Subgroups — #25253408

Order 11008.

636 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147746	C₂	1	2	yes	maximal, complemented
2	#19146400	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136847	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136718	C₂	1	2	yes	maximal, complemented
5	#19136676	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136673	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136573	C₂	1	2	yes	maximal, complemented
8	#19136571	C₂	1	2	yes	maximal, complemented
9	#19136474	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136432	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#3293	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583716	C₂²	1	2	yes	—
13	#14583713	—	2	1	no	—
14	#14583340	—	2	2	no	—
15	#14583340	C₂²	1	2	yes	—
16	#14583339	—	2	2	no	—
17	#14583339	C₂²	1	2	yes	—
18	#14583329	—	2	2	no	—
19	#14583329	C₂²	1	1	yes	characteristic
20	C₂×#13063271	—	2	2	no	—
21	C₂×#13063271	C₂²	1	2	yes	—
22	C₂×#13063271	C₂²	1	2	yes	—
23	#14583316	C₂²	1	2	yes	—
24	#14583279	C₂²	1	1	yes	characteristic
25	#14583175	—	2	1	no	—
26	#14583174	C₂²	1	2	yes	—
27	#14583083	—	2	2	no	—
28	#14583017	C₂²	1	1	yes	characteristic
29	#14582728	C₂²	1	1	yes	characteristic
30	#14582720	C₂²	1	1	yes	characteristic
31	#14582715	C₂²	1	2	yes	—
32	#14582714	C₂²	1	2	yes	—
33	#14582706	C₂²	1	1	yes	characteristic
34	#14582705	C₂²	1	1	yes	characteristic
35	#14582683	C₂²	1	2	yes	—
36	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₄₃×#578	—	2	1	no	—
38	C₄₃×#578	C₂²	1	2	yes	—
39	C₄₃×#494	C₂²	1	1	yes	characteristic
40	C₄₃×#473	—	2	1	no	—
41	C₄₃×#473	C₂²	1	2	yes	—
42	C₄₃×#456	—	2	2	no	—
43	C₄₃×#443	C₂²	1	1	yes	characteristic
44	C₄₃×#443	C₂²	1	1	yes	characteristic
45	#13063372	—	2	2	no	—
46	#13063358	—	2	2	no	—
47	#13063358	—	4	2	no	—
48	#13063355	—	2	2	no	—
49	#13063354	—	4	2	no	—
50	#13063347	—	2	2	no	—
51	#13063347	C₂³	1	1	yes	characteristic
52	#13063344	—	4	1	no	—
53	C₂×Dic₁₇₂	—	2	2	no	—
54	C₂×Dic₁₇₂	—	2	1	no	—
55	C₂×Dic₁₇₂	C₂³	1	2	yes	—
56	Q₁₆×D₄₃	—	4	1	no	—
57	#13063331	—	2	1	no	—
58	#13063325	—	2	1	no	—
59	#13063271	—	2	2	no	—
60	#13063271	—	2	2	no	—
61	#13063271	—	2	2	no	—
62	#13063271	—	2	2	no	—
63	#13063271	—	4	2	no	—
64	#13063271	D₄	1	1	yes	characteristic
65	#13063269	—	2	2	no	—
66	#13063269	D₄	1	1	yes	characteristic
67	#13063268	—	2	2	no	—
68	#13063268	D₄	1	1	yes	characteristic
69	#13063266	D₄	1	1	yes	characteristic
70	#13063265	—	2	2	no	—
71	#13063265	C₂³	1	2	yes	—
72	C₂×C₄₃⋊C₁₆	—	2	2	no	—
73	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
75	#13063261	C₂³	1	1	yes	characteristic
76	#13063253	—	2	1	no	—
77	#13063243	—	2	2	no	—
78	2₋¹⁺⁴×C₄₃	—	2	2	no	—
79	#137×C₄₃	—	2	2	no	—
80	#137×C₄₃	—	2	2	no	—
81	#137×C₄₃	—	4	2	no	—
82	#137×C₄₃	—	4	2	no	—
83	#135×C₄₃	—	2	2	no	—
84	#135×C₄₃	—	4	1	no	—
85	#135×C₄₃	D₄	1	1	yes	characteristic
86	#135×C₄₃	C₂³	1	1	yes	characteristic
87	C₂×Q₁₆×C₄₃	—	2	2	no	—
88	C₂×Q₁₆×C₄₃	—	2	1	no	—
89	C₂×Q₁₆×C₄₃	—	4	1	no	—
90	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
91	#131×C₄₃	—	2	1	no	—
92	#131×C₄₃	C₂³	1	2	yes	—
93	C₄⋊Q₈×C₄₃	—	2	1	no	—
94	#108×C₄₃	D₄	1	1	yes	characteristic
95	#108×C₄₃	C₂³	1	1	yes	characteristic
96	#104×C₄₃	—	2	1	no	—
97	#104×C₄₃	D₄	1	2	yes	—
98	#101×C₄₃	—	2	2	no	—
99	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
100	#41×C₄₃	—	2	1	no	—
101	#41×C₄₃	—	4	2	no	—
102	#41×C₄₃	—	4	2	no	—
103	#41×C₄₃	—	8	2	no	—
104	#41×C₄₃	C₂×D₄	1	2	yes	—
105	C₂×Q₈×C₄₃	—	2	2	no	—
106	C₂×Q₈×C₄₃	—	4	2	no	—
107	C₂×Q₈×C₄₃	—	4	2	no	—
108	Q₈×D₄₃	—	4	2	no	—
109	#10621900	—	4	2	no	—
110	#10621900	—	8	2	no	—
111	#10621898	—	2	1	no	—
112	#10621898	—	4	2	no	—
113	C₂×Dic₈₆	—	2	2	no	—
114	C₂×Dic₈₆	—	4	2	no	—
115	Q₁₆×C₄₃	—	2	2	no	—
116	Q₁₆×C₄₃	—	2	2	no	—
117	Q₁₆×C₄₃	—	4	2	no	—
118	Q₁₆×C₄₃	—	4	2	no	—
119	Q₁₆×C₄₃	—	4	1	no	—
120	Q₁₆×C₄₃	—	4	1	no	—
121	Q₁₆×C₄₃	D₈	1	1	yes	characteristic
122	Q₁₆×C₄₃	C₂×D₄	1	1	yes	characteristic
123	SD₁₆×C₄₃	—	2	1	no	—
124	SD₁₆×C₄₃	—	4	2	no	—
125	SD₁₆×C₄₃	—	4	2	no	—
126	SD₁₆×C₄₃	D₈	1	2	yes	—
127	D₈×C₄₃	D₈	1	1	yes	characteristic
128	D₈×C₄₃	C₂×D₄	1	1	yes	characteristic
129	M₁₆×C₄₃	—	2	2	no	—
130	M₁₆×C₄₃	—	2	1	no	—
131	M₁₆×C₄₃	—	4	1	no	—
132	M₁₆×C₄₃	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₄₃	—	2	2	no	—
134	C₂×C₈×C₄₃	—	4	1	no	—
135	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
137	C₄⋊C₄×C₄₃	—	8	1	no	—
138	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
139	#10621885	—	4	2	no	—
140	#10621883	—	4	2	no	—
141	C₄⋊Dic₄₃	—	8	1	no	—
142	#10621877	—	2	1	no	—
143	Dic₁₇₂	—	2	2	no	—
144	Dic₁₇₂	—	2	2	no	—
145	Dic₁₇₂	—	4	2	no	—
146	Dic₁₇₂	—	4	1	no	—
147	Dic₁₇₂	—	4	1	no	—
148	Dic₁₇₂	C₂×D₄	1	1	yes	characteristic
149	D₃₄₄	C₂×D₄	1	1	yes	characteristic
150	#10621873	—	2	1	no	—
151	#10621873	—	4	2	no	—
152	#10621872	—	4	1	no	—
153	C₈×D₄₃	—	4	1	no	—
154	C₄₃⋊C₁₆	—	2	2	no	—
155	C₄₃⋊C₁₆	—	2	1	no	—
156	C₄₃⋊C₁₆	C₂×D₄	1	1	yes	characteristic
157	C₄₃⋊C₁₆	C₂×D₄	1	1	yes	characteristic
158	Q₈×C₄₃	—	2	1	no	—
159	Q₈×C₄₃	—	4	2	no	—
160	Q₈×C₄₃	—	4	2	no	—
161	Q₈×C₄₃	—	8	2	no	—
162	Q₈×C₄₃	C₂×D₈	1	2	yes	—
163	D₄×C₄₃	—	2	2	no	—
164	D₄×C₄₃	—	2	1	no	—
165	D₄×C₄₃	—	4	1	no	—
166	D₄×C₄₃	—	8	2	no	—
167	D₄×C₄₃	C₂×D₈	1	2	yes	—
168	C₂×C₄×C₄₃	—	2	1	no	—
169	C₂×C₄×C₄₃	—	2	2	no	—
170	C₂×C₄×C₄₃	—	4	1	no	—
171	C₂×C₄×C₄₃	—	4	2	no	—
172	C₂×C₄×C₄₃	—	8	2	no	—
173	C₂×C₄×C₄₃	—	8	2	no	—
174	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
175	C₂²⋊D₄₃	—	4	1	no	—
176	C₂²⋊D₄₃	—	8	2	no	—
177	C₂×Dic₄₃	—	4	2	no	—
178	C₂×Dic₄₃	—	8	2	no	—
179	D₁₇₂	—	2	1	no	—
180	C₄×D₄₃	—	4	1	no	—
181	C₄×D₄₃	—	8	2	no	—
182	Dic₈₆	—	2	1	no	—
183	Dic₈₆	—	4	2	no	—
184	Dic₈₆	—	4	2	no	—
185	Dic₈₆	—	8	2	no	—
186	C₈×C₄₃	—	2	2	no	—
187	C₈×C₄₃	—	4	1	no	—
188	C₈×C₄₃	C₂×D₈	1	1	yes	characteristic
189	C₈×C₄₃	C₂×D₈	1	1	yes	characteristic
190	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
191	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
192	C₄₃⋊C₈	—	4	1	no	—
193	#33860	—	43	1	no	maximal, Sylow
194	C₂²×C₄₃	—	2	2	no	—
195	C₂²×C₄₃	—	4	1	no	—
196	C₂²×C₄₃	#576	1	1	yes	characteristic
197	C₂×D₄₃	—	4	1	no	—
198	C₄×C₄₃	—	2	2	no	—
199	C₄×C₄₃	—	2	2	no	—
200	C₄×C₄₃	—	4	1	no	—
201	C₄×C₄₃	—	8	2	no	—
202	C₄×C₄₃	D₄²	1	1	yes	characteristic
203	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
204	Dic₄₃	—	4	1	no	—
205	Dic₄₃	—	8	2	no	—
206	#3417	—	43	2	no	—
207	#3293	—	43	1	no	—
208	#3293	D₄₃	1	1	yes	characteristic, complemented
209	#2247	—	43	1	no	—
210	#2222	—	43	2	no	—
211	#2214	—	43	1	no	—
212	#2214	—	43	1	no	—
213	#2195	—	43	2	no	—
214	#2195	—	43	2	no	—
215	#2178	—	43	1	no	—
216	#2170	—	43	1	no	—
217	C₂×C₄₃	—	2	1	no	—
218	C₂×C₄₃	—	4	2	no	—
219	C₂×C₄₃	—	8	1	no	—
220	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
221	D₄₃	—	8	1	no	—
222	#578	—	2	1	no	—
223	#578	—	43	2	no	—
224	#578	—	86	1	no	—
225	#578	C₂×D₄₃	1	2	yes	—
226	#510	—	43	2	no	—
227	#510	—	43	2	no	—
228	#510	—	86	2	no	—
229	#510	—	86	2	no	—
230	#508	—	43	1	no	—
231	#508	—	86	2	no	—
232	C₂×Q₃₂	—	43	2	no	—
233	C₂×Q₃₂	—	43	2	no	—
234	C₂×Q₃₂	—	86	2	no	—
235	#504	—	43	2	no	—
236	#494	—	43	1	no	—
237	#494	C₂×D₄₃	1	1	yes	characteristic
238	#473	—	2	1	no	—
239	#473	—	43	2	no	—
240	#473	—	86	1	no	—
241	#473	C₂×D₄₃	1	2	yes	—
242	#456	—	2	2	no	—
243	#456	—	86	2	no	—
244	#443	—	43	1	no	—
245	#443	C₂×D₄₃	1	1	yes	characteristic
246	#443	C₂×D₄₃	1	1	yes	characteristic
247	#368	—	43	1	no	—
248	#364	—	43	1	no	—
249	#362	—	43	2	no	—
250	#362	—	43	2	no	—
251	#359	—	43	1	no	—
252	#359	—	43	1	no	—
253	#350	—	43	2	no	—
254	C₄×C₁₆	—	43	1	no	—
255	C₄₃	#33860	1	1	yes	characteristic, complemented, Sylow
256	2₋¹⁺⁴	—	2	2	no	—
257	2₋¹⁺⁴	—	86	2	no	—
258	#137	—	2	2	no	—
259	#137	—	2	2	no	—
260	#137	—	4	2	no	—
261	#137	—	4	2	no	—
262	#137	—	86	2	no	—
263	#137	—	86	2	no	—
264	#137	—	172	2	no	—
265	#137	—	172	2	no	—
266	#135	—	2	2	no	—
267	#135	—	4	1	no	—
268	#135	—	43	1	no	—
269	#135	—	86	2	no	—
270	#135	—	172	1	no	—
271	#135	C₂²⋊D₄₃	1	1	yes	characteristic
272	#135	C₂²×D₄₃	1	1	yes	characteristic
273	C₂×Q₁₆	—	2	2	no	—
274	C₂×Q₁₆	—	2	1	no	—
275	C₂×Q₁₆	—	4	1	no	—
276	C₂×Q₁₆	—	43	2	no	—
277	C₂×Q₁₆	—	86	2	no	—
278	C₂×Q₁₆	—	86	1	no	—
279	C₂×Q₁₆	—	172	1	no	—
280	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
281	#131	—	2	1	no	—
282	#131	—	86	1	no	—
283	#131	C₂²×D₄₃	1	2	yes	—
284	C₄⋊Q₈	—	2	1	no	—
285	C₄⋊Q₈	—	86	1	no	—
286	Q₃₂	—	43	1	no	—
287	Q₃₂	—	86	2	no	—
288	Q₃₂	—	86	2	no	—
289	Q₃₂	—	86	2	no	—
290	Q₃₂	—	86	2	no	—
291	Q₃₂	—	172	2	no	—
292	SD₃₂	—	43	1	no	—
293	SD₃₂	—	43	1	no	—
294	SD₃₂	—	86	2	no	—
295	SD₃₂	—	86	2	no	—
296	D₁₆	—	43	1	no	—
297	M₃₂	—	43	2	no	—
298	M₃₂	—	86	2	no	—
299	C₂×C₁₆	—	43	1	no	—
300	C₂×C₁₆	—	43	1	no	—
301	C₂×C₁₆	—	86	2	no	—
302	#108	—	43	1	no	—
303	#108	C₂²⋊D₄₃	1	1	yes	characteristic
304	#108	C₂²×D₄₃	1	1	yes	characteristic
305	#104	—	2	1	no	—
306	#104	—	86	1	no	—
307	#104	C₂²⋊D₄₃	1	2	yes	—
308	#101	—	2	2	no	—
309	#101	—	86	2	no	—
310	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
311	#41	—	2	1	no	—
312	#41	—	4	2	no	—
313	#41	—	4	2	no	—
314	#41	—	8	2	no	—
315	#41	—	86	1	no	—
316	#41	—	344	2	no	—
317	#41	—	172	2	no	—
318	#41	—	172	2	no	—
319	#41	C₂×C₂²⋊D₄₃	1	2	yes	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×Q₈	—	4	2	no	—
323	C₂×Q₈	—	86	2	no	—
324	C₂×Q₈	—	172	2	no	—
325	C₂×Q₈	—	172	2	no	—
326	Q₁₆	—	2	2	no	—
327	Q₁₆	—	2	2	no	—
328	Q₁₆	—	4	2	no	—
329	Q₁₆	—	4	2	no	—
330	Q₁₆	—	4	1	no	—
331	Q₁₆	—	4	1	no	—
332	Q₁₆	—	43	1	no	—
333	Q₁₆	—	86	2	no	—
334	Q₁₆	—	86	2	no	—
335	Q₁₆	—	172	2	no	—
336	Q₁₆	—	172	2	no	—
337	Q₁₆	—	172	1	no	—
338	Q₁₆	—	172	1	no	—
339	Q₁₆	#10621882	1	1	yes	characteristic
340	Q₁₆	D₄×D₄₃	1	1	yes	characteristic
341	SD₁₆	—	2	1	no	—
342	SD₁₆	—	4	2	no	—
343	SD₁₆	—	4	2	no	—
344	SD₁₆	—	86	1	no	—
345	SD₁₆	—	172	2	no	—
346	SD₁₆	—	172	2	no	—
347	SD₁₆	#10621882	1	2	yes	—
348	D₈	—	43	1	no	—
349	D₈	#10621882	1	1	yes	characteristic
350	D₈	D₄×D₄₃	1	1	yes	characteristic
351	M₁₆	—	2	2	no	—
352	M₁₆	—	2	1	no	—
353	M₁₆	—	4	1	no	—
354	M₁₆	—	86	1	no	—
355	M₁₆	—	172	1	no	—
356	M₁₆	C₂×C₂²⋊D₄₃	1	2	yes	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	4	1	no	—
359	C₂×C₈	—	172	1	no	—
360	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
361	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
362	C₄⋊C₄	—	8	1	no	—
363	C₄⋊C₄	—	344	1	no	—
364	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
365	C₁₆	—	43	1	no	—
366	C₁₆	—	43	1	no	—
367	C₁₆	—	86	2	no	—
368	C₁₆	—	86	1	no	—
369	Q₈	—	2	1	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	8	2	no	—
373	Q₈	—	86	1	no	—
374	Q₈	—	344	2	no	—
375	Q₈	—	172	2	no	—
376	Q₈	—	172	2	no	—
377	Q₈	C₂×#10621882	1	2	yes	—
378	D₄	—	2	2	no	—
379	D₄	—	2	1	no	—
380	D₄	—	4	1	no	—
381	D₄	—	8	2	no	—
382	D₄	—	86	1	no	—
383	D₄	—	344	2	no	—
384	D₄	—	172	1	no	—
385	D₄	C₂×#10621882	1	2	yes	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	8	2	no	—
391	C₂×C₄	—	8	2	no	—
392	C₂×C₄	—	344	2	no	—
393	C₂×C₄	—	344	2	no	—
394	C₂×C₄	—	172	2	no	—
395	C₂×C₄	—	172	1	no	—
396	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	4	1	no	—
399	C₈	—	172	1	no	—
400	C₈	C₂×#10621882	1	1	yes	characteristic
401	C₈	C₂×#10621882	1	1	yes	characteristic
402	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
403	C₈	#13063370	1	1	yes	characteristic
404	C₂²	—	2	2	no	—
405	C₂²	—	4	1	no	—
406	C₂²	—	172	1	no	—
407	C₂²	#14583702	1	1	yes	characteristic
408	C₄	—	2	2	no	—
409	C₄	—	2	2	no	—
410	C₄	—	4	1	no	—
411	C₄	—	8	2	no	—
412	C₄	—	344	2	no	—
413	C₄	—	172	1	no	—
414	C₄	D₄×C₂²⋊D₄₃	1	1	yes	characteristic
415	C₄	C₂²×#10621882	1	1	yes	characteristic
416	C₂	—	2	1	no	—
417	C₂	—	4	2	no	—
418	C₂	—	8	1	no	—
419	C₂	—	344	1	no	—
420	C₂	D₄×#10621882	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3293	C₂
socle	C₂×C₄₃	D₄×D₈
hypercenter	#3293	D₄₃
hypocenter	C₄₃	#33860
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ #3293

Complements

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

N	H ≅ G/N	type
1. #19147746	C₂	semidirect
2. #19146400	C₂	semidirect
3. #19136847	C₂	semidirect
4. #19136718	C₂	semidirect
5. #19136676	C₂	semidirect
6. #19136673	C₂	semidirect
7. #19136573	C₂	semidirect
8. #19136571	C₂	semidirect
9. #19136474	C₂	semidirect
10. #19136432	C₂	semidirect
11. C₄₃×#3293	C₂	semidirect
208. #3293	D₄₃	semidirect
255. C₄₃	#33860	semidirect

Maximal subgroups

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

subgroup	index
#19147746	2
#19146400	2
#19136847	2
#19136718	2
#19136676	2
#19136673	2
#19136573	2
#19136571	2
#19136474	2
#19136432	2
C₄₃×#3293	2
#33860	43