Subgroups — #25252034

Order 11008.

814 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×#14583342	C₂	1	4	yes	maximal, complemented
2	#19145023	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583279	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136866	C₂	1	4	yes	maximal, complemented
5	#19136851	C₂	1	2	yes	maximal, complemented
6	#19136848	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	—	2	1	no	—
9	C₂×#13063347	C₂²	1	2	yes	—
10	C₂²×Dic₁₇₂	—	2	1	no	—
11	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
12	#14583449	—	2	1	no	—
13	C₂×#13063325	—	2	1	no	—
14	#14583342	—	2	16	no	—
15	C₂×Dic₃₄₄	C₂²	1	8	yes	—
16	C₂×#13063270	C₂²	1	8	yes	—
17	#14583283	—	2	1	no	—
18	#14583283	C₂²	1	2	yes	—
19	#14583279	—	2	1	no	—
20	#14583279	—	2	1	no	—
21	#14583279	C₂²	1	2	yes	—
22	#14583279	C₂²	1	2	yes	—
23	#14583279	C₂²	1	1	yes	characteristic
24	#14583276	C₂²	1	1	yes	characteristic
25	#14583272	—	2	1	no	—
26	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#346	C₂²	1	2	yes	—
29	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
30	C₂×#10621898	—	2	2	no	—
31	C₂²×Dic₈₆	—	2	2	no	—
32	#13063347	—	2	8	no	—
33	#13063347	—	4	4	no	—
34	C₂×Dic₁₇₂	—	2	8	no	—
35	C₂×Dic₁₇₂	—	2	2	no	—
36	C₂×Dic₁₇₂	—	2	2	no	—
37	C₂×Dic₁₇₂	—	2	2	no	—
38	C₂×Dic₁₇₂	—	2	1	no	—
39	C₂×Dic₁₇₂	—	4	4	no	—
40	C₂×Dic₁₇₂	C₂³	1	4	yes	—
41	C₂×Dic₁₇₂	C₂³	1	2	yes	—
42	C₂×#10621873	—	2	2	no	—
43	C₂×#10621873	—	2	2	no	—
44	C₂×D₃₄₄	—	2	1	no	—
45	C₂×D₃₄₄	C₂³	1	2	yes	—
46	#13063325	—	2	2	no	—
47	#13063325	—	2	2	no	—
48	#13063325	—	2	1	no	—
49	#13063325	—	4	2	no	—
50	#13063322	—	2	1	no	—
51	#13063313	—	2	2	no	—
52	#13063299	—	4	2	no	—
53	C₄×D₁₇₂	—	4	2	no	—
54	C₂×C₄⋊Dic₄₃	—	4	2	no	—
55	Dic₃₄₄	—	2	16	no	—
56	#13063270	—	2	16	no	—
57	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
60	M₃₂×C₄₃	D₄	1	8	yes	complemented
61	C₂×C₁₆×C₄₃	C₂³	1	4	yes	—
62	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	D₈	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₄₃	—	4	2	no	—
69	C₂²×Dic₄₃	—	4	2	no	—
70	#10621898	—	4	4	no	—
71	#10621898	—	4	4	no	—
72	C₂×D₁₇₂	—	2	2	no	—
73	C₂×D₁₇₂	—	4	2	no	—
74	C₂×C₄×D₄₃	—	4	2	no	—
75	C₂×Dic₈₆	—	2	2	no	—
76	C₂×Dic₈₆	—	2	4	no	—
77	C₂×Dic₈₆	—	4	4	no	—
78	C₂×Dic₈₆	—	4	4	no	—
79	C₂×Dic₈₆	—	4	2	no	—
80	C₂×C₈×C₄₃	D₈	1	4	yes	—
81	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₄²×C₄₃	D₈	1	2	yes	—
87	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
89	(C₂×D₄₃)⋊C₄	—	4	4	no	—
90	C₄⋊Dic₄₃	—	4	4	no	—
91	C₄⋊Dic₄₃	—	4	4	no	—
92	C₄⋊Dic₄₃	—	4	2	no	—
93	Dic₁₇₂	—	2	8	no	—
94	Dic₁₇₂	—	2	4	no	—
95	Dic₁₇₂	—	4	4	no	—
96	Dic₁₇₂	—	4	4	no	—
97	Dic₁₇₂	—	4	4	no	—
98	Dic₁₇₂	—	4	2	no	—
99	D₃₄₄	—	2	4	no	—
100	D₃₄₄	—	4	2	no	—
101	#10621873	—	4	4	no	—
102	#10621873	—	4	4	no	—
103	C₁₆×C₄₃	C₂×D₄	1	8	yes	—
104	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
105	C₂²×D₄₃	—	4	2	no	—
106	C₂×C₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	#127	1	2	yes	—
108	C₂×C₄×C₄₃	C₂×D₈	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	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
113	C₂²⋊D₄₃	—	8	4	no	—
114	C₂×Dic₄₃	—	4	4	no	—
115	C₂×Dic₄₃	—	4	2	no	—
116	C₂×Dic₄₃	—	8	4	no	—
117	D₁₇₂	—	4	2	no	—
118	D₁₇₂	—	4	2	no	—
119	D₁₇₂	—	8	4	no	—
120	C₄×D₄₃	—	8	4	no	—
121	Dic₈₆	—	4	2	no	—
122	Dic₈₆	—	4	4	no	—
123	Dic₈₆	—	4	2	no	—
124	Dic₈₆	—	4	4	no	—
125	Dic₈₆	—	8	4	no	—
126	C₈×C₄₃	#127	1	2	yes	—
127	C₈×C₄₃	C₂×D₈	1	2	yes	—
128	C₈×C₄₃	C₂×D₈	1	2	yes	—
129	C₈×C₄₃	C₂²×D₄	1	2	yes	—
130	#33718	—	43	1	no	maximal, Sylow
131	C₂²×C₄₃	—	2	2	no	—
132	C₂²×C₄₃	#493	1	2	yes	—
133	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
134	C₂×D₄₃	—	8	2	no	—
135	C₂×D₄₃	—	8	4	no	—
136	C₄×C₄₃	—	2	2	no	—
137	C₄×C₄₃	#493	1	2	yes	—
138	C₄×C₄₃	#510	1	4	yes	complemented
139	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
140	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
141	Dic₄₃	—	8	4	no	—
142	Dic₄₃	—	8	2	no	—
143	C₂×#510	—	43	4	no	—
144	#3147	—	43	1	no	—
145	C₂×#494	—	43	1	no	—
146	#2251	—	43	4	no	—
147	#2248	—	43	2	no	—
148	#2247	—	43	2	no	—
149	C₄×M₃₂	D₄₃	1	1	yes	characteristic, complemented
150	C₂×C₄₃	—	2	2	no	—
151	C₂×C₄₃	C₂×#493	1	1	yes	characteristic
152	C₂×C₄₃	C₂×#510	1	2	yes	—
153	D₄₃	—	16	2	no	—
154	C₂×#135	—	43	2	no	—
155	C₂×#135	—	86	1	no	—
156	C₂²×Q₁₆	—	43	2	no	—
157	C₂²×Q₁₆	—	86	1	no	—
158	#532	—	86	1	no	—
159	C₂×C₄⋊Q₈	—	86	1	no	—
160	#510	—	86	16	no	—
161	C₂×Q₃₂	—	43	8	no	—
162	C₂×SD₃₂	—	43	8	no	—
163	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
164	#495	—	43	2	no	—
165	#495	—	86	1	no	—
166	#494	—	43	2	no	—
167	#494	—	43	2	no	—
168	#494	—	43	1	no	—
169	#494	—	86	1	no	—
170	#494	—	86	1	no	—
171	#493	—	43	1	no	—
172	#492	—	86	1	no	—
173	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
174	#346	C₂×D₄₃	1	2	yes	—
175	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
176	C₄₃	#33718	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	—	86	2	no	—
178	C₂²×Q₈	—	86	2	no	—
179	#135	—	86	8	no	—
180	#135	—	172	4	no	—
181	C₂×Q₁₆	—	43	4	no	—
182	C₂×Q₁₆	—	43	2	no	—
183	C₂×Q₁₆	—	86	8	no	—
184	C₂×Q₁₆	—	86	2	no	—
185	C₂×Q₁₆	—	86	2	no	—
186	C₂×Q₁₆	—	86	2	no	—
187	C₂×Q₁₆	—	86	1	no	—
188	C₂×Q₁₆	—	172	4	no	—
189	C₂×SD₁₆	—	86	2	no	—
190	C₂×SD₁₆	—	86	2	no	—
191	C₂×D₈	—	43	2	no	—
192	C₂×D₈	—	86	1	no	—
193	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
195	C₄⋊Q₈	—	86	2	no	—
196	C₄⋊Q₈	—	86	2	no	—
197	C₄⋊Q₈	—	86	1	no	—
198	C₄⋊Q₈	—	172	2	no	—
199	#127	—	86	1	no	—
200	#124	—	86	2	no	—
201	#121	—	172	2	no	—
202	C₄×D₄	—	172	2	no	—
203	C₂×C₄⋊C₄	—	172	2	no	—
204	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
205	Q₃₂	—	86	16	no	—
206	SD₃₂	—	86	16	no	—
207	M₃₂	D₁₇₂	1	8	yes	complemented
208	C₂×C₁₆	C₂²×D₄₃	1	4	yes	—
209	C₄×C₈	D₁₇₂	1	1	yes	characteristic
210	C₄×C₈	D₁₇₂	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
213	#41	—	172	4	no	—
214	#41	—	172	4	no	—
215	C₂×Q₈	—	86	2	no	—
216	C₂×Q₈	—	86	4	no	—
217	C₂×Q₈	—	172	4	no	—
218	C₂×Q₈	—	172	4	no	—
219	C₂×Q₈	—	172	2	no	—
220	C₂×D₄	—	86	2	no	—
221	C₂×D₄	—	172	2	no	—
222	C₂×D₄	—	172	2	no	—
223	C₂²×C₄	—	172	2	no	—
224	C₂²×C₄	—	172	2	no	—
225	C₂²×C₄	D₃₄₄	1	2	yes	—
226	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
227	Q₁₆	—	86	8	no	—
228	Q₁₆	—	86	4	no	—
229	Q₁₆	—	172	4	no	—
230	Q₁₆	—	172	4	no	—
231	Q₁₆	—	172	4	no	—
232	Q₁₆	—	172	2	no	—
233	SD₁₆	—	172	4	no	—
234	SD₁₆	—	172	4	no	—
235	D₈	—	86	4	no	—
236	D₈	—	172	2	no	—
237	C₂×C₈	D₃₄₄	1	4	yes	—
238	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
239	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
240	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
241	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
242	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
243	C₄⋊C₄	—	172	4	no	—
244	C₄⋊C₄	—	172	4	no	—
245	C₄⋊C₄	—	172	2	no	—
246	C₂²⋊C₄	—	172	4	no	—
247	C₄²	D₃₄₄	1	2	yes	—
248	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
249	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
250	C₁₆	C₂×D₁₇₂	1	8	yes	—
251	C₂³	—	172	2	no	—
252	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
253	Q₈	—	344	4	no	—
254	Q₈	—	172	2	no	—
255	Q₈	—	172	4	no	—
256	Q₈	—	172	2	no	—
257	Q₈	—	172	4	no	—
258	D₄	—	344	4	no	—
259	D₄	—	344	4	no	—
260	D₄	—	172	2	no	—
261	D₄	—	172	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	344	4	no	—
264	C₂×C₄	—	344	4	no	—
265	C₂×C₄	—	172	2	no	—
266	C₂×C₄	—	172	4	no	—
267	C₂×C₄	#13063322	1	2	yes	—
268	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
269	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
270	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
271	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
273	C₈	#13063322	1	2	yes	—
274	C₈	C₂×D₃₄₄	1	2	yes	—
275	C₈	C₂×D₃₄₄	1	2	yes	—
276	C₈	C₂²×D₁₇₂	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	344	2	no	—
279	C₂²	—	344	4	no	—
280	C₂²	#14583276	1	2	yes	—
281	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	344	2	no	—
284	C₄	—	344	4	no	—
285	C₄	#14583276	1	2	yes	—
286	C₄	#14583342	1	4	yes	complemented
287	C₄	C₂×#13063322	1	1	yes	characteristic
288	C₄	C₂²×D₃₄₄	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	688	2	no	—
291	C₂	C₂×#14583276	1	1	yes	characteristic
292	C₂	C₂×#14583342	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄×M₃₂×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19145023	C₂
hypercenter	C₄×M₃₂	D₄₃
hypocenter	C₄₃	#33718
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#14583342	C₂	semidirect
3. C₂×#14583279	C₂	semidirect
4. #19136866	C₂	semidirect
5. #19136851	C₂	semidirect
6. #19136848	C₂	semidirect
7. C₄×M₃₂×C₄₃	C₂	semidirect
60. M₃₂×C₄₃	D₄	semidirect
138. C₄×C₄₃	#510	semidirect
149. C₄×M₃₂	D₄₃	semidirect
176. C₄₃	#33718	semidirect
207. M₃₂	D₁₇₂	semidirect
286. C₄	#14583342	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583342	2
#19145023	2
C₂×#14583279	2
#19136866	2
#19136851	2
#19136848	2
C₄×M₃₂×C₄₃	2
#33718	43