Subgroups — #25079874

Order 11008.

646 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	#19145748	C₂	1	1	yes	maximal, characteristic, complemented
2	#19145538	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583246	C₂	1	1	yes	maximal, characteristic
4	#19143529	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134136	C₂	1	4	yes	maximal, complemented
6	#19134012	C₂	1	4	yes	maximal, complemented
7	#19133275	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133262	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583456	C₂²	1	1	yes	characteristic
11	C₂×#13063325	C₂²	1	1	yes	characteristic
12	C₂×#13063322	C₂²	1	1	yes	characteristic
13	#14583304	—	2	1	no	—
14	#14583272	—	2	1	no	—
15	#14583250	—	2	1	no	—
16	#14583246	—	2	1	no	—
17	#14583246	—	2	1	no	—
18	#14583246	—	2	1	no	—
19	#14583246	C₂²	1	2	yes	—
20	#14583246	C₂²	1	2	yes	—
21	#14583235	—	2	2	no	—
22	#14583195	C₂²	1	4	yes	complemented
23	#14583100	C₂²	1	4	yes	—
24	#14583052	—	2	2	no	—
25	#14583029	—	2	1	no	—
26	#14583029	C₂²	1	2	yes	complemented
27	C₈×Dic₈₆	—	2	1	no	—
28	C₈×Dic₈₆	C₂²	1	2	yes	complemented
29	#14582925	—	2	1	no	—
30	#14582923	C₂²	1	1	yes	characteristic
31	C₂×#13063247	—	2	1	no	—
32	C₂×#13063247	—	2	1	no	—
33	#14582854	—	2	1	no	—
34	#14582647	C₂²	1	4	yes	complemented
35	#14582642	C₂²	1	4	yes	—
36	C₄₃×#431	C₂²	1	1	yes	characteristic
37	C₄₃×#431	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#335	C₂²	1	2	yes	complemented
40	C₄₃×#334	C₂²	1	2	yes	complemented
41	C₂²×Dic₈₆	—	2	1	no	—
42	C₂²×D₁₇₂	—	2	1	no	—
43	C₂²×D₁₇₂	—	2	1	no	—
44	C₂²×D₁₇₂	—	2	1	no	—
45	C₂×#10621873	—	2	4	no	—
46	C₂×#10621877	—	2	1	no	—
47	#13063325	—	2	1	no	—
48	#13063325	—	2	1	no	—
49	#13063325	—	2	1	no	—
50	#13063325	D₄	1	1	yes	characteristic
51	#13063325	D₄	1	1	yes	characteristic
52	#13063325	C₂³	1	2	yes	—
53	#13063322	—	2	1	no	—
54	#13063322	—	2	1	no	—
55	#13063322	—	2	1	no	—
56	#13063322	C₂³	1	2	yes	—
57	#13063317	—	2	1	no	—
58	#13063304	—	2	1	no	—
59	#13063304	D₄	1	2	yes	—
60	C₄×Dic₈₆	—	2	1	no	—
61	C₄×Dic₈₆	C₂³	1	2	yes	—
62	#13063284	—	2	1	no	—
63	C₂×C₄⋊Dic₄₃	—	2	1	no	—
64	C₂×C₄⋊Dic₄₃	—	2	1	no	—
65	#13063259	—	2	4	no	—
66	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
67	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
68	Dic₄₃⋊C₈	—	2	2	no	—
69	Dic₄₃⋊C₈	—	2	2	no	—
70	#13063250	—	2	4	no	—
71	#13063247	—	2	2	no	—
72	#13063247	—	2	2	no	—
73	#13063247	—	2	2	no	—
74	#13063247	—	2	2	no	—
75	#13063247	—	2	4	no	—
76	#13063235	—	2	2	no	—
77	C₈×Dic₄₃	—	2	2	no	—
78	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
82	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
83	C₂²⋊C₈×C₄₃	—	2	1	no	—
84	C₂²⋊C₈×C₄₃	—	2	1	no	—
85	C₄×C₈×C₄₃	—	2	1	no	—
86	C₄×C₈×C₄₃	—	2	1	no	—
87	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
88	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
91	C₂³×D₄₃	—	4	1	no	—
92	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
95	C₂²×Dic₄₃	—	4	1	no	—
96	C₂×D₁₇₂	—	2	2	no	—
97	C₂×D₁₇₂	—	2	2	no	—
98	C₂×D₁₇₂	—	2	2	no	—
99	C₂×D₁₇₂	—	4	1	no	—
100	C₂×D₁₇₂	—	4	1	no	—
101	C₂×D₁₇₂	—	4	1	no	—
102	C₂×D₁₇₂	—	4	2	no	—
103	C₂×D₁₇₂	—	4	2	no	—
104	C₂×D₁₇₂	—	8	2	no	—
105	C₂×D₁₇₂	—	8	2	no	—
106	C₂×Dic₈₆	—	2	1	no	—
107	C₂×Dic₈₆	—	2	2	no	—
108	C₂×Dic₈₆	—	4	1	no	—
109	C₂×Dic₈₆	—	8	2	no	—
110	C₂×Dic₈₆	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₄₃	—	2	2	no	—
112	C₂×C₈×C₄₃	—	2	2	no	—
113	C₂×C₈×C₄₃	—	2	2	no	—
114	C₂×C₈×C₄₃	—	2	2	no	—
115	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₄₃	#41	1	1	yes	characteristic
122	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
123	C₂²⋊Dic₄₃	—	2	2	no	—
124	C₄⋊Dic₄₃	—	2	1	no	—
125	C₄⋊Dic₄₃	—	2	1	no	—
126	C₄⋊Dic₄₃	—	2	1	no	—
127	C₄⋊Dic₄₃	—	2	2	no	—
128	C₄⋊Dic₄₃	—	2	2	no	—
129	C₄⋊Dic₄₃	—	4	1	no	—
130	C₄⋊Dic₄₃	—	4	1	no	—
131	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
132	Dic₄₃⋊C₄	—	2	2	no	—
133	Dic₄₃⋊C₄	—	2	2	no	—
134	C₄×Dic₄₃	—	2	2	no	—
135	#10621877	—	4	2	no	—
136	C₂×C₄₃⋊C₈	—	2	2	no	—
137	#10621873	—	8	4	no	—
138	C₂³×C₄₃	#121	1	1	yes	characteristic
139	C₂²×D₄₃	—	4	2	no	—
140	C₂²×D₄₃	—	8	2	no	—
141	C₂²×D₄₃	—	8	2	no	—
142	C₂²×D₄₃	—	8	2	no	—
143	C₂×C₄×C₄₃	—	2	2	no	—
144	C₂×C₄×C₄₃	—	2	2	no	—
145	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
147	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
148	C₂×C₄×C₄₃	#135	1	2	yes	—
149	C₂×C₄×C₄₃	#137	1	2	yes	—
150	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
153	C₂×Dic₄₃	—	2	2	no	—
154	C₂×Dic₄₃	—	2	2	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	C₂×Dic₄₃	—	8	2	no	—
157	D₁₇₂	—	4	2	no	—
158	D₁₇₂	—	4	2	no	—
159	D₁₇₂	—	4	4	no	—
160	D₁₇₂	—	8	2	no	—
161	D₁₇₂	—	8	2	no	—
162	D₁₇₂	—	8	2	no	—
163	D₁₇₂	—	8	2	no	—
164	Dic₈₆	—	2	4	no	—
165	Dic₈₆	—	4	2	no	—
166	Dic₈₆	—	8	2	no	—
167	Dic₈₆	—	8	2	no	—
168	C₈×C₄₃	—	2	1	no	—
169	C₈×C₄₃	—	2	1	no	—
170	C₈×C₄₃	—	2	1	no	—
171	C₈×C₄₃	—	2	1	no	—
172	C₈×C₄₃	—	4	2	no	—
173	C₈×C₄₃	—	4	2	no	—
174	C₄₃⋊C₈	—	4	2	no	—
175	#19212	—	43	1	no	maximal, Sylow
176	C₂²×C₄₃	—	2	1	no	—
177	C₂²×C₄₃	—	2	1	no	—
178	C₂²×C₄₃	#464	1	2	yes	—
179	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
180	C₂×D₄₃	—	8	2	no	—
181	C₂×D₄₃	—	8	2	no	—
182	C₂×D₄₃	—	8	2	no	—
183	C₂×D₄₃	—	16	4	no	—
184	C₄×C₄₃	—	2	1	no	—
185	C₄×C₄₃	—	2	1	no	—
186	C₄×C₄₃	#464	1	2	yes	—
187	C₄×C₄₃	#472	1	2	yes	—
188	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
189	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
190	C₄×C₄₃	#577	1	2	yes	—
191	Dic₄₃	—	4	2	no	—
192	Dic₄₃	—	4	2	no	—
193	Dic₄₃	—	8	2	no	—
194	#3230	—	43	1	no	—
195	#3208	—	43	1	no	—
196	C₂×#486	—	43	1	no	—
197	#2960	—	43	1	no	—
198	#1698	—	43	4	no	—
199	#1678	—	43	4	no	—
200	#1564	D₄₃	1	1	yes	characteristic, complemented
201	#1548	—	43	1	no	—
202	#1545	—	43	1	no	—
203	C₂×C₄₃	—	2	2	no	—
204	C₂×C₄₃	C₂×#464	1	1	yes	characteristic
205	C₂×C₄₃	#3057	1	1	yes	characteristic
206	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
207	D₄₃	—	16	2	no	—
208	#533	—	43	1	no	—
209	C₂×C₄⋊Q₈	—	43	1	no	—
210	C₂×#127	—	43	1	no	—
211	#500	—	86	1	no	—
212	#492	—	86	1	no	—
213	#487	—	86	1	no	—
214	#486	—	43	2	no	—
215	#486	—	43	2	no	—
216	#486	—	86	1	no	—
217	#486	—	86	1	no	—
218	#486	—	86	1	no	—
219	#484	—	86	2	no	—
220	#477	—	43	4	no	—
221	#460	—	43	4	no	—
222	#450	—	86	2	no	—
223	#446	—	43	2	no	—
224	#446	—	86	1	no	—
225	C₈×Q₈	—	43	2	no	—
226	C₈×Q₈	—	86	1	no	—
227	#431	C₂×D₄₃	1	1	yes	characteristic
228	#431	C₂×D₄₃	1	1	yes	characteristic
229	#424	—	86	1	no	—
230	#423	—	43	1	no	—
231	C₂×#102	—	86	1	no	—
232	C₂×#102	—	86	1	no	—
233	#405	—	86	1	no	—
234	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
235	#335	C₂×D₄₃	1	2	yes	complemented
236	#334	C₂×D₄₃	1	2	yes	complemented
237	#333	—	43	4	no	—
238	#331	—	43	4	no	—
239	C₄₃	#19212	1	1	yes	characteristic, complemented, Sylow
240	C₂²×Q₈	—	86	1	no	—
241	C₂²×D₄	—	86	1	no	—
242	C₂²×D₄	—	86	1	no	—
243	C₂²×D₄	—	86	1	no	—
244	C₂×SD₁₆	—	86	4	no	—
245	C₂×M₁₆	—	86	1	no	—
246	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
247	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
248	C₄⋊Q₈	—	43	1	no	—
249	C₄⋊Q₈	—	43	1	no	—
250	C₄⋊Q₈	—	43	2	no	—
251	C₄⋊Q₈	—	86	1	no	—
252	C₄⋊Q₈	—	86	1	no	—
253	C₄⋊Q₈	—	86	1	no	—
254	#127	—	43	2	no	—
255	#127	—	86	1	no	—
256	#127	—	86	1	no	—
257	#127	—	86	1	no	—
258	#125	—	86	1	no	—
259	C₂²⋊Q₈	—	43	2	no	—
260	C₂²⋊Q₈	—	86	1	no	—
261	C₄×Q₈	—	43	2	no	—
262	C₄×Q₈	—	86	1	no	—
263	#117	—	86	1	no	—
264	C₂×C₄⋊C₄	—	86	1	no	—
265	C₂×C₄⋊C₄	—	86	1	no	—
266	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
267	#107	—	86	4	no	—
268	C₄⋊C₈	—	43	2	no	—
269	C₄⋊C₈	—	86	2	no	—
270	C₄⋊C₈	—	86	2	no	—
271	C₄⋊C₈	—	86	1	no	—
272	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
273	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
274	#103	—	86	4	no	—
275	#102	—	86	4	no	—
276	#102	—	86	2	no	—
277	#102	—	86	2	no	—
278	#102	—	86	2	no	—
279	#102	—	86	2	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	#97	—	86	2	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	—	2	1	no	—
285	C₄×C₈	—	86	2	no	—
286	C₄×C₈	D₁₇₂	1	1	yes	characteristic
287	C₄×C₈	D₁₇₂	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
290	C₂⁴	—	172	1	no	—
291	C₂×Q₈	—	43	2	no	—
292	C₂×Q₈	—	86	1	no	—
293	C₂×Q₈	—	86	2	no	—
294	C₂×Q₈	—	344	2	no	—
295	C₂×Q₈	—	172	1	no	—
296	C₂×D₄	—	86	2	no	—
297	C₂×D₄	—	86	2	no	—
298	C₂×D₄	—	86	2	no	—
299	C₂×D₄	—	344	2	no	—
300	C₂×D₄	—	344	2	no	—
301	C₂×D₄	—	172	1	no	—
302	C₂×D₄	—	172	1	no	—
303	C₂×D₄	—	172	1	no	—
304	C₂×D₄	—	172	2	no	—
305	C₂×D₄	—	172	2	no	—
306	C₂²×C₄	—	172	1	no	—
307	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
308	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
309	C₂²×C₄	#10621902	1	1	yes	characteristic
310	SD₁₆	—	344	4	no	—
311	M₁₆	—	172	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	86	2	no	—
317	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
318	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
319	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
320	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
321	C₄⋊C₄	—	43	2	no	—
322	C₄⋊C₄	—	86	1	no	—
323	C₄⋊C₄	—	86	1	no	—
324	C₄⋊C₄	—	86	2	no	—
325	C₄⋊C₄	—	86	1	no	—
326	C₄⋊C₄	—	86	2	no	—
327	C₄⋊C₄	—	86	2	no	—
328	C₄⋊C₄	—	86	2	no	—
329	C₄⋊C₄	—	172	1	no	—
330	C₄⋊C₄	—	172	1	no	—
331	C₂²⋊C₄	—	86	2	no	—
332	C₄²	—	86	2	no	—
333	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
334	C₄²	D₄×D₄₃	1	1	yes	characteristic
335	C₄²	#10621902	1	1	yes	characteristic
336	C₄²	C₂³×D₄₃	1	1	yes	characteristic
337	C₂³	—	344	2	no	—
338	C₂³	—	344	2	no	—
339	C₂³	—	344	2	no	—
340	C₂³	—	172	2	no	—
341	C₂³	#13063300	1	1	yes	characteristic
342	Q₈	—	86	4	no	—
343	Q₈	—	344	2	no	—
344	Q₈	—	344	2	no	—
345	Q₈	—	172	2	no	—
346	D₄	—	344	2	no	—
347	D₄	—	344	2	no	—
348	D₄	—	344	2	no	—
349	D₄	—	344	2	no	—
350	D₄	—	172	2	no	—
351	D₄	—	172	2	no	—
352	D₄	—	172	4	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	86	2	no	—
356	C₂×C₄	—	86	2	no	—
357	C₂×C₄	—	344	2	no	—
358	C₂×C₄	—	172	2	no	—
359	C₂×C₄	#13063300	1	1	yes	characteristic
360	C₂×C₄	#13063300	1	1	yes	characteristic
361	C₂×C₄	#13063300	1	1	yes	characteristic
362	C₂×C₄	#13063347	1	2	yes	—
363	C₂×C₄	#13063358	1	2	yes	—
364	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
365	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
366	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	4	2	no	—
372	C₈	—	4	2	no	—
373	C₈	—	172	2	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	344	2	no	—
377	C₂²	—	344	2	no	—
378	C₂²	—	344	2	no	—
379	C₂²	—	688	4	no	—
380	C₂²	#14583122	1	2	yes	—
381	C₂²	C₂×#13063300	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	344	2	no	—
385	C₄	—	172	2	no	—
386	C₄	—	172	2	no	—
387	C₄	#14583122	1	2	yes	—
388	C₄	#14583170	1	2	yes	—
389	C₄	C₂×#13063347	1	1	yes	characteristic
390	C₄	C₂×#13063358	1	1	yes	characteristic
391	C₄	#14583706	1	2	yes	—
392	C₂	—	2	2	no	—
393	C₂	—	688	2	no	—
394	C₂	C₂×#14583122	1	1	yes	characteristic
395	C₂	#19144235	1	1	yes	characteristic
396	C₂	C₂×#14583170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063300
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1564	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂×#13063322	C₂²
hypercenter	#1564	D₄₃
hypocenter	C₄₃	#19212
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #19145748	C₂	semidirect
4. #19143529	C₂	semidirect
5. #19134136	C₂	semidirect
6. #19134012	C₂	semidirect
7. #19133275	C₂	semidirect
9. C₄₃×#1564	C₂	semidirect
22. #14583195	C₂²	semidirect
26. #14583029	C₂²	semidirect
28. C₈×Dic₈₆	C₂²	semidirect
34. #14582647	C₂²	semidirect
39. C₄₃×#335	C₂²	semidirect
40. C₄₃×#334	C₂²	semidirect
200. #1564	D₄₃	semidirect
235. #335	C₂×D₄₃	semidirect
236. #334	C₂×D₄₃	semidirect
239. C₄₃	#19212	semidirect

Maximal subgroups

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

subgroup	index
#19145748	2
#19145538	2
C₂×#14583246	2
#19143529	2
#19134136	2
#19134012	2
#19133275	2
#19133262	2
C₄₃×#1564	2
#19212	43