Subgroups — #25250896

Order 11008.

702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583339	C₂	1	2	yes	maximal
2	C₂×#14583279	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136645	C₂	1	4	yes	maximal, complemented
4	#19136640	C₂	1	4	yes	maximal, complemented
5	#19136351	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136314	C₂	1	2	yes	maximal, complemented
7	C₄×#135×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₇₂	—	2	1	no	—
9	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
10	C₂×#13063325	—	2	1	no	—
11	#14583339	—	2	8	no	—
12	C₂×#13063271	C₂²	1	4	yes	—
13	C₂×#13063269	C₂²	1	4	yes	—
14	C₂×#13063265	C₂²	1	2	yes	—
15	#14583279	—	2	2	no	—
16	#14583279	C₂²	1	4	yes	—
17	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	—
18	#14582702	C₂²	1	8	yes	—
19	C₂×#135×C₄₃	C₂²	1	1	yes	characteristic
20	C₄×#41×C₄₃	—	2	1	no	—
21	C₄×Q₁₆×C₄₃	C₂²	1	2	yes	—
22	C₄×SD₁₆×C₄₃	—	2	2	no	—
23	C₄×D₈×C₄₃	C₂²	1	2	yes	—
24	C₄₃×#427	C₂²	1	1	yes	characteristic
25	C₄₃×#416	—	2	1	no	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×Dic₈₆	—	2	2	no	—
28	C₂×Dic₁₇₂	—	2	4	no	—
29	C₂×Dic₁₇₂	—	2	4	no	—
30	C₂×Dic₁₇₂	—	2	4	no	—
31	C₂×Dic₁₇₂	—	2	2	no	—
32	C₂×Dic₁₇₂	—	4	4	no	—
33	C₂×Dic₁₇₂	C₂³	1	4	yes	—
34	#13063325	—	2	4	no	—
35	#13063325	—	4	1	no	—
36	#13063325	—	4	1	no	—
37	C₂×C₄⋊Dic₄₃	—	4	2	no	—
38	#13063271	—	2	8	no	—
39	#13063269	—	2	8	no	—
40	#13063265	—	2	4	no	—
41	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
42	C₂×#41×C₄₃	—	2	1	no	—
43	#135×C₄₃	—	2	2	no	—
44	#135×C₄₃	D₄	1	4	yes	—
45	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×SD₁₆×C₄₃	—	2	1	no	—
47	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₄₃	—	2	2	no	—
51	C₄×D₄×C₄₃	—	2	2	no	—
52	C₄×D₄×C₄₃	—	4	2	no	—
53	#117×C₄₃	—	2	1	no	—
54	#117×C₄₃	—	4	1	no	—
55	C₂×C₄²×C₄₃	—	4	1	no	—
56	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
57	#107×C₄₃	C₂³	1	2	yes	—
58	#106×C₄₃	—	2	1	no	—
59	#103×C₄₃	—	2	2	no	—
60	#102×C₄₃	—	2	2	no	—
61	C₄×C₈×C₄₃	D₄	1	2	yes	—
62	C₄×C₈×C₄₃	C₂³	1	2	yes	—
63	#41×C₄₃	—	2	4	no	—
64	#41×C₄₃	—	4	2	no	—
65	C₂×Q₈×C₄₃	—	2	1	no	—
66	C₂×D₄×C₄₃	—	2	1	no	—
67	C₂×D₄×C₄₃	—	4	1	no	—
68	C₂²×C₄×C₄₃	—	4	1	no	—
69	C₂²×C₄×C₄₃	—	4	2	no	—
70	C₂²×C₄×C₄₃	D₈	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₄₃	—	4	2	no	—
73	C₂×Dic₈₆	—	2	4	no	—
74	C₂×Dic₈₆	—	4	4	no	—
75	C₂×Dic₈₆	—	4	4	no	—
76	C₂×Dic₈₆	—	4	2	no	—
77	Q₁₆×C₄₃	—	2	1	no	—
78	Q₁₆×C₄₃	C₂×D₄	1	2	yes	—
79	SD₁₆×C₄₃	—	2	4	no	—
80	D₈×C₄₃	—	2	1	no	—
81	D₈×C₄₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₄₃	—	2	2	no	—
83	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	#41	1	2	yes	—
87	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₄₃	—	2	2	no	—
89	C₄⋊C₄×C₄₃	—	4	2	no	—
90	C₂²⋊C₄×C₄₃	—	4	2	no	—
91	C₂²⋊C₄×C₄₃	—	4	1	no	—
92	C₄²×C₄₃	—	4	1	no	—
93	C₄²×C₄₃	—	4	2	no	—
94	C₄²×C₄₃	D₈	1	1	yes	characteristic
95	C₄²×C₄₃	D₈	1	1	yes	characteristic
96	C₄²×C₄₃	C₂×D₄	1	2	yes	—
97	C₄⋊Dic₄₃	—	4	4	no	—
98	C₄⋊Dic₄₃	—	4	4	no	—
99	Dic₁₇₂	—	2	4	no	—
100	Dic₁₇₂	—	2	4	no	—
101	Dic₁₇₂	—	4	8	no	—
102	Dic₁₇₂	—	4	4	no	—
103	C₄₃⋊C₁₆	—	2	4	no	—
104	C₂³×C₄₃	—	4	1	no	—
105	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
106	Q₈×C₄₃	—	2	2	no	—
107	Q₈×C₄₃	—	4	1	no	—
108	D₄×C₄₃	—	2	2	no	—
109	D₄×C₄₃	—	4	4	no	—
110	D₄×C₄₃	—	4	1	no	—
111	C₂×C₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	—	4	1	no	—
113	C₂×C₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	—	4	2	no	—
115	C₂×C₄×C₄₃	—	8	4	no	—
116	C₂×C₄×C₄₃	#121	1	2	yes	—
117	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₄₃	—	4	4	no	—
123	C₂×Dic₄₃	—	8	4	no	—
124	Dic₈₆	—	4	4	no	—
125	Dic₈₆	—	4	4	no	—
126	Dic₈₆	—	8	4	no	—
127	C₈×C₄₃	—	2	2	no	—
128	C₈×C₄₃	#121	1	2	yes	—
129	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
131	#33583	—	43	1	no	maximal, Sylow
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	—	4	2	no	—
134	C₂²×C₄₃	—	8	2	no	—
135	C₂²×C₄₃	#459	1	2	yes	—
136	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
137	C₄×C₄₃	—	2	2	no	—
138	C₄×C₄₃	—	4	2	no	—
139	C₄×C₄₃	—	8	2	no	—
140	C₄×C₄₃	#459	1	2	yes	—
141	C₄×C₄₃	#510	1	4	yes	complemented
142	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
143	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
144	Dic₄₃	—	8	4	no	—
145	C₂×#510	—	43	2	no	—
146	C₂×#494	—	43	1	no	—
147	C₄×#135	D₄₃	1	1	yes	characteristic
148	#2209	—	43	4	no	—
149	#2208	—	43	4	no	—
150	#2150	—	43	1	no	—
151	#2141	—	43	2	no	—
152	C₂×C₄₃	—	2	2	no	—
153	C₂×C₄₃	—	8	2	no	—
154	C₂×C₄₃	C₂×#459	1	1	yes	characteristic
155	C₂×C₄₃	C₂×#510	1	2	yes	—
156	C₂×#135	C₂×D₄₃	1	1	yes	characteristic
157	C₂²×Q₁₆	—	43	2	no	—
158	C₂²×Q₁₆	—	86	1	no	—
159	C₂×C₄⋊Q₈	—	86	1	no	—
160	C₄×#41	—	2	1	no	—
161	#510	—	86	8	no	—
162	C₂×Q₃₂	—	43	4	no	—
163	C₂×SD₃₂	—	43	4	no	—
164	C₂×M₃₂	—	43	2	no	—
165	#494	—	43	4	no	—
166	#494	—	86	2	no	—
167	C₄×Q₁₆	C₂×D₄₃	1	2	yes	—
168	C₄×SD₁₆	—	2	2	no	—
169	C₄×D₈	C₂×D₄₃	1	2	yes	—
170	#427	C₂×D₄₃	1	1	yes	characteristic
171	#416	—	2	1	no	—
172	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
173	C₄⋊C₁₆	—	43	4	no	—
174	#358	—	43	8	no	—
175	C₄₃	#33583	1	1	yes	characteristic, complemented, Sylow
176	C₂×#41	—	2	1	no	—
177	C₂²×Q₈	—	86	2	no	—
178	#135	—	2	2	no	—
179	#135	D₁₇₂	1	4	yes	—
180	C₂×Q₁₆	—	43	4	no	—
181	C₂×Q₁₆	—	86	4	no	—
182	C₂×Q₁₆	—	86	4	no	—
183	C₂×Q₁₆	—	86	2	no	—
184	C₂×Q₁₆	—	86	4	no	—
185	C₂×Q₁₆	—	172	4	no	—
186	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
187	C₂×SD₁₆	—	2	1	no	—
188	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
191	C₄⋊Q₈	—	86	4	no	—
192	C₄⋊Q₈	—	172	1	no	—
193	C₄⋊Q₈	—	172	1	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	C₄×D₄	—	4	2	no	—
197	#117	—	2	1	no	—
198	#117	—	4	1	no	—
199	C₂×C₄⋊C₄	—	172	2	no	—
200	C₂×C₄²	—	4	1	no	—
201	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
202	Q₃₂	—	86	8	no	—
203	SD₃₂	—	86	8	no	—
204	M₃₂	—	86	4	no	—
205	C₂×C₁₆	—	43	4	no	—
206	#107	C₂²×D₄₃	1	2	yes	—
207	#106	—	2	1	no	—
208	#103	—	2	2	no	—
209	#102	—	2	2	no	—
210	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
211	C₄×C₈	C₂²×D₄₃	1	2	yes	—
212	#41	—	2	4	no	—
213	#41	—	4	2	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	86	4	no	—
216	C₂×Q₈	—	172	4	no	—
217	C₂×Q₈	—	172	4	no	—
218	C₂×Q₈	—	172	2	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	172	2	no	—
224	C₂²×C₄	#10621882	1	2	yes	—
225	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
226	Q₁₆	—	2	1	no	—
227	Q₁₆	—	86	4	no	—
228	Q₁₆	—	86	4	no	—
229	Q₁₆	—	172	8	no	—
230	Q₁₆	—	172	4	no	—
231	Q₁₆	C₂×D₁₇₂	1	2	yes	—
232	SD₁₆	—	2	4	no	—
233	D₈	—	2	1	no	—
234	D₈	C₂×D₁₇₂	1	2	yes	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
237	C₂×C₈	#10621898	1	2	yes	—
238	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
240	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	172	4	no	—
244	C₄⋊C₄	—	172	4	no	—
245	C₂²⋊C₄	—	4	1	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	2	no	—
249	C₄²	#10621882	1	1	yes	characteristic
250	C₄²	#10621882	1	1	yes	characteristic
251	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
252	C₁₆	—	86	4	no	—
253	C₂³	—	4	1	no	—
254	C₂³	C₂×#10621882	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	—	4	1	no	—
257	Q₈	—	344	4	no	—
258	Q₈	—	172	4	no	—
259	Q₈	—	172	4	no	—
260	D₄	—	2	2	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	344	4	no	—
269	C₂×C₄	—	172	4	no	—
270	C₂×C₄	#13063299	1	2	yes	—
271	C₂×C₄	C₂×#10621882	1	2	yes	—
272	C₂×C₄	C₂×#10621882	1	2	yes	—
273	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
274	C₂×C₄	#13063350	1	2	yes	—
275	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	#13063299	1	2	yes	—
278	C₈	C₂²×D₁₇₂	1	1	yes	characteristic
279	C₈	C₂×#10621898	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	#14583094	1	2	yes	—
284	C₂²	C₂²×#10621882	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	8	2	no	—
288	C₄	—	344	4	no	—
289	C₄	#14583094	1	2	yes	—
290	C₄	#14583339	1	4	yes	complemented
291	C₄	C₂×#13063299	1	1	yes	characteristic
292	C₄	C₂×#13063350	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	8	2	no	—
295	C₂	C₂×#14583094	1	1	yes	characteristic
296	C₂	C₂×#14583339	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄×#135×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#135×C₄₃	C₂²
hypercenter	C₄×#135	D₄₃
hypocenter	C₄₃	#33583
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₂×#14583279	C₂	semidirect
3. #19136645	C₂	semidirect
4. #19136640	C₂	semidirect
5. #19136351	C₂	semidirect
6. #19136314	C₂	semidirect
141. C₄×C₄₃	#510	semidirect
175. C₄₃	#33583	semidirect
290. C₄	#14583339	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583339	2
C₂×#14583279	2
#19136645	2
#19136640	2
#19136351	2
#19136314	2
C₄×#135×C₄₃	2
#33583	43