Subgroups — #25100083

Order 11008.

1022 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582941	C₂	1	2	yes	maximal
2	C₂×#14582765	C₂	1	1	yes	maximal, characteristic
3	C₄×#13063284	C₂	1	1	yes	maximal, characteristic
4	#19135365	C₂	1	8	yes	maximal, complemented
5	#19134315	C₂	1	2	yes	maximal
6	C₂×C₄₃×#423	C₂	1	1	yes	maximal, characteristic
7	C₂×#13063284	C₂²	1	1	yes	characteristic
8	C₂×#13063284	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
10	#14582941	—	2	8	no	—
11	C₂×#13063259	C₂²	1	4	yes	—
12	C₂×#13063257	C₂²	1	4	yes	—
13	#14582765	—	2	1	no	—
14	#14582765	—	2	1	no	—
15	#14582765	C₂²	1	4	yes	—
16	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	—
17	C₄×C₂²⋊Dic₄₃	—	2	2	no	—
18	#14582745	—	2	1	no	—
19	C₂×C₄₃⋊#95	—	2	2	no	—
20	C₄²×Dic₄₃	—	2	1	no	—
21	#14582662	C₂²	1	8	yes	—
22	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
23	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#423	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₈×C₄₃	C₂²	1	4	yes	—
26	C₂³×Dic₄₃	—	2	2	no	—
27	#13063284	—	2	2	no	—
28	#13063284	—	2	2	no	—
29	#13063284	D₄	1	4	yes	—
30	#13063284	Q₈	1	4	yes	—
31	C₂×C₄⋊Dic₄₃	—	2	4	no	—
32	C₂×C₄⋊Dic₄₃	—	2	4	no	—
33	C₂×C₄⋊Dic₄₃	—	2	2	no	—
34	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
37	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
38	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
39	C₂×C₄×Dic₄₃	—	2	1	no	—
40	C₂×C₄×Dic₄₃	—	2	1	no	—
41	C₂×C₄×Dic₄₃	—	2	4	no	—
42	#13063259	—	2	8	no	—
43	#13063257	—	2	8	no	—
44	C₄₃⋊#95	—	2	4	no	—
45	C₄₃⋊#95	—	2	4	no	—
46	C₄₃⋊#95	—	2	4	no	—
47	C₂³×C₄×C₄₃	D₄	1	2	yes	—
48	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₄₃	—	2	4	no	—
50	C₂×M₁₆×C₄₃	—	2	4	no	—
51	C₂×M₁₆×C₄₃	C₂×C₄	1	8	yes	—
52	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
53	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
58	C₄⋊C₈×C₄₃	C₂×C₄	1	16	yes	complemented
59	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	—	4	4	no	—
61	C₂²×C₄×C₄₃	C₄⋊C₄	1	8	yes	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₄₃	#41	1	2	yes	—
70	C₂²×C₄×C₄₃	#41	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₃	—	2	2	no	—
73	C₂²×Dic₄₃	—	2	2	no	—
74	C₂²×Dic₄₃	—	2	4	no	—
75	C₂²×Dic₄₃	—	4	8	no	—
76	C₂²×Dic₄₃	—	4	4	no	—
77	M₁₆×C₄₃	—	2	16	no	—
78	C₂×C₈×C₄₃	—	2	4	no	—
79	C₂×C₈×C₄₃	—	2	4	no	—
80	C₂×C₈×C₄₃	C₂²×C₄	1	8	yes	—
81	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
82	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
83	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
84	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
85	C₂²⋊Dic₄₃	—	2	4	no	—
86	C₂²⋊Dic₄₃	—	2	4	no	—
87	C₄⋊Dic₄₃	—	2	2	no	—
88	C₄⋊Dic₄₃	—	2	2	no	—
89	C₄⋊Dic₄₃	—	2	4	no	—
90	C₄⋊Dic₄₃	—	2	4	no	—
91	C₄⋊Dic₄₃	—	4	4	no	—
92	C₄⋊Dic₄₃	C₂×D₄	1	4	yes	—
93	C₄⋊Dic₄₃	C₂×Q₈	1	4	yes	—
94	C₄×Dic₄₃	—	2	4	no	—
95	C₄×Dic₄₃	—	2	4	no	—
96	C₄×Dic₄₃	—	4	8	no	—
97	C₂³×C₄₃	—	2	4	no	—
98	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₄₃	#121	1	2	yes	—
100	C₂³×C₄₃	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	2	4	no	—
104	C₂×C₄×C₄₃	—	4	8	no	—
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
109	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
110	C₂×C₄×C₄₃	#121	1	2	yes	—
111	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₃	#125	1	4	yes	—
113	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
114	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₄₃	—	2	4	no	—
121	C₂×Dic₄₃	—	2	4	no	—
122	C₂×Dic₄₃	—	4	4	no	—
123	C₂×Dic₄₃	—	4	4	no	—
124	C₂×Dic₄₃	—	4	4	no	—
125	C₂×Dic₄₃	—	4	8	no	—
126	C₂×Dic₄₃	—	8	8	no	—
127	C₈×C₄₃	—	2	8	no	—
128	#21172	—	43	1	no	maximal, Sylow
129	C₂²×C₄₃	—	2	4	no	—
130	C₂²×C₄₃	—	2	4	no	—
131	C₂²×C₄₃	—	2	4	no	—
132	C₂²×C₄₃	#389	1	4	yes	—
133	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
135	C₂²×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
136	C₂²×C₄₃	#575	1	4	yes	—
137	C₄×C₄₃	—	2	4	no	—
138	C₄×C₄₃	—	2	4	no	—
139	C₄×C₄₃	#389	1	4	yes	—
140	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₄×C₄₃	C₂×#125	1	1	yes	characteristic
143	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	Dic₄₃	—	4	4	no	—
145	Dic₄₃	—	4	4	no	—
146	Dic₄₃	—	8	4	no	—
147	C₂×#428	—	43	2	no	—
148	C₂×#423	D₄₃	1	1	yes	characteristic
149	C₂×#382	—	43	1	no	—
150	C₄×#117	—	43	1	no	—
151	#1949	—	43	8	no	—
152	#1736	—	43	2	no	—
153	C₂×C₄₃	—	2	4	no	—
154	C₂×C₄₃	C₂×#389	1	1	yes	characteristic
155	C₂×C₄₃	#2911	1	2	yes	—
156	C₂×C₄₃	#3040	1	2	yes	—
157	C₂×C₄₃	#3080	1	2	yes	—
158	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
159	C₂×#117	—	43	1	no	—
160	C₂×#117	—	43	1	no	—
161	C₂²×C₄⋊C₄	—	43	2	no	—
162	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
163	#428	—	86	8	no	—
164	C₂×#107	—	43	4	no	—
165	C₂×#106	—	43	4	no	—
166	#423	Dic₄₃	1	8	yes	complemented
167	C₂×C₄⋊C₈	C₂×D₄₃	1	4	yes	—
168	#382	—	43	4	no	—
169	#382	—	86	1	no	—
170	#382	—	86	1	no	—
171	C₄×C₄⋊C₄	—	43	4	no	—
172	C₄×C₂²⋊C₄	—	86	2	no	—
173	#376	—	86	1	no	—
174	C₂×#95	—	86	2	no	—
175	C₄³	—	86	1	no	—
176	#340	—	43	8	no	—
177	C₄₃	#21172	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	86	2	no	—
179	C₂³×C₄	D₁₇₂	1	2	yes	—
180	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	C₂×Dic₄₃	1	8	yes	—
184	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
185	#117	—	43	4	no	—
186	#117	—	43	4	no	—
187	#117	—	86	2	no	—
188	#117	—	86	2	no	—
189	C₂×C₄⋊C₄	—	43	2	no	—
190	C₂×C₄⋊C₄	—	43	2	no	—
191	C₂×C₄⋊C₄	—	43	4	no	—
192	C₂×C₄⋊C₄	—	86	4	no	—
193	C₂×C₄⋊C₄	—	86	4	no	—
194	C₂×C₄⋊C₄	—	86	2	no	—
195	C₂×C₂²⋊C₄	—	86	1	no	—
196	C₂×C₂²⋊C₄	—	86	1	no	—
197	C₂×C₄²	—	86	1	no	—
198	C₂×C₄²	—	86	1	no	—
199	C₂×C₄²	—	86	4	no	—
200	C₂×C₄²	Dic₈₆	1	4	yes	—
201	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
202	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
203	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
204	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
205	#107	—	86	8	no	—
206	#106	—	86	8	no	—
207	C₄⋊C₈	C₂×Dic₄₃	1	16	yes	complemented
208	#95	—	86	4	no	—
209	#95	—	86	4	no	—
210	#95	—	86	4	no	—
211	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	86	2	no	—
214	C₂²×C₄	—	86	2	no	—
215	C₂²×C₄	—	86	4	no	—
216	C₂²×C₄	—	172	8	no	—
217	C₂²×C₄	—	172	4	no	—
218	C₂²×C₄	C₄⋊Dic₄₃	1	8	yes	—
219	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
220	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
221	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
222	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
223	C₂²×C₄	D₄×D₄₃	1	2	yes	—
224	C₂²×C₄	#10621900	1	2	yes	—
225	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
226	C₂²×C₄	#10621902	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
228	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
229	M₁₆	—	2	16	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₂²×Dic₄₃	1	8	yes	—
233	C₄⋊C₄	—	43	4	no	—
234	C₄⋊C₄	—	43	4	no	—
235	C₄⋊C₄	—	86	2	no	—
236	C₄⋊C₄	—	86	2	no	—
237	C₄⋊C₄	—	86	4	no	—
238	C₄⋊C₄	—	86	4	no	—
239	C₄⋊C₄	—	172	4	no	—
240	C₂²⋊C₄	—	86	4	no	—
241	C₂²⋊C₄	—	86	4	no	—
242	C₄²	—	86	4	no	—
243	C₄²	—	86	4	no	—
244	C₄²	—	172	8	no	—
245	C₄²	C₄⋊Dic₄₃	1	4	yes	—
246	C₄²	C₄⋊Dic₄₃	1	4	yes	—
247	C₄²	C₂×Dic₈₆	1	4	yes	—
248	C₄²	C₂²×Dic₄₃	1	4	yes	—
249	C₂³	—	2	4	no	—
250	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
251	C₂³	#13063300	1	2	yes	—
252	C₂³	#13063305	1	2	yes	—
253	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	86	4	no	—
258	C₂×C₄	—	86	4	no	—
259	C₂×C₄	—	344	8	no	—
260	C₂×C₄	—	172	4	no	—
261	C₂×C₄	—	172	8	no	—
262	C₂×C₄	—	172	4	no	—
263	C₂×C₄	—	172	4	no	—
264	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
265	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
266	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
267	C₂×C₄	D₄×Dic₄₃	1	4	yes	—
268	C₂×C₄	Q₈×Dic₄₃	1	4	yes	—
269	C₂×C₄	#13063300	1	2	yes	—
270	C₂×C₄	#13063305	1	2	yes	—
271	C₂×C₄	#13063316	1	4	yes	—
272	C₂×C₄	#13063324	1	4	yes	—
273	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
274	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
275	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
277	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
278	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
279	C₈	—	2	8	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	#14582798	1	4	yes	—
284	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
285	C₂²	C₂×#13063300	1	1	yes	characteristic
286	C₂²	C₂×#13063305	1	1	yes	characteristic
287	C₂²	#14583699	1	4	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	2	4	no	—
290	C₄	—	344	4	no	—
291	C₄	—	172	4	no	—
292	C₄	—	172	4	no	—
293	C₄	#14582798	1	4	yes	—
294	C₄	C₂×D₄×Dic₄₃	1	1	yes	characteristic
295	C₄	C₂×Q₈×Dic₄₃	1	1	yes	characteristic
296	C₄	C₂×#13063316	1	1	yes	characteristic
297	C₄	C₂×#13063324	1	1	yes	characteristic
298	C₂	—	2	4	no	—
299	C₂	C₂×#14582798	1	1	yes	characteristic
300	C₂	#19143185	1	2	yes	—
301	C₂	#19144090	1	2	yes	—
302	C₂	#19144418	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#423	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#423	D₄₃
hypocenter	C₄₃	#21172
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #19135365	C₂	semidirect
24. C₄₃×#423	C₄	semidirect
58. C₄⋊C₈×C₄₃	C₂×C₄	semidirect
166. #423	Dic₄₃	semidirect
177. C₄₃	#21172	semidirect
207. C₄⋊C₈	C₂×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582941	2
C₂×#14582765	2
C₄×#13063284	2
#19135365	2
#19134315	2
C₂×C₄₃×#423	2
#21172	43