Subgroups — #19146432

Order 5504.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583589	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583545	C₂	1	1	yes	maximal, characteristic
3	#14583254	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583230	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583224	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583200	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583140	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583128	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583122	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583063	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583052	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#10621873	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582896	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#13063247	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621898	C₂²	1	1	yes	characteristic
17	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic, complemented
19	C₂×#10621873	C₂²	1	1	yes	characteristic
20	C₂×#10621873	C₂²	1	1	yes	characteristic
21	#13063318	—	2	1	no	—
22	#13063317	C₂²	1	1	yes	characteristic, complemented
23	#13063315	—	2	1	no	—
24	#13063313	C₂²	1	1	yes	characteristic
25	#13063310	—	2	1	no	—
26	#13063306	—	2	1	no	—
27	#13063304	C₂²	1	1	yes	characteristic
28	#13063304	C₂²	1	1	yes	characteristic
29	#13063304	C₂²	1	1	yes	characteristic, complemented
30	#13063299	C₂²	1	1	yes	characteristic
31	#13063298	—	2	1	no	—
32	#13063293	—	2	1	no	—
33	C₄×Dic₈₆	C₂²	1	1	yes	characteristic, complemented
34	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
35	#13063284	C₂²	1	1	yes	characteristic, complemented
36	#13063281	—	2	1	no	—
37	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
38	C₂²⋊C₄×D₄₃	—	2	1	no	—
39	#13063259	C₂²	1	1	yes	characteristic, complemented
40	#13063257	C₂²	1	1	yes	characteristic, complemented
41	#13063257	C₂²	1	1	yes	characteristic, complemented
42	#13063250	C₂²	1	1	yes	characteristic, complemented
43	#13063250	C₂²	1	1	yes	characteristic, complemented
44	#13063250	C₂²	1	1	yes	characteristic, complemented
45	#13063250	C₂²	1	1	yes	characteristic, complemented
46	#13063247	C₂²	1	1	yes	characteristic
47	#13063247	C₂²	1	1	yes	characteristic
48	#13063247	C₂²	1	1	yes	characteristic
49	#13063247	C₂²	1	1	yes	characteristic
50	#13063247	C₂²	1	1	yes	characteristic
51	#13063247	C₂²	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
54	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
56	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
58	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₄₃	—	2	1	no	—
60	C₂×D₄×C₄₃	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	C₂×C₂²⋊D₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	—	2	1	no	—
66	#10621898	—	2	2	no	—
67	#10621898	—	4	1	no	—
68	C₂×D₁₇₂	—	2	1	no	—
69	C₂×D₁₇₂	—	2	2	no	—
70	C₂×D₁₇₂	—	4	1	no	—
71	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
72	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₄₃	—	2	1	no	—
74	C₂×C₄×D₄₃	—	2	1	no	—
75	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
76	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₄₃	—	4	1	no	—
78	C₂×C₈×C₄₃	—	4	1	no	—
79	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
84	C₂²⋊C₄×C₄₃	D₄	1	1	yes	characteristic
85	C₂²⋊C₄×C₄₃	D₄	1	1	yes	characteristic
86	C₄²×C₄₃	C₂³	1	1	yes	characteristic
87	C₂²⋊Dic₄₃	—	2	1	no	—
88	C₂²⋊Dic₄₃	—	2	1	no	—
89	C₂²⋊Dic₄₃	—	2	1	no	—
90	C₂²⋊Dic₄₃	—	2	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	C₄⋊Dic₄₃	—	2	1	no	—
96	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
106	C₄×Dic₄₃	—	2	1	no	—
107	C₄×Dic₄₃	—	2	1	no	—
108	Dic₁₇₂	—	4	1	no	—
109	#10621873	—	2	2	no	—
110	#10621873	—	4	1	no	—
111	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₄₃	—	2	1	no	—
114	C₂²×D₄₃	—	2	1	no	—
115	C₂²×D₄₃	—	4	2	no	—
116	C₂²×D₄₃	—	4	2	no	—
117	C₂²×D₄₃	—	8	1	no	—
118	D₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₄₃	—	4	2	no	—
129	C₂²⋊D₄₃	—	8	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	—	4	2	no	—
137	D₁₇₂	—	2	1	no	—
138	D₁₇₂	—	2	1	no	—
139	D₁₇₂	—	2	1	no	—
140	D₁₇₂	—	4	2	no	—
141	D₁₇₂	#41	1	2	yes	—
142	C₄×D₄₃	—	4	2	no	—
143	C₄×D₄₃	—	8	1	no	—
144	Dic₈₆	—	2	1	no	—
145	Dic₈₆	—	2	1	no	—
146	Dic₈₆	—	2	1	no	—
147	Dic₈₆	—	2	1	no	—
148	C₈×C₄₃	—	2	1	no	—
149	C₈×C₄₃	—	2	1	no	—
150	C₈×C₄₃	—	2	1	no	—
151	C₈×C₄₃	—	2	1	no	—
152	C₂²×C₄₃	—	2	1	no	—
153	C₂²×C₄₃	—	2	1	no	—
154	C₂²×C₄₃	—	2	1	no	—
155	C₂²×C₄₃	—	2	1	no	—
156	C₂²×C₄₃	—	2	1	no	—
157	C₂²×C₄₃	#135	1	2	yes	—
158	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₄₃	—	2	2	no	—
160	C₂×D₄₃	—	4	1	no	—
161	C₂×D₄₃	—	4	1	no	—
162	C₂×D₄₃	—	4	1	no	—
163	C₂×D₄₃	—	4	1	no	—
164	C₂×D₄₃	—	4	1	no	—
165	C₂×D₄₃	—	8	4	no	—
166	C₄×C₄₃	—	2	1	no	—
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
171	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₄₃	—	4	1	no	—
173	Dic₄₃	—	4	1	no	—
174	Dic₄₃	—	4	1	no	—
175	Dic₄₃	—	4	1	no	—
176	Dic₄₃	—	4	1	no	—
177	Dic₄₃	—	4	1	no	—
178	#3296	—	43	1	no	maximal, Sylow
179	C₂×C₄₃	—	2	2	no	—
180	C₂×C₄₃	—	4	1	no	—
181	C₂×C₄₃	#546	1	1	yes	characteristic
182	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
183	C₂×C₄₃	#577	1	1	yes	characteristic
184	D₄₃	—	4	2	no	—
185	D₄₃	—	8	1	no	—
186	#552	—	43	1	no	—
187	#546	—	43	1	no	—
188	#488	—	43	1	no	—
189	#483	—	43	1	no	—
190	#482	—	43	1	no	—
191	#478	—	43	1	no	—
192	#467	—	43	1	no	—
193	#465	—	43	1	no	—
194	#464	—	43	1	no	—
195	#452	—	43	1	no	—
196	#450	—	43	1	no	—
197	C₄×SD₁₆	—	43	1	no	—
198	C₈×D₄	D₄₃	1	1	yes	characteristic, complemented
199	#416	—	43	1	no	—
200	C₂×#102	—	43	1	no	—
201	C₄₃	#3296	1	1	yes	characteristic, complemented, Sylow
202	C₂×#41	—	43	1	no	—
203	C₂²×D₄	—	43	1	no	—
204	C₂×Q₁₆	—	43	1	no	—
205	C₂×SD₁₆	—	43	1	no	—
206	C₂×SD₁₆	—	43	1	no	—
207	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
208	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic, complemented
209	#126	—	86	1	no	—
210	#125	—	43	1	no	—
211	#125	—	86	1	no	—
212	#124	—	43	1	no	—
213	#123	—	86	1	no	—
214	#123	—	86	1	no	—
215	C₂²⋊Q₈	—	43	1	no	—
216	C₂²⋊Q₈	—	43	1	no	—
217	C₂²⋊Q₈	—	43	1	no	—
218	#121	—	43	1	no	—
219	#121	—	86	1	no	—
220	#120	—	86	1	no	—
221	C₄×Q₈	—	43	1	no	—
222	C₄×D₄	—	43	1	no	—
223	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
224	#117	—	43	1	no	—
225	#117	—	86	1	no	—
226	C₂×C₄⋊C₄	—	43	1	no	—
227	C₂×C₂²⋊C₄	—	86	1	no	—
228	#107	—	43	1	no	—
229	#106	—	43	1	no	—
230	#106	—	43	1	no	—
231	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic, complemented
232	#103	—	43	1	no	—
233	#103	—	43	1	no	—
234	#103	—	43	1	no	—
235	#103	—	43	1	no	—
236	#102	—	43	1	no	—
237	#102	—	43	1	no	—
238	#102	—	43	1	no	—
239	#102	—	43	1	no	—
240	#102	—	43	1	no	—
241	#102	—	43	1	no	—
242	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
243	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic, complemented
244	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic, complemented
245	C₂⁴	—	86	1	no	—
246	#41	—	86	2	no	—
247	#41	—	172	1	no	—
248	C₂×Q₈	—	43	1	no	—
249	C₂×Q₈	—	43	1	no	—
250	C₂×D₄	—	43	1	no	—
251	C₂×D₄	—	43	1	no	—
252	C₂×D₄	—	86	2	no	—
253	C₂×D₄	—	86	1	no	—
254	C₂×D₄	—	86	1	no	—
255	C₂×D₄	—	86	1	no	—
256	C₂×D₄	—	172	1	no	—
257	C₂×D₄	D₁₇₂	1	1	yes	characteristic
258	C₂²×C₄	—	86	1	no	—
259	C₂²×C₄	—	86	1	no	—
260	C₂²×C₄	—	86	1	no	—
261	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
262	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
263	Q₁₆	—	172	1	no	—
264	SD₁₆	—	86	2	no	—
265	SD₁₆	—	172	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
269	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
272	C₄⋊C₄	—	43	1	no	—
273	C₄⋊C₄	—	43	1	no	—
274	C₄⋊C₄	—	43	1	no	—
275	C₄⋊C₄	—	43	1	no	—
276	C₄⋊C₄	—	86	1	no	—
277	C₄⋊C₄	—	86	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	D₁₇₂	1	1	yes	characteristic
284	C₂²⋊C₄	—	86	1	no	—
285	C₂²⋊C₄	—	86	1	no	—
286	C₂²⋊C₄	—	86	1	no	—
287	C₂²⋊C₄	—	86	1	no	—
288	C₂²⋊C₄	—	86	1	no	—
289	C₂²⋊C₄	—	86	1	no	—
290	C₂²⋊C₄	—	86	1	no	—
291	C₂²⋊C₄	—	86	1	no	—
292	C₂²⋊C₄	D₁₇₂	1	1	yes	characteristic
293	C₂²⋊C₄	D₁₇₂	1	1	yes	characteristic
294	C₄²	—	86	1	no	—
295	C₄²	—	86	1	no	—
296	C₄²	C₂²×D₄₃	1	1	yes	characteristic
297	C₂³	—	86	1	no	—
298	C₂³	—	86	1	no	—
299	C₂³	—	344	1	no	—
300	C₂³	—	172	2	no	—
301	C₂³	—	172	2	no	—
302	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
303	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
304	Q₈	—	86	1	no	—
305	Q₈	—	86	1	no	—
306	Q₈	—	86	1	no	—
307	Q₈	—	86	1	no	—
308	D₄	—	2	2	no	—
309	D₄	—	43	2	no	—
310	D₄	—	86	1	no	—
311	D₄	—	86	1	no	—
312	D₄	—	86	1	no	—
313	D₄	—	344	1	no	—
314	D₄	—	172	2	no	—
315	D₄	—	172	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	86	1	no	—
321	C₂×C₄	—	86	1	no	—
322	C₂×C₄	—	86	1	no	—
323	C₂×C₄	—	86	1	no	—
324	C₂×C₄	—	86	1	no	—
325	C₂×C₄	—	86	1	no	—
326	C₂×C₄	—	344	1	no	—
327	C₂×C₄	—	172	2	no	—
328	C₂×C₄	—	172	2	no	—
329	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
330	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
331	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
332	C₂×C₄	C₂×D₁₇₂	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	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	86	2	no	—
344	C₂²	—	344	4	no	—
345	C₂²	—	172	1	no	—
346	C₂²	—	172	1	no	—
347	C₂²	—	172	1	no	—
348	C₂²	—	172	1	no	—
349	C₂²	—	172	1	no	—
350	C₂²	#13063347	1	2	yes	—
351	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	172	1	no	—
357	C₄	—	172	1	no	—
358	C₄	—	172	1	no	—
359	C₄	—	172	1	no	—
360	C₄	—	172	1	no	—
361	C₄	—	172	1	no	—
362	C₄	#41×D₄₃	1	1	yes	characteristic
363	C₄	#13063370	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	4	1	no	—
366	C₂	—	344	1	no	—
367	C₂	—	172	2	no	—
368	C₂	#14583544	1	1	yes	characteristic
369	C₂	C₂×#13063347	1	1	yes	characteristic
370	C₂	#14583711	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₈×D₄×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₈×D₄	D₄₃
hypocenter	C₄₃	#3296
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #14583589	C₂	semidirect
3. #14583254	C₂	semidirect
4. #14583230	C₂	semidirect
5. #14583224	C₂	semidirect
6. #14583200	C₂	semidirect
7. #14583140	C₂	semidirect
8. #14583128	C₂	semidirect
9. #14583122	C₂	semidirect
10. #14583063	C₂	semidirect
11. #14583052	C₂	semidirect
12. C₄×#10621873	C₂	semidirect
13. #14582896	C₂	semidirect
14. C₂×#13063247	C₂	semidirect
15. C₈×D₄×C₄₃	C₂	semidirect
18. C₂×Dic₁₇₂	C₂²	semidirect
22. #13063317	C₂²	semidirect
29. #13063304	C₂²	semidirect
33. C₄×Dic₈₆	C₂²	semidirect
35. #13063284	C₂²	semidirect
39. #13063259	C₂²	semidirect
40. #13063257	C₂²	semidirect
41. #13063257	C₂²	semidirect
42. #13063250	C₂²	semidirect
43. #13063250	C₂²	semidirect
44. #13063250	C₂²	semidirect
45. #13063250	C₂²	semidirect
53. C₂²×C₈×C₄₃	C₂²	semidirect
55. C₄⋊C₈×C₄₃	C₂²	semidirect
57. C₂²⋊C₈×C₄₃	C₂²	semidirect
58. C₄×C₈×C₄₃	C₂²	semidirect
198. C₈×D₄	D₄₃	semidirect
201. C₄₃	#3296	semidirect
208. C₂²×C₈	C₂×D₄₃	semidirect
231. C₄⋊C₈	C₂×D₄₃	semidirect
243. C₂²⋊C₈	C₂×D₄₃	semidirect
244. C₄×C₈	C₂×D₄₃	semidirect