Subgroups — #25252170

Order 11008.

554 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	#19145174	C₂	1	1	yes	maximal, characteristic, complemented
2	#19145165	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136881	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136861	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136837	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136832	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136708	C₂	1	2	yes	maximal, complemented
8	#19136698	C₂	1	2	yes	maximal, complemented
9	C₄×#13063270	C₂	1	2	yes	maximal, complemented
10	C₄×D₆₈₈	C₂	1	2	yes	maximal, complemented
11	C₄×M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583449	—	2	1	no	—
13	#14583433	—	2	1	no	—
14	C₂×#13063270	C₂²	1	2	yes	—
15	C₂×D₆₈₈	C₂²	1	2	yes	—
16	#14583304	C₂²	1	1	yes	characteristic
17	#14583301	C₂²	1	1	yes	characteristic
18	#14583283	C₂²	1	1	yes	characteristic
19	#14583276	C₂²	1	1	yes	characteristic
20	#14583254	—	2	1	no	—
21	#14583246	—	2	1	no	—
22	#14583140	C₂²	1	1	yes	characteristic
23	#14583135	—	2	1	no	—
24	#14583135	C₂²	1	1	yes	characteristic
25	#14583135	C₂²	1	2	yes	—
26	#14583128	—	2	1	no	—
27	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
28	C₄×#10621873	—	2	1	no	—
29	C₄×D₃₄₄	—	2	1	no	—
30	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
31	C₄×D₃₄₄	C₂²	1	2	yes	—
32	#14582727	C₂²	1	2	yes	—
33	#14582725	C₂²	1	2	yes	—
34	#14582704	C₂²	1	2	yes	—
35	#14582701	C₂²	1	2	yes	—
36	#14582701	C₂²	1	2	yes	—
37	#14582701	C₂²	1	2	yes	—
38	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
40	C₄₃×#346	C₂²	1	1	yes	characteristic
41	C₄₃×#346	C₂²	1	1	yes	characteristic
42	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
43	C₂×#10621898	—	4	1	no	—
44	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
45	C₂×#10621873	—	2	1	no	—
46	C₂×D₃₄₄	—	2	1	no	—
47	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
48	C₂×D₃₄₄	C₂³	1	2	yes	—
49	#13063325	—	2	1	no	—
50	#13063322	—	2	1	no	—
51	#13063319	—	4	1	no	—
52	#13063317	—	2	1	no	—
53	#13063313	—	2	1	no	—
54	#13063313	—	4	1	no	—
55	#13063308	—	4	1	no	—
56	#13063304	—	2	1	no	—
57	#13063299	—	2	1	no	—
58	#13063299	—	2	2	no	—
59	#13063299	—	4	1	no	—
60	C₄×Dic₈₆	—	2	1	no	—
61	C₄×D₁₇₂	—	2	1	no	—
62	C₄×D₁₇₂	—	2	2	no	—
63	C₄×D₁₇₂	—	4	1	no	—
64	C₄×C₂²⋊D₄₃	—	4	1	no	—
65	#13063282	—	4	1	no	—
66	#13063270	—	2	4	no	—
67	D₆₈₈	—	2	4	no	—
68	#13063259	—	2	1	no	—
69	#13063259	C₂³	1	1	yes	characteristic
70	#13063259	C₂³	1	1	yes	characteristic
71	#13063259	C₂³	1	2	yes	—
72	#13063257	—	2	1	no	—
73	#13063250	—	2	1	no	—
74	#13063250	—	2	1	no	—
75	#13063247	—	2	1	no	—
76	#13063247	—	2	2	no	—
77	#13063247	—	2	2	no	—
78	#13063247	—	2	1	no	—
79	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
82	M₃₂×C₄₃	—	2	4	no	—
83	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
84	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
85	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
86	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
87	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₄₃	—	4	1	no	—
93	C₂×C₂²⋊D₄₃	—	4	2	no	—
94	#10621898	—	8	4	no	—
95	C₂×D₁₇₂	—	2	1	no	—
96	C₂×D₁₇₂	—	2	2	no	—
97	C₂×D₁₇₂	—	4	1	no	—
98	C₂×D₁₇₂	—	4	1	no	—
99	C₂×C₄×D₄₃	—	4	1	no	—
100	C₂×C₄×D₄₃	—	4	2	no	—
101	C₂×Dic₈₆	—	2	1	no	—
102	C₂×Dic₈₆	—	4	1	no	—
103	C₂×C₈×C₄₃	—	2	2	no	—
104	C₂×C₈×C₄₃	—	2	2	no	—
105	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
109	C₄²×C₄₃	D₈	1	1	yes	characteristic
110	C₄²×C₄₃	D₈	1	1	yes	characteristic
111	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₂²⋊Dic₄₃	—	4	1	no	—
114	(C₂×D₄₃)⋊C₄	—	4	1	no	—
115	(C₂×D₄₃)⋊C₄	—	4	1	no	—
116	(C₂×D₄₃)⋊C₄	—	4	1	no	—
117	(C₂×D₄₃)⋊C₄	—	4	2	no	—
118	(C₂×D₄₃)⋊C₄	—	4	1	no	—
119	(C₂×D₄₃)⋊C₄	—	4	2	no	—
120	C₄⋊Dic₄₃	—	2	2	no	—
121	C₄⋊Dic₄₃	—	2	1	no	—
122	C₄⋊Dic₄₃	—	2	1	no	—
123	C₄⋊Dic₄₃	—	4	1	no	—
124	Dic₄₃⋊C₄	—	4	1	no	—
125	Dic₄₃⋊C₄	—	4	1	no	—
126	Dic₄₃⋊C₄	—	4	1	no	—
127	Dic₄₃⋊C₄	—	4	1	no	—
128	C₄×Dic₄₃	—	4	1	no	—
129	Dic₁₇₂	—	2	2	no	—
130	D₃₄₄	—	2	2	no	—
131	D₃₄₄	—	2	2	no	—
132	D₃₄₄	—	2	2	no	—
133	D₃₄₄	—	4	2	no	—
134	#10621873	—	4	2	no	—
135	C₁₆×C₄₃	—	2	2	no	—
136	C₁₆×C₄₃	#41	1	4	yes	complemented
137	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
138	C₂²×D₄₃	—	4	1	no	—
139	C₂²×D₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	2	1	no	—
141	C₂×C₄×C₄₃	—	2	1	no	—
142	C₂×C₄×C₄₃	—	2	2	no	—
143	C₂×C₄×C₄₃	—	4	1	no	—
144	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
150	C₂²⋊D₄₃	—	8	4	no	—
151	C₂²⋊D₄₃	—	8	2	no	—
152	C₂×Dic₄₃	—	4	1	no	—
153	C₂×Dic₄₃	—	4	1	no	—
154	C₂×Dic₄₃	—	4	1	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	D₁₇₂	—	4	1	no	—
157	D₁₇₂	—	4	2	no	—
158	D₁₇₂	—	4	1	no	—
159	D₁₇₂	—	4	2	no	—
160	D₁₇₂	—	8	2	no	—
161	D₁₇₂	—	8	2	no	—
162	C₄×D₄₃	—	8	2	no	—
163	C₄×D₄₃	—	8	4	no	—
164	Dic₈₆	—	4	1	no	—
165	Dic₈₆	—	4	1	no	—
166	Dic₈₆	—	8	2	no	—
167	C₈×C₄₃	—	2	1	no	—
168	C₈×C₄₃	—	2	1	no	—
169	C₈×C₄₃	#135	1	2	yes	—
170	C₈×C₄₃	C₂×#41	1	2	yes	—
171	#33737	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	—	2	1	no	—
173	C₂²×C₄₃	—	2	1	no	—
174	C₂²×C₄₃	—	2	1	no	—
175	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
176	C₂×D₄₃	—	8	1	no	—
177	C₂×D₄₃	—	8	2	no	—
178	C₂×D₄₃	—	8	2	no	—
179	C₂×D₄₃	—	8	1	no	—
180	C₂×D₄₃	—	8	1	no	—
181	C₂×D₄₃	—	8	2	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	—	2	1	no	—
184	C₄×C₄₃	—	2	1	no	—
185	C₄×C₄₃	—	2	1	no	—
186	C₄×C₄₃	#509	1	2	yes	—
187	C₄×C₄₃	#532	1	1	yes	characteristic
188	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
189	Dic₄₃	—	8	2	no	—
190	Dic₄₃	—	8	1	no	—
191	Dic₄₃	—	8	1	no	—
192	Dic₄₃	—	8	1	no	—
193	#3169	—	43	1	no	—
194	#3168	—	43	1	no	—
195	#2255	—	43	1	no	—
196	#2250	—	43	1	no	—
197	#2244	—	43	1	no	—
198	#2243	—	43	1	no	—
199	#2220	—	43	2	no	—
200	#2218	—	43	2	no	—
201	C₄×SD₃₂	—	43	2	no	—
202	C₄×D₁₆	—	43	2	no	—
203	C₄×M₃₂	D₄₃	1	1	yes	characteristic, complemented
204	C₂×C₄₃	—	4	1	no	—
205	C₂×C₄₃	#3169	1	1	yes	characteristic
206	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
207	C₂×C₄₃	#3414	1	1	yes	characteristic
208	D₄₃	—	16	1	no	—
209	D₄₃	—	16	2	no	—
210	#532	—	86	1	no	—
211	#529	—	86	1	no	—
212	C₂×SD₃₂	—	43	2	no	—
213	C₂×D₁₆	—	43	2	no	—
214	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
215	#500	—	43	1	no	—
216	#499	—	43	1	no	—
217	#495	—	43	1	no	—
218	#493	—	43	1	no	—
219	#488	—	86	1	no	—
220	#486	—	86	1	no	—
221	#467	—	43	1	no	—
222	#466	—	43	1	no	—
223	#466	—	43	2	no	—
224	#466	—	86	1	no	—
225	#465	—	86	1	no	—
226	C₄×Q₁₆	—	43	1	no	—
227	C₄×SD₁₆	—	86	1	no	—
228	C₄×D₈	—	43	1	no	—
229	C₄×D₈	—	43	2	no	—
230	C₄×D₈	—	86	1	no	—
231	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
232	#367	—	43	2	no	—
233	#366	—	43	2	no	—
234	#358	—	43	2	no	—
235	#357	—	43	2	no	—
236	#357	—	43	2	no	—
237	#357	—	43	2	no	—
238	#346	C₂×D₄₃	1	1	yes	characteristic
239	#346	C₂×D₄₃	1	1	yes	characteristic
240	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
241	C₄₃	#33737	1	1	yes	characteristic, complemented, Sylow
242	C₂×#41	—	172	1	no	—
243	C₂×Q₁₆	—	43	1	no	—
244	C₂×SD₁₆	—	86	1	no	—
245	C₂×D₈	—	43	2	no	—
246	C₂×D₈	—	43	1	no	—
247	C₂×D₈	—	86	1	no	—
248	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
249	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
250	C₄⋊Q₈	—	86	1	no	—
251	#127	—	86	1	no	—
252	#126	—	172	1	no	—
253	#125	—	86	1	no	—
254	#124	—	86	1	no	—
255	#124	—	172	1	no	—
256	#123	—	172	1	no	—
257	C₂²⋊Q₈	—	86	1	no	—
258	#121	—	86	1	no	—
259	#121	—	86	2	no	—
260	#121	—	172	1	no	—
261	C₄×Q₈	—	86	1	no	—
262	C₄×D₄	—	86	1	no	—
263	C₄×D₄	—	86	2	no	—
264	C₄×D₄	—	172	1	no	—
265	C₄×D₄	—	172	1	no	—
266	#117	—	172	1	no	—
267	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
268	SD₃₂	—	86	4	no	—
269	D₁₆	—	86	4	no	—
270	M₃₂	—	2	4	no	—
271	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
272	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
273	#107	—	43	1	no	—
274	#107	—	43	1	no	—
275	#107	—	43	2	no	—
276	#107	—	86	1	no	—
277	#106	—	86	1	no	—
278	#103	—	86	1	no	—
279	#103	—	86	1	no	—
280	#102	—	86	1	no	—
281	#102	—	86	2	no	—
282	#102	—	86	2	no	—
283	#102	—	86	1	no	—
284	C₄×C₈	D₁₇₂	1	1	yes	characteristic
285	C₄×C₈	D₁₇₂	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
288	#41	—	344	4	no	—
289	C₂×Q₈	—	86	1	no	—
290	C₂×Q₈	—	172	1	no	—
291	C₂×D₄	—	86	1	no	—
292	C₂×D₄	—	86	2	no	—
293	C₂×D₄	—	172	1	no	—
294	C₂×D₄	—	172	2	no	—
295	C₂×D₄	—	172	1	no	—
296	C₂×D₄	—	172	1	no	—
297	C₂²×C₄	—	172	1	no	—
298	C₂²×C₄	—	172	2	no	—
299	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
300	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
302	Q₁₆	—	86	2	no	—
303	SD₁₆	—	172	2	no	—
304	D₈	—	86	2	no	—
305	D₈	—	86	2	no	—
306	D₈	—	86	2	no	—
307	D₈	—	172	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
311	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
312	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
313	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
314	C₄⋊C₄	—	86	2	no	—
315	C₄⋊C₄	—	86	1	no	—
316	C₄⋊C₄	—	86	1	no	—
317	C₄⋊C₄	—	172	1	no	—
318	C₄⋊C₄	—	172	1	no	—
319	C₄⋊C₄	—	172	1	no	—
320	C₄⋊C₄	—	172	1	no	—
321	C₄⋊C₄	—	172	1	no	—
322	C₂²⋊C₄	—	172	1	no	—
323	C₂²⋊C₄	—	172	1	no	—
324	C₂²⋊C₄	—	172	1	no	—
325	C₂²⋊C₄	—	172	2	no	—
326	C₂²⋊C₄	—	172	1	no	—
327	C₂²⋊C₄	—	172	2	no	—
328	C₂²⋊C₄	—	172	1	no	—
329	C₄²	—	172	1	no	—
330	C₄²	D₃₄₄	1	1	yes	characteristic
331	C₄²	D₃₄₄	1	1	yes	characteristic
332	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
333	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
334	C₁₆	—	2	2	no	—
335	C₁₆	#10621898	1	4	yes	complemented
336	C₂³	—	172	1	no	—
337	C₂³	—	172	2	no	—
338	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
339	Q₈	—	344	2	no	—
340	Q₈	—	172	1	no	—
341	Q₈	—	172	1	no	—
342	D₄	—	344	2	no	—
343	D₄	—	344	2	no	—
344	D₄	—	344	4	no	—
345	D₄	—	344	2	no	—
346	D₄	—	172	2	no	—
347	D₄	—	172	1	no	—
348	D₄	—	172	1	no	—
349	D₄	—	172	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	344	4	no	—
355	C₂×C₄	—	344	2	no	—
356	C₂×C₄	—	172	1	no	—
357	C₂×C₄	—	172	1	no	—
358	C₂×C₄	—	172	1	no	—
359	C₂×C₄	—	172	2	no	—
360	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
361	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
362	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
363	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
364	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
365	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	#13063347	1	2	yes	—
369	C₈	C₂×#10621898	1	2	yes	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	344	2	no	—
374	C₂²	—	344	1	no	—
375	C₂²	—	344	2	no	—
376	C₂²	—	344	1	no	—
377	C₂²	—	344	2	no	—
378	C₂²	—	344	1	no	—
379	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	344	2	no	—
385	C₄	—	344	1	no	—
386	C₄	—	344	1	no	—
387	C₄	—	344	1	no	—
388	C₄	#14583337	1	2	yes	—
389	C₄	#14583449	1	1	yes	characteristic
390	C₄	C₂×#13063347	1	1	yes	characteristic
391	C₂	—	4	1	no	—
392	C₂	—	688	2	no	—
393	C₂	—	688	1	no	—
394	C₂	#19145174	1	1	yes	characteristic
395	C₂	C₂×#14583337	1	1	yes	characteristic
396	C₂	#19147731	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄×M₃₂×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	C₄×M₃₂	D₄₃
hypocenter	C₄₃	#33737
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
1. #19145174	C₂	semidirect
2. #19145165	C₂	semidirect
3. #19136881	C₂	semidirect
4. #19136861	C₂	semidirect
5. #19136837	C₂	semidirect
6. #19136832	C₂	semidirect
7. #19136708	C₂	semidirect
8. #19136698	C₂	semidirect
9. C₄×#13063270	C₂	semidirect
10. C₄×D₆₈₈	C₂	semidirect
11. C₄×M₃₂×C₄₃	C₂	semidirect
136. C₁₆×C₄₃	#41	semidirect
203. C₄×M₃₂	D₄₃	semidirect
241. C₄₃	#33737	semidirect
335. C₁₆	#10621898	semidirect

Maximal subgroups

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

subgroup	index
#19145174	2
#19145165	2
#19136881	2
#19136861	2
#19136837	2
#19136832	2
#19136708	2
#19136698	2
C₄×#13063270	2
C₄×D₆₈₈	2
C₄×M₃₂×C₄₃	2
#33737	43