Subgroups — #19147922

Order 5504.

1556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₄₃	C₂	1	2	yes	maximal, complemented
2	#14583476	C₂	1	4	yes	maximal, complemented
3	#14583474	C₂	1	2	yes	maximal, complemented
4	#14583469	C₂	1	2	yes	maximal, complemented
5	#14583467	C₂	1	2	yes	maximal, complemented
6	#14583462	C₂	1	2	yes	maximal, complemented
7	#14583457	C₂	1	4	yes	maximal, complemented
8	#14583449	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583445	C₂	1	2	yes	maximal, complemented
10	#14583408	C₂	1	8	yes	maximal, complemented
11	C₂×#13063282	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#10621902	C₂²	1	2	yes	—
14	C₂×#10621898	C₂²	1	4	yes	—
15	C₂×#10621898	C₂²	1	2	yes	complemented
16	C₂×#10621900	C₂²	1	2	yes	—
17	C₂×#10621900	C₂²	1	4	yes	complemented
18	#41×D₄₃	—	2	16	no	—
19	C₂×Q₈×D₄₃	C₂²	1	2	yes	complemented
20	C₂×D₄×D₄₃	C₂²	1	4	yes	—
21	C₂×D₄×D₄₃	C₂²	1	2	yes	complemented
22	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄×D₄₃	C₂²	1	4	yes	complemented
24	#13063325	C₂²	1	1	yes	characteristic, complemented
25	#13063323	C₂²	1	2	yes	complemented
26	#13063322	C₂²	1	1	yes	characteristic, complemented
27	#13063321	C₂²	1	2	yes	complemented
28	#13063314	C₂²	1	2	yes	complemented
29	#13063313	C₂²	1	2	yes	complemented
30	#13063312	C₂²	1	2	yes	complemented
31	#13063310	C₂²	1	8	yes	complemented
32	#13063307	C₂²	1	8	yes	complemented
33	#13063303	C₂²	1	8	yes	complemented
34	#13063302	C₂²	1	8	yes	complemented
35	#13063299	C₂²	1	4	yes	complemented
36	#13063298	C₂²	1	8	yes	complemented
37	#13063297	C₂²	1	8	yes	complemented
38	#13063297	C₂²	1	8	yes	complemented
39	#13063295	C₂²	1	8	yes	complemented
40	#13063293	C₂²	1	8	yes	complemented
41	C₄×D₁₇₂	C₂²	1	4	yes	complemented
42	C₄×C₂²⋊D₄₃	C₂²	1	8	yes	complemented
43	#13063282	C₂²	1	4	yes	complemented
44	#13063282	C₂²	1	2	yes	complemented
45	#13063282	C₂²	1	2	yes	complemented
46	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	complemented
47	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
48	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
49	C₂×#41×C₄₃	C₂²	1	2	yes	complemented
50	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
51	#127×C₄₃	C₂²	1	1	yes	characteristic, complemented
52	#124×C₄₃	C₂²	1	2	yes	complemented
53	#121×C₄₃	C₂²	1	4	yes	complemented
54	C₄×D₄×C₄₃	C₂²	1	4	yes	complemented
55	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic, complemented
56	C₂³×D₄₃	C₂³	1	4	yes	—
57	C₂³×D₄₃	C₂³	1	1	yes	characteristic
58	#41×C₄₃	—	2	8	no	—
59	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
60	C₂×D₄×C₄₃	C₂³	1	4	yes	—
61	C₂×D₄×C₄₃	C₂³	1	2	yes	—
62	C₂×D₄×C₄₃	C₂³	1	4	yes	—
63	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
64	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
67	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
68	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
69	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
70	C₂²×Dic₄₃	C₂³	1	4	yes	—
71	C₂²×Dic₄₃	C₂³	1	2	yes	—
72	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
73	#10621902	—	2	8	no	—
74	Q₈×D₄₃	—	2	8	no	—
75	#10621900	—	2	16	no	—
76	#10621900	—	2	8	no	—
77	D₄×D₄₃	—	2	8	no	—
78	D₄×D₄₃	—	2	16	no	—
79	#10621898	—	2	8	no	—
80	#10621898	—	2	16	no	—
81	C₂×D₁₇₂	C₂³	1	4	yes	—
82	C₂×D₁₇₂	C₂³	1	2	yes	—
83	C₂×D₁₇₂	C₂³	1	4	yes	—
84	C₂×C₄×D₄₃	—	2	16	no	—
85	C₂×C₄×D₄₃	D₄	1	8	yes	complemented
86	C₂×C₄×D₄₃	C₂³	1	4	yes	—
87	C₂×C₄×D₄₃	C₂³	1	4	yes	—
88	C₂×C₄×D₄₃	C₂³	1	4	yes	—
89	C₂×C₄×D₄₃	C₂³	1	2	yes	—
90	C₂×C₄×D₄₃	C₂³	1	2	yes	—
91	C₂×Dic₈₆	C₂³	1	4	yes	—
92	C₂×Dic₈₆	C₂³	1	2	yes	—
93	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
94	C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
95	C₄²×C₄₃	C₂³	1	2	yes	—
96	C₄²×C₄₃	C₂³	1	1	yes	characteristic
97	C₄²×C₄₃	C₂³	1	1	yes	characteristic
98	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
99	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
100	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
101	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
102	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
103	C₄⋊Dic₄₃	C₂³	1	4	yes	—
104	Dic₄₃⋊C₄	C₂³	1	4	yes	—
105	Dic₄₃⋊C₄	C₂³	1	4	yes	—
106	Dic₄₃⋊C₄	C₂³	1	8	yes	—
107	C₄×Dic₄₃	C₂³	1	2	yes	—
108	C₄×Dic₄₃	C₂³	1	2	yes	—
109	C₂³×C₄₃	C₂⁴	1	4	yes	—
110	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
111	C₂²×D₄₃	—	2	8	no	—
112	C₂²×D₄₃	—	4	2	no	—
113	C₂²×D₄₃	—	4	8	no	—
114	C₂²×D₄₃	C₂×D₄	1	4	yes	—
115	C₂²×D₄₃	C₂⁴	1	4	yes	—
116	C₂²×D₄₃	C₂⁴	1	2	yes	—
117	Q₈×C₄₃	—	2	4	no	—
118	D₄×C₄₃	—	2	8	no	—
119	D₄×C₄₃	—	2	4	no	—
120	D₄×C₄₃	—	4	4	no	—
121	C₂×C₄×C₄₃	—	2	8	no	—
122	C₂×C₄×C₄₃	—	4	2	no	—
123	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
124	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
125	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
126	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
127	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₄₃	—	2	8	no	—
130	C₂²⋊D₄₃	—	2	8	no	—
131	C₂²⋊D₄₃	—	2	8	no	—
132	C₂²⋊D₄₃	—	4	8	no	—
133	C₂×Dic₄₃	—	2	8	no	—
134	C₂×Dic₄₃	—	4	2	no	—
135	C₂×Dic₄₃	C₂×D₄	1	4	yes	—
136	C₂×Dic₄₃	C₂⁴	1	4	yes	—
137	C₂×Dic₄₃	C₂⁴	1	4	yes	—
138	C₂×Dic₄₃	C₂⁴	1	2	yes	—
139	D₁₇₂	—	2	8	no	—
140	D₁₇₂	—	2	4	no	—
141	D₁₇₂	—	4	4	no	—
142	C₄×D₄₃	—	2	8	no	—
143	C₄×D₄₃	—	2	8	no	—
144	C₄×D₄₃	—	2	8	no	—
145	C₄×D₄₃	C₂×D₄	1	8	yes	—
146	C₄×D₄₃	C₂×D₄	1	8	yes	—
147	Dic₈₆	—	2	4	no	—
148	Dic₈₆	—	2	8	no	—
149	C₂²×C₄₃	—	2	2	no	—
150	C₂²×C₄₃	—	2	4	no	—
151	C₂²×C₄₃	—	2	8	no	—
152	C₂²×C₄₃	C₂²×D₄	1	2	yes	—
153	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
154	C₂×D₄₃	—	2	4	no	—
155	C₂×D₄₃	—	2	4	no	—
156	C₂×D₄₃	—	2	8	no	—
157	C₂×D₄₃	—	4	4	no	—
158	C₂×D₄₃	—	4	16	no	—
159	C₂×D₄₃	C₂²×D₄	1	4	yes	—
160	C₄×C₄₃	—	2	4	no	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	2	2	no	—
163	C₄×C₄₃	C₂²×D₄	1	2	yes	—
164	C₄×C₄₃	C₂²×D₄	1	2	yes	—
165	Dic₄₃	—	2	4	no	—
166	Dic₄₃	—	2	4	no	—
167	Dic₄₃	C₂²×D₄	1	4	yes	—
168	#3449	—	43	1	no	maximal, Sylow
169	C₂×C₄₃	—	2	2	no	—
170	C₂×C₄₃	—	4	4	no	—
171	C₂×C₄₃	C₂³×D₄	1	1	yes	characteristic
172	C₂×C₄₃	#585	1	2	yes	—
173	D₄₃	—	2	4	no	—
174	D₄₃	—	4	4	no	—
175	C₂²×#41	—	43	2	no	—
176	#537	—	43	4	no	—
177	#536	—	43	2	no	—
178	#536	—	43	2	no	—
179	#535	—	43	2	no	—
180	#535	—	43	2	no	—
181	#534	—	43	4	no	—
182	#532	—	43	2	no	—
183	#532	—	43	1	no	—
184	#532	D₄₃	1	1	yes	characteristic, complemented
185	#525	—	43	8	no	—
186	C₂×#117	—	43	1	no	—
187	C₄₃	#3449	1	1	yes	characteristic, complemented, Sylow
188	C₂×#41	—	43	2	no	—
189	C₂×#41	—	43	2	no	—
190	C₂×#41	—	43	4	no	—
191	C₂×#41	—	43	4	no	—
192	C₂×#41	—	43	2	no	—
193	C₂×#41	—	86	16	no	—
194	C₂×#41	C₂×D₄₃	1	2	yes	complemented
195	C₂²×Q₈	—	43	2	no	—
196	C₂²×D₄	—	43	4	no	—
197	C₂²×D₄	—	43	2	no	—
198	C₂³×C₄	—	43	1	no	—
199	C₂³×C₄	—	43	4	no	—
200	C₄⋊Q₈	—	43	2	no	—
201	C₄⋊Q₈	—	43	1	no	—
202	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
203	#127	—	43	2	no	—
204	#127	—	43	1	no	—
205	#127	C₂×D₄₃	1	1	yes	characteristic, complemented
206	#124	—	43	2	no	—
207	#124	—	43	2	no	—
208	#124	—	43	2	no	—
209	#124	C₂×D₄₃	1	2	yes	complemented
210	#123	—	43	8	no	—
211	#123	—	43	8	no	—
212	C₂²⋊Q₈	—	43	8	no	—
213	C₂²⋊Q₈	—	43	8	no	—
214	#121	—	43	8	no	—
215	#121	—	43	8	no	—
216	#121	—	43	4	no	—
217	#121	—	43	8	no	—
218	#121	C₂×D₄₃	1	4	yes	complemented
219	#120	—	43	8	no	—
220	#120	—	43	8	no	—
221	C₄×D₄	—	43	8	no	—
222	C₄×D₄	—	43	4	no	—
223	C₄×D₄	C₂×D₄₃	1	4	yes	complemented
224	#117	—	43	4	no	—
225	#117	—	43	2	no	—
226	#117	—	43	2	no	—
227	C₂×C₄⋊C₄	—	43	2	no	—
228	C₂×C₂²⋊C₄	—	43	2	no	—
229	C₂×C₄²	—	43	1	no	—
230	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic, complemented
231	C₂⁴	—	43	1	no	—
232	C₂⁴	—	43	4	no	—
233	#41	—	2	8	no	—
234	#41	—	86	16	no	—
235	#41	—	86	16	no	—
236	#41	—	86	8	no	—
237	#41	—	86	8	no	—
238	#41	—	86	8	no	—
239	C₂×Q₈	—	43	4	no	—
240	C₂×Q₈	—	43	2	no	—
241	C₂×Q₈	—	86	8	no	—
242	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
243	C₂×D₄	—	43	4	no	—
244	C₂×D₄	—	43	4	no	—
245	C₂×D₄	—	43	4	no	—
246	C₂×D₄	—	43	8	no	—
247	C₂×D₄	—	43	4	no	—
248	C₂×D₄	—	43	2	no	—
249	C₂×D₄	—	43	4	no	—
250	C₂×D₄	—	86	8	no	—
251	C₂×D₄	—	86	16	no	—
252	C₂×D₄	C₂²×D₄₃	1	4	yes	—
253	C₂×D₄	C₂²×D₄₃	1	2	yes	—
254	C₂×D₄	C₂²×D₄₃	1	4	yes	—
255	C₂²×C₄	—	43	1	no	—
256	C₂²×C₄	—	43	8	no	—
257	C₂²×C₄	—	43	2	no	—
258	C₂²×C₄	—	43	2	no	—
259	C₂²×C₄	—	43	4	no	—
260	C₂²×C₄	—	43	2	no	—
261	C₂²×C₄	—	43	4	no	—
262	C₂²×C₄	—	43	4	no	—
263	C₂²×C₄	—	43	4	no	—
264	C₂²×C₄	—	86	16	no	—
265	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
266	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
267	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
268	C₄⋊C₄	—	43	4	no	—
269	C₄⋊C₄	—	43	4	no	—
270	C₄⋊C₄	—	43	8	no	—
271	C₄⋊C₄	—	43	4	no	—
272	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
273	C₂²⋊C₄	—	43	8	no	—
274	C₂²⋊C₄	—	43	4	no	—
275	C₂²⋊C₄	—	43	4	no	—
276	C₂²⋊C₄	—	43	8	no	—
277	C₂²⋊C₄	—	43	8	no	—
278	C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
279	C₄²	—	43	2	no	—
280	C₄²	—	43	2	no	—
281	C₄²	C₂²×D₄₃	1	1	yes	characteristic
282	C₄²	C₂²×D₄₃	1	1	yes	characteristic
283	C₄²	C₂²×D₄₃	1	2	yes	—
284	C₂³	—	43	2	no	—
285	C₂³	—	43	4	no	—
286	C₂³	—	43	4	no	—
287	C₂³	—	86	8	no	—
288	C₂³	—	172	2	no	—
289	C₂³	—	172	8	no	—
290	C₂³	C₂³×D₄₃	1	4	yes	—
291	C₂³	C₂³×D₄₃	1	1	yes	characteristic
292	Q₈	—	2	4	no	—
293	Q₈	—	86	8	no	—
294	Q₈	—	86	4	no	—
295	D₄	—	2	8	no	—
296	D₄	—	2	4	no	—
297	D₄	—	4	4	no	—
298	D₄	—	86	8	no	—
299	D₄	—	86	8	no	—
300	D₄	—	86	8	no	—
301	D₄	—	86	8	no	—
302	D₄	—	86	4	no	—
303	D₄	—	172	4	no	—
304	D₄	—	172	8	no	—
305	C₂×C₄	—	2	8	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	43	4	no	—
308	C₂×C₄	—	43	2	no	—
309	C₂×C₄	—	43	8	no	—
310	C₂×C₄	—	43	8	no	—
311	C₂×C₄	—	43	4	no	—
312	C₂×C₄	—	43	4	no	—
313	C₂×C₄	—	86	8	no	—
314	C₂×C₄	—	86	8	no	—
315	C₂×C₄	—	86	8	no	—
316	C₂×C₄	—	86	8	no	—
317	C₂×C₄	—	172	2	no	—
318	C₂×C₄	D₄×D₄₃	1	4	yes	—
319	C₂×C₄	C₂³×D₄₃	1	2	yes	—
320	C₂×C₄	C₂³×D₄₃	1	2	yes	—
321	C₂×C₄	C₂³×D₄₃	1	4	yes	—
322	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
324	C₂²	—	2	2	no	—
325	C₂²	—	2	4	no	—
326	C₂²	—	2	8	no	—
327	C₂²	—	43	4	no	—
328	C₂²	—	86	4	no	—
329	C₂²	—	86	4	no	—
330	C₂²	—	86	8	no	—
331	C₂²	—	172	4	no	—
332	C₂²	—	172	16	no	—
333	C₂²	C₂×D₄×D₄₃	1	2	yes	—
334	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
335	C₄	—	2	4	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	43	4	no	—
339	C₄	—	86	4	no	—
340	C₄	—	86	4	no	—
341	C₄	C₂×D₄×D₄₃	1	2	yes	—
342	C₄	C₂×D₄×D₄₃	1	2	yes	—
343	C₂	—	2	2	no	—
344	C₂	—	4	4	no	—
345	C₂	—	86	4	no	—
346	C₂	—	172	4	no	—
347	C₂	C₂²×D₄×D₄₃	1	1	yes	characteristic
348	C₂	#14583735	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₄₃×#532	C₂
hypercenter	#532	D₄₃
hypocenter	C₄₃	#3449
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₄₃	C₂	semidirect
2. #14583476	C₂	semidirect
3. #14583474	C₂	semidirect
4. #14583469	C₂	semidirect
5. #14583467	C₂	semidirect
6. #14583462	C₂	semidirect
7. #14583457	C₂	semidirect
8. #14583449	C₂	semidirect
9. #14583445	C₂	semidirect
10. #14583408	C₂	semidirect
11. C₂×#13063282	C₂	semidirect
12. C₄₃×#532	C₂	semidirect
15. C₂×#10621898	C₂²	semidirect
17. C₂×#10621900	C₂²	semidirect
19. C₂×Q₈×D₄₃	C₂²	semidirect
21. C₂×D₄×D₄₃	C₂²	semidirect
23. C₂²×C₄×D₄₃	C₂²	semidirect
24. #13063325	C₂²	semidirect
25. #13063323	C₂²	semidirect
26. #13063322	C₂²	semidirect
27. #13063321	C₂²	semidirect
28. #13063314	C₂²	semidirect
29. #13063313	C₂²	semidirect
30. #13063312	C₂²	semidirect
31. #13063310	C₂²	semidirect
32. #13063307	C₂²	semidirect
33. #13063303	C₂²	semidirect
34. #13063302	C₂²	semidirect
35. #13063299	C₂²	semidirect
36. #13063298	C₂²	semidirect
37. #13063297	C₂²	semidirect
38. #13063297	C₂²	semidirect
39. #13063295	C₂²	semidirect
40. #13063293	C₂²	semidirect
41. C₄×D₁₇₂	C₂²	semidirect
42. C₄×C₂²⋊D₄₃	C₂²	semidirect
43. #13063282	C₂²	semidirect
44. #13063282	C₂²	semidirect
45. #13063282	C₂²	semidirect
46. C₂×Dic₄₃⋊C₄	C₂²	semidirect
48. C₂×C₄×Dic₄₃	C₂²	semidirect
49. C₂×#41×C₄₃	C₂²	semidirect
50. C₄⋊Q₈×C₄₃	C₂²	semidirect
51. #127×C₄₃	C₂²	semidirect
52. #124×C₄₃	C₂²	semidirect
53. #121×C₄₃	C₂²	semidirect
54. C₄×D₄×C₄₃	C₂²	semidirect
55. C₂×C₄²×C₄₃	C₂²	semidirect
85. C₂×C₄×D₄₃	D₄	semidirect
184. #532	D₄₃	semidirect
187. C₄₃	#3449	semidirect
194. C₂×#41	C₂×D₄₃	semidirect
202. C₄⋊Q₈	C₂×D₄₃	semidirect
205. #127	C₂×D₄₃	semidirect
209. #124	C₂×D₄₃	semidirect
218. #121	C₂×D₄₃	semidirect
223. C₄×D₄	C₂×D₄₃	semidirect
230. C₂×C₄²	C₂×D₄₃	semidirect