃⋊C₈	—	2	4	no	—
33	C₂²×C₄₃⋊C₈	—	2	4	no	—
34	C₂²×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
35	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
37	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
38	#13063236	C₂×C₄	1	16	yes	complemented
39	C₄×C₄₃⋊C₈	C₂×C₄	1	16	yes	complemented
40	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	#117×C₄₃	—	2	2	no	—
43	#117×C₄₃	—	2	2	no	—
44	#117×C₄₃	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₃	—	2	4	no	—
51	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
58	#95×C₄₃	C₂×C₄	1	4	yes	—
59	#95×C₄₃	C₂×C₄	1	4	yes	—
60	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	4	no	—
62	C₂²×C₄×C₄₃	—	2	4	no	—
63	C₂²×C₄×C₄₃	—	4	2	no	—
64	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	#41	1	4	yes	—
72	C₂²×C₄×C₄₃	#41	1	2	yes	—
73	C₂²×C₄×C₄₃	#41	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₄₃	—	2	4	no	—
76	C₄⋊C₄×C₄₃	C₄²	1	4	yes	—
77	C₄⋊C₄×C₄₃	C₄²	1	4	yes	—
78	C₂²⋊C₄×C₄₃	—	2	4	no	—
79	C₂²⋊C₄×C₄₃	C₄²	1	8	yes	—
80	C₄²×C₄₃	—	2	4	no	—
81	C₄²×C₄₃	—	2	8	no	—
82	C₄²×C₄₃	—	2	2	no	—
83	C₄²×C₄₃	—	2	2	no	—
84	C₄²×C₄₃	—	2	2	no	—
85	C₄²×C₄₃	—	2	2	no	—
86	C₄²×C₄₃	—	4	4	no	—
87	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
88	#10621877	—	2	16	no	—
89	C₂×C₄₃⋊C₈	—	2	16	no	—
90	C₂×C₄₃⋊C₈	—	2	2	no	—
91	C₂×C₄₃⋊C₈	—	2	2	no	—
92	C₂×C₄₃⋊C₈	—	2	2	no	—
93	C₂×C₄₃⋊C₈	—	2	2	no	—
94	C₂×C₄₃⋊C₈	—	2	4	no	—
95	C₂×C₄₃⋊C₈	—	2	4	no	—
96	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
97	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
98	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
99	C₂³×C₄₃	—	2	2	no	—
100	C₂³×C₄₃	—	2	2	no	—
101	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
102	C₂³×C₄₃	#117	1	2	yes	—
103	C₂³×C₄₃	#117	1	2	yes	—
104	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	—	2	4	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	4	no	—
111	C₂×C₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	4	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	4	8	no	—
116	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
117	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
118	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₄₃	#117	1	8	yes	—
120	C₂×C₄×C₄₃	#117	1	8	yes	—
121	C₂×C₄×C₄₃	#117	1	2	yes	—
122	C₂×C₄×C₄₃	#117	1	2	yes	—
123	C₂×C₄×C₄₃	#131	1	4	yes	—
124	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
128	C₄₃⋊C₈	—	2	4	no	—
129	C₄₃⋊C₈	—	2	4	no	—
130	C₄₃⋊C₈	—	2	8	no	—
131	#20368	—	43	1	no	maximal, Sylow
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	—	2	2	no	—
134	C₂²×C₄₃	—	2	4	no	—
135	C₂²×C₄₃	—	2	4	no	—
136	C₂²×C₄₃	#376	1	4	yes	—
137	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
138	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
139	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
140	C₂²×C₄₃	C₂×#131	1	2	yes	—
141	C₂²×C₄₃	#568	1	2	yes	—
142	C₄×C₄₃	—	2	4	no	—
143	C₄×C₄₃	—	2	4	no	—
144	C₄×C₄₃	—	2	4	no	—
145	C₄×C₄₃	—	4	2	no	—
146	C₄×C₄₃	—	4	4	no	—
147	C₄×C₄₃	#376	1	4	yes	—
148	C₄×C₄₃	C₂×#117	1	2	yes	—
149	C₄×C₄₃	C₂×#117	1	2	yes	—
150	C₂×#405	—	43	2	no	—
151	C₄×#117	D₄₃	1	1	yes	characteristic
152	#1746	—	43	4	no	—
153	#1745	—	43	2	no	—
154	#1744	—	43	2	no	—
155	#1729	—	43	2	no	—
156	#1728	—	43	1	no	—
157	C₂×#336	—	43	1	no	—
158	C₂×C₄₃	—	2	4	no	—
159	C₂×C₄₃	C₂×#376	1	1	yes	characteristic
160	C₂×C₄₃	C₄×#131	1	1	yes	characteristic
161	C₂×C₄₃	#2875	1	1	yes	characteristic
162	C₂×C₄₃	#2924	1	2	yes	—
163	C₂×C₄₃	#2925	1	2	yes	—
164	C₂²×M₁₆	—	43	2	no	—
165	C₂³×C₈	—	43	2	no	—
166	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
167	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
168	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
169	#405	—	43	16	no	—
170	C₂×#97	—	43	4	no	—
171	C₂×C₄×C₈	—	43	4	no	—
172	C₄×C₄⋊C₄	Dic₄₃	1	2	yes	—
173	C₄×C₄⋊C₄	Dic₄₃	1	2	yes	—
174	C₄×C₂²⋊C₄	Dic₄₃	1	4	yes	—
175	#376	C₂×D₄₃	1	2	yes	—
176	C₄³	C₂×D₄₃	1	2	yes	—
177	#336	—	43	1	no	—
178	#336	—	43	1	no	—
179	#336	—	43	2	no	—
180	#336	—	43	2	no	—
181	#336	—	43	2	no	—
182	#336	—	43	4	no	—
183	#336	—	43	4	no	—
184	C₄₃	#20368	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
186	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
187	C₂×M₁₆	—	43	8	no	—
188	C₂×M₁₆	—	86	4	no	—
189	C₂×M₁₆	—	86	4	no	—
190	C₂²×C₈	—	43	8	no	—
191	C₂²×C₈	—	43	2	no	—
192	C₂²×C₈	—	43	2	no	—
193	C₂²×C₈	—	43	4	no	—
194	C₂²×C₈	—	86	4	no	—
195	C₂²×C₈	—	86	4	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	C₄×D₄₃	1	8	yes	—
199	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
200	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	2	yes	—
203	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	2	yes	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
206	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
207	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
208	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
209	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
210	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
212	#97	—	43	16	no	—
213	C₄×C₈	—	43	16	no	—
214	#95	C₂×Dic₄₃	1	4	yes	—
215	#95	C₂×Dic₄₃	1	4	yes	—
216	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
221	C₂²×C₄	#10621898	1	4	yes	—
222	C₂²×C₄	#10621898	1	2	yes	—
223	C₂²×C₄	#10621898	1	2	yes	—
224	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
225	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
226	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
228	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
229	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
230	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
231	M₁₆	—	86	16	no	—
232	C₂×C₈	—	43	4	no	—
233	C₂×C₈	—	43	8	no	—
234	C₂×C₈	—	43	4	no	—
235	C₂×C₈	—	86	16	no	—
236	C₂×C₈	—	86	2	no	—
237	C₂×C₈	—	86	2	no	—
238	C₂×C₈	—	86	2	no	—
239	C₂×C₈	—	86	2	no	—
240	C₂×C₈	—	86	4	no	—
241	C₂×C₈	—	86	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	C₄×Dic₄₃	1	4	yes	—
244	C₄⋊C₄	C₄×Dic₄₃	1	4	yes	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	C₄×Dic₄₃	1	8	yes	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	8	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	4	no	—
254	C₄²	C₂×C₄×D₄₃	1	8	yes	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
258	C₂³	#13063284	1	2	yes	—
259	C₂³	#13063284	1	2	yes	—
260	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	8	no	—
272	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
273	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
274	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
275	C₂×C₄	#13063282	1	8	yes	—
276	C₂×C₄	#13063282	1	8	yes	—
277	C₂×C₄	#13063284	1	2	yes	—
278	C₂×C₄	#13063284	1	2	yes	—
279	C₂×C₄	#13063332	1	4	yes	—
280	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
281	C₂×C₄	C₂×#10621898	1	2	yes	—
282	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
283	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
284	C₈	—	86	4	no	—
285	C₈	—	86	4	no	—
286	C₈	—	86	8	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	#14582745	1	4	yes	—
292	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
293	C₂²	C₂×#13063284	1	1	yes	characteristic
294	C₂²	C₂×#13063284	1	1	yes	characteristic
295	C₂²	C₂×#13063332	1	2	yes	—
296	C₂²	#14583666	1	2	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	2	4	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	4	no	—
302	C₄	#14582745	1	4	yes	—
303	C₄	C₂×#13063282	1	2	yes	—
304	C₄	C₂×#13063282	1	2	yes	—
305	C₂	—	2	4	no	—
306	C₂	C₂×#14582745	1	1	yes	characteristic
307	C₂	C₄×#13063332	1	1	yes	characteristic
308	C₂	#19143023	1	1	yes	characteristic
309	C₂	#19143253	1	2	yes	—
310	C₂	#19143261	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Fitting	C₄×#117×C₄₃	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#117	D₄₃
hypocenter	C₄₃	#20368
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #19134347	C₂	semidirect
3. #19134344	C₂	semidirect
4. #19134341	C₂	semidirect
11. #14582855	C₄	semidirect
38. #13063236	C₂×C₄	semidirect
39. C₄×C₄₃⋊C₈	C₂×C₄	semidirect
184. C₄₃	#20368	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582855	2
#19134347	2
#19134344	2
#19134341	2
#19134292	2
#19134289	2
C₂×#14582654	2
C₄×#117×C₄₃	2
#20368	43