Subgroups — #25253305

Order 11008.

686 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583329	C₂	1	2	yes	maximal, complemented
2	#19146317	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136850	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136689	C₂	1	2	yes	maximal, complemented
5	#19136645	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136627	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136535	C₂	1	2	yes	maximal, complemented
8	#19136516	C₂	1	2	yes	maximal, complemented
9	C₄×#13063268	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#3287	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	2	yes	—
13	#14583554	—	2	1	no	—
14	#14583329	—	2	8	no	—
15	C₂×#13063271	C₂²	1	2	yes	—
16	C₂×#13063269	C₂²	1	2	yes	—
17	C₂×#13063269	C₂²	1	2	yes	—
18	C₂×#13063268	C₂²	1	1	yes	characteristic
19	C₂×#13063266	C₂²	1	2	yes	—
20	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
21	#14583279	C₂²	1	1	yes	characteristic
22	#14583140	C₂²	1	2	yes	—
23	#14583105	—	2	1	no	—
24	#14583063	—	2	2	no	—
25	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
26	#14582726	C₂²	1	1	yes	characteristic
27	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
28	#14582703	C₂²	1	1	yes	characteristic
29	#14582702	C₂²	1	2	yes	—
30	#14582700	C₂²	1	2	yes	—
31	#14582699	C₂²	1	1	yes	characteristic
32	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
33	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₂×#135×C₄₃	C₂²	1	2	yes	—
35	C₄₃×#550	—	2	1	no	—
36	C₄₃×#493	C₂²	1	1	yes	characteristic
37	C₄₃×#466	C₂²	1	2	yes	—
38	C₄₃×#460	—	2	1	no	—
39	C₄₃×#449	—	2	2	no	—
40	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
42	C₂×#10621898	—	2	2	no	—
43	#13063347	—	2	4	no	—
44	#13063347	—	4	2	no	—
45	C₂×Dic₁₇₂	—	2	1	no	—
46	C₂×Dic₁₇₂	C₂³	1	2	yes	—
47	C₂×#10621873	—	2	2	no	—
48	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
49	#13063325	—	2	1	no	—
50	#13063307	—	4	2	no	—
51	#13063304	—	2	2	no	—
52	#13063296	—	4	1	no	—
53	C₄×D₁₇₂	—	2	1	no	—
54	C₄⋊C₄×D₄₃	—	4	1	no	—
55	#13063271	—	2	4	no	—
56	#13063269	—	2	4	no	—
57	#13063269	—	4	2	no	—
58	#13063268	D₄	1	4	yes	complemented
59	#13063266	—	2	4	no	—
60	C₂×C₄₃⋊C₁₆	—	2	4	no	—
61	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
62	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
64	#13063259	C₂³	1	1	yes	characteristic
65	#13063250	—	2	2	no	—
66	#13063247	—	2	1	no	—
67	C₂×#41×C₄₃	—	2	2	no	—
68	#135×C₄₃	—	2	4	no	—
69	#135×C₄₃	—	4	2	no	—
70	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×SD₁₆×C₄₃	—	2	2	no	—
72	C₂×D₈×C₄₃	—	2	1	no	—
73	C₂×D₈×C₄₃	C₂³	1	2	yes	—
74	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
75	#127×C₄₃	—	2	1	no	—
76	#127×C₄₃	—	4	1	no	—
77	#121×C₄₃	—	2	2	no	—
78	#121×C₄₃	—	4	2	no	—
79	C₄×Q₈×C₄₃	—	2	1	no	—
80	C₄×D₄×C₄₃	—	4	1	no	—
81	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
82	#107×C₄₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
84	#103×C₄₃	—	2	1	no	—
85	#102×C₄₃	—	2	2	no	—
86	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
87	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
88	#41×C₄₃	—	4	4	no	—
89	#41×C₄₃	—	8	2	no	—
90	C₂×Q₈×C₄₃	—	2	1	no	—
91	C₂×D₄×C₄₃	—	2	2	no	—
92	C₂×D₄×C₄₃	—	4	1	no	—
93	C₂×D₄×C₄₃	—	4	1	no	—
94	C₂×D₄×C₄₃	—	4	2	no	—
95	C₂×D₄×C₄₃	—	4	2	no	—
96	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₄₃	—	4	2	no	—
98	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
99	C₂×C₂²⋊D₄₃	—	4	2	no	—
100	#10621898	—	4	4	no	—
101	#10621898	—	8	2	no	—
102	C₂×D₁₇₂	—	2	1	no	—
103	C₂×C₄×D₄₃	—	4	1	no	—
104	C₂×C₄×D₄₃	—	4	2	no	—
105	C₂×Dic₈₆	—	2	2	no	—
106	Q₁₆×C₄₃	—	2	1	no	—
107	Q₁₆×C₄₃	C₂×D₄	1	2	yes	—
108	SD₁₆×C₄₃	—	4	4	no	—
109	D₈×C₄₃	—	2	2	no	—
110	D₈×C₄₃	—	2	2	no	—
111	D₈×C₄₃	—	8	1	no	—
112	C₂×C₈×C₄₃	—	2	2	no	—
113	C₂×C₈×C₄₃	—	2	2	no	—
114	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₄₃	—	2	1	no	—
118	C₄⋊C₄×C₄₃	—	4	1	no	—
119	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
120	C₂²⋊C₄×C₄₃	—	4	2	no	—
121	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
122	C₄²×C₄₃	—	4	1	no	—
123	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
124	C₂²⋊Dic₄₃	—	4	2	no	—
125	(C₂×D₄₃)⋊C₄	—	4	1	no	—
126	C₄⋊Dic₄₃	—	2	1	no	—
127	C₄⋊Dic₄₃	—	4	1	no	—
128	C₄⋊Dic₄₃	—	4	1	no	—
129	Dic₄₃⋊C₄	—	4	2	no	—
130	Dic₁₇₂	—	2	2	no	—
131	Dic₁₇₂	—	2	2	no	—
132	Dic₁₇₂	—	8	1	no	—
133	D₃₄₄	—	2	1	no	—
134	D₃₄₄	C₂×D₄	1	2	yes	—
135	#10621873	—	4	4	no	—
136	C₄₃⋊C₁₆	—	2	2	no	—
137	C₄₃⋊C₁₆	—	2	1	no	—
138	C₄₃⋊C₁₆	C₂×D₄	1	2	yes	—
139	C₂³×C₄₃	—	4	2	no	—
140	C₂³×C₄₃	C₂×D₈	1	2	yes	—
141	C₂²×D₄₃	—	4	1	no	—
142	Q₈×C₄₃	—	2	2	no	—
143	Q₈×C₄₃	—	4	1	no	—
144	D₄×C₄₃	—	2	2	no	—
145	D₄×C₄₃	—	4	2	no	—
146	D₄×C₄₃	—	4	2	no	—
147	D₄×C₄₃	—	8	4	no	—
148	D₄×C₄₃	—	8	2	no	—
149	D₄×C₄₃	—	16	1	no	—
150	D₄×C₄₃	—	16	2	no	—
151	C₂×C₄×C₄₃	—	2	2	no	—
152	C₂×C₄×C₄₃	—	2	2	no	—
153	C₂×C₄×C₄₃	—	4	1	no	—
154	C₂×C₄×C₄₃	—	4	1	no	—
155	C₂×C₄×C₄₃	—	8	4	no	—
156	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
157	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
160	C₂²⋊D₄₃	—	8	4	no	—
161	C₂×Dic₄₃	—	4	1	no	—
162	C₂×Dic₄₃	—	4	2	no	—
163	D₁₇₂	—	2	2	no	—
164	D₁₇₂	—	4	1	no	—
165	C₄×D₄₃	—	8	4	no	—
166	C₄×D₄₃	—	8	2	no	—
167	Dic₈₆	—	4	2	no	—
168	Dic₈₆	—	4	2	no	—
169	C₈×C₄₃	—	2	2	no	—
170	C₈×C₄₃	—	2	1	no	—
171	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
172	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
173	#33850	—	43	1	no	maximal, Sylow
174	C₂²×C₄₃	—	2	2	no	—
175	C₂²×C₄₃	—	2	2	no	—
176	C₂²×C₄₃	—	2	2	no	—
177	C₂²×C₄₃	—	8	2	no	—
178	C₂²×C₄₃	—	8	2	no	—
179	C₂²×C₄₃	—	8	2	no	—
180	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
181	C₂×D₄₃	—	4	2	no	—
182	C₂×D₄₃	—	8	1	no	—
183	C₂×D₄₃	—	8	1	no	—
184	C₄×C₄₃	—	2	1	no	—
185	C₄×C₄₃	—	2	2	no	—
186	C₄×C₄₃	—	4	2	no	—
187	C₄×C₄₃	—	8	1	no	—
188	C₄×C₄₃	#508	1	2	yes	complemented
189	C₄×C₄₃	D₄²	1	1	yes	characteristic
190	C₄×C₄₃	#576	1	1	yes	characteristic
191	Dic₄₃	—	8	1	no	—
192	Dic₄₃	—	8	2	no	—
193	C₂×#508	—	43	2	no	—
194	#3287	D₄₃	1	1	yes	characteristic, complemented
195	#3283	—	43	1	no	—
196	#2248	—	43	1	no	—
197	#2217	—	43	2	no	—
198	#2209	—	43	1	no	—
199	#2206	—	43	1	no	—
200	#2189	—	43	2	no	—
201	#2186	—	43	2	no	—
202	C₄×SD₃₂	—	43	1	no	—
203	D₄×C₁₆	—	43	1	no	—
204	C₂×C₄₃	—	4	2	no	—
205	C₂×C₄₃	—	16	2	no	—
206	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
207	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
208	C₂×C₄₃	#3416	1	1	yes	characteristic
209	D₄₃	—	8	2	no	—
210	C₂×#135	—	43	2	no	—
211	C₂×#135	C₂×D₄₃	1	2	yes	—
212	#550	—	2	1	no	—
213	#547	—	86	1	no	—
214	#508	—	86	8	no	—
215	C₂×Q₃₂	—	43	2	no	—
216	C₂×SD₃₂	—	43	1	no	—
217	C₂×SD₃₂	—	43	2	no	—
218	C₂×SD₃₂	—	43	2	no	—
219	C₂×D₁₆	—	43	2	no	—
220	C₂²×C₁₆	—	43	2	no	—
221	#494	—	43	1	no	—
222	#493	C₂×D₄₃	1	1	yes	characteristic
223	#467	—	43	2	no	—
224	#466	C₂×D₄₃	1	2	yes	—
225	#461	—	86	1	no	—
226	#460	—	2	1	no	—
227	#452	—	86	2	no	—
228	#449	—	2	2	no	—
229	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
230	C₄×D₈	—	43	1	no	—
231	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
232	#367	—	43	1	no	—
233	C₄⋊C₁₆	—	43	1	no	—
234	#358	—	43	1	no	—
235	#358	—	43	2	no	—
236	#357	—	43	2	no	—
237	#357	—	43	1	no	—
238	C₂²⋊C₁₆	—	43	2	no	—
239	C₄×C₁₆	—	43	1	no	—
240	C₄₃	#33850	1	1	yes	characteristic, complemented, Sylow
241	C₂×#41	—	2	2	no	—
242	C₂×#41	—	86	2	no	—
243	#135	—	2	4	no	—
244	#135	—	4	2	no	—
245	#135	—	86	4	no	—
246	#135	—	172	2	no	—
247	C₂×Q₁₆	—	43	2	no	—
248	C₂×Q₁₆	—	86	1	no	—
249	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
250	C₂×SD₁₆	—	2	2	no	—
251	C₂×SD₁₆	—	86	2	no	—
252	C₂×D₈	—	2	1	no	—
253	C₂×D₈	—	43	1	no	—
254	C₂×D₈	C₂²×D₄₃	1	2	yes	—
255	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
256	C₄⋊Q₈	—	86	1	no	—
257	#127	—	2	1	no	—
258	#127	—	4	1	no	—
259	#123	—	172	2	no	—
260	C₂²⋊Q₈	—	86	2	no	—
261	#121	—	2	2	no	—
262	#121	—	4	2	no	—
263	#121	—	172	1	no	—
264	C₄×Q₈	—	2	1	no	—
265	C₄×D₄	—	4	1	no	—
266	C₄×D₄	—	86	1	no	—
267	C₄×D₄	C₂²⋊D₄₃	1	1	yes	characteristic
268	C₂×C₄⋊C₄	—	172	1	no	—
269	Q₃₂	—	86	4	no	—
270	SD₃₂	—	43	4	no	—
271	SD₃₂	—	86	4	no	—
272	SD₃₂	—	172	2	no	—
273	D₁₆	—	86	4	no	—
274	C₂×C₁₆	—	43	1	no	—
275	C₂×C₁₆	—	43	1	no	—
276	C₂×C₁₆	—	43	2	no	—
277	C₂×C₁₆	—	86	4	no	—
278	#107	—	43	1	no	—
279	#107	C₂²×D₄₃	1	1	yes	characteristic
280	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
281	#103	—	2	1	no	—
282	#103	—	86	2	no	—
283	#102	—	2	2	no	—
284	#102	—	86	1	no	—
285	C₂²⋊C₈	C₂²⋊D₄₃	1	2	yes	—
286	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
287	#41	—	4	4	no	—
288	#41	—	8	2	no	—
289	#41	—	344	2	no	—
290	#41	—	172	4	no	—
291	C₂×Q₈	—	2	1	no	—
292	C₂×Q₈	—	86	2	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	4	2	no	—
295	C₂×D₄	—	4	2	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	86	1	no	—
299	C₂×D₄	—	172	2	no	—
300	C₂×D₄	#10621882	1	1	yes	characteristic
301	C₂²×C₄	—	4	2	no	—
302	C₂²×C₄	—	172	1	no	—
303	C₂²×C₄	—	172	2	no	—
304	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
305	Q₁₆	—	2	1	no	—
306	Q₁₆	—	86	2	no	—
307	Q₁₆	—	86	2	no	—
308	Q₁₆	—	344	1	no	—
309	Q₁₆	D₄×D₄₃	1	2	yes	—
310	SD₁₆	—	4	4	no	—
311	SD₁₆	—	172	4	no	—
312	D₈	—	2	2	no	—
313	D₈	—	2	2	no	—
314	D₈	—	8	1	no	—
315	D₈	—	43	2	no	—
316	D₈	—	86	1	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
320	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
321	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	86	1	no	—
325	C₄⋊C₄	—	172	1	no	—
326	C₄⋊C₄	—	172	2	no	—
327	C₄⋊C₄	—	172	1	no	—
328	C₄⋊C₄	#10621882	1	1	yes	characteristic
329	C₂²⋊C₄	—	4	2	no	—
330	C₂²⋊C₄	—	172	2	no	—
331	C₂²⋊C₄	—	172	1	no	—
332	C₂²⋊C₄	#10621882	1	2	yes	—
333	C₄²	—	4	1	no	—
334	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
335	C₁₆	—	43	2	no	—
336	C₁₆	—	86	2	no	—
337	C₁₆	—	86	1	no	—
338	C₂³	—	4	2	no	—
339	C₂³	—	172	1	no	—
340	C₂³	C₂×#10621882	1	2	yes	—
341	Q₈	—	2	2	no	—
342	Q₈	—	4	1	no	—
343	Q₈	—	172	2	no	—
344	Q₈	—	172	2	no	—
345	D₄	—	2	2	no	—
346	D₄	—	4	2	no	—
347	D₄	—	4	2	no	—
348	D₄	—	8	4	no	—
349	D₄	—	8	2	no	—
350	D₄	—	16	1	no	—
351	D₄	—	16	2	no	—
352	D₄	—	86	2	no	—
353	D₄	—	344	4	no	—
354	D₄	—	172	1	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	8	4	no	—
360	C₂×C₄	—	344	4	no	—
361	C₂×C₄	—	344	2	no	—
362	C₂×C₄	—	172	1	no	—
363	C₂×C₄	—	172	2	no	—
364	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
365	C₂×C₄	C₂×#10621882	1	2	yes	—
366	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
367	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	2	2	no	—
370	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
371	C₈	#13063370	1	1	yes	characteristic
372	C₂²	—	2	2	no	—
373	C₂²	—	2	2	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	8	2	no	—
376	C₂²	—	8	2	no	—
377	C₂²	—	8	2	no	—
378	C₂²	—	344	1	no	—
379	C₂²	—	344	1	no	—
380	C₂²	—	172	2	no	—
381	C₂²	C₂²×#10621882	1	1	yes	characteristic
382	C₄	—	2	2	no	—
383	C₄	—	2	1	no	—
384	C₄	—	4	2	no	—
385	C₄	—	8	1	no	—
386	C₄	—	344	2	no	—
387	C₄	—	344	1	no	—
388	C₄	#14583329	1	2	yes	complemented
389	C₄	D₄×C₂²⋊D₄₃	1	1	yes	characteristic
390	C₄	#14583702	1	1	yes	characteristic
391	C₂	—	4	2	no	—
392	C₂	—	16	2	no	—
393	C₂	—	344	2	no	—
394	C₂	D₄×#10621882	1	1	yes	characteristic
395	C₂	C₂×#14583329	1	1	yes	characteristic
396	C₂	#19147742	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3287	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#3287	D₄₃
hypocenter	C₄₃	#33850
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3287

Complements

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

N	H ≅ G/N	type
1. C₂×#14583329	C₂	semidirect
2. #19146317	C₂	semidirect
3. #19136850	C₂	semidirect
4. #19136689	C₂	semidirect
5. #19136645	C₂	semidirect
6. #19136627	C₂	semidirect
7. #19136535	C₂	semidirect
8. #19136516	C₂	semidirect
9. C₄×#13063268	C₂	semidirect
10. D₄×C₄₃⋊C₁₆	C₂	semidirect
11. C₄₃×#3287	C₂	semidirect
58. #13063268	D₄	semidirect
188. C₄×C₄₃	#508	semidirect
194. #3287	D₄₃	semidirect
240. C₄₃	#33850	semidirect
388. C₄	#14583329	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583329	2
#19146317	2
#19136850	2
#19136689	2
#19136645	2
#19136627	2
#19136535	2
#19136516	2
C₄×#13063268	2
D₄×C₄₃⋊C₁₆	2
C₄₃×#3287	2
#33850	43