Subgroups — #25251125

Order 11008.

674 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×#14583331	C₂	1	1	yes	maximal, characteristic, complemented
2	#19144265	C₂	1	2	yes	maximal, complemented
3	#19136703	C₂	1	2	yes	maximal, complemented
4	#19136698	C₂	1	2	yes	maximal, complemented
5	#19136691	C₂	1	4	yes	maximal, complemented
6	C₂×#14582727	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136308	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063347	C₂²	1	2	yes	—
10	#14583477	—	2	2	no	—
11	#14583331	—	2	4	no	—
12	C₂×Dic₃₄₄	C₂²	1	1	yes	characteristic
13	C₂×#13063270	C₂²	1	2	yes	—
14	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
15	#14583145	—	2	4	no	—
16	#14583140	C₂²	1	4	yes	—
17	#14583135	C₂²	1	4	yes	—
18	C₂×#13063259	C₂²	1	2	yes	—
19	#14582901	—	2	2	no	—
20	#14582727	C₂²	1	2	yes	—
21	#14582727	C₂²	1	2	yes	—
22	#14582704	C₂²	1	4	yes	—
23	#14582701	C₂²	1	4	yes	—
24	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#504	—	2	4	no	—
26	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
28	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
29	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×#10621898	—	2	2	no	—
31	#13063347	—	2	4	no	—
32	#13063347	—	2	4	no	—
33	C₂×Dic₁₇₂	C₂³	1	2	yes	—
34	C₂×#10621873	—	2	2	no	—
35	C₂×D₃₄₄	C₂³	1	2	yes	—
36	#13063304	—	2	4	no	—
37	#13063301	—	4	2	no	—
38	#13063299	—	2	4	no	—
39	#13063297	—	4	4	no	—
40	#13063296	—	4	2	no	—
41	C₂×C₄⋊Dic₄₃	—	2	2	no	—
42	Dic₃₄₄	—	2	2	no	—
43	#13063270	—	2	4	no	—
44	D₆₈₈	—	2	2	no	—
45	#13063259	—	2	2	no	—
46	#13063259	C₂³	1	4	yes	—
47	#13063250	—	2	4	no	—
48	#13063247	—	2	4	no	—
49	C₂×#41×C₄₃	D₄	1	1	yes	characteristic
50	#131×C₄₃	—	2	2	no	—
51	#131×C₄₃	—	2	2	no	—
52	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
53	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	M₃₂×C₄₃	—	2	2	no	—
57	M₃₂×C₄₃	—	2	4	no	—
58	C₂×C₁₆×C₄₃	—	2	4	no	—
59	C₂×C₁₆×C₄₃	D₄	1	4	yes	complemented
60	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
61	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
63	#41×C₄₃	—	2	2	no	—
64	#41×C₄₃	—	2	2	no	—
65	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
66	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
67	C₂×D₄×C₄₃	D₈	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₄₃	—	4	2	no	—
71	C₂×C₂²⋊D₄₃	—	4	4	no	—
72	C₂²×Dic₄₃	—	4	2	no	—
73	#10621898	—	4	4	no	—
74	#10621898	—	4	4	no	—
75	C₂×D₁₇₂	—	2	2	no	—
76	C₂×C₄×D₄₃	—	4	2	no	—
77	C₂×Dic₈₆	—	2	2	no	—
78	M₁₆×C₄₃	—	2	2	no	—
79	M₁₆×C₄₃	—	2	4	no	—
80	C₂×C₈×C₄₃	—	2	2	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₃	#41	1	2	yes	—
86	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
87	C₂²⋊Dic₄₃	—	4	4	no	—
88	(C₂×D₄₃)⋊C₄	—	4	4	no	—
89	C₄⋊Dic₄₃	—	2	4	no	—
90	C₄⋊Dic₄₃	—	4	2	no	—
91	Dic₄₃⋊C₄	—	4	4	no	—
92	Dic₁₇₂	—	2	2	no	—
93	Dic₁₇₂	—	2	2	no	—
94	D₃₄₄	—	2	2	no	—
95	D₃₄₄	—	2	2	no	—
96	#10621873	—	4	4	no	—
97	C₁₆×C₄₃	—	2	2	no	—
98	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
99	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
100	C₂³×C₄₃	C₂×D₈	1	2	yes	—
101	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₄₃	—	4	2	no	—
103	Q₈×C₄₃	—	2	2	no	—
104	D₄×C₄₃	—	2	4	no	—
105	D₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	#121	1	2	yes	—
109	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#135	1	2	yes	—
112	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
113	C₂²⋊D₄₃	—	8	4	no	—
114	C₂²⋊D₄₃	—	16	4	no	—
115	C₂×Dic₄₃	—	4	2	no	—
116	C₂×Dic₄₃	—	4	4	no	—
117	C₂×Dic₄₃	—	16	2	no	—
118	D₁₇₂	—	4	2	no	—
119	D₁₇₂	—	4	2	no	—
120	C₄×D₄₃	—	8	4	no	—
121	Dic₈₆	—	4	2	no	—
122	Dic₈₆	—	4	2	no	—
123	C₈×C₄₃	—	2	2	no	—
124	C₈×C₄₃	#121	1	2	yes	—
125	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
127	#33611	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	2	2	no	—
129	C₂²×C₄₃	—	2	4	no	—
130	C₂²×C₄₃	—	2	2	no	—
131	C₂²×C₄₃	#466	1	2	yes	—
132	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
133	C₂×D₄₃	—	8	2	no	—
134	C₂×D₄₃	—	8	2	no	—
135	C₂×D₄₃	—	8	2	no	—
136	C₄×C₄₃	—	2	2	no	—
137	C₄×C₄₃	#466	1	2	yes	—
138	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
139	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
140	Dic₄₃	—	8	4	no	—
141	Dic₄₃	—	8	2	no	—
142	C₂×#508	—	43	1	no	—
143	C₂×#504	D₄₃	1	1	yes	characteristic, complemented
144	#3061	—	43	2	no	—
145	#2219	—	43	2	no	—
146	#2218	—	43	2	no	—
147	#2217	—	43	4	no	—
148	C₂×#367	—	43	1	no	—
149	#2139	—	43	2	no	—
150	C₂×C₄₃	—	2	2	no	—
151	C₂×C₄₃	—	4	2	no	—
152	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
153	C₂×C₄₃	#3416	1	2	yes	—
154	D₄₃	—	16	2	no	—
155	C₂×#135	—	43	2	no	—
156	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
157	#537	—	86	2	no	—
158	#508	—	86	4	no	—
159	C₂×Q₃₂	—	43	1	no	—
160	C₂×SD₃₂	—	43	2	no	—
161	C₂×D₁₆	—	43	1	no	—
162	#504	—	2	4	no	—
163	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
164	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
165	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
166	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
167	#468	—	86	4	no	—
168	#467	—	43	4	no	—
169	#466	—	43	4	no	—
170	C₂×#107	—	43	2	no	—
171	#417	—	86	2	no	—
172	#367	—	43	2	no	—
173	#367	—	43	2	no	—
174	#358	—	43	4	no	—
175	#357	—	43	4	no	—
176	C₄₃	#33611	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	—	86	2	no	—
178	C₂×#41	D₁₇₂	1	1	yes	characteristic
179	#135	—	86	4	no	—
180	#135	—	86	4	no	—
181	C₂×Q₁₆	—	43	2	no	—
182	C₂×SD₁₆	—	86	2	no	—
183	C₂×D₈	—	43	2	no	—
184	#131	—	2	2	no	—
185	#131	—	2	2	no	—
186	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
187	C₂×M₁₆	D₁₇₂	1	2	yes	—
188	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
189	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
190	C₂²⋊Q₈	—	86	4	no	—
191	C₂²⋊Q₈	—	172	2	no	—
192	#121	—	86	4	no	—
193	#121	—	172	2	no	—
194	#121	—	172	4	no	—
195	C₂×C₄⋊C₄	—	86	2	no	—
196	Q₃₂	—	86	2	no	—
197	SD₃₂	—	86	4	no	—
198	D₁₆	—	86	2	no	—
199	M₃₂	—	2	2	no	—
200	M₃₂	—	2	4	no	—
201	C₂×C₁₆	—	2	4	no	—
202	C₂×C₁₆	C₂²⋊D₄₃	1	4	yes	complemented
203	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
204	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
205	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
206	#107	—	43	4	no	—
207	#107	—	86	2	no	—
208	#103	—	86	4	no	—
209	#102	—	86	4	no	—
210	#41	—	2	2	no	—
211	#41	—	2	2	no	—
212	#41	—	172	4	no	—
213	#41	—	172	4	no	—
214	C₂×Q₈	—	86	2	no	—
215	C₂×Q₈	D₃₄₄	1	1	yes	characteristic
216	C₂×D₄	—	86	2	no	—
217	C₂×D₄	—	172	2	no	—
218	C₂×D₄	—	172	4	no	—
219	C₂×D₄	D₃₄₄	1	2	yes	—
220	C₂×D₄	D₃₄₄	1	1	yes	characteristic
221	C₂²×C₄	—	172	2	no	—
222	C₂²×C₄	—	172	2	no	—
223	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
224	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
225	Q₁₆	—	86	2	no	—
226	Q₁₆	—	86	2	no	—
227	SD₁₆	—	172	4	no	—
228	D₈	—	86	2	no	—
229	D₈	—	86	2	no	—
230	M₁₆	—	2	2	no	—
231	M₁₆	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
235	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
236	C₂×C₈	#10621898	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
238	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
239	C₄⋊C₄	—	86	4	no	—
240	C₄⋊C₄	—	172	2	no	—
241	C₄⋊C₄	—	172	4	no	—
242	C₂²⋊C₄	—	172	4	no	—
243	C₂²⋊C₄	—	172	4	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	C₂×C₂²⋊D₄₃	1	2	yes	—
246	C₁₆	C₂×C₂²⋊D₄₃	1	2	yes	—
247	C₂³	—	172	2	no	—
248	C₂³	C₂×D₃₄₄	1	2	yes	—
249	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
250	Q₈	—	2	2	no	—
251	Q₈	—	172	2	no	—
252	Q₈	—	172	2	no	—
253	D₄	—	2	4	no	—
254	D₄	—	2	2	no	—
255	D₄	—	344	4	no	—
256	D₄	—	172	2	no	—
257	D₄	—	172	2	no	—
258	D₄	—	688	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	344	4	no	—
262	C₂×C₄	—	172	4	no	—
263	C₂×C₄	—	172	2	no	—
264	C₂×C₄	—	688	2	no	—
265	C₂×C₄	#13063299	1	2	yes	—
266	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
267	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
268	C₂×C₄	#13063347	1	2	yes	—
269	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	#13063299	1	2	yes	—
272	C₈	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
273	C₈	C₂×#10621898	1	1	yes	characteristic
274	C₂²	—	2	2	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	344	2	no	—
278	C₂²	—	344	2	no	—
279	C₂²	—	344	2	no	—
280	C₂²	#14583135	1	2	yes	—
281	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	344	4	no	—
284	C₄	—	344	2	no	—
285	C₄	#14583135	1	2	yes	—
286	C₄	C₂×#13063299	1	1	yes	characteristic
287	C₄	C₂×#13063347	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	4	2	no	—
290	C₂	—	688	2	no	—
291	C₂	C₂×#14583135	1	1	yes	characteristic
292	C₂	#19147743	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#504	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	C₂×#504	D₄₃
hypocenter	C₄₃	#33611
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#14583331	C₂	semidirect
2. #19144265	C₂	semidirect
3. #19136703	C₂	semidirect
4. #19136698	C₂	semidirect
5. #19136691	C₂	semidirect
6. C₂×#14582727	C₂	semidirect
7. #19136308	C₂	semidirect
8. C₂×C₄₃×#504	C₂	semidirect
59. C₂×C₁₆×C₄₃	D₄	semidirect
143. C₂×#504	D₄₃	semidirect
176. C₄₃	#33611	semidirect
202. C₂×C₁₆	C₂²⋊D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583331	2
#19144265	2
#19136703	2
#19136698	2
#19136691	2
C₂×#14582727	2
#19136308	2
C₂×C₄₃×#504	2
#33611	43