Subgroups — #25080442

Order 11008.

710 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145174	C₂	1	1	yes	maximal, characteristic
2	#19144069	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134040	C₂	1	4	yes	maximal, complemented
5	#19133990	C₂	1	2	yes	maximal, complemented
6	#19133956	C₂	1	2	yes	maximal, complemented
7	#19133273	C₂	1	2	yes	maximal, complemented
8	C₄₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063358	—	2	2	no	—
10	#14583449	C₂²	1	2	yes	—
11	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
12	#14583304	C₂²	1	1	yes	characteristic
13	#14583276	C₂²	1	1	yes	characteristic
14	#14583246	C₂²	1	2	yes	—
15	#14583135	—	2	1	no	—
16	#14583135	—	2	1	no	—
17	#14583122	—	2	4	no	—
18	#14583110	C₂²	1	4	yes	—
19	#14583100	C₂²	1	4	yes	—
20	C₈×Dic₈₆	—	2	1	no	—
21	C₈×Dic₈₆	—	2	1	no	—
22	C₈×Dic₈₆	C₂²	1	2	yes	—
23	C₈×Dic₈₆	C₂²	1	2	yes	—
24	C₄×D₃₄₄	—	2	1	no	—
25	C₄×D₃₄₄	—	2	1	no	—
26	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
27	C₂×Dic₄₃⋊C₈	—	2	1	no	—
28	#14582901	—	2	2	no	—
29	C₂×C₈×Dic₄₃	—	2	1	no	—
30	#14582646	C₂²	1	4	yes	—
31	#14582642	C₂²	1	4	yes	—
32	C₄₃×#423	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₄₃×#335	C₂²	1	2	yes	—
35	C₄₃×#335	C₂²	1	2	yes	—
36	C₂×#10621898	—	2	2	no	—
37	C₂²×Dic₈₆	D₄	1	1	yes	characteristic
38	#13063358	—	4	8	no	—
39	C₂×Dic₁₇₂	—	2	4	no	—
40	C₂×#10621873	—	2	4	no	—
41	C₂×D₃₄₄	—	2	1	no	—
42	C₂×D₃₄₄	—	2	1	no	—
43	C₂²×C₄₃⋊C₈	—	2	1	no	—
44	#13063325	C₂³	1	2	yes	—
45	#13063322	C₂³	1	2	yes	—
46	#13063313	—	2	2	no	—
47	#13063299	—	2	2	no	—
48	#13063299	—	2	2	no	—
49	C₄×Dic₈₆	—	2	1	no	—
50	C₄×Dic₈₆	—	2	1	no	—
51	C₄×Dic₈₆	D₄	1	2	yes	—
52	C₄×Dic₈₆	C₂³	1	2	yes	—
53	C₄×D₁₇₂	—	2	2	no	—
54	C₄×D₁₇₂	—	2	2	no	—
55	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
56	C₂×Dic₄₃⋊C₄	—	2	1	no	—
57	C₂×C₄×Dic₄₃	—	2	1	no	—
58	#13063259	—	2	1	no	—
59	#13063259	—	2	1	no	—
60	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
61	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
62	Dic₄₃⋊C₈	—	2	2	no	—
63	Dic₄₃⋊C₈	—	2	2	no	—
64	#13063250	—	2	4	no	—
65	#13063247	—	2	2	no	—
66	#13063247	—	2	2	no	—
67	#13063247	—	2	4	no	—
68	C₈×Dic₄₃	—	2	2	no	—
69	C₈×Dic₄₃	—	2	2	no	—
70	C₂×M₁₆×C₄₃	—	2	2	no	—
71	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
72	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₄₃	—	2	2	no	—
75	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
76	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
83	C₂×C₂²⋊D₄₃	—	4	2	no	—
84	C₂²×Dic₄₃	—	2	1	no	—
85	C₂²×Dic₄₃	—	2	1	no	—
86	#10621898	—	4	4	no	—
87	#10621898	—	4	4	no	—
88	C₂×D₁₇₂	—	2	2	no	—
89	C₂×D₁₇₂	—	2	2	no	—
90	C₂×D₁₇₂	—	2	2	no	—
91	C₂×C₄×D₄₃	—	4	2	no	—
92	C₂×Dic₈₆	—	2	1	no	—
93	C₂×Dic₈₆	—	2	2	no	—
94	C₂×Dic₈₆	—	2	2	no	—
95	C₂×Dic₈₆	D₈	1	4	yes	complemented
96	C₂×Dic₈₆	C₂×D₄	1	2	yes	—
97	M₁₆×C₄₃	—	4	4	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	—	2	4	no	—
100	C₂×C₈×C₄₃	—	4	1	no	—
101	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₃	#41	1	1	yes	characteristic
108	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
109	(C₂×D₄₃)⋊C₄	—	4	2	no	—
110	(C₂×D₄₃)⋊C₄	—	4	2	no	—
111	C₄⋊Dic₄₃	—	2	1	no	—
112	C₄⋊Dic₄₃	—	2	2	no	—
113	C₄⋊Dic₄₃	—	2	2	no	—
114	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
115	Dic₄₃⋊C₄	—	2	2	no	—
116	Dic₄₃⋊C₄	—	2	2	no	—
117	C₄×Dic₄₃	—	2	2	no	—
118	C₄×Dic₄₃	—	2	2	no	—
119	C₂×C₄₃⋊C₈	—	2	2	no	—
120	C₂×C₄₃⋊C₈	—	4	2	no	—
121	Dic₁₇₂	—	4	8	no	—
122	D₃₄₄	—	4	2	no	—
123	D₃₄₄	—	4	2	no	—
124	#10621873	—	4	8	no	—
125	C₂³×C₄₃	#121	1	1	yes	characteristic
126	C₂²×D₄₃	—	4	2	no	—
127	C₂×C₄×C₄₃	—	2	1	no	—
128	C₂×C₄×C₄₃	—	2	1	no	—
129	C₂×C₄×C₄₃	—	4	1	no	—
130	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₄₃	#135	1	2	yes	—
135	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
138	C₂²⋊D₄₃	—	8	4	no	—
139	C₂×Dic₄₃	—	2	2	no	—
140	C₂×Dic₄₃	—	2	4	no	—
141	C₂×Dic₄₃	—	2	2	no	—
142	C₂×Dic₄₃	—	4	2	no	—
143	C₂×Dic₄₃	—	4	2	no	—
144	D₁₇₂	—	4	2	no	—
145	D₁₇₂	—	4	2	no	—
146	D₁₇₂	—	4	2	no	—
147	D₁₇₂	—	4	2	no	—
148	D₁₇₂	—	8	2	no	—
149	C₄×D₄₃	—	8	4	no	—
150	Dic₈₆	—	2	2	no	—
151	Dic₈₆	—	2	4	no	—
152	Dic₈₆	—	4	2	no	—
153	Dic₈₆	—	4	2	no	—
154	Dic₈₆	C₂×D₈	1	4	yes	—
155	C₈×C₄₃	—	2	1	no	—
156	C₈×C₄₃	—	2	1	no	—
157	C₈×C₄₃	—	2	1	no	—
158	C₈×C₄₃	—	4	4	no	—
159	C₈×C₄₃	#137	1	2	yes	complemented
160	C₄₃⋊C₈	—	4	2	no	—
161	#19261	—	43	1	no	maximal, Sylow
162	C₂²×C₄₃	—	2	1	no	—
163	C₂²×C₄₃	—	2	1	no	—
164	C₂²×C₄₃	#466	1	2	yes	—
165	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
166	C₂×D₄₃	—	8	2	no	—
167	C₂×D₄₃	—	8	2	no	—
168	C₂×D₄₃	—	8	2	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	—	2	1	no	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	#466	1	2	yes	—
173	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
174	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
175	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
176	C₄×C₄₃	#575	1	1	yes	characteristic
177	Dic₄₃	—	2	4	no	—
178	Dic₄₃	—	4	2	no	—
179	Dic₄₃	—	8	2	no	—
180	#3169	—	43	1	no	—
181	#3037	—	43	2	no	—
182	C₂×C₈×Q₈	—	43	1	no	—
183	#1682	—	43	4	no	—
184	#1674	—	43	2	no	—
185	#1668	—	43	2	no	—
186	#1567	D₄₃	1	1	yes	characteristic, complemented
187	#1547	—	43	2	no	—
188	C₂×C₄₃	—	2	2	no	—
189	C₂×C₄₃	#3037	1	1	yes	characteristic
190	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
191	C₂×C₄₃	#3067	1	1	yes	characteristic
192	D₄₃	—	16	2	no	—
193	C₂×#137	—	86	2	no	—
194	#532	—	43	2	no	—
195	C₂×C₄×Q₈	—	43	1	no	—
196	#500	—	43	1	no	—
197	#493	—	43	1	no	—
198	#486	—	43	2	no	—
199	#466	—	86	1	no	—
200	#466	—	86	1	no	—
201	#464	—	86	4	no	—
202	#462	—	43	4	no	—
203	#460	—	43	4	no	—
204	C₈×Q₈	—	43	2	no	—
205	C₈×Q₈	—	43	2	no	—
206	C₈×Q₈	—	86	1	no	—
207	C₈×Q₈	—	86	1	no	—
208	C₄×D₈	—	86	1	no	—
209	C₄×D₈	—	86	1	no	—
210	#423	C₂×D₄₃	1	2	yes	—
211	C₂×C₄⋊C₈	—	43	1	no	—
212	C₂×C₄⋊C₈	—	86	1	no	—
213	#417	—	86	2	no	—
214	C₂×C₄×C₈	—	86	1	no	—
215	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
216	#335	C₂×D₄₃	1	2	yes	—
217	#335	C₂×D₄₃	1	2	yes	—
218	#332	—	43	4	no	—
219	#331	—	43	4	no	—
220	C₄₃	#19261	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	86	2	no	—
222	C₂²×Q₈	—	43	1	no	—
223	#137	—	172	8	no	—
224	C₂×Q₁₆	—	86	4	no	—
225	C₂×SD₁₆	—	86	4	no	—
226	C₂×D₈	—	86	1	no	—
227	C₂×D₈	—	86	1	no	—
228	C₂×M₁₆	—	2	2	no	—
229	C₂²×C₈	—	86	1	no	—
230	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
231	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
232	C₄⋊Q₈	—	43	2	no	—
233	#127	—	43	2	no	—
234	#124	—	86	2	no	—
235	#121	—	86	2	no	—
236	#121	—	86	2	no	—
237	C₄×Q₈	—	43	2	no	—
238	C₄×Q₈	—	43	2	no	—
239	C₄×Q₈	—	86	1	no	—
240	C₄×Q₈	—	86	1	no	—
241	C₄×D₄	—	86	2	no	—
242	C₄×D₄	—	86	2	no	—
243	C₂×C₄⋊C₄	—	43	1	no	—
244	C₂×C₄⋊C₄	—	86	1	no	—
245	C₂×C₄²	—	86	1	no	—
246	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
247	#107	—	86	1	no	—
248	#107	—	86	1	no	—
249	C₄⋊C₈	—	2	2	no	—
250	C₄⋊C₈	—	43	2	no	—
251	C₄⋊C₈	—	86	2	no	—
252	C₄⋊C₈	—	86	2	no	—
253	C₄⋊C₈	—	86	1	no	—
254	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
255	#103	—	86	4	no	—
256	#102	—	86	4	no	—
257	#102	—	86	2	no	—
258	#102	—	86	2	no	—
259	C₄×C₈	—	86	2	no	—
260	C₄×C₈	—	86	2	no	—
261	C₄×C₈	D₁₇₂	1	1	yes	characteristic
262	C₄×C₈	D₁₇₂	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
265	#41	—	172	4	no	—
266	#41	—	172	4	no	—
267	C₂×Q₈	—	43	4	no	—
268	C₂×Q₈	—	43	2	no	—
269	C₂×Q₈	—	86	1	no	—
270	C₂×Q₈	—	86	2	no	—
271	C₂×Q₈	—	86	2	no	—
272	C₂×D₄	—	86	2	no	—
273	C₂×D₄	—	86	2	no	—
274	C₂×D₄	—	86	2	no	—
275	C₂×D₄	—	172	2	no	—
276	C₂²×C₄	—	86	1	no	—
277	C₂²×C₄	—	86	1	no	—
278	C₂²×C₄	—	172	2	no	—
279	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
281	C₂²×C₄	#10621902	1	1	yes	characteristic
282	Q₁₆	—	172	8	no	—
283	SD₁₆	—	172	8	no	—
284	D₈	—	172	2	no	—
285	D₈	—	172	2	no	—
286	M₁₆	—	4	4	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	4	1	no	—
290	C₂×C₈	—	86	2	no	—
291	C₂×C₈	—	172	2	no	—
292	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
293	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
294	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
295	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
296	C₄⋊C₄	—	43	2	no	—
297	C₄⋊C₄	—	86	1	no	—
298	C₄⋊C₄	—	86	2	no	—
299	C₄⋊C₄	—	86	2	no	—
300	C₄⋊C₄	—	86	2	no	—
301	C₄⋊C₄	—	86	2	no	—
302	C₂²⋊C₄	—	172	2	no	—
303	C₂²⋊C₄	—	172	2	no	—
304	C₄²	—	86	2	no	—
305	C₄²	—	86	2	no	—
306	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
307	C₄²	D₄×D₄₃	1	1	yes	characteristic
308	C₄²	#10621902	1	1	yes	characteristic
309	C₄²	C₂³×D₄₃	1	1	yes	characteristic
310	C₂³	—	172	2	no	—
311	C₂³	#13063300	1	1	yes	characteristic
312	Q₈	—	43	4	no	—
313	Q₈	—	86	4	no	—
314	Q₈	—	86	2	no	—
315	Q₈	—	172	2	no	—
316	Q₈	—	172	2	no	—
317	D₄	—	344	2	no	—
318	D₄	—	344	4	no	—
319	D₄	—	172	2	no	—
320	D₄	—	172	2	no	—
321	D₄	—	172	2	no	—
322	D₄	—	172	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	86	4	no	—
327	C₂×C₄	—	86	2	no	—
328	C₂×C₄	—	86	2	no	—
329	C₂×C₄	—	344	4	no	—
330	C₂×C₄	—	172	2	no	—
331	C₂×C₄	—	172	2	no	—
332	C₂×C₄	#13063300	1	1	yes	characteristic
333	C₂×C₄	#13063300	1	1	yes	characteristic
334	C₂×C₄	#13063300	1	1	yes	characteristic
335	C₂×C₄	D₈×D₄₃	1	2	yes	—
336	C₂×C₄	#13063348	1	2	yes	—
337	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
338	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
339	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	4	4	no	—
344	C₈	—	172	2	no	—
345	C₈	#13063358	1	2	yes	complemented
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	344	2	no	—
349	C₂²	—	344	2	no	—
350	C₂²	—	344	2	no	—
351	C₂²	#14583136	1	2	yes	—
352	C₂²	C₂×#13063300	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	86	4	no	—
357	C₄	—	344	2	no	—
358	C₄	—	172	2	no	—
359	C₄	#14583136	1	2	yes	—
360	C₄	C₂×D₈×D₄₃	1	1	yes	characteristic
361	C₄	C₂×#13063348	1	1	yes	characteristic
362	C₄	C₂×#13063358	1	1	yes	characteristic
363	C₄	#14583699	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	688	2	no	—
366	C₂	#19144069	1	1	yes	characteristic
367	C₂	C₂×#14583136	1	1	yes	characteristic
368	C₂	#19144316	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063300
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1567	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	#19145174	C₂
hypercenter	#1567	D₄₃
hypocenter	C₄₃	#19261
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
2. #19144069	C₂	semidirect
3. C₂×C₈×Dic₈₆	C₂	semidirect
4. #19134040	C₂	semidirect
5. #19133990	C₂	semidirect
6. #19133956	C₂	semidirect
7. #19133273	C₂	semidirect
8. C₄₃×#1567	C₂	semidirect
95. C₂×Dic₈₆	D₈	semidirect
159. C₈×C₄₃	#137	semidirect
186. #1567	D₄₃	semidirect
220. C₄₃	#19261	semidirect
345. C₈	#13063358	semidirect

Maximal subgroups

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

subgroup	index
#19145174	2
#19144069	2
C₂×C₈×Dic₈₆	2
#19134040	2
#19133990	2
#19133956	2
#19133273	2
C₄₃×#1567	2
#19261	43