Subgroups — #25114913

Order 11008.

914 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583005	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
3	#19137507	C₂	1	2	yes	maximal
4	#19135331	C₂	1	2	yes	maximal
5	#19134860	C₂	1	4	yes	maximal
6	#19134597	C₂	1	2	yes	maximal
7	#19134462	C₂	1	1	yes	maximal, characteristic
8	C₄₃×#1844	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
11	C₂×#13063290	C₂²	1	4	yes	—
12	#14583005	—	2	8	no	—
13	C₄×Dic₁₇₂	C₄	1	8	yes	complemented
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×#13063250	C₂²	1	2	yes	—
17	C₂×#13063248	C₂²	1	4	yes	—
18	C₂×#13063235	C₂²	1	2	yes	—
19	#14582795	C₂²	1	4	yes	—
20	#14582794	—	2	4	no	—
21	#14582768	—	2	2	no	—
22	C₄⋊C₄×Dic₄₃	—	2	2	no	—
23	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
24	#14582661	C₂²	1	4	yes	—
25	#14582653	C₂²	1	2	yes	—
26	C₂×#106×C₄₃	C₂²	1	2	yes	—
27	C₂×#106×C₄₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#384	C₂²	1	2	yes	—
30	C₂²×Dic₈₆	C₂³	1	2	yes	—
31	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
32	C₂×Dic₁₇₂	D₄	1	4	yes	—
33	C₂×Dic₁₇₂	Q₈	1	4	yes	—
34	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
35	C₄×Dic₈₆	—	2	4	no	—
36	C₄×Dic₈₆	C₂×C₄	1	8	yes	—
37	#13063290	—	2	8	no	—
38	#13063290	—	4	4	no	—
39	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
41	C₂×Dic₄₃⋊C₄	—	2	2	no	—
42	C₂×Dic₄₃⋊C₄	—	2	4	no	—
43	C₂×Dic₄₃⋊C₄	—	2	4	no	—
44	C₂×C₄×Dic₄₃	—	2	2	no	—
45	C₂×C₄×Dic₄₃	—	2	4	no	—
46	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
47	#13063259	C₂×C₄	1	4	yes	—
48	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
49	#13063250	C₂×C₄	1	8	yes	—
50	#13063248	—	2	8	no	—
51	#13063235	—	2	4	no	—
52	C₄₃⋊#95	—	2	4	no	—
53	C₄₃⋊#95	D₄	1	2	yes	—
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	#106×C₄₃	—	2	4	no	—
61	#106×C₄₃	—	4	2	no	—
62	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
63	C₂²×C₄×C₄₃	—	2	4	no	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₃	—	2	2	no	—
68	C₂²×Dic₄₃	—	2	2	no	—
69	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
70	C₂×Dic₈₆	—	2	2	no	—
71	C₂×Dic₈₆	—	2	2	no	—
72	C₂×Dic₈₆	—	2	2	no	—
73	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
74	C₂×Dic₈₆	C₂×D₄	1	4	yes	—
75	C₂×Dic₈₆	C₂×Q₈	1	4	yes	—
76	C₂×C₈×C₄₃	—	2	1	no	—
77	C₂×C₈×C₄₃	—	2	1	no	—
78	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
82	C₄⋊C₄×C₄₃	—	2	4	no	—
83	C₄⋊C₄×C₄₃	—	2	4	no	—
84	C₄⋊C₄×C₄₃	—	4	4	no	—
85	C₄⋊C₄×C₄₃	—	8	2	no	—
86	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₃	#41	1	2	yes	—
88	C₄⋊Dic₄₃	—	2	4	no	—
89	C₄⋊Dic₄₃	—	2	2	no	—
90	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
91	Dic₄₃⋊C₄	—	2	8	no	—
92	Dic₄₃⋊C₄	—	4	8	no	—
93	Dic₄₃⋊C₄	—	8	4	no	—
94	C₄×Dic₄₃	—	2	8	no	—
95	C₄×Dic₄₃	—	2	2	no	—
96	C₄×Dic₄₃	—	2	2	no	—
97	C₄×Dic₄₃	—	4	8	no	—
98	C₂×C₄₃⋊C₈	—	2	2	no	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	Dic₁₇₂	C₄⋊C₄	1	16	yes	complemented
102	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	4	4	no	—
105	C₂×C₄×C₄₃	—	4	4	no	—
106	C₂×C₄×C₄₃	—	4	4	no	—
107	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₄₃	#137	1	4	yes	—
110	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×Dic₄₃	—	2	4	no	—
117	C₂×Dic₄₃	—	2	4	no	—
118	C₂×Dic₄₃	—	2	4	no	—
119	C₂×Dic₄₃	—	2	4	no	—
120	C₂×Dic₄₃	—	2	4	no	—
121	C₂×Dic₄₃	—	2	4	no	—
122	C₂×Dic₄₃	—	2	4	no	—
123	C₂×Dic₄₃	—	4	2	no	—
124	C₂×Dic₄₃	—	4	2	no	—
125	Dic₈₆	—	2	4	no	—
126	Dic₈₆	—	2	4	no	—
127	Dic₈₆	—	2	4	no	—
128	Dic₈₆	C₂×C₄⋊C₄	1	8	yes	—
129	C₈×C₄₃	—	2	2	no	—
130	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
131	C₄₃⋊C₈	—	4	2	no	—
132	#22674	—	43	1	no	maximal, Sylow
133	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₄₃	D₄²	1	1	yes	characteristic
135	C₂²×C₄₃	D₄×Q₈	1	1	yes	characteristic
136	C₂²×C₄₃	C₂×#137	1	2	yes	—
137	C₂²×C₄₃	#577	1	2	yes	—
138	C₄×C₄₃	—	2	2	no	—
139	C₄×C₄₃	—	8	4	no	—
140	C₄×C₄₃	#441	1	4	yes	complemented
141	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₄₃	#520	1	1	yes	characteristic
143	C₄×C₄₃	#547	1	1	yes	characteristic
144	C₄×C₄₃	#554	1	1	yes	characteristic
145	Dic₄₃	—	2	8	no	—
146	Dic₄₃	—	4	4	no	—
147	Dic₄₃	—	4	4	no	—
148	C₂×#441	—	43	2	no	—
149	C₂×C₄×Q₁₆	—	43	1	no	—
150	Q₈×C₄⋊C₄	—	43	2	no	—
151	#1943	—	43	2	no	—
152	#1865	—	43	4	no	—
153	#1844	D₄₃	1	1	yes	characteristic
154	#1803	—	43	2	no	—
155	#1770	—	43	1	no	—
156	C₂×C₄₃	D₄×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄₃	C₂×#441	1	1	yes	characteristic
158	C₂×C₄₃	#2952	1	1	yes	characteristic
159	C₂×C₄₃	#3278	1	2	yes	—
160	C₂×C₄₃	#3387	1	2	yes	—
161	C₂²×Q₁₆	—	43	1	no	—
162	C₂×C₄×Q₈	—	43	4	no	—
163	C₂×C₄×Q₈	—	43	2	no	—
164	#441	—	86	8	no	—
165	C₄×Q₁₆	—	43	8	no	—
166	C₂×#107	—	43	1	no	—
167	C₂×#106	C₂×D₄₃	1	2	yes	—
168	C₂×#106	C₂×D₄₃	1	2	yes	—
169	C₂×C₄⋊C₈	—	43	1	no	—
170	C₂×#103	—	43	2	no	—
171	C₂×#103	—	43	4	no	—
172	C₂×#97	—	43	2	no	—
173	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
174	#389	—	43	4	no	—
175	#389	—	86	4	no	—
176	#384	—	86	2	no	—
177	#384	C₂×D₄₃	1	2	yes	—
178	C₄×C₄⋊C₄	—	43	1	no	—
179	C₄×C₄⋊C₄	—	86	2	no	—
180	#340	—	43	4	no	—
181	#336	—	43	2	no	—
182	C₄₃	#22674	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	—	43	2	no	—
184	C₂×Q₁₆	—	43	4	no	—
185	C₂×Q₁₆	—	43	4	no	—
186	C₂×Q₁₆	—	43	4	no	—
187	C₂²×C₈	—	43	2	no	—
188	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
190	C₄×Q₈	—	43	8	no	—
191	C₄×Q₈	—	86	4	no	—
192	C₄×Q₈	—	86	8	no	—
193	C₄×Q₈	—	172	4	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	43	2	no	—
197	C₂×C₄⋊C₄	—	43	2	no	—
198	C₂×C₄⋊C₄	—	86	2	no	—
199	C₂×C₄⋊C₄	—	86	4	no	—
200	C₂×C₄⋊C₄	—	86	4	no	—
201	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
202	C₂×C₄²	—	43	2	no	—
203	C₂×C₄²	—	86	2	no	—
204	C₂×C₄²	—	86	4	no	—
205	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
206	#107	—	43	4	no	—
207	#106	—	2	4	no	—
208	#106	—	4	2	no	—
209	C₄⋊C₈	—	86	2	no	—
210	#103	—	43	8	no	—
211	#103	—	86	8	no	—
212	#97	—	86	4	no	—
213	C₄×C₈	C₄×D₄₃	1	4	yes	—
214	#95	—	43	2	no	—
215	#95	—	86	4	no	—
216	C₂×Q₈	—	43	4	no	—
217	C₂×Q₈	—	43	4	no	—
218	C₂×Q₈	—	43	4	no	—
219	C₂×Q₈	—	86	2	no	—
220	C₂×Q₈	—	86	2	no	—
221	C₂×Q₈	—	86	2	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	43	4	no	—
224	C₂²×C₄	—	86	2	no	—
225	C₂²×C₄	—	86	2	no	—
226	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
229	Q₁₆	—	43	16	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	86	2	no	—
233	C₂×C₈	—	86	2	no	—
234	C₂×C₈	—	86	2	no	—
235	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
237	C₂×C₈	D₄×D₄₃	1	2	yes	—
238	C₂×C₈	Q₈×D₄₃	1	2	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	8	2	no	—
243	C₄⋊C₄	—	43	4	no	—
244	C₄⋊C₄	—	86	8	no	—
245	C₄⋊C₄	—	86	4	no	—
246	C₄⋊C₄	—	86	2	no	—
247	C₄⋊C₄	—	344	4	no	—
248	C₄⋊C₄	—	172	8	no	—
249	C₄²	—	86	8	no	—
250	C₄²	—	86	2	no	—
251	C₄²	—	86	2	no	—
252	C₄²	—	172	8	no	—
253	C₄²	C₂×C₄×D₄₃	1	2	yes	—
254	C₄²	#10621902	1	2	yes	—
255	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
256	Q₈	—	43	8	no	—
257	Q₈	—	86	4	no	—
258	Q₈	—	86	4	no	—
259	Q₈	—	86	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	86	4	no	—
265	C₂×C₄	—	86	4	no	—
266	C₂×C₄	—	86	4	no	—
267	C₂×C₄	—	86	4	no	—
268	C₂×C₄	—	86	4	no	—
269	C₂×C₄	—	86	4	no	—
270	C₂×C₄	—	86	4	no	—
271	C₂×C₄	—	172	2	no	—
272	C₂×C₄	—	172	2	no	—
273	C₂×C₄	#13063287	1	2	yes	—
274	C₂×C₄	#13063287	1	2	yes	—
275	C₂×C₄	#13063354	1	4	yes	—
276	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
278	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
280	C₂×C₄	#13063370	1	1	yes	characteristic
281	C₂×C₄	#13063373	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	172	2	no	—
284	C₈	C₄⋊C₄×D₄₃	1	4	yes	—
285	C₂²	C₂×#13063287	1	1	yes	characteristic
286	C₂²	#14583534	1	1	yes	characteristic
287	C₂²	#14583564	1	1	yes	characteristic
288	C₂²	C₂×#13063354	1	2	yes	—
289	C₂²	#14583705	1	2	yes	—
290	C₄	—	2	2	no	—
291	C₄	—	8	4	no	—
292	C₄	—	86	8	no	—
293	C₄	—	172	4	no	—
294	C₄	—	172	4	no	—
295	C₄	#14583005	1	4	yes	complemented
296	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
297	C₄	#14583380	1	1	yes	characteristic
298	C₄	#14583552	1	1	yes	characteristic
299	C₄	#14583610	1	1	yes	characteristic
300	C₂	#19137495	1	1	yes	characteristic
301	C₂	C₂×#14583005	1	1	yes	characteristic
302	C₂	#19143478	1	1	yes	characteristic
303	C₂	#19146260	1	2	yes	—
304	C₂	#19147532	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1844	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1844	D₄₃
hypocenter	C₄₃	#22674
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
13. C₄×Dic₁₇₂	C₄	semidirect
101. Dic₁₇₂	C₄⋊C₄	semidirect
140. C₄×C₄₃	#441	semidirect
182. C₄₃	#22674	semidirect
295. C₄	#14583005	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583005	2
C₂×C₄×Dic₁₇₂	2
#19137507	2
#19135331	2
#19134860	2
#19134597	2
#19134462	2
C₄₃×#1844	2
#22674	43