Subgroups — #25114875

Order 11008.

1122 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583010	C₂	1	2	yes	maximal
2	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
3	#19137495	C₂	1	2	yes	maximal, complemented
4	#19135331	C₂	1	2	yes	maximal, complemented
5	#19134853	C₂	1	4	yes	maximal, complemented
6	#19134592	C₂	1	2	yes	maximal, complemented
7	#19134462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
11	C₂×#13063287	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×D₄₃	—	2	2	no	—
13	#14583010	—	2	8	no	—
14	C₄×D₃₄₄	C₄	1	8	yes	complemented
15	C₂×#13063259	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×#13063247	C₂²	1	2	yes	—
18	C₂×#13063245	C₂²	1	4	yes	—
19	C₂×#13063235	C₂²	1	2	yes	—
20	#14582795	C₂²	1	4	yes	—
21	#14582770	—	2	4	no	—
22	#14582756	—	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×D₁₇₂	C₂³	1	2	yes	—
31	C₂²×C₄×D₄₃	—	2	2	no	—
32	C₂²×C₄×D₄₃	—	2	4	no	—
33	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
34	C₂×D₃₄₄	D₄	1	4	yes	—
35	C₂×D₃₄₄	Q₈	1	4	yes	—
36	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
37	C₄×D₁₇₂	—	2	4	no	—
38	C₄×D₁₇₂	C₂×C₄	1	8	yes	complemented
39	#13063287	—	2	8	no	—
40	#13063287	—	4	4	no	—
41	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
42	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
43	C₂×Dic₄₃⋊C₄	—	2	4	no	—
44	C₄⋊C₄×D₄₃	—	4	8	no	—
45	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
46	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
47	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
48	#13063259	C₂×C₄	1	4	yes	complemented
49	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
50	#13063247	C₂×C₄	1	8	yes	complemented
51	#13063245	—	2	8	no	—
52	#13063235	—	2	4	no	—
53	C₄₃⋊#95	—	2	4	no	—
54	C₄₃⋊#95	D₄	1	2	yes	—
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	#106×C₄₃	—	2	4	no	—
62	#106×C₄₃	—	4	2	no	—
63	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
64	C₂³×D₄₃	—	2	2	no	—
65	C₂²×C₄×C₄₃	—	2	4	no	—
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₃	—	2	2	no	—
70	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
71	C₂×D₁₇₂	—	2	2	no	—
72	C₂×D₁₇₂	—	2	2	no	—
73	C₂×D₁₇₂	—	2	2	no	—
74	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
75	C₂×D₁₇₂	C₂×D₄	1	4	yes	—
76	C₂×D₁₇₂	C₂×Q₈	1	4	yes	—
77	C₂×C₄×D₄₃	—	2	8	no	—
78	C₂×C₄×D₄₃	—	4	8	no	—
79	C₂×C₄×D₄₃	—	4	4	no	—
80	C₂×C₄×D₄₃	—	4	4	no	—
81	C₂×C₄×D₄₃	—	4	8	no	—
82	C₂×C₄×D₄₃	—	8	4	no	—
83	C₂×C₈×C₄₃	—	2	1	no	—
84	C₂×C₈×C₄₃	—	2	1	no	—
85	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
89	C₄⋊C₄×C₄₃	—	2	4	no	—
90	C₄⋊C₄×C₄₃	—	2	4	no	—
91	C₄⋊C₄×C₄₃	—	4	4	no	—
92	C₄⋊C₄×C₄₃	—	8	2	no	—
93	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₄₃	#41	1	2	yes	—
95	(C₂×D₄₃)⋊C₄	—	2	8	no	—
96	(C₂×D₄₃)⋊C₄	—	4	8	no	—
97	C₄⋊Dic₄₃	—	2	4	no	—
98	C₄⋊Dic₄₃	—	2	2	no	—
99	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
100	Dic₄₃⋊C₄	—	4	8	no	—
101	C₄×Dic₄₃	—	2	2	no	—
102	C₄×Dic₄₃	—	2	2	no	—
103	C₂×C₄₃⋊C₈	—	2	2	no	—
104	C₂×C₄₃⋊C₈	—	2	2	no	—
105	C₂×C₄₃⋊C₈	—	2	2	no	—
106	D₃₄₄	C₄⋊C₄	1	16	yes	complemented
107	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
108	C₂²×D₄₃	—	2	4	no	—
109	C₂²×D₄₃	—	2	4	no	—
110	C₂²×D₄₃	—	2	4	no	—
111	C₂²×D₄₃	—	4	2	no	—
112	C₂²×D₄₃	—	4	2	no	—
113	C₂²×D₄₃	—	4	2	no	—
114	C₂²×D₄₃	—	4	2	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	4	4	no	—
117	C₂×C₄×C₄₃	—	4	4	no	—
118	C₂×C₄×C₄₃	—	4	4	no	—
119	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×Dic₄₃	—	2	4	no	—
129	C₂×Dic₄₃	—	2	4	no	—
130	C₂×Dic₄₃	—	2	4	no	—
131	C₂×Dic₄₃	—	2	4	no	—
132	C₂×Dic₄₃	—	4	2	no	—
133	C₂×Dic₄₃	—	4	2	no	—
134	D₁₇₂	—	2	4	no	—
135	D₁₇₂	—	2	4	no	—
136	D₁₇₂	—	2	4	no	—
137	D₁₇₂	C₂×C₄⋊C₄	1	8	yes	complemented
138	C₄×D₄₃	—	4	16	no	—
139	C₄×D₄₃	—	8	16	no	—
140	C₈×C₄₃	—	2	2	no	—
141	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	complemented
142	C₄₃⋊C₈	—	4	2	no	—
143	#22670	—	43	1	no	maximal, Sylow
144	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂²×C₄₃	D₄²	1	1	yes	characteristic
146	C₂²×C₄₃	D₄×Q₈	1	1	yes	characteristic
147	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
148	C₂²×C₄₃	#577	1	2	yes	—
149	C₂×D₄₃	—	2	8	no	—
150	C₂×D₄₃	—	4	4	no	—
151	C₂×D₄₃	—	4	4	no	—
152	C₂×D₄₃	—	4	4	no	—
153	C₂×D₄₃	—	4	4	no	—
154	C₂×D₄₃	—	4	4	no	—
155	C₂×D₄₃	—	4	4	no	—
156	C₄×C₄₃	—	2	2	no	—
157	C₄×C₄₃	—	8	4	no	—
158	C₄×C₄₃	#442	1	4	yes	complemented
159	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₄×C₄₃	#520	1	1	yes	characteristic
161	C₄×C₄₃	#547	1	1	yes	characteristic
162	C₄×C₄₃	#554	1	1	yes	characteristic
163	Dic₄₃	—	4	4	no	—
164	Dic₄₃	—	4	4	no	—
165	C₂×#442	—	43	2	no	—
166	C₂×C₄×D₈	—	43	1	no	—
167	D₄×C₄⋊C₄	—	43	2	no	—
168	#1943	—	43	2	no	—
169	#1864	—	43	4	no	—
170	#1844	D₄₃	1	1	yes	characteristic, complemented
171	#1802	—	43	2	no	—
172	#1770	—	43	1	no	—
173	C₂×C₄₃	D₄×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄₃	C₂×#442	1	1	yes	characteristic
175	C₂×C₄₃	#2952	1	1	yes	characteristic
176	C₂×C₄₃	#3272	1	2	yes	—
177	C₂×C₄₃	#3384	1	2	yes	—
178	D₄₃	—	4	8	no	—
179	C₂²×D₈	—	43	1	no	—
180	C₂×C₄×D₄	—	43	4	no	—
181	C₂×C₄×D₄	—	43	2	no	—
182	C₂²×C₄⋊C₄	—	86	2	no	—
183	#442	—	86	8	no	—
184	C₄×D₈	—	43	8	no	—
185	C₂×#107	—	43	1	no	—
186	C₂×#106	C₂×D₄₃	1	2	yes	—
187	C₂×#106	C₂×D₄₃	1	2	yes	—
188	C₂×C₄⋊C₈	—	43	1	no	—
189	C₂×#102	—	43	2	no	—
190	C₂×#102	—	43	4	no	—
191	C₂×#97	—	43	2	no	—
192	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
193	#389	—	43	4	no	—
194	#385	—	86	4	no	—
195	#384	C₂×D₄₃	1	2	yes	—
196	#380	—	86	2	no	—
197	C₄×C₄⋊C₄	—	43	1	no	—
198	#340	—	43	4	no	—
199	#336	—	43	2	no	—
200	C₄₃	#22670	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₄	—	43	2	no	—
202	C₂³×C₄	—	86	4	no	—
203	C₂³×C₄	—	86	2	no	—
204	C₂×D₈	—	43	4	no	—
205	C₂×D₈	—	43	4	no	—
206	C₂×D₈	—	43	4	no	—
207	C₂²×C₈	—	43	2	no	—
208	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
210	C₄×D₄	—	43	8	no	—
211	C₄×D₄	—	86	8	no	—
212	C₄×D₄	—	86	4	no	—
213	C₄×D₄	—	172	4	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	43	2	no	—
217	C₂×C₄⋊C₄	—	43	2	no	—
218	C₂×C₄⋊C₄	—	86	4	no	—
219	C₂×C₄⋊C₄	—	172	8	no	—
220	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
221	C₂×C₂²⋊C₄	—	86	2	no	—
222	C₂×C₂²⋊C₄	—	86	4	no	—
223	C₂×C₄²	—	43	2	no	—
224	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
225	#107	—	43	4	no	—
226	#106	—	2	4	no	—
227	#106	—	4	2	no	—
228	C₄⋊C₈	—	86	2	no	—
229	#102	—	43	8	no	—
230	#102	—	86	8	no	—
231	#97	—	86	4	no	—
232	C₄×C₈	C₄×D₄₃	1	4	yes	complemented
233	#95	—	43	2	no	—
234	#95	—	86	4	no	—
235	C₂⁴	—	86	2	no	—
236	C₂×D₄	—	43	4	no	—
237	C₂×D₄	—	43	4	no	—
238	C₂×D₄	—	43	4	no	—
239	C₂×D₄	—	86	2	no	—
240	C₂×D₄	—	86	2	no	—
241	C₂×D₄	—	86	2	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	43	4	no	—
244	C₂²×C₄	—	86	2	no	—
245	C₂²×C₄	—	86	8	no	—
246	C₂²×C₄	—	344	4	no	—
247	C₂²×C₄	—	172	8	no	—
248	C₂²×C₄	—	172	8	no	—
249	C₂²×C₄	—	172	4	no	—
250	C₂²×C₄	—	172	4	no	—
251	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
253	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
254	D₈	—	43	16	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	86	2	no	—
258	C₂×C₈	—	86	2	no	—
259	C₂×C₈	—	86	2	no	—
260	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
261	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
262	C₂×C₈	D₄×D₄₃	1	2	yes	—
263	C₂×C₈	Q₈×D₄₃	1	2	yes	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	8	2	no	—
268	C₄⋊C₄	—	43	4	no	—
269	C₄⋊C₄	—	86	4	no	—
270	C₄⋊C₄	—	86	2	no	—
271	C₄⋊C₄	—	172	8	no	—
272	C₂²⋊C₄	—	86	8	no	—
273	C₂²⋊C₄	—	172	8	no	—
274	C₄²	—	86	2	no	—
275	C₄²	—	86	2	no	—
276	C₄²	C₂×C₄×D₄₃	1	2	yes	—
277	C₄²	#10621902	1	2	yes	—
278	C₂³	—	86	4	no	—
279	C₂³	—	86	4	no	—
280	C₂³	—	86	4	no	—
281	C₂³	—	172	2	no	—
282	C₂³	—	172	2	no	—
283	C₂³	—	172	2	no	—
284	C₂³	—	172	2	no	—
285	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
286	D₄	—	43	8	no	—
287	D₄	—	86	4	no	—
288	D₄	—	86	4	no	—
289	D₄	—	86	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	86	4	no	—
295	C₂×C₄	—	86	4	no	—
296	C₂×C₄	—	86	4	no	—
297	C₂×C₄	—	86	4	no	—
298	C₂×C₄	—	344	16	no	—
299	C₂×C₄	—	172	2	no	—
300	C₂×C₄	—	172	2	no	—
301	C₂×C₄	—	172	16	no	—
302	C₂×C₄	#13063287	1	2	yes	—
303	C₂×C₄	#13063287	1	2	yes	—
304	C₂×C₄	#13063352	1	4	yes	—
305	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
307	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
308	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
309	C₂×C₄	#13063370	1	1	yes	characteristic
310	C₂×C₄	#13063373	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	172	2	no	—
313	C₈	C₄⋊C₄×D₄₃	1	4	yes	complemented
314	C₂²	—	86	8	no	—
315	C₂²	—	172	4	no	—
316	C₂²	—	172	4	no	—
317	C₂²	—	172	4	no	—
318	C₂²	—	172	4	no	—
319	C₂²	—	172	4	no	—
320	C₂²	—	172	4	no	—
321	C₂²	C₂×#13063287	1	1	yes	characteristic
322	C₂²	#14583534	1	1	yes	characteristic
323	C₂²	#14583564	1	1	yes	characteristic
324	C₂²	C₂×#13063352	1	2	yes	—
325	C₂²	#14583705	1	2	yes	—
326	C₄	—	2	2	no	—
327	C₄	—	8	4	no	—
328	C₄	—	172	4	no	—
329	C₄	—	172	4	no	—
330	C₄	#14583010	1	4	yes	complemented
331	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
332	C₄	#14583380	1	1	yes	characteristic
333	C₄	#14583552	1	1	yes	characteristic
334	C₄	#14583610	1	1	yes	characteristic
335	C₂	—	172	8	no	—
336	C₂	#19137495	1	1	yes	characteristic
337	C₂	C₂×#14583010	1	1	yes	characteristic
338	C₂	#19143478	1	1	yes	characteristic
339	C₂	#19146194	1	2	yes	—
340	C₂	#19147499	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	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1844	D₄₃
hypocenter	C₄₃	#22670
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
3. #19137495	C₂	semidirect
4. #19135331	C₂	semidirect
5. #19134853	C₂	semidirect
6. #19134592	C₂	semidirect
7. #19134462	C₂	semidirect
8. C₄₃×#1844	C₂	semidirect
14. C₄×D₃₄₄	C₄	semidirect
38. C₄×D₁₇₂	C₂×C₄	semidirect
48. #13063259	C₂×C₄	semidirect
50. #13063247	C₂×C₄	semidirect
63. C₄×C₈×C₄₃	C₂×C₄	semidirect
106. D₃₄₄	C₄⋊C₄	semidirect
137. D₁₇₂	C₂×C₄⋊C₄	semidirect
141. C₈×C₄₃	C₂×C₄⋊C₄	semidirect
158. C₄×C₄₃	#442	semidirect
170. #1844	D₄₃	semidirect
200. C₄₃	#22670	semidirect
232. C₄×C₈	C₄×D₄₃	semidirect
313. C₈	C₄⋊C₄×D₄₃	semidirect
330. C₄	#14583010	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583010	2
C₂×C₄×D₃₄₄	2
#19137495	2
#19135331	2
#19134853	2
#19134592	2
#19134462	2
C₄₃×#1844	2
#22670	43