Subgroups — #19147986

Order 5504.

1040 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583522	C₂	1	2	yes	maximal, complemented
2	#14583521	C₂	1	2	yes	maximal, complemented
3	#14583510	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583505	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583504	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583497	C₂	1	4	yes	maximal, complemented
7	#14583488	C₂	1	4	yes	maximal, complemented
8	#14583455	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583450	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583434	C₂	1	4	yes	maximal, complemented
11	#14583432	C₂	1	4	yes	maximal, complemented
12	#14583412	C₂	1	2	yes	maximal, complemented
13	C₄×#10621902	C₂	1	2	yes	maximal, complemented
14	#117×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621902	C₂²	1	2	yes	—
17	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
18	#13063324	C₂²	1	2	yes	complemented
19	#13063322	C₂²	1	1	yes	characteristic, complemented
20	#13063320	C₂²	1	4	yes	—
21	#13063320	C₂²	1	4	yes	complemented
22	#13063319	C₂²	1	4	yes	complemented
23	#13063317	C₂²	1	1	yes	characteristic, complemented
24	#13063315	C₂²	1	4	yes	complemented
25	#13063314	C₂²	1	2	yes	complemented
26	#13063313	C₂²	1	4	yes	—
27	#13063311	C₂²	1	4	yes	—
28	#13063310	C₂²	1	4	yes	—
29	#13063310	C₂²	1	4	yes	—
30	#13063310	C₂²	1	4	yes	complemented
31	#13063310	C₂²	1	4	yes	complemented
32	#13063307	C₂²	1	4	yes	—
33	#13063305	C₂²	1	4	yes	complemented
34	#13063303	C₂²	1	4	yes	—
35	#13063303	C₂²	1	4	yes	complemented
36	#13063301	C₂²	1	2	yes	—
37	#13063300	C₂²	1	2	yes	—
38	#13063300	C₂²	1	4	yes	—
39	#13063300	C₂²	1	4	yes	—
40	#13063299	C₂²	1	2	yes	—
41	#13063298	C₂²	1	4	yes	—
42	#13063295	C₂²	1	4	yes	—
43	Q₈×Dic₄₃	C₂²	1	2	yes	complemented
44	#13063290	C₂²	1	4	yes	complemented
45	C₄×D₁₇₂	C₂²	1	4	yes	—
46	C₄×D₁₇₂	C₂²	1	2	yes	complemented
47	#13063287	C₂²	1	4	yes	—
48	#13063287	C₂²	1	4	yes	—
49	#13063287	C₂²	1	2	yes	complemented
50	#13063287	C₂²	1	4	yes	complemented
51	(C₂²⋊D₄₃)⋊C₄	C₂²	1	4	yes	—
52	#13063284	C₂²	1	1	yes	characteristic
53	#13063283	C₂²	1	2	yes	—
54	#13063283	C₂²	1	2	yes	—
55	#13063283	C₂²	1	4	yes	complemented
56	#13063282	C₂²	1	1	yes	characteristic
57	#13063282	C₂²	1	1	yes	characteristic
58	#13063281	C₂²	1	2	yes	—
59	C₄⋊C₄×D₄₃	C₂²	1	2	yes	—
60	C₂²⋊C₄×D₄₃	C₂²	1	2	yes	—
61	C₄²×D₄₃	C₂²	1	1	yes	characteristic
62	C₄²×D₄₃	C₂²	1	1	yes	characteristic
63	C₄²×D₄₃	C₂²	1	4	yes	complemented
64	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
65	#126×C₄₃	C₂²	1	4	yes	—
66	#125×C₄₃	C₂²	1	4	yes	complemented
67	#125×C₄₃	C₂²	1	1	yes	characteristic, complemented
68	C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	—
69	C₄×Q₈×C₄₃	C₂²	1	2	yes	complemented
70	#117×C₄₃	C₂²	1	1	yes	characteristic
71	C₂³×D₄₃	C₂³	1	1	yes	characteristic
72	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
73	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
75	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
76	#10621902	—	2	8	no	—
77	C₂×D₁₇₂	C₂³	1	4	yes	—
78	C₂×D₁₇₂	C₂³	1	2	yes	—
79	C₂×D₁₇₂	C₂³	1	4	yes	—
80	C₂×D₁₇₂	C₂³	1	4	yes	—
81	C₂×C₄×D₄₃	—	2	4	no	—
82	C₂×C₄×D₄₃	C₂³	1	4	yes	—
83	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
84	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
85	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
86	C₂×C₄×D₄₃	C₂³	1	2	yes	—
87	C₂×C₄×D₄₃	C₂³	1	2	yes	—
88	C₂×C₄×D₄₃	C₂³	1	4	yes	—
89	C₂×C₄×D₄₃	C₂³	1	4	yes	—
90	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
91	C₂×Dic₈₆	C₂³	1	4	yes	—
92	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
93	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
94	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
95	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
96	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
97	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
98	C₂²⋊C₄×C₄₃	C₂³	1	2	yes	—
99	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
100	C₄²×C₄₃	C₂³	1	1	yes	characteristic
101	C₄²×C₄₃	C₂³	1	1	yes	characteristic
102	C₄²×C₄₃	C₂³	1	4	yes	—
103	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
104	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
105	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
106	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
107	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
108	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
109	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
110	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
111	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
112	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
113	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
114	C₄⋊Dic₄₃	C₂³	1	2	yes	—
115	C₄⋊Dic₄₃	C₂³	1	2	yes	—
116	C₄⋊Dic₄₃	C₂³	1	4	yes	—
117	Dic₄₃⋊C₄	C₂³	1	2	yes	—
118	Dic₄₃⋊C₄	C₂³	1	2	yes	—
119	Dic₄₃⋊C₄	C₂³	1	4	yes	—
120	Dic₄₃⋊C₄	C₂³	1	4	yes	—
121	Dic₄₃⋊C₄	C₂³	1	4	yes	—
122	C₄×Dic₄₃	C₂³	1	2	yes	—
123	C₄×Dic₄₃	C₂³	1	2	yes	—
124	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
125	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
126	C₄×Dic₄₃	C₂³	1	4	yes	—
127	C₄×Dic₄₃	C₂³	1	4	yes	—
128	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
129	C₂²×D₄₃	—	2	2	no	—
130	C₂²×D₄₃	—	4	2	no	—
131	C₂²×D₄₃	C₂⁴	1	1	yes	characteristic
132	C₂²×D₄₃	C₂⁴	1	4	yes	—
133	C₂²×D₄₃	C₂⁴	1	1	yes	characteristic
134	Q₈×C₄₃	—	2	4	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
137	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
138	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
139	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
140	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
142	C₂²⋊D₄₃	—	4	4	no	—
143	C₂×Dic₄₃	—	2	2	no	—
144	C₂×Dic₄₃	C₂⁴	1	2	yes	—
145	C₂×Dic₄₃	C₂⁴	1	4	yes	—
146	C₂×Dic₄₃	C₂⁴	1	2	yes	—
147	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
148	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
149	D₁₇₂	—	2	8	no	—
150	D₁₇₂	—	2	4	no	—
151	D₁₇₂	—	4	4	no	—
152	D₁₇₂	—	4	4	no	—
153	C₄×D₄₃	—	2	2	no	—
154	C₄×D₄₃	—	2	2	no	—
155	C₄×D₄₃	—	2	2	no	—
156	C₄×D₄₃	—	2	8	no	—
157	C₄×D₄₃	—	4	2	no	—
158	C₄×D₄₃	—	4	4	no	—
159	C₄×D₄₃	—	4	4	no	—
160	C₄×D₄₃	—	4	2	no	—
161	C₄×D₄₃	#41	1	4	yes	—
162	Dic₈₆	—	4	4	no	—
163	C₂²×C₄₃	—	2	1	no	—
164	C₂²×C₄₃	—	2	2	no	—
165	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
166	C₂×D₄₃	—	2	2	no	—
167	C₂×D₄₃	—	2	1	no	—
168	C₂×D₄₃	—	2	4	no	—
169	C₂×D₄₃	—	2	2	no	—
170	C₂×D₄₃	—	2	4	no	—
171	C₂×D₄₃	—	2	4	no	—
172	C₂×D₄₃	—	4	4	no	—
173	C₂×D₄₃	C₂×#41	1	2	yes	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	2	4	no	—
176	C₄×C₄₃	—	2	4	no	—
177	C₄×C₄₃	—	2	2	no	—
178	C₄×C₄₃	—	2	2	no	—
179	C₄×C₄₃	C₂×#41	1	2	yes	—
180	Dic₄₃	—	2	1	no	—
181	Dic₄₃	—	2	2	no	—
182	Dic₄₃	—	2	4	no	—
183	Dic₄₃	—	2	2	no	—
184	Dic₄₃	C₂×#41	1	2	yes	—
185	#3455	—	43	1	no	maximal, Sylow
186	C₂×C₄₃	—	4	1	no	—
187	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
188	C₂×C₄₃	#585	1	2	yes	—
189	D₄₃	—	2	2	no	—
190	D₄₃	—	4	4	no	—
191	D₄₃	—	4	1	no	—
192	#542	—	43	2	no	—
193	#542	—	43	2	no	—
194	#541	—	43	1	no	—
195	#541	D₄₃	1	1	yes	characteristic, complemented
196	#540	—	43	1	no	—
197	#540	—	43	1	no	—
198	#539	—	43	4	no	—
199	#538	—	43	4	no	—
200	#533	—	43	1	no	—
201	#532	—	43	1	no	—
202	#529	—	43	4	no	—
203	#529	—	43	4	no	—
204	#525	—	43	2	no	—
205	C₄×#41	—	43	2	no	—
206	C₂×#117	—	43	1	no	—
207	C₄₃	#3455	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	—	43	2	no	—
209	C₂³×C₄	—	43	1	no	—
210	C₄⋊Q₈	—	43	2	no	—
211	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
212	#127	—	43	1	no	—
213	#126	—	43	4	no	—
214	#126	—	43	4	no	—
215	#126	—	43	4	no	—
216	#126	C₂×D₄₃	1	4	yes	—
217	#125	—	43	4	no	—
218	#125	—	43	1	no	—
219	#125	C₂×D₄₃	1	4	yes	complemented
220	#125	C₂×D₄₃	1	1	yes	characteristic, complemented
221	#124	—	43	2	no	—
222	#124	—	43	4	no	—
223	#124	—	43	4	no	—
224	#123	—	43	4	no	—
225	#123	—	43	4	no	—
226	#123	—	43	4	no	—
227	#123	—	43	4	no	—
228	#123	—	43	4	no	—
229	C₂²⋊Q₈	—	43	2	no	—
230	C₂²⋊Q₈	—	43	4	no	—
231	C₂²⋊Q₈	—	43	4	no	—
232	C₂²⋊Q₈	—	43	4	no	—
233	C₂²⋊Q₈	C₂×D₄₃	1	2	yes	—
234	#121	—	43	2	no	—
235	#121	—	43	2	no	—
236	#121	—	43	4	no	—
237	#121	—	43	4	no	—
238	#121	—	43	4	no	—
239	#120	—	43	4	no	—
240	C₄×Q₈	—	43	2	no	—
241	C₄×Q₈	—	43	4	no	—
242	C₄×Q₈	C₂×D₄₃	1	2	yes	complemented
243	C₄×D₄	—	43	4	no	—
244	C₄×D₄	—	43	2	no	—
245	C₄×D₄	—	43	4	no	—
246	C₄×D₄	—	43	4	no	—
247	C₄×D₄	—	43	4	no	—
248	C₄×D₄	—	43	4	no	—
249	C₄×D₄	—	43	2	no	—
250	#117	—	43	1	no	—
251	#117	—	43	2	no	—
252	#117	—	43	2	no	—
253	#117	—	43	4	no	—
254	#117	—	43	2	no	—
255	#117	—	43	1	no	—
256	#117	—	43	1	no	—
257	#117	C₂×D₄₃	1	1	yes	characteristic
258	C₂×C₄⋊C₄	—	43	2	no	—
259	C₂×C₂²⋊C₄	—	43	2	no	—
260	C₂×C₄²	—	43	1	no	—
261	C₂×C₄²	—	43	1	no	—
262	C₂×C₄²	—	43	4	no	—
263	C₂⁴	—	43	1	no	—
264	#41	—	86	8	no	—
265	C₂×Q₈	—	43	4	no	—
266	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
267	C₂×D₄	—	43	4	no	—
268	C₂×D₄	—	43	4	no	—
269	C₂×D₄	—	43	2	no	—
270	C₂×D₄	—	43	4	no	—
271	C₂×D₄	—	43	4	no	—
272	C₂²×C₄	—	43	1	no	—
273	C₂²×C₄	—	43	1	no	—
274	C₂²×C₄	—	43	4	no	—
275	C₂²×C₄	—	43	1	no	—
276	C₂²×C₄	—	43	1	no	—
277	C₂²×C₄	—	43	1	no	—
278	C₂²×C₄	—	43	2	no	—
279	C₂²×C₄	—	43	2	no	—
280	C₂²×C₄	—	43	4	no	—
281	C₂²×C₄	—	43	4	no	—
282	C₂²×C₄	—	86	4	no	—
283	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
284	C₄⋊C₄	—	43	2	no	—
285	C₄⋊C₄	—	43	2	no	—
286	C₄⋊C₄	—	43	2	no	—
287	C₄⋊C₄	—	43	2	no	—
288	C₄⋊C₄	—	43	4	no	—
289	C₄⋊C₄	—	43	4	no	—
290	C₄⋊C₄	—	43	4	no	—
291	C₄⋊C₄	—	43	4	no	—
292	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
293	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
294	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
295	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
296	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
297	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
298	C₂²⋊C₄	—	43	2	no	—
299	C₂²⋊C₄	—	43	2	no	—
300	C₂²⋊C₄	—	43	2	no	—
301	C₂²⋊C₄	—	43	4	no	—
302	C₂²⋊C₄	—	43	4	no	—
303	C₂²⋊C₄	—	43	4	no	—
304	C₂²⋊C₄	—	43	4	no	—
305	C₂²⋊C₄	—	43	4	no	—
306	C₂²⋊C₄	—	43	4	no	—
307	C₂²⋊C₄	—	43	4	no	—
308	C₂²⋊C₄	—	43	4	no	—
309	C₂²⋊C₄	C₂²×D₄₃	1	2	yes	—
310	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
311	C₄²	—	43	2	no	—
312	C₄²	—	43	2	no	—
313	C₄²	—	43	4	no	—
314	C₄²	—	43	1	no	—
315	C₄²	—	43	1	no	—
316	C₄²	—	43	4	no	—
317	C₄²	C₂²×D₄₃	1	1	yes	characteristic
318	C₄²	C₂²×D₄₃	1	1	yes	characteristic
319	C₄²	C₂²×D₄₃	1	4	yes	—
320	C₂³	—	43	1	no	—
321	C₂³	—	43	1	no	—
322	C₂³	—	43	4	no	—
323	C₂³	—	86	2	no	—
324	C₂³	—	172	2	no	—
325	C₂³	C₂³×D₄₃	1	1	yes	characteristic
326	Q₈	—	2	4	no	—
327	Q₈	—	172	4	no	—
328	D₄	—	86	8	no	—
329	D₄	—	86	4	no	—
330	D₄	—	172	4	no	—
331	D₄	—	172	4	no	—
332	D₄	—	172	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	43	1	no	—
335	C₂×C₄	—	43	4	no	—
336	C₂×C₄	—	43	2	no	—
337	C₂×C₄	—	43	2	no	—
338	C₂×C₄	—	43	4	no	—
339	C₂×C₄	—	43	1	no	—
340	C₂×C₄	—	86	2	no	—
341	C₂×C₄	—	86	8	no	—
342	C₂×C₄	—	86	2	no	—
343	C₂×C₄	—	86	2	no	—
344	C₂×C₄	—	86	2	no	—
345	C₂×C₄	—	172	2	no	—
346	C₂×C₄	—	172	4	no	—
347	C₂×C₄	—	172	2	no	—
348	C₂×C₄	—	172	4	no	—
349	C₂×C₄	C₂³×D₄₃	1	2	yes	—
350	C₂×C₄	C₂³×D₄₃	1	4	yes	—
351	C₂×C₄	C₂³×D₄₃	1	4	yes	—
352	C₂×C₄	C₂³×D₄₃	1	2	yes	—
353	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
354	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
355	C₂²	—	2	1	no	—
356	C₂²	—	2	2	no	—
357	C₂²	—	43	2	no	—
358	C₂²	—	86	2	no	—
359	C₂²	—	86	1	no	—
360	C₂²	—	86	4	no	—
361	C₂²	—	86	4	no	—
362	C₂²	—	86	4	no	—
363	C₂²	—	86	2	no	—
364	C₂²	—	172	4	no	—
365	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	4	no	—
368	C₄	—	2	2	no	—
369	C₄	—	2	2	no	—
370	C₄	—	2	4	no	—
371	C₄	—	43	2	no	—
372	C₄	—	86	1	no	—
373	C₄	—	86	2	no	—
374	C₄	—	86	4	no	—
375	C₄	—	86	2	no	—
376	C₄	#41×D₄₃	1	2	yes	—
377	C₂	—	4	1	no	—
378	C₂	—	86	2	no	—
379	C₂	—	172	1	no	—
380	C₂	—	172	4	no	—
381	C₂	C₂×#41×D₄₃	1	1	yes	characteristic
382	C₂	#14583736	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#541	C₂
hypercenter	#541	D₄₃
hypocenter	C₄₃	#3455
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #541

Complements

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

N	H ≅ G/N	type
1. #14583522	C₂	semidirect
2. #14583521	C₂	semidirect
3. #14583510	C₂	semidirect
4. #14583505	C₂	semidirect
5. #14583504	C₂	semidirect
6. #14583497	C₂	semidirect
7. #14583488	C₂	semidirect
8. #14583455	C₂	semidirect
9. #14583450	C₂	semidirect
10. #14583434	C₂	semidirect
11. #14583432	C₂	semidirect
12. #14583412	C₂	semidirect
13. C₄×#10621902	C₂	semidirect
14. #117×D₄₃	C₂	semidirect
15. C₄₃×#541	C₂	semidirect
18. #13063324	C₂²	semidirect
19. #13063322	C₂²	semidirect
21. #13063320	C₂²	semidirect
22. #13063319	C₂²	semidirect
23. #13063317	C₂²	semidirect
24. #13063315	C₂²	semidirect
25. #13063314	C₂²	semidirect
30. #13063310	C₂²	semidirect
31. #13063310	C₂²	semidirect
33. #13063305	C₂²	semidirect
35. #13063303	C₂²	semidirect
43. Q₈×Dic₄₃	C₂²	semidirect
44. #13063290	C₂²	semidirect
46. C₄×D₁₇₂	C₂²	semidirect
49. #13063287	C₂²	semidirect
50. #13063287	C₂²	semidirect
55. #13063283	C₂²	semidirect
63. C₄²×D₄₃	C₂²	semidirect
64. C₄⋊Q₈×C₄₃	C₂²	semidirect
66. #125×C₄₃	C₂²	semidirect
67. #125×C₄₃	C₂²	semidirect
69. C₄×Q₈×C₄₃	C₂²	semidirect
195. #541	D₄₃	semidirect
207. C₄₃	#3455	semidirect
211. C₄⋊Q₈	C₂×D₄₃	semidirect
219. #125	C₂×D₄₃	semidirect
220. #125	C₂×D₄₃	semidirect
242. C₄×Q₈	C₂×D₄₃	semidirect