Subgroups — #25068029

Order 11008.

656 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143610	C₂	1	2	yes	maximal, complemented
2	C₄×#13063347	C₂	1	1	yes	maximal, characteristic
3	#19133493	C₂	1	2	yes	maximal, complemented
4	#19133433	C₂	1	4	yes	maximal, complemented
5	C₈×D₃₄₄	C₂	1	2	yes	maximal, complemented
6	#19133358	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133030	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063347	C₄	1	1	yes	characteristic
10	C₄×#10621898	C₂²	1	2	yes	—
11	#14583029	C₂²	1	4	yes	—
12	C₄×Dic₁₇₂	C₄	1	1	yes	characteristic
13	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
14	C₄×#10621873	C₄	1	2	yes	—
15	C₄×#10621873	C₂²	1	2	yes	—
16	C₄×D₃₄₄	C₄	1	1	yes	characteristic
17	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
18	#14582968	—	2	4	no	—
19	C₈×D₁₇₂	C₂²	1	4	yes	—
20	#14582956	C₂²	1	2	yes	—
21	#14582954	—	2	2	no	—
22	#14582937	C₄	1	1	yes	characteristic
23	#14582896	C₄	1	2	yes	—
24	#97×D₄₃	—	2	2	no	—
25	#14582652	C₂²	1	2	yes	—
26	#14582650	C₂²	1	2	yes	—
27	#14582631	C₂²	1	4	yes	—
28	#14582628	C₂²	1	4	yes	—
29	C₄₃×#431	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₄₃×#322	C₂²	1	2	yes	—
33	C₈²×C₄₃	C₂²	1	2	yes	—
34	C₂×#10621898	C₂×C₄	1	2	yes	—
35	#13063347	—	2	4	no	—
36	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
37	C₂×#10621873	C₂×C₄	1	2	yes	—
38	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₄₃	—	2	4	no	—
40	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
41	C₄×Dic₈₆	C₂³	1	2	yes	—
42	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
43	C₄×D₁₇₂	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₄₃	—	2	2	no	—
45	C₄×C₂²⋊D₄₃	—	2	2	no	—
46	#13063284	C₂×C₄	1	2	yes	—
47	#13063282	—	2	2	no	—
48	C₄²×D₄₃	—	2	2	no	—
49	#13063259	C₂×C₄	1	1	yes	characteristic
50	#13063259	C₂×C₄	1	1	yes	characteristic
51	#13063257	C₂×C₄	1	2	yes	—
52	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
53	Dic₄₃⋊C₈	—	2	4	no	—
54	#13063250	C₂×C₄	1	2	yes	—
55	#13063250	C₂×C₄	1	2	yes	—
56	#13063247	C₂×C₄	1	2	yes	—
57	#13063247	C₂×C₄	1	2	yes	—
58	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
59	(C₂×D₄₃)⋊C₈	—	2	4	no	—
60	#13063236	—	2	2	no	—
61	#13063235	—	2	4	no	—
62	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
63	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₄₃	—	2	1	no	—
67	C₂²⋊C₈×C₄₃	—	2	1	no	—
68	#97×C₄₃	D₄	1	1	yes	characteristic
69	#97×C₄₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₄₃	C₂³	1	2	yes	—
73	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₄₃	C₂³	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₄₃	—	2	2	no	—
80	#10621898	—	2	4	no	—
81	#10621898	—	4	2	no	—
82	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₄₃	—	2	2	no	—
84	C₂×C₄×D₄₃	—	2	2	no	—
85	C₂×C₄×D₄₃	—	2	2	no	—
86	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
87	M₁₆×C₄₃	—	2	4	no	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	4	1	no	—
91	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
97	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₃	#41	1	1	yes	characteristic
100	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
101	C₂²⋊Dic₄₃	—	2	2	no	—
102	(C₂×D₄₃)⋊C₄	—	2	2	no	—
103	(C₂×D₄₃)⋊C₄	—	2	2	no	—
104	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
105	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
106	Dic₄₃⋊C₄	—	2	2	no	—
107	Dic₄₃⋊C₄	—	2	2	no	—
108	C₄×Dic₄₃	—	2	2	no	—
109	C₄×Dic₄₃	—	2	2	no	—
110	C₄×Dic₄₃	—	2	2	no	—
111	C₂×C₄₃⋊C₈	—	2	4	no	—
112	Dic₁₇₂	—	2	2	no	—
113	D₃₄₄	M₁₆	1	4	yes	complemented
114	#10621873	—	2	4	no	—
115	C₈×D₄₃	—	4	8	no	—
116	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	2	no	—
121	C₂×C₄×C₄₃	—	4	1	no	—
122	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
128	C₂²⋊D₄₃	—	4	4	no	—
129	C₂×Dic₄₃	—	2	2	no	—
130	C₂×Dic₄₃	—	2	2	no	—
131	C₂×Dic₄₃	—	2	2	no	—
132	D₁₇₂	—	2	2	no	—
133	D₁₇₂	C₂×M₁₆	1	4	yes	—
134	C₄×D₄₃	—	4	4	no	—
135	C₄×D₄₃	—	4	4	no	—
136	C₄×D₄₃	—	8	2	no	—
137	Dic₈₆	—	2	2	no	—
138	Dic₈₆	—	2	2	no	—
139	C₈×C₄₃	—	2	1	no	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	—	2	1	no	—
142	C₈×C₄₃	—	2	2	no	—
143	C₈×C₄₃	—	4	2	no	—
144	C₈×C₄₃	C₂×M₁₆	1	2	yes	—
145	C₈×C₄₃	#135	1	4	yes	complemented
146	C₄₃⋊C₈	—	4	4	no	—
147	#18035	—	43	1	no	maximal, Sylow
148	C₂²×C₄₃	—	2	1	no	—
149	C₂²×C₄₃	—	2	1	no	—
150	C₂²×C₄₃	—	2	1	no	—
151	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂×D₄₃	—	2	4	no	—
153	C₂×D₄₃	—	4	2	no	—
154	C₂×D₄₃	—	4	2	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
160	C₄×C₄₃	#568	1	1	yes	characteristic
161	C₄×C₄₃	C₂×#135	1	2	yes	—
162	Dic₄₃	—	4	2	no	—
163	Dic₄₃	—	4	2	no	—
164	Dic₄₃	—	4	2	no	—
165	#2970	—	43	2	no	—
166	C₄×#135	—	43	1	no	—
167	#1591	—	43	2	no	—
168	#1582	—	43	4	no	—
169	C₈×D₈	—	43	2	no	—
170	#1568	—	43	1	no	—
171	#1488	—	43	2	no	—
172	C₈×M₁₆	D₄₃	1	1	yes	characteristic, complemented
173	C₂×C₄₃	—	4	1	no	—
174	C₂×C₄₃	D₄×M₁₆	1	1	yes	characteristic
175	C₂×C₄₃	C₄×#135	1	1	yes	characteristic
176	C₂×C₄₃	#2957	1	1	yes	characteristic
177	D₄₃	—	4	4	no	—
178	C₂×#135	—	43	1	no	—
179	C₄×#41	—	43	2	no	—
180	#446	—	43	4	no	—
181	C₄×Q₁₆	—	43	1	no	—
182	C₄×Q₁₆	—	43	1	no	—
183	C₄×SD₁₆	—	43	2	no	—
184	C₄×SD₁₆	—	43	2	no	—
185	C₄×D₈	—	43	1	no	—
186	C₄×D₈	—	43	1	no	—
187	#435	—	86	4	no	—
188	C₈×D₄	—	43	4	no	—
189	#432	—	43	2	no	—
190	#432	—	86	2	no	—
191	#431	C₂×D₄₃	1	1	yes	characteristic
192	#427	—	43	1	no	—
193	#416	—	43	2	no	—
194	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
195	C₂×#97	—	86	2	no	—
196	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
197	#335	—	43	2	no	—
198	#334	—	43	2	no	—
199	Q₈⋊C₈	—	43	4	no	—
200	D₄⋊C₈	—	43	4	no	—
201	#322	C₂×D₄₃	1	2	yes	—
202	C₈²	C₂×D₄₃	1	2	yes	—
203	C₄₃	#18035	1	1	yes	characteristic, complemented, Sylow
204	C₂×#41	—	43	2	no	—
205	#135	—	86	4	no	—
206	C₂×Q₁₆	—	43	1	no	—
207	C₂×SD₁₆	—	43	2	no	—
208	C₂×D₈	—	43	1	no	—
209	C₂×M₁₆	D₁₇₂	1	2	yes	—
210	C₂²×C₈	—	86	4	no	—
211	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
213	C₄×Q₈	—	43	2	no	—
214	C₄×Q₈	—	43	2	no	—
215	C₄×D₄	—	43	2	no	—
216	C₄×D₄	—	43	2	no	—
217	C₄×D₄	—	86	2	no	—
218	C₄×D₄	—	86	2	no	—
219	#117	—	43	2	no	—
220	#117	—	86	2	no	—
221	C₂×C₄²	—	86	2	no	—
222	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
223	#107	—	43	1	no	—
224	#107	—	43	1	no	—
225	#106	—	43	2	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	43	4	no	—
228	C₄⋊C₈	—	86	4	no	—
229	#103	—	43	2	no	—
230	#103	—	43	2	no	—
231	#102	—	43	2	no	—
232	#102	—	43	2	no	—
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	86	2	no	—
235	C₂²⋊C₈	—	86	4	no	—
236	#97	—	86	4	no	—
237	#97	—	86	2	no	—
238	#97	D₁₇₂	1	1	yes	characteristic
239	#97	D₁₇₂	1	1	yes	characteristic
240	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
241	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₄₃	1	2	yes	—
243	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₄₃	1	2	yes	—
246	#41	—	86	4	no	—
247	#41	—	172	2	no	—
248	C₂×Q₈	—	43	2	no	—
249	C₂×D₄	—	43	2	no	—
250	C₂×D₄	—	86	2	no	—
251	C₂²×C₄	—	86	2	no	—
252	C₂²×C₄	—	86	2	no	—
253	C₂²×C₄	—	86	2	no	—
254	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
256	C₂²×C₄	#10621898	1	1	yes	characteristic
257	Q₁₆	—	86	2	no	—
258	SD₁₆	—	86	4	no	—
259	D₈	—	43	4	no	—
260	M₁₆	—	2	4	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	86	4	no	—
265	C₂×C₈	—	172	8	no	—
266	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
270	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
271	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
272	C₄⋊C₄	—	43	2	no	—
273	C₄⋊C₄	—	43	2	no	—
274	C₄⋊C₄	—	86	2	no	—
275	C₄⋊C₄	—	86	2	no	—
276	C₂²⋊C₄	—	86	2	no	—
277	C₂²⋊C₄	—	86	2	no	—
278	C₂²⋊C₄	—	86	2	no	—
279	C₄²	—	86	2	no	—
280	C₄²	—	86	2	no	—
281	C₄²	—	86	2	no	—
282	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
283	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
284	C₄²	#10621898	1	1	yes	characteristic
285	C₄²	C₂³×D₄₃	1	1	yes	characteristic
286	C₂³	—	86	2	no	—
287	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
288	Q₈	—	86	2	no	—
289	Q₈	—	86	2	no	—
290	D₄	—	43	4	no	—
291	D₄	—	86	2	no	—
292	D₄	—	172	4	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	86	2	no	—
298	C₂×C₄	—	86	2	no	—
299	C₂×C₄	—	86	2	no	—
300	C₂×C₄	—	344	2	no	—
301	C₂×C₄	—	172	4	no	—
302	C₂×C₄	—	172	4	no	—
303	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
304	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
305	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
308	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	172	4	no	—
315	C₈	M₁₆×D₄₃	1	2	yes	—
316	C₈	#13063347	1	4	yes	complemented
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	86	4	no	—
321	C₂²	—	172	2	no	—
322	C₂²	—	172	2	no	—
323	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	172	2	no	—
329	C₄	—	172	2	no	—
330	C₄	—	172	2	no	—
331	C₄	C₂×M₁₆×D₄₃	1	1	yes	characteristic
332	C₄	#14583665	1	1	yes	characteristic
333	C₄	C₂×#13063347	1	2	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	172	4	no	—
336	C₂	M₁₆×D₁₇₂	1	1	yes	characteristic
337	C₂	C₄×#13063347	1	1	yes	characteristic
338	C₂	#19143513	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₈×M₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂×#13063347	C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	C₈×M₁₆	D₄₃
hypocenter	C₄₃	#18035
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #19143610	C₂	semidirect
3. #19133493	C₂	semidirect
4. #19133433	C₂	semidirect
5. C₈×D₃₄₄	C₂	semidirect
6. #19133358	C₂	semidirect
7. #19133030	C₂	semidirect
8. C₈×M₁₆×C₄₃	C₂	semidirect
113. D₃₄₄	M₁₆	semidirect
145. C₈×C₄₃	#135	semidirect
172. C₈×M₁₆	D₄₃	semidirect
203. C₄₃	#18035	semidirect
316. C₈	#13063347	semidirect

Maximal subgroups

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

subgroup	index
#19143610	2
C₄×#13063347	2
#19133493	2
#19133433	2
C₈×D₃₄₄	2
#19133358	2
#19133030	2
C₈×M₁₆×C₄₃	2
#18035	43