Subgroups — #19146456

Order 5504.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₂²⋊D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583550	C₂	1	1	yes	maximal, characteristic
3	#14583248	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583229	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583228	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583182	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583131	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583128	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583103	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583039	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583038	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582984	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₂²⋊D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#103×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621898	—	2	1	no	—
17	C₂×Q₈×D₄₃	C₂²	1	1	yes	characteristic
18	C₂×#10621873	C₂²	1	1	yes	characteristic
19	C₂×#10621884	C₂²	1	1	yes	characteristic
20	C₂×#10621884	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₄₃	C₂²	1	1	yes	characteristic
22	#13063324	C₂²	1	1	yes	characteristic
23	#13063323	—	2	1	no	—
24	#13063323	C₂²	1	1	yes	characteristic
25	#13063310	—	2	1	no	—
26	#13063308	—	2	1	no	—
27	#13063304	—	2	1	no	—
28	#13063304	C₂²	1	1	yes	characteristic
29	#13063303	—	2	1	no	—
30	#13063301	—	2	1	no	—
31	#13063301	C₂²	1	1	yes	characteristic
32	#13063300	C₂²	1	1	yes	characteristic
33	#13063299	C₂²	1	1	yes	characteristic
34	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
35	#13063287	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₄₃	—	2	1	no	—
37	C₄×C₂²⋊D₄₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
39	#13063257	C₂²	1	1	yes	characteristic
40	#13063256	C₂²	1	1	yes	characteristic
41	#13063256	C₂²	1	1	yes	characteristic
42	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
43	#13063250	C₂²	1	1	yes	characteristic
44	#13063249	C₂²	1	1	yes	characteristic
45	#13063249	C₂²	1	1	yes	characteristic
46	#13063248	C₂²	1	1	yes	characteristic
47	#13063248	C₂²	1	1	yes	characteristic
48	#13063247	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
55	#103×C₄₃	C₂²	1	1	yes	characteristic
56	#103×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#103×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	—	2	1	no	—
60	C₂×Q₈×C₄₃	—	2	2	no	—
61	C₂×Q₈×C₄₃	—	4	1	no	—
62	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	—	2	1	no	—
65	C₂²×C₄×C₄₃	—	2	1	no	—
66	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₄₃	—	2	1	no	—
68	C₂×C₂²⋊D₄₃	D₄	1	1	yes	characteristic
69	Q₈×D₄₃	—	2	2	no	—
70	Q₈×D₄₃	—	4	1	no	—
71	#10621898	—	4	4	no	—
72	C₂×D₁₇₂	—	2	1	no	—
73	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₄₃	—	2	1	no	—
75	C₂×C₄×D₄₃	—	2	1	no	—
76	C₂×C₄×D₄₃	—	2	1	no	—
77	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
78	C₂×Dic₈₆	—	2	1	no	—
79	C₂×Dic₈₆	—	2	1	no	—
80	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₄₃	—	4	1	no	—
82	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₄₃	—	2	1	no	—
85	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
87	C₂²⋊Dic₄₃	—	2	1	no	—
88	C₂²⋊Dic₄₃	D₄	1	1	yes	characteristic
89	#10621884	—	2	2	no	—
90	#10621884	—	4	1	no	—
91	(C₂×D₄₃)⋊C₄	—	2	1	no	—
92	(C₂×D₄₃)⋊C₄	—	2	1	no	—
93	(C₂×D₄₃)⋊C₄	—	2	1	no	—
94	(C₂×D₄₃)⋊C₄	—	2	1	no	—
95	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
96	C₄⋊Dic₄₃	—	2	1	no	—
97	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
99	Dic₄₃⋊C₄	—	2	1	no	—
100	Dic₄₃⋊C₄	—	2	1	no	—
101	Dic₄₃⋊C₄	—	2	1	no	—
102	Dic₄₃⋊C₄	—	2	1	no	—
103	Dic₄₃⋊C₄	—	2	1	no	—
104	Dic₄₃⋊C₄	—	2	1	no	—
105	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
106	C₄×Dic₄₃	—	2	1	no	—
107	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
108	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
110	#10621873	—	4	1	no	—
111	C₈×D₄₃	—	4	1	no	—
112	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₄₃	—	2	1	no	—
114	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₄₃	—	2	1	no	—
116	Q₈×C₄₃	—	2	1	no	—
117	Q₈×C₄₃	—	2	1	no	—
118	Q₈×C₄₃	—	4	2	no	—
119	Q₈×C₄₃	#41	1	2	yes	complemented
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	—	4	2	no	—
127	C₂×C₄×C₄₃	—	4	2	no	—
128	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₄₃	—	4	2	no	—
131	C₂²⋊D₄₃	SD₁₆	1	4	yes	complemented
132	C₂×Dic₄₃	—	2	1	no	—
133	C₂×Dic₄₃	—	2	1	no	—
134	C₂×Dic₄₃	—	2	1	no	—
135	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
138	D₁₇₂	—	2	1	no	—
139	D₁₇₂	—	2	1	no	—
140	D₁₇₂	—	4	2	no	—
141	C₄×D₄₃	—	2	1	no	—
142	C₄×D₄₃	—	2	1	no	—
143	C₄×D₄₃	—	4	2	no	—
144	C₄×D₄₃	—	4	2	no	—
145	C₄×D₄₃	—	4	2	no	—
146	Dic₈₆	—	2	1	no	—
147	Dic₈₆	—	2	1	no	—
148	Dic₈₆	—	4	2	no	—
149	Dic₈₆	—	4	2	no	—
150	C₈×C₄₃	—	2	1	no	—
151	C₈×C₄₃	—	2	1	no	—
152	C₄₃⋊C₈	—	2	1	no	—
153	C₄₃⋊C₈	—	2	1	no	—
154	C₂²×C₄₃	—	2	1	no	—
155	C₂²×C₄₃	—	2	1	no	—
156	C₂²×C₄₃	C₂×SD₁₆	1	2	yes	—
157	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₄₃	—	2	1	no	—
159	C₂×D₄₃	—	2	1	no	—
160	C₂×D₄₃	—	4	1	no	—
161	C₂×D₄₃	—	4	1	no	—
162	C₂×D₄₃	—	4	1	no	—
163	C₂×D₄₃	C₂×SD₁₆	1	2	yes	—
164	C₄×C₄₃	—	2	2	no	—
165	C₄×C₄₃	—	2	1	no	—
166	C₄×C₄₃	—	4	1	no	—
167	C₄×C₄₃	—	4	1	no	—
168	C₄×C₄₃	—	4	1	no	—
169	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
170	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₄₃	—	2	1	no	—
172	Dic₄₃	—	2	1	no	—
173	Dic₄₃	—	4	1	no	—
174	Dic₄₃	—	4	1	no	—
175	Dic₄₃	—	4	1	no	—
176	Dic₄₃	C₂×SD₁₆	1	2	yes	—
177	#3298	—	43	1	no	maximal, Sylow
178	C₂×C₄₃	—	2	2	no	—
179	C₂×C₄₃	#546	1	1	yes	characteristic
180	C₂×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
181	C₂×C₄₃	#578	1	1	yes	characteristic
182	D₄₃	—	2	2	no	—
183	D₄₃	—	8	1	no	—
184	D₄×Q₈	—	43	1	no	—
185	#547	—	43	1	no	—
186	#487	—	43	1	no	—
187	#483	—	43	1	no	—
188	#483	—	43	1	no	—
189	#475	—	43	1	no	—
190	#465	—	43	1	no	—
191	#465	—	43	1	no	—
192	#461	—	43	1	no	—
193	#448	—	43	1	no	—
194	#448	—	43	1	no	—
195	C₄×SD₁₆	—	43	1	no	—
196	C₈×D₄	—	43	1	no	—
197	C₂×#103	—	43	1	no	—
198	C₂×#103	D₄₃	1	1	yes	characteristic, complemented
199	C₄₃	#3298	1	1	yes	characteristic, complemented, Sylow
200	C₂×#41	—	86	1	no	—
201	C₂²×Q₈	—	43	1	no	—
202	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
203	C₂×SD₁₆	—	43	1	no	—
204	C₂×SD₁₆	—	43	1	no	—
205	C₂×SD₁₆	—	43	1	no	—
206	C₂²×C₈	—	43	1	no	—
207	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
208	C₄⋊Q₈	—	43	1	no	—
209	C₄⋊Q₈	—	43	1	no	—
210	C₄⋊Q₈	—	86	1	no	—
211	#123	—	86	1	no	—
212	#123	—	86	1	no	—
213	C₂²⋊Q₈	—	43	1	no	—
214	C₂²⋊Q₈	—	43	1	no	—
215	C₂²⋊Q₈	—	86	1	no	—
216	C₂²⋊Q₈	—	86	1	no	—
217	C₂²⋊Q₈	—	86	1	no	—
218	#121	—	43	1	no	—
219	#121	—	43	1	no	—
220	C₄×Q₈	—	43	1	no	—
221	C₄×D₄	—	43	1	no	—
222	C₄×D₄	—	43	1	no	—
223	C₄×D₄	—	86	1	no	—
224	C₂×C₄⋊C₄	—	43	1	no	—
225	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
226	#106	—	43	1	no	—
227	#106	—	43	1	no	—
228	#106	—	43	1	no	—
229	C₄⋊C₈	—	43	1	no	—
230	#103	—	43	1	no	—
231	#103	—	43	1	no	—
232	#103	—	43	1	no	—
233	#103	—	43	1	no	—
234	#103	—	43	1	no	—
235	#103	C₂×D₄₃	1	1	yes	characteristic
236	#103	C₂×D₄₃	1	1	yes	characteristic
237	#103	C₂×D₄₃	1	1	yes	characteristic
238	#103	C₂×D₄₃	1	1	yes	characteristic
239	#102	—	43	1	no	—
240	C₂²⋊C₈	—	43	1	no	—
241	C₂²⋊C₈	—	43	1	no	—
242	C₄×C₈	—	43	1	no	—
243	#41	—	172	4	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	43	1	no	—
248	C₂×Q₈	—	86	1	no	—
249	C₂×Q₈	—	86	1	no	—
250	C₂×Q₈	—	86	2	no	—
251	C₂×Q₈	—	172	1	no	—
252	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
253	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
254	C₂×D₄	—	43	1	no	—
255	C₂×D₄	—	43	1	no	—
256	C₂×D₄	—	86	1	no	—
257	C₂×D₄	—	86	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	43	1	no	—
261	C₂²×C₄	—	86	1	no	—
262	C₂²×C₄	—	86	1	no	—
263	C₂²×C₄	—	86	1	no	—
264	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
265	SD₁₆	—	86	2	no	—
266	SD₁₆	—	172	1	no	—
267	SD₁₆	—	172	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	43	1	no	—
270	C₂×C₈	—	43	1	no	—
271	C₂×C₈	—	172	1	no	—
272	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	43	1	no	—
276	C₄⋊C₄	—	43	1	no	—
277	C₄⋊C₄	—	43	1	no	—
278	C₄⋊C₄	—	86	1	no	—
279	C₄⋊C₄	—	86	1	no	—
280	C₄⋊C₄	—	86	1	no	—
281	C₄⋊C₄	—	86	1	no	—
282	C₄⋊C₄	—	86	1	no	—
283	C₄⋊C₄	—	86	1	no	—
284	C₄⋊C₄	—	86	1	no	—
285	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
286	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
287	C₂²⋊C₄	—	43	1	no	—
288	C₂²⋊C₄	—	43	1	no	—
289	C₂²⋊C₄	—	86	1	no	—
290	C₂²⋊C₄	—	86	1	no	—
291	C₂²⋊C₄	—	86	1	no	—
292	C₂²⋊C₄	—	86	1	no	—
293	C₂²⋊C₄	—	86	1	no	—
294	C₄²	—	43	1	no	—
295	C₄²	—	86	1	no	—
296	C₂³	—	43	1	no	—
297	C₂³	—	86	1	no	—
298	C₂³	D₄×D₄₃	1	1	yes	characteristic
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	86	1	no	—
304	Q₈	—	86	1	no	—
305	Q₈	—	172	2	no	—
306	Q₈	—	172	2	no	—
307	Q₈	#10621898	1	2	yes	complemented
308	D₄	—	43	4	no	—
309	D₄	—	86	1	no	—
310	D₄	—	86	1	no	—
311	D₄	—	172	2	no	—
312	D₄	—	172	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	43	1	no	—
322	C₂×C₄	—	43	1	no	—
323	C₂×C₄	—	43	1	no	—
324	C₂×C₄	—	86	1	no	—
325	C₂×C₄	—	86	1	no	—
326	C₂×C₄	—	86	1	no	—
327	C₂×C₄	—	86	1	no	—
328	C₂×C₄	—	86	1	no	—
329	C₂×C₄	—	172	2	no	—
330	C₂×C₄	—	172	2	no	—
331	C₂×C₄	—	172	2	no	—
332	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	86	1	no	—
337	C₈	—	86	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	43	2	no	—
341	C₂²	—	86	1	no	—
342	C₂²	—	86	1	no	—
343	C₂²	—	172	1	no	—
344	C₂²	—	172	1	no	—
345	C₂²	—	172	1	no	—
346	C₂²	SD₁₆×D₄₃	1	2	yes	—
347	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	43	2	no	—
354	C₄	—	86	1	no	—
355	C₄	—	86	1	no	—
356	C₄	—	172	1	no	—
357	C₄	—	172	1	no	—
358	C₄	—	172	1	no	—
359	C₄	C₂×#10621898	1	1	yes	characteristic
360	C₄	#13063370	1	1	yes	characteristic
361	C₂	—	2	2	no	—
362	C₂	—	86	2	no	—
363	C₂	—	344	1	no	—
364	C₂	#14583539	1	1	yes	characteristic
365	C₂	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
366	C₂	#14583712	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×#103×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₂×#103	D₄₃
hypocenter	C₄₃	#3298
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
1. Q₈×C₂²⋊D₄₃	C₂	semidirect
3. #14583248	C₂	semidirect
4. #14583229	C₂	semidirect
5. #14583228	C₂	semidirect
6. #14583182	C₂	semidirect
7. #14583131	C₂	semidirect
8. #14583128	C₂	semidirect
9. #14583103	C₂	semidirect
10. #14583039	C₂	semidirect
11. #14583038	C₂	semidirect
12. #14582984	C₂	semidirect
13. C₈×C₂²⋊D₄₃	C₂	semidirect
14. #103×D₄₃	C₂	semidirect
15. C₂×#103×C₄₃	C₂	semidirect
119. Q₈×C₄₃	#41	semidirect
131. C₂²⋊D₄₃	SD₁₆	semidirect
198. C₂×#103	D₄₃	semidirect
199. C₄₃	#3298	semidirect
307. Q₈	#10621898	semidirect