Subgroups — #19137374

Order 5504.

476 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14582798	C₂	1	1	yes	maximal, characteristic, complemented
2	#14582798	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582792	C₂	1	1	yes	maximal, characteristic
4	#14582792	C₂	1	1	yes	maximal, characteristic
5	#14582792	C₂	1	1	yes	maximal, characteristic
6	#14582785	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582785	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582785	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582774	C₂	1	1	yes	maximal, characteristic
10	#14582774	C₂	1	1	yes	maximal, characteristic
11	#14582765	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
15	C₂×#126×C₄₃	C₂	1	1	yes	maximal, characteristic
16	C₂³×Dic₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
32	C₄₃⋊#95	C₂²	1	1	yes	characteristic
33	C₄₃⋊#95	C₂²	1	1	yes	characteristic
34	C₄₃⋊#95	C₂²	1	1	yes	characteristic
35	C₄₃⋊#95	C₂²	1	1	yes	characteristic
36	C₄₃⋊#95	C₂²	1	1	yes	characteristic
37	C₄₃⋊#95	C₂²	1	1	yes	characteristic
38	C₄₃⋊#95	C₂²	1	1	yes	characteristic
39	C₄₃⋊#95	C₂²	1	1	yes	characteristic
40	C₄₃⋊#95	C₂²	1	1	yes	characteristic
41	C₄₃⋊#95	C₂²	1	1	yes	characteristic
42	C₄₃⋊#95	C₂²	1	1	yes	characteristic
43	C₄₃⋊#95	C₂²	1	1	yes	characteristic
44	#126×C₄₃	C₄	1	8	yes	complemented
45	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
52	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
59	C₂²×Dic₄₃	—	2	2	no	—
60	C₂²×Dic₄₃	—	4	1	no	—
61	C₂²×Dic₄₃	—	4	1	no	—
62	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
63	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
64	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
65	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
66	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
67	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
68	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
69	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	complemented
71	C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	complemented
72	C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	complemented
73	C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
74	C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
75	C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
76	C₄²×C₄₃	C₂×C₄	1	4	yes	complemented
77	C₂²⋊Dic₄₃	—	2	2	no	—
78	C₂²⋊Dic₄₃	—	4	1	no	—
79	C₂²⋊Dic₄₃	—	4	1	no	—
80	C₄⋊Dic₄₃	—	2	2	no	—
81	C₄⋊Dic₄₃	—	2	2	no	—
82	C₄⋊Dic₄₃	—	2	2	no	—
83	C₄⋊Dic₄₃	—	4	1	no	—
84	C₄⋊Dic₄₃	—	4	1	no	—
85	C₄⋊Dic₄₃	—	4	1	no	—
86	C₄⋊Dic₄₃	—	4	1	no	—
87	C₄×Dic₄₃	—	2	2	no	—
88	C₄×Dic₄₃	—	4	1	no	—
89	C₄×Dic₄₃	—	4	1	no	—
90	C₄×Dic₄₃	—	4	1	no	—
91	C₄×Dic₄₃	—	4	1	no	—
92	C₂³×C₄₃	—	2	1	no	—
93	C₂³×C₄₃	—	2	1	no	—
94	C₂³×C₄₃	—	2	1	no	—
95	C₂³×C₄₃	—	2	1	no	—
96	C₂³×C₄₃	—	2	1	no	—
97	C₂³×C₄₃	—	2	1	no	—
98	C₂³×C₄₃	C₂²×C₄	1	2	yes	—
99	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₄₃	—	2	1	no	—
101	C₂×C₄×C₄₃	—	2	1	no	—
102	C₂×C₄×C₄₃	—	2	1	no	—
103	C₂×C₄×C₄₃	—	2	1	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	1	no	—
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
113	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
114	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
116	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
117	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
118	C₂×Dic₄₃	—	2	1	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	2	1	no	—
121	C₂×Dic₄₃	—	2	1	no	—
122	C₂×Dic₄₃	—	2	1	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	—	2	1	no	—
130	C₂×Dic₄₃	—	2	1	no	—
131	C₂×Dic₄₃	—	2	1	no	—
132	C₂×Dic₄₃	—	2	1	no	—
133	C₂×Dic₄₃	—	2	1	no	—
134	C₂×Dic₄₃	—	2	1	no	—
135	C₂×Dic₄₃	—	2	1	no	—
136	C₂×Dic₄₃	—	2	1	no	—
137	C₂×Dic₄₃	—	2	1	no	—
138	C₂×Dic₄₃	—	2	1	no	—
139	C₂×Dic₄₃	—	2	1	no	—
140	C₂×Dic₄₃	—	4	4	no	—
141	C₂×Dic₄₃	#41	1	2	yes	—
142	C₂×Dic₄₃	#41	1	2	yes	—
143	C₂²×C₄₃	—	2	2	no	—
144	C₂²×C₄₃	—	2	2	no	—
145	C₂²×C₄₃	—	2	2	no	—
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	—	4	1	no	—
148	C₂²×C₄₃	—	4	1	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
151	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
152	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
153	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
156	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	—	2	2	no	—
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	—	2	2	no	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	2	2	no	—
163	Dic₄₃	—	2	2	no	—
164	Dic₄₃	—	2	2	no	—
165	Dic₄₃	—	4	1	no	—
166	Dic₄₃	—	4	1	no	—
167	Dic₄₃	—	4	1	no	—
168	Dic₄₃	—	4	1	no	—
169	Dic₄₃	—	4	1	no	—
170	Dic₄₃	—	4	1	no	—
171	#2334	—	43	1	no	maximal, Sylow
172	C₂×C₄₃	—	4	2	no	—
173	C₂×C₄₃	C₄×#41	1	1	yes	characteristic
174	C₂×C₄₃	#520	1	1	yes	characteristic
175	C₂×C₄₃	#520	1	1	yes	characteristic
176	C₂×C₄₃	#538	1	1	yes	characteristic
177	C₂×C₄₃	#539	1	1	yes	characteristic
178	C₂×C₄₃	#541	1	1	yes	characteristic
179	C₂×C₄₃	#542	1	1	yes	characteristic
180	C₂×#126	D₄₃	1	1	yes	characteristic
181	#389	—	43	1	no	—
182	#389	—	43	1	no	—
183	#388	—	43	1	no	—
184	#388	—	43	1	no	—
185	#388	—	43	1	no	—
186	#387	—	43	1	no	—
187	#387	—	43	1	no	—
188	#387	—	43	1	no	—
189	#385	—	43	1	no	—
190	#385	—	43	1	no	—
191	#382	—	43	1	no	—
192	C₄×C₄⋊C₄	—	43	1	no	—
193	C₄×C₄⋊C₄	—	43	1	no	—
194	C₄×C₂²⋊C₄	—	43	1	no	—
195	C₄₃	#2334	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	—	43	1	no	—
197	#126	Dic₄₃	1	8	yes	complemented
198	C₂×C₄⋊C₄	—	43	1	no	—
199	C₂×C₄⋊C₄	—	43	1	no	—
200	C₂×C₄⋊C₄	—	43	1	no	—
201	C₂×C₄⋊C₄	—	43	1	no	—
202	C₂×C₄⋊C₄	—	43	1	no	—
203	C₂×C₄⋊C₄	—	43	1	no	—
204	C₂×C₄⋊C₄	—	43	1	no	—
205	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	—	43	1	no	—
209	C₂×C₂²⋊C₄	—	43	1	no	—
210	C₂×C₂²⋊C₄	—	43	1	no	—
211	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
214	C₂×C₄²	—	43	1	no	—
215	C₂×C₄²	—	43	1	no	—
216	C₂×C₄²	—	43	1	no	—
217	C₂×C₄²	—	43	1	no	—
218	C₂×C₄²	—	43	1	no	—
219	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
220	#95	—	43	1	no	—
221	#95	—	43	1	no	—
222	#95	—	43	1	no	—
223	#95	—	43	1	no	—
224	#95	—	43	1	no	—
225	#95	—	43	1	no	—
226	#95	—	43	1	no	—
227	#95	—	43	1	no	—
228	#95	—	43	1	no	—
229	#95	—	43	1	no	—
230	#95	—	43	1	no	—
231	#95	—	43	1	no	—
232	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
233	C₂²×C₄	—	43	1	no	—
234	C₂²×C₄	—	43	1	no	—
235	C₂²×C₄	—	43	1	no	—
236	C₂²×C₄	—	43	1	no	—
237	C₂²×C₄	—	43	1	no	—
238	C₂²×C₄	—	43	1	no	—
239	C₂²×C₄	—	43	1	no	—
240	C₂²×C₄	—	43	1	no	—
241	C₂²×C₄	—	86	2	no	—
242	C₂²×C₄	—	172	1	no	—
243	C₂²×C₄	—	172	1	no	—
244	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
245	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
246	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
247	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
250	C₄⋊C₄	—	86	2	no	—
251	C₄⋊C₄	—	86	2	no	—
252	C₄⋊C₄	—	86	2	no	—
253	C₄⋊C₄	—	172	1	no	—
254	C₄⋊C₄	—	172	1	no	—
255	C₄⋊C₄	—	172	1	no	—
256	C₄⋊C₄	—	172	1	no	—
257	C₄⋊C₄	C₂×Dic₄₃	1	4	yes	complemented
258	C₄⋊C₄	C₂×Dic₄₃	1	4	yes	complemented
259	C₄⋊C₄	C₂×Dic₄₃	1	4	yes	complemented
260	C₂²⋊C₄	—	86	2	no	—
261	C₂²⋊C₄	—	172	1	no	—
262	C₂²⋊C₄	—	172	1	no	—
263	C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
264	C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
265	C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
266	C₄²	—	86	2	no	—
267	C₄²	—	172	1	no	—
268	C₄²	—	172	1	no	—
269	C₄²	—	172	1	no	—
270	C₄²	—	172	1	no	—
271	C₄²	C₂×Dic₄₃	1	4	yes	complemented
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	C₂²×Dic₄₃	1	2	yes	—
279	C₂³	C₂³×D₄₃	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	43	2	no	—
293	C₂×C₄	—	43	2	no	—
294	C₂×C₄	—	86	1	no	—
295	C₂×C₄	—	86	1	no	—
296	C₂×C₄	—	86	1	no	—
297	C₂×C₄	—	86	1	no	—
298	C₂×C₄	—	86	1	no	—
299	C₂×C₄	—	86	1	no	—
300	C₂×C₄	—	86	1	no	—
301	C₂×C₄	—	86	1	no	—
302	C₂×C₄	—	86	1	no	—
303	C₂×C₄	—	86	1	no	—
304	C₂×C₄	—	86	1	no	—
305	C₂×C₄	—	86	1	no	—
306	C₂×C₄	—	86	1	no	—
307	C₂×C₄	—	86	1	no	—
308	C₂×C₄	—	86	1	no	—
309	C₂×C₄	—	86	1	no	—
310	C₂×C₄	—	86	1	no	—
311	C₂×C₄	—	86	1	no	—
312	C₂×C₄	—	86	1	no	—
313	C₂×C₄	—	86	1	no	—
314	C₂×C₄	—	86	1	no	—
315	C₂×C₄	—	86	1	no	—
316	C₂×C₄	—	172	4	no	—
317	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
318	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
319	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
320	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
321	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
322	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	C₂³×Dic₄₃	1	1	yes	characteristic
331	C₂²	#41×D₄₃	1	1	yes	characteristic
332	C₂²	#41×D₄₃	1	1	yes	characteristic
333	C₂²	#13063371	1	1	yes	characteristic
334	C₂²	#13063371	1	1	yes	characteristic
335	C₂²	#13063372	1	1	yes	characteristic
336	C₂²	#13063372	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	86	2	no	—
344	C₄	—	86	2	no	—
345	C₄	—	172	1	no	—
346	C₄	—	172	1	no	—
347	C₄	—	172	1	no	—
348	C₄	—	172	1	no	—
349	C₄	—	172	1	no	—
350	C₄	—	172	1	no	—
351	C₂	—	4	2	no	—
352	C₂	#41×Dic₄₃	1	1	yes	characteristic
353	C₂	#14583382	1	1	yes	characteristic
354	C₂	#14583382	1	1	yes	characteristic
355	C₂	#14583488	1	1	yes	characteristic
356	C₂	#14583490	1	1	yes	characteristic
357	C₂	#14583506	1	1	yes	characteristic
358	C₂	#14583515	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₂×#126×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×#126	D₄₃
hypocenter	C₄₃	#2334
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×#126

Complements

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

N	H ≅ G/N	type
1. #14582798	C₂	semidirect
2. #14582798	C₂	semidirect
6. #14582785	C₂	semidirect
7. #14582785	C₂	semidirect
8. #14582785	C₂	semidirect
11. #14582765	C₂	semidirect
12. C₄⋊C₄×Dic₄₃	C₂	semidirect
13. C₄×C₄⋊Dic₄₃	C₂	semidirect
44. #126×C₄₃	C₄	semidirect
70. C₄⋊C₄×C₄₃	C₂×C₄	semidirect
71. C₄⋊C₄×C₄₃	C₂×C₄	semidirect
72. C₄⋊C₄×C₄₃	C₂×C₄	semidirect
76. C₄²×C₄₃	C₂×C₄	semidirect
195. C₄₃	#2334	semidirect
197. #126	Dic₄₃	semidirect
257. C₄⋊C₄	C₂×Dic₄₃	semidirect
258. C₄⋊C₄	C₂×Dic₄₃	semidirect
259. C₄⋊C₄	C₂×Dic₄₃	semidirect
271. C₄²	C₂×Dic₄₃	semidirect