Subgroups — #25253529

Order 11008.

558 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19146317	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136824	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136803	C₂	1	2	yes	maximal, complemented
4	#19136739	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136689	C₂	1	2	yes	maximal, complemented
6	#19136640	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136516	C₂	1	2	yes	maximal, complemented
8	C₄×#13063266	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136300	C₂	1	2	yes	maximal, complemented
11	C₄₃×#3341	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	2	yes	—
13	#14583554	—	2	1	no	—
14	C₂×#13063269	C₂²	1	2	yes	—
15	C₂×#13063266	C₂²	1	1	yes	characteristic
16	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
17	#14583279	C₂²	1	1	yes	characteristic
18	#14583140	C₂²	1	2	yes	—
19	#14583105	—	2	1	no	—
20	#14583063	—	2	2	no	—
21	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
22	#14582726	C₂²	1	2	yes	—
23	#14582724	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#14582702	C₂²	1	2	yes	—
26	#14582702	C₂²	1	2	yes	—
27	#14582702	C₂²	1	2	yes	—
28	#14582700	C₂²	1	1	yes	characteristic
29	#14582699	C₂²	1	2	yes	—
30	#14582699	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄₃×#555	—	2	1	no	—
34	C₄₃×#500	C₂²	1	1	yes	characteristic
35	C₄₃×#482	—	2	2	no	—
36	C₄₃×#476	—	2	1	no	—
37	C₄₃×#466	C₂²	1	2	yes	—
38	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
40	C₄₃×#427	C₂²	1	2	yes	—
41	C₂×#10621898	—	2	2	no	—
42	#13063347	—	2	4	no	—
43	#13063347	—	4	2	no	—
44	C₂×Dic₁₇₂	—	2	1	no	—
45	C₂×Dic₁₇₂	C₂³	1	2	yes	—
46	C₂×#10621873	—	2	2	no	—
47	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
48	#13063325	—	2	1	no	—
49	#13063307	—	4	2	no	—
50	#13063304	—	2	2	no	—
51	#13063296	—	4	1	no	—
52	C₄×D₁₇₂	—	2	1	no	—
53	C₄⋊C₄×D₄₃	—	4	1	no	—
54	#13063269	—	4	2	no	—
55	#13063266	—	2	2	no	—
56	C₂×C₄₃⋊C₁₆	—	4	2	no	—
57	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
60	#13063259	C₂³	1	1	yes	characteristic
61	#13063250	—	2	2	no	—
62	#13063247	—	2	1	no	—
63	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₄₃	—	2	1	no	—
66	#124×C₄₃	—	4	2	no	—
67	#121×C₄₃	—	2	2	no	—
68	#121×C₄₃	—	4	1	no	—
69	C₄×D₄×C₄₃	—	2	1	no	—
70	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
71	#117×C₄₃	—	2	2	no	—
72	#117×C₄₃	—	4	1	no	—
73	#107×C₄₃	—	2	1	no	—
74	#107×C₄₃	C₂³	1	2	yes	—
75	#107×C₄₃	C₂³	1	1	yes	characteristic
76	#106×C₄₃	—	2	2	no	—
77	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
78	#102×C₄₃	—	2	1	no	—
79	#102×C₄₃	—	2	2	no	—
80	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
81	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₃	—	4	1	no	—
83	C₂×D₄×C₄₃	—	2	1	no	—
84	C₂×D₄×C₄₃	—	4	2	no	—
85	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	—	4	1	no	—
87	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
88	C₂×C₂²⋊D₄₃	—	4	2	no	—
89	#10621898	—	4	4	no	—
90	#10621898	—	8	2	no	—
91	C₂×D₁₇₂	—	2	1	no	—
92	C₂×C₄×D₄₃	—	4	1	no	—
93	C₂×C₄×D₄₃	—	4	2	no	—
94	C₂×Dic₈₆	—	2	2	no	—
95	D₈×C₄₃	—	2	1	no	—
96	D₈×C₄₃	—	2	1	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₄₃	—	2	1	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	4	1	no	—
105	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
106	C₂²⋊C₄×C₄₃	—	4	2	no	—
107	C₂²⋊C₄×C₄₃	—	4	1	no	—
108	C₂²⋊C₄×C₄₃	—	4	2	no	—
109	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
110	C₄²×C₄₃	—	4	2	no	—
111	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₂²⋊Dic₄₃	—	4	2	no	—
113	(C₂×D₄₃)⋊C₄	—	4	1	no	—
114	C₄⋊Dic₄₃	—	2	1	no	—
115	C₄⋊Dic₄₃	—	4	1	no	—
116	C₄⋊Dic₄₃	—	4	1	no	—
117	Dic₄₃⋊C₄	—	4	2	no	—
118	Dic₁₇₂	—	2	2	no	—
119	Dic₁₇₂	—	2	2	no	—
120	Dic₁₇₂	—	8	1	no	—
121	D₃₄₄	—	2	1	no	—
122	D₃₄₄	#41	1	2	yes	—
123	#10621873	—	4	4	no	—
124	C₄₃⋊C₁₆	—	2	1	no	—
125	C₄₃⋊C₁₆	—	2	1	no	—
126	C₄₃⋊C₁₆	—	2	2	no	—
127	C₂³×C₄₃	—	4	1	no	—
128	C₂³×C₄₃	C₂×D₈	1	2	yes	—
129	C₂²×D₄₃	—	4	1	no	—
130	Q₈×C₄₃	—	16	1	no	—
131	D₄×C₄₃	—	2	2	no	—
132	D₄×C₄₃	—	4	1	no	—
133	D₄×C₄₃	—	4	1	no	—
134	D₄×C₄₃	—	16	2	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	4	1	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	—	8	2	no	—
140	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂²⋊D₄₃	—	8	4	no	—
145	C₂×Dic₄₃	—	4	1	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	D₁₇₂	—	2	2	no	—
148	D₁₇₂	—	4	1	no	—
149	C₄×D₄₃	—	8	4	no	—
150	C₄×D₄₃	—	8	2	no	—
151	Dic₈₆	—	4	2	no	—
152	Dic₈₆	—	4	2	no	—
153	C₈×C₄₃	—	2	2	no	—
154	C₈×C₄₃	—	2	1	no	—
155	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
156	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
157	#33870	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	—	2	2	no	—
159	C₂²×C₄₃	—	2	2	no	—
160	C₂²×C₄₃	—	2	2	no	—
161	C₂²×C₄₃	—	8	1	no	—
162	C₂²×C₄₃	—	8	1	no	—
163	C₂²×C₄₃	—	8	1	no	—
164	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
165	C₂×D₄₃	—	4	2	no	—
166	C₂×D₄₃	—	8	1	no	—
167	C₂×D₄₃	—	8	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	2	no	—
170	C₄×C₄₃	—	8	1	no	—
171	C₄×C₄₃	—	8	2	no	—
172	C₄×C₄₃	#508	1	2	yes	—
173	C₄×C₄₃	#546	1	1	yes	characteristic
174	C₄×C₄₃	#577	1	1	yes	characteristic
175	Dic₄₃	—	8	1	no	—
176	Dic₄₃	—	8	2	no	—
177	#3341	D₄₃	1	1	yes	characteristic, complemented
178	#3283	—	43	1	no	—
179	#2241	—	43	1	no	—
180	#2237	—	43	2	no	—
181	#2226	—	43	1	no	—
182	#2217	—	43	2	no	—
183	#2208	—	43	1	no	—
184	#2186	—	43	2	no	—
185	C₄×D₁₆	—	43	1	no	—
186	D₄×C₁₆	—	43	1	no	—
187	#2138	—	43	2	no	—
188	C₂×C₄₃	—	4	2	no	—
189	C₂×C₄₃	—	16	1	no	—
190	C₂×C₄₃	#3294	1	1	yes	characteristic
191	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
192	C₂×C₄₃	#3416	1	1	yes	characteristic
193	D₄₃	—	8	2	no	—
194	C₂×#135	—	43	2	no	—
195	#555	—	2	1	no	—
196	#547	—	86	1	no	—
197	C₂×SD₃₂	—	43	2	no	—
198	C₂×D₁₆	—	43	1	no	—
199	C₂²×C₁₆	—	43	2	no	—
200	#500	C₂×D₄₃	1	1	yes	characteristic
201	#494	—	43	1	no	—
202	#482	—	2	2	no	—
203	#476	—	2	1	no	—
204	#467	—	43	2	no	—
205	#466	C₂×D₄₃	1	2	yes	—
206	#461	—	86	1	no	—
207	#452	—	86	2	no	—
208	C₄×D₈	—	43	1	no	—
209	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
210	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
211	#427	C₂×D₄₃	1	2	yes	—
212	#367	—	43	2	no	—
213	#366	—	43	1	no	—
214	C₄⋊C₁₆	—	43	1	no	—
215	#358	—	43	2	no	—
216	#358	—	43	2	no	—
217	#358	—	43	2	no	—
218	#357	—	43	1	no	—
219	#357	—	43	1	no	—
220	#357	—	43	2	no	—
221	C₂²⋊C₁₆	—	43	2	no	—
222	C₄×C₁₆	—	43	1	no	—
223	C₄₃	#33870	1	1	yes	characteristic, complemented, Sylow
224	C₂×#41	—	86	2	no	—
225	#135	—	86	4	no	—
226	#135	—	172	2	no	—
227	C₂×Q₁₆	—	43	2	no	—
228	C₂×Q₁₆	—	86	1	no	—
229	C₂×SD₁₆	—	86	2	no	—
230	C₂×D₈	—	43	1	no	—
231	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	86	1	no	—
235	#124	—	4	2	no	—
236	#123	—	172	2	no	—
237	C₂²⋊Q₈	—	86	2	no	—
238	#121	—	2	2	no	—
239	#121	—	4	1	no	—
240	#121	—	172	1	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	86	1	no	—
243	C₄×D₄	C₂²⋊D₄₃	1	1	yes	characteristic
244	#117	—	2	2	no	—
245	#117	—	4	1	no	—
246	C₂×C₄⋊C₄	—	172	1	no	—
247	SD₃₂	—	172	2	no	—
248	D₁₆	—	86	2	no	—
249	C₂×C₁₆	—	43	1	no	—
250	C₂×C₁₆	—	43	1	no	—
251	C₂×C₁₆	—	43	2	no	—
252	C₂×C₁₆	—	172	2	no	—
253	#107	—	2	1	no	—
254	#107	—	43	1	no	—
255	#107	C₂²×D₄₃	1	1	yes	characteristic
256	#107	C₂²×D₄₃	1	2	yes	—
257	#106	—	2	2	no	—
258	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
259	#103	—	86	2	no	—
260	#102	—	2	1	no	—
261	#102	—	2	2	no	—
262	#102	—	86	1	no	—
263	C₂²⋊C₈	C₂²⋊D₄₃	1	2	yes	—
264	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
265	#41	—	344	2	no	—
266	#41	—	172	4	no	—
267	C₂×Q₈	—	4	1	no	—
268	C₂×Q₈	—	86	2	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	86	1	no	—
272	C₂×D₄	—	172	2	no	—
273	C₂×D₄	#10621882	1	1	yes	characteristic
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	172	1	no	—
276	C₂²×C₄	—	172	2	no	—
277	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
278	Q₁₆	—	86	2	no	—
279	Q₁₆	—	86	2	no	—
280	Q₁₆	—	344	1	no	—
281	SD₁₆	—	172	4	no	—
282	D₈	—	2	1	no	—
283	D₈	—	2	1	no	—
284	D₈	—	43	2	no	—
285	D₈	—	86	1	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
290	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	86	1	no	—
295	C₄⋊C₄	—	172	1	no	—
296	C₄⋊C₄	—	172	2	no	—
297	C₄⋊C₄	—	172	1	no	—
298	C₄⋊C₄	#10621882	1	1	yes	characteristic
299	C₂²⋊C₄	—	4	2	no	—
300	C₂²⋊C₄	—	4	2	no	—
301	C₂²⋊C₄	—	4	1	no	—
302	C₂²⋊C₄	—	172	2	no	—
303	C₂²⋊C₄	—	172	1	no	—
304	C₂²⋊C₄	#10621882	1	2	yes	—
305	C₄²	—	4	2	no	—
306	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
307	C₁₆	—	86	1	no	—
308	C₁₆	—	86	1	no	—
309	C₁₆	—	86	2	no	—
310	C₂³	—	4	1	no	—
311	C₂³	—	172	1	no	—
312	C₂³	C₂×#10621882	1	2	yes	—
313	Q₈	—	16	1	no	—
314	Q₈	—	172	2	no	—
315	Q₈	—	172	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	16	2	no	—
320	D₄	—	86	2	no	—
321	D₄	—	344	4	no	—
322	D₄	—	172	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	8	2	no	—
328	C₂×C₄	—	344	4	no	—
329	C₂×C₄	—	344	2	no	—
330	C₂×C₄	—	172	1	no	—
331	C₂×C₄	—	172	2	no	—
332	C₂×C₄	C₂×#10621882	1	2	yes	—
333	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
334	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
335	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	#41×D₄₃	1	1	yes	characteristic
339	C₈	#13063371	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	8	1	no	—
344	C₂²	—	8	1	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	344	1	no	—
347	C₂²	—	344	1	no	—
348	C₂²	—	172	2	no	—
349	C₂²	C₂²×#10621882	1	1	yes	characteristic
350	C₄	—	2	2	no	—
351	C₄	—	2	1	no	—
352	C₄	—	8	1	no	—
353	C₄	—	8	2	no	—
354	C₄	—	344	2	no	—
355	C₄	—	344	1	no	—
356	C₄	#14583329	1	2	yes	—
357	C₄	#14583537	1	1	yes	characteristic
358	C₄	#14583707	1	1	yes	characteristic
359	C₂	—	4	2	no	—
360	C₂	—	16	1	no	—
361	C₂	—	344	2	no	—
362	C₂	#19146402	1	1	yes	characteristic
363	C₂	C₂×#14583329	1	1	yes	characteristic
364	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×#3341	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#3341	D₄₃
hypocenter	C₄₃	#33870
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3341

Complements

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

N	H ≅ G/N	type
1. #19146317	C₂	semidirect
2. #19136824	C₂	semidirect
3. #19136803	C₂	semidirect
4. #19136739	C₂	semidirect
5. #19136689	C₂	semidirect
6. #19136640	C₂	semidirect
7. #19136516	C₂	semidirect
8. C₄×#13063266	C₂	semidirect
9. D₄×C₄₃⋊C₁₆	C₂	semidirect
10. #19136300	C₂	semidirect
11. C₄₃×#3341	C₂	semidirect
177. #3341	D₄₃	semidirect
223. C₄₃	#33870	semidirect

Maximal subgroups

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

subgroup	index
#19146317	2
#19136824	2
#19136803	2
#19136739	2
#19136689	2
#19136640	2
#19136516	2
C₄×#13063266	2
D₄×C₄₃⋊C₁₆	2
#19136300	2
C₄₃×#3341	2
#33870	43