Subgroups — #25100824

Order 11008.

822 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
2	#19137072	C₂	1	2	yes	maximal
3	#19135456	C₂	1	2	yes	maximal
4	#19135328	C₂	1	2	yes	maximal
5	#19135271	C₂	1	4	yes	maximal
6	C₄×#13063259	C₂	1	1	yes	maximal, characteristic
7	C₄×#13063250	C₂	1	2	yes	maximal
8	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
11	C₄×Dic₁₇₂	C₄	1	8	yes	complemented
12	C₂×#13063259	C₂²	1	2	yes	—
13	C₂×#13063259	C₂²	1	1	yes	characteristic
14	C₂×#13063250	C₂²	1	4	yes	—
15	C₂×#13063250	C₂²	1	2	yes	—
16	#14582805	C₂²	1	4	yes	—
17	#14582797	C₂²	1	4	yes	—
18	#14582783	—	2	4	no	—
19	C₄×Dic₄₃⋊C₄	—	2	2	no	—
20	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
21	#14582745	—	2	2	no	—
22	#14582662	C₂²	1	2	yes	—
23	#14582662	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄³×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₈₆	C₂³	1	2	yes	—
30	C₂×Dic₁₇₂	—	2	4	no	—
31	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
32	C₄×Dic₈₆	—	2	4	no	—
33	C₄×Dic₈₆	C₂×C₄	1	8	yes	—
34	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
36	C₂×Dic₄₃⋊C₄	—	2	2	no	—
37	C₂×Dic₄₃⋊C₄	—	2	4	no	—
38	C₂×C₄×Dic₄₃	—	2	2	no	—
39	C₂×C₄×Dic₄₃	—	2	2	no	—
40	#13063259	—	2	4	no	—
41	#13063259	C₂×C₄	1	4	yes	—
42	#13063250	—	2	8	no	—
43	#13063250	C₂×C₄	1	8	yes	—
44	C₄₃⋊#95	—	2	4	no	—
45	C₄₃⋊#95	—	2	4	no	—
46	C₄₃⋊#95	—	2	4	no	—
47	C₄₃⋊#95	—	2	2	no	—
48	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	D₄	1	2	yes	—
53	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₄₃	—	2	4	no	—
57	C₄×C₈×C₄₃	—	2	4	no	—
58	C₄×C₈×C₄₃	—	2	2	no	—
59	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₄₃	—	2	2	no	—
66	C₂²×Dic₄₃	—	2	2	no	—
67	C₂²×Dic₄₃	—	2	4	no	—
68	C₂×Dic₈₆	—	2	4	no	—
69	C₂×Dic₈₆	—	2	4	no	—
70	C₂×Dic₈₆	—	2	2	no	—
71	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	—	2	2	no	—
74	C₂×C₈×C₄₃	—	2	2	no	—
75	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₃	#41	1	4	yes	—
78	C₂×C₈×C₄₃	#41	1	4	yes	—
79	C₄²×C₄₃	—	2	4	no	—
80	C₄²×C₄₃	—	2	2	no	—
81	C₄²×C₄₃	—	2	2	no	—
82	C₄²×C₄₃	—	2	2	no	—
83	C₄²×C₄₃	Q₁₆	1	4	yes	complemented
84	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
85	C₄²×C₄₃	#41	1	2	yes	—
86	C₄⋊Dic₄₃	—	2	2	no	—
87	C₄⋊Dic₄₃	—	2	4	no	—
88	C₄⋊Dic₄₃	—	2	4	no	—
89	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
90	Dic₄₃⋊C₄	—	2	8	no	—
91	Dic₄₃⋊C₄	—	4	8	no	—
92	C₄×Dic₄₃	—	2	8	no	—
93	C₄×Dic₄₃	—	4	4	no	—
94	Dic₁₇₂	—	2	8	no	—
95	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₃	—	2	2	no	—
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
104	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₄₃	#135	1	4	yes	—
106	C₂×C₄×C₄₃	#135	1	2	yes	—
107	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
109	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
110	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
111	C₂×Dic₄₃	—	2	4	no	—
112	C₂×Dic₄₃	—	2	4	no	—
113	C₂×Dic₄₃	—	4	4	no	—
114	C₂×Dic₄₃	—	4	4	no	—
115	C₂×Dic₄₃	—	4	4	no	—
116	C₂×Dic₄₃	—	4	4	no	—
117	C₂×Dic₄₃	—	4	4	no	—
118	Dic₈₆	—	2	4	no	—
119	Dic₈₆	—	2	4	no	—
120	Dic₈₆	—	4	4	no	—
121	C₈×C₄₃	—	2	2	no	—
122	C₈×C₄₃	—	2	4	no	—
123	C₈×C₄₃	#117	1	4	yes	—
124	#21276	—	43	1	no	maximal, Sylow
125	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂²×C₄₃	#532	1	2	yes	—
127	C₂²×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
128	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
129	C₂²×C₄₃	C₂×#135	1	2	yes	—
130	C₄×C₄₃	—	2	4	no	—
131	C₄×C₄₃	—	2	4	no	—
132	C₄×C₄₃	—	2	2	no	—
133	C₄×C₄₃	C₄×Q₁₆	1	4	yes	—
134	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
135	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
136	C₄×C₄₃	#529	1	2	yes	—
137	Dic₄₃	—	4	4	no	—
138	Dic₄₃	—	4	4	no	—
139	Dic₄₃	—	8	4	no	—
140	C₂×C₄×Q₁₆	—	43	1	no	—
141	#2295	—	43	2	no	—
142	#1969	—	43	2	no	—
143	#1942	—	43	2	no	—
144	#1933	—	43	4	no	—
145	C₄×#107	—	43	1	no	—
146	C₄×#103	—	43	2	no	—
147	C₄²×C₈	D₄₃	1	1	yes	characteristic
148	C₂×C₄₃	#2293	1	1	yes	characteristic
149	C₂×C₄₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
150	C₂×C₄₃	C₄×#135	1	1	yes	characteristic
151	C₂×C₄₃	#3168	1	2	yes	—
152	C₂×C₄₃	#3170	1	2	yes	—
153	C₂²×Q₁₆	—	43	1	no	—
154	C₂×C₄×Q₈	—	43	2	no	—
155	C₄×Q₁₆	—	43	8	no	—
156	C₂×#107	—	43	1	no	—
157	C₂×#107	—	43	2	no	—
158	C₂×#103	—	43	2	no	—
159	C₂×#103	—	43	4	no	—
160	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
161	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
162	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
164	#391	—	43	4	no	—
165	#389	—	43	4	no	—
166	#387	—	86	4	no	—
167	C₄×C₄⋊C₄	—	43	2	no	—
168	C₄×C₄⋊C₄	—	86	2	no	—
169	#376	—	86	2	no	—
170	C₄³	C₂×D₄₃	1	1	yes	characteristic
171	#340	—	43	2	no	—
172	#340	—	43	4	no	—
173	C₄₃	#21276	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₈	—	43	2	no	—
175	C₂×Q₁₆	—	43	4	no	—
176	C₂×Q₁₆	—	86	4	no	—
177	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
179	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
180	C₄×Q₈	—	43	8	no	—
181	C₄×Q₈	—	86	4	no	—
182	C₂×C₄⋊C₄	—	43	2	no	—
183	C₂×C₄⋊C₄	—	43	4	no	—
184	C₂×C₄⋊C₄	—	86	2	no	—
185	C₂×C₄⋊C₄	—	86	4	no	—
186	C₂×C₄²	—	86	2	no	—
187	C₂×C₄²	—	86	2	no	—
188	C₂×C₄²	D₁₇₂	1	2	yes	—
189	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
190	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
192	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
193	#107	—	43	4	no	—
194	#107	—	86	4	no	—
195	#103	—	43	8	no	—
196	#103	—	86	8	no	—
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	—	2	4	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	C₄×D₄₃	1	4	yes	—
201	#95	—	86	4	no	—
202	#95	—	86	4	no	—
203	#95	—	86	2	no	—
204	#95	—	86	4	no	—
205	C₂×Q₈	—	43	4	no	—
206	C₂×Q₈	—	86	2	no	—
207	C₂×Q₈	—	86	4	no	—
208	C₂×Q₈	—	86	4	no	—
209	C₂²×C₄	—	86	2	no	—
210	C₂²×C₄	—	86	2	no	—
211	C₂²×C₄	—	86	4	no	—
212	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
214	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
215	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
217	Q₁₆	—	86	8	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
222	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
223	C₂×C₈	#10621898	1	4	yes	—
224	C₂×C₈	#10621898	1	4	yes	—
225	C₄⋊C₄	—	43	4	no	—
226	C₄⋊C₄	—	86	8	no	—
227	C₄⋊C₄	—	86	2	no	—
228	C₄⋊C₄	—	86	4	no	—
229	C₄⋊C₄	—	86	4	no	—
230	C₄⋊C₄	—	172	8	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	86	8	no	—
236	C₄²	—	172	4	no	—
237	C₄²	Dic₁₇₂	1	4	yes	complemented
238	C₄²	C₂×C₄×D₄₃	1	2	yes	—
239	C₄²	#10621898	1	2	yes	—
240	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
241	Q₈	—	86	4	no	—
242	Q₈	—	86	4	no	—
243	Q₈	—	172	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	86	4	no	—
250	C₂×C₄	—	86	4	no	—
251	C₂×C₄	—	172	4	no	—
252	C₂×C₄	—	172	4	no	—
253	C₂×C₄	—	172	4	no	—
254	C₂×C₄	—	172	4	no	—
255	C₂×C₄	—	172	4	no	—
256	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
257	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
258	C₂×C₄	C₂×Dic₁₇₂	1	4	yes	—
259	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
260	C₂×C₄	#13063347	1	4	yes	—
261	C₂×C₄	#13063347	1	2	yes	—
262	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
263	C₂×C₄	C₂×#10621898	1	2	yes	—
264	C₂×C₄	C₂×#10621898	1	2	yes	—
265	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	4	no	—
268	C₈	#13063282	1	4	yes	—
269	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
270	C₂²	#14583449	1	2	yes	—
271	C₂²	C₂²×Dic₁₇₂	1	1	yes	characteristic
272	C₂²	C₂×#13063347	1	1	yes	characteristic
273	C₂²	C₂×#13063347	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	4	no	—
277	C₄	—	344	4	no	—
278	C₄	—	172	4	no	—
279	C₄	—	172	4	no	—
280	C₄	C₄×Dic₁₇₂	1	4	yes	—
281	C₄	C₂×#13063282	1	1	yes	characteristic
282	C₄	C₄×#10621898	1	1	yes	characteristic
283	C₄	#14583433	1	2	yes	—
284	C₂	#19137059	1	1	yes	characteristic
285	C₂	C₂×C₄×Dic₁₇₂	1	1	yes	characteristic
286	C₂	C₄×#13063347	1	1	yes	characteristic
287	C₂	#19145165	1	2	yes	—
288	C₂	#19145181	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄²×C₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	#21276
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₁₇₂	C₄	semidirect
83. C₄²×C₄₃	Q₁₆	semidirect
173. C₄₃	#21276	semidirect
237. C₄²	Dic₁₇₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₇₂	2
#19137072	2
#19135456	2
#19135328	2
#19135271	2
C₄×#13063259	2
C₄×#13063250	2
C₄²×C₈×C₄₃	2
#21276	43