Subgroups — #19144048

Order 5504.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063347	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13063345	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583445	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583367	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583124	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583120	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583117	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583113	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583108	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583103	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583098	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583093	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×Dic₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582893	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#416	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621898	C₂²	1	1	yes	characteristic
17	C₂×#10621898	C₂²	1	1	yes	characteristic
18	#13063347	—	2	4	no	—
19	#13063345	—	2	4	no	—
20	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
21	C₂×#10621885	C₂²	1	1	yes	characteristic
22	C₂×#10621873	C₂²	1	1	yes	characteristic
23	C₂×#10621873	C₂²	1	1	yes	characteristic
24	C₂×#10621883	C₂²	1	1	yes	characteristic
25	C₂×#10621884	C₂²	1	1	yes	characteristic
26	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
27	C₂×#10621882	C₂²	1	1	yes	characteristic
28	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
29	#13063323	C₂²	1	1	yes	characteristic
30	#13063321	C₂²	1	1	yes	characteristic
31	#13063314	C₂²	1	1	yes	characteristic
32	#13063312	C₂²	1	1	yes	characteristic
33	#13063297	—	2	1	no	—
34	#13063297	—	2	1	no	—
35	#13063290	C₂²	1	1	yes	characteristic
36	#13063290	C₂²	1	1	yes	characteristic
37	#13063287	C₂²	1	1	yes	characteristic
38	#13063287	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊D₄₃	—	2	1	no	—
40	C₄×C₂²⋊D₄₃	—	2	1	no	—
41	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
42	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
43	#13063282	—	2	1	no	—
44	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
45	C₄²×D₄₃	—	2	1	no	—
46	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
47	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
48	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
49	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
50	#13063248	C₂²	1	1	yes	characteristic
51	#13063248	C₂²	1	1	yes	characteristic
52	#13063245	C₂²	1	1	yes	characteristic
53	#13063245	C₂²	1	1	yes	characteristic
54	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
56	#117×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#103×C₄₃	C₂²	1	1	yes	characteristic
59	#102×C₄₃	C₂²	1	1	yes	characteristic
60	#102×C₄₃	C₂²	1	1	yes	characteristic
61	#41×C₄₃	—	2	2	no	—
62	#41×C₄₃	—	2	2	no	—
63	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₄₃	—	2	1	no	—
65	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	—	2	1	no	—
67	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₄₃	—	2	1	no	—
69	C₂×C₂²⋊D₄₃	—	2	1	no	—
70	C₂×C₂²⋊D₄₃	—	2	1	no	—
71	C₂×C₂²⋊D₄₃	—	2	1	no	—
72	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
73	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
74	#10621898	—	2	2	no	—
75	#10621898	—	2	2	no	—
76	#10621898	—	2	2	no	—
77	#10621898	D₄	1	4	yes	complemented
78	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
79	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₄₃	—	2	1	no	—
81	C₂×C₄×D₄₃	—	2	1	no	—
82	C₂×C₄×D₄₃	—	2	1	no	—
83	C₂×C₄×D₄₃	—	2	1	no	—
84	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
85	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
88	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
91	C₂²⋊C₄×C₄₃	—	2	1	no	—
92	C₄²×C₄₃	—	2	1	no	—
93	C₂²⋊Dic₄₃	—	2	1	no	—
94	C₂²⋊Dic₄₃	—	2	1	no	—
95	#10621885	—	2	2	no	—
96	#10621884	—	2	2	no	—
97	#10621883	—	2	2	no	—
98	#10621882	—	2	2	no	—
99	(C₂×D₄₃)⋊C₄	—	2	1	no	—
100	(C₂×D₄₃)⋊C₄	—	2	1	no	—
101	(C₂×D₄₃)⋊C₄	—	2	1	no	—
102	(C₂×D₄₃)⋊C₄	—	2	1	no	—
103	Dic₄₃⋊C₄	—	2	1	no	—
104	Dic₄₃⋊C₄	—	2	1	no	—
105	Dic₄₃⋊C₄	—	2	1	no	—
106	Dic₄₃⋊C₄	—	2	1	no	—
107	C₄×Dic₄₃	—	2	1	no	—
108	C₄×Dic₄₃	—	2	1	no	—
109	C₄×Dic₄₃	D₄	1	1	yes	characteristic
110	C₄×Dic₄₃	D₄	1	1	yes	characteristic
111	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
112	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
113	C₂×C₄₃⋊C₈	—	2	2	no	—
114	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
115	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
116	Dic₁₇₂	—	2	2	no	—
117	D₃₄₄	—	2	2	no	—
118	#10621873	—	2	2	no	—
119	#10621873	—	2	2	no	—
120	C₂³×C₄₃	—	2	1	no	—
121	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×D₄₃	—	2	1	no	—
123	C₂²×D₄₃	—	2	1	no	—
124	Q₈×C₄₃	—	2	1	no	—
125	Q₈×C₄₃	—	2	1	no	—
126	D₄×C₄₃	—	2	1	no	—
127	D₄×C₄₃	—	2	1	no	—
128	D₄×C₄₃	—	4	2	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	2	1	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
134	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	#41	1	2	yes	—
136	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
137	C₂²⋊D₄₃	—	2	4	no	—
138	C₂²⋊D₄₃	—	4	2	no	—
139	C₂²⋊D₄₃	—	4	2	no	—
140	C₂²⋊D₄₃	—	4	2	no	—
141	C₂×Dic₄₃	—	2	1	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	C₂×Dic₄₃	—	2	1	no	—
144	C₂×Dic₄₃	—	4	1	no	—
145	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
146	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
147	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
148	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
149	D₁₇₂	—	2	1	no	—
150	D₁₇₂	—	2	1	no	—
151	D₁₇₂	—	2	1	no	—
152	D₁₇₂	C₂×D₄	1	2	yes	—
153	C₄×D₄₃	—	2	4	no	—
154	C₄×D₄₃	—	4	2	no	—
155	C₄×D₄₃	—	4	2	no	—
156	C₄×D₄₃	—	4	2	no	—
157	Dic₈₆	—	2	1	no	—
158	Dic₈₆	—	2	1	no	—
159	Dic₈₆	—	2	1	no	—
160	Dic₈₆	C₂×D₄	1	2	yes	—
161	C₈×C₄₃	—	2	1	no	—
162	C₈×C₄₃	—	2	1	no	—
163	C₄₃⋊C₈	—	2	1	no	—
164	C₄₃⋊C₈	—	2	1	no	—
165	C₂²×C₄₃	—	2	1	no	—
166	C₂²×C₄₃	—	2	1	no	—
167	C₂²×C₄₃	—	4	1	no	—
168	C₂²×C₄₃	—	4	1	no	—
169	C₂²×C₄₃	—	4	1	no	—
170	C₂²×C₄₃	#121	1	2	yes	—
171	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
172	C₂×D₄₃	—	2	2	no	—
173	C₂×D₄₃	—	4	1	no	—
174	C₂×D₄₃	—	4	1	no	—
175	C₂×D₄₃	—	4	1	no	—
176	C₂×D₄₃	—	4	1	no	—
177	C₂×D₄₃	—	4	1	no	—
178	C₄×C₄₃	—	4	1	no	—
179	C₄×C₄₃	—	4	1	no	—
180	C₄×C₄₃	—	4	1	no	—
181	C₄×C₄₃	#121	1	2	yes	—
182	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
183	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
184	Dic₄₃	—	2	2	no	—
185	Dic₄₃	—	2	1	no	—
186	Dic₄₃	—	2	1	no	—
187	Dic₄₃	—	2	1	no	—
188	Dic₄₃	—	4	1	no	—
189	Dic₄₃	#135	1	2	yes	complemented
190	#3035	—	43	1	no	maximal, Sylow
191	C₂×C₄₃	—	2	2	no	—
192	C₂×C₄₃	—	8	1	no	—
193	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
194	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
195	C₂×C₄₃	#575	1	1	yes	characteristic
196	D₄₃	—	4	2	no	—
197	D₄₃	—	8	1	no	—
198	C₂×#135	—	43	1	no	—
199	C₂×#135	—	43	1	no	—
200	#532	—	43	1	no	—
201	C₄×#41	—	43	1	no	—
202	#464	—	43	1	no	—
203	#464	—	43	1	no	—
204	#463	—	43	1	no	—
205	#463	—	43	1	no	—
206	#462	—	43	1	no	—
207	#461	—	43	1	no	—
208	#460	—	43	1	no	—
209	#459	—	43	1	no	—
210	C₂×C₄⋊C₈	—	43	1	no	—
211	#416	—	43	1	no	—
212	#416	D₄₃	1	1	yes	characteristic, complemented
213	C₄₃	#3035	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	43	1	no	—
215	C₂×#41	—	43	1	no	—
216	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
217	#135	—	86	4	no	—
218	#135	—	86	4	no	—
219	C₂×Q₁₆	—	43	1	no	—
220	C₂×Q₁₆	—	43	1	no	—
221	C₂×SD₁₆	—	43	1	no	—
222	C₂×SD₁₆	—	43	1	no	—
223	C₂×SD₁₆	—	43	1	no	—
224	C₂×SD₁₆	—	43	1	no	—
225	C₂×D₈	—	43	1	no	—
226	C₂×D₈	—	43	1	no	—
227	C₂²×C₈	—	43	1	no	—
228	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
229	C₄⋊Q₈	—	43	1	no	—
230	#127	—	43	1	no	—
231	#124	—	43	1	no	—
232	#124	—	43	1	no	—
233	#121	—	86	1	no	—
234	#121	—	86	1	no	—
235	C₄×Q₈	—	43	1	no	—
236	C₄×Q₈	—	43	1	no	—
237	C₄×D₄	—	43	1	no	—
238	C₄×D₄	—	43	1	no	—
239	C₄×D₄	—	86	1	no	—
240	C₄×D₄	—	86	1	no	—
241	C₄×D₄	—	86	1	no	—
242	C₄×D₄	—	86	1	no	—
243	#117	—	86	1	no	—
244	#117	C₂×D₄₃	1	1	yes	characteristic
245	C₂×C₄²	—	43	1	no	—
246	C₂×C₄²	—	86	1	no	—
247	C₄⋊C₈	—	43	1	no	—
248	C₄⋊C₈	—	43	1	no	—
249	C₄⋊C₈	—	43	1	no	—
250	C₄⋊C₈	—	43	1	no	—
251	#103	—	43	1	no	—
252	#103	—	43	1	no	—
253	#103	C₂×D₄₃	1	1	yes	characteristic
254	#103	C₂×D₄₃	1	1	yes	characteristic
255	#102	—	43	1	no	—
256	#102	—	43	1	no	—
257	#102	C₂×D₄₃	1	1	yes	characteristic
258	#102	C₂×D₄₃	1	1	yes	characteristic
259	#41	—	2	2	no	—
260	#41	—	2	2	no	—
261	#41	—	43	4	no	—
262	#41	—	86	2	no	—
263	#41	—	86	2	no	—
264	#41	—	86	2	no	—
265	C₂×Q₈	—	43	1	no	—
266	C₂×Q₈	—	43	1	no	—
267	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	43	1	no	—
270	C₂×D₄	—	43	1	no	—
271	C₂×D₄	—	86	1	no	—
272	C₂×D₄	—	86	1	no	—
273	C₂×D₄	—	86	1	no	—
274	C₂×D₄	—	86	1	no	—
275	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	43	1	no	—
278	C₂²×C₄	—	43	1	no	—
279	C₂²×C₄	—	86	1	no	—
280	C₂²×C₄	—	86	1	no	—
281	C₂²×C₄	—	86	1	no	—
282	C₂²×C₄	—	86	1	no	—
283	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
284	Q₁₆	—	86	2	no	—
285	Q₁₆	—	86	2	no	—
286	SD₁₆	—	86	2	no	—
287	SD₁₆	—	86	2	no	—
288	SD₁₆	—	86	2	no	—
289	SD₁₆	—	86	2	no	—
290	D₈	—	86	2	no	—
291	D₈	—	86	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	43	1	no	—
294	C₂×C₈	—	43	1	no	—
295	C₂×C₈	—	86	2	no	—
296	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
297	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
298	C₄⋊C₄	—	86	1	no	—
299	C₄⋊C₄	—	86	1	no	—
300	C₄⋊C₄	—	86	1	no	—
301	C₄⋊C₄	—	86	1	no	—
302	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
303	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
304	C₂²⋊C₄	—	2	1	no	—
305	C₂²⋊C₄	—	86	1	no	—
306	C₂²⋊C₄	—	86	1	no	—
307	C₂²⋊C₄	—	86	1	no	—
308	C₂²⋊C₄	—	86	1	no	—
309	C₂²⋊C₄	—	86	1	no	—
310	C₂²⋊C₄	—	86	1	no	—
311	C₄²	—	2	1	no	—
312	C₄²	—	43	1	no	—
313	C₄²	—	43	1	no	—
314	C₄²	—	43	1	no	—
315	C₄²	—	43	1	no	—
316	C₄²	—	86	1	no	—
317	C₄²	—	86	1	no	—
318	C₂³	—	2	1	no	—
319	C₂³	—	86	1	no	—
320	C₂³	—	86	1	no	—
321	C₂³	D₄×D₄₃	1	1	yes	characteristic
322	Q₈	—	2	1	no	—
323	Q₈	—	2	1	no	—
324	Q₈	—	43	2	no	—
325	Q₈	—	86	1	no	—
326	Q₈	—	86	1	no	—
327	Q₈	—	86	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	2	1	no	—
330	D₄	—	4	2	no	—
331	D₄	—	43	2	no	—
332	D₄	—	86	4	no	—
333	D₄	—	86	1	no	—
334	D₄	—	86	1	no	—
335	D₄	—	86	1	no	—
336	D₄	—	172	2	no	—
337	D₄	—	172	2	no	—
338	D₄	—	172	2	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	43	1	no	—
344	C₂×C₄	—	43	1	no	—
345	C₂×C₄	—	43	1	no	—
346	C₂×C₄	—	43	1	no	—
347	C₂×C₄	—	86	2	no	—
348	C₂×C₄	—	86	1	no	—
349	C₂×C₄	—	86	4	no	—
350	C₂×C₄	—	86	1	no	—
351	C₂×C₄	—	172	2	no	—
352	C₂×C₄	—	172	2	no	—
353	C₂×C₄	—	172	1	no	—
354	C₂×C₄	—	172	2	no	—
355	C₂×C₄	#10621898	1	2	yes	—
356	C₂×C₄	D₄×D₄₃	1	2	yes	—
357	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
358	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	86	1	no	—
362	C₈	—	86	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	4	1	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	86	2	no	—
369	C₂²	—	172	1	no	—
370	C₂²	—	172	1	no	—
371	C₂²	—	172	1	no	—
372	C₂²	—	172	1	no	—
373	C₂²	—	172	1	no	—
374	C₂²	#13063298	1	2	yes	—
375	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
376	C₄	—	4	1	no	—
377	C₄	—	4	1	no	—
378	C₄	—	4	1	no	—
379	C₄	—	43	2	no	—
380	C₄	—	86	2	no	—
381	C₄	—	86	1	no	—
382	C₄	—	86	1	no	—
383	C₄	—	86	1	no	—
384	C₄	—	172	1	no	—
385	C₄	#13063298	1	2	yes	—
386	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
387	C₄	C₂×#10621898	1	1	yes	characteristic
388	C₂	—	2	2	no	—
389	C₂	—	8	1	no	—
390	C₂	—	344	1	no	—
391	C₂	—	172	2	no	—
392	C₂	C₂×#13063298	1	1	yes	characteristic
393	C₂	#135×D₄₃	1	1	yes	characteristic
394	C₂	#14583696	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#416	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#416	D₄₃
hypocenter	C₄₃	#3035
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
1. C₂×#13063347	C₂	semidirect
2. C₂×#13063345	C₂	semidirect
3. #14583445	C₂	semidirect
4. #14583367	C₂	semidirect
5. #14583124	C₂	semidirect
6. #14583120	C₂	semidirect
7. #14583117	C₂	semidirect
8. #14583113	C₂	semidirect
9. #14583108	C₂	semidirect
10. #14583103	C₂	semidirect
11. #14583098	C₂	semidirect
12. #14583093	C₂	semidirect
13. C₂×Dic₄₃⋊C₈	C₂	semidirect
14. #14582893	C₂	semidirect
15. C₄₃×#416	C₂	semidirect
77. #10621898	D₄	semidirect
189. Dic₄₃	#135	semidirect
212. #416	D₄₃	semidirect
213. C₄₃	#3035	semidirect