Subgroups — #19144210

Order 5504.

548 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063351	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#10621877	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583410	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583150	C₂	1	2	yes	maximal, complemented
5	#14583143	C₂	1	2	yes	maximal, complemented
6	#14583133	C₂	1	2	yes	maximal, complemented
7	#14583126	C₂	1	2	yes	maximal, complemented
8	#14582939	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582904	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582898	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#121×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621898	C₂²	1	1	yes	characteristic
13	#13063351	—	2	4	no	—
14	C₂×#10621883	C₂²	1	2	yes	complemented
15	C₂×#10621882	C₂²	1	2	yes	complemented
16	C₂×#10621877	—	2	4	no	—
17	C₂×#10621877	C₂²	1	1	yes	characteristic
18	C₂×#10621877	C₂²	1	1	yes	characteristic
19	C₂×#10621877	C₂²	1	2	yes	complemented
20	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	complemented
21	#13063308	—	2	1	no	—
22	#13063304	C₂²	1	1	yes	characteristic, complemented
23	#13063304	C₂²	1	1	yes	characteristic, complemented
24	#13063299	C₂²	1	1	yes	characteristic, complemented
25	#13063299	C₂²	1	1	yes	characteristic, complemented
26	#13063294	—	2	1	no	—
27	C₄×C₂²⋊D₄₃	—	2	1	no	—
28	C₄×C₂²⋊D₄₃	—	2	1	no	—
29	#13063284	C₂²	1	1	yes	characteristic
30	#13063258	C₂²	1	2	yes	complemented
31	#13063256	C₂²	1	2	yes	complemented
32	#13063248	C₂²	1	2	yes	complemented
33	#13063246	C₂²	1	2	yes	complemented
34	#13063246	C₂²	1	2	yes	complemented
35	#13063245	C₂²	1	2	yes	complemented
36	C₂²×D₄×C₄₃	—	2	1	no	—
37	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
38	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
39	#121×C₄₃	—	2	1	no	—
40	#121×C₄₃	—	2	1	no	—
41	#121×C₄₃	C₂²	1	2	yes	—
42	#121×C₄₃	C₂²	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
45	C₂⁴×C₄₃	—	2	1	no	—
46	C₂⁴×C₄₃	D₄	1	1	yes	characteristic
47	C₂×D₄×C₄₃	—	2	1	no	—
48	C₂×D₄×C₄₃	—	2	2	no	—
49	C₂×D₄×C₄₃	—	2	2	no	—
50	C₂×D₄×C₄₃	—	2	2	no	—
51	C₂×D₄×C₄₃	—	2	2	no	—
52	C₂×D₄×C₄₃	—	4	4	no	—
53	C₂×D₄×C₄₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	—	2	1	no	—
57	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂³	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	C₂×C₂²⋊D₄₃	—	2	1	no	—
66	#10621898	—	2	2	no	—
67	#10621898	—	2	2	no	—
68	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₄₃	—	2	1	no	—
72	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
73	C₂²⋊C₄×C₄₃	—	2	2	no	—
74	C₂²⋊C₄×C₄₃	—	2	2	no	—
75	C₂²⋊Dic₄₃	—	2	1	no	—
76	C₂²⋊Dic₄₃	—	2	1	no	—
77	C₂²⋊Dic₄₃	—	2	1	no	—
78	#10621883	—	2	4	no	—
79	#10621882	—	2	4	no	—
80	(C₂×D₄₃)⋊C₄	—	2	1	no	—
81	(C₂×D₄₃)⋊C₄	—	2	1	no	—
82	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
84	Dic₄₃⋊C₄	—	2	1	no	—
85	Dic₄₃⋊C₄	—	2	1	no	—
86	C₄×Dic₄₃	—	2	1	no	—
87	#10621877	—	2	2	no	—
88	#10621877	—	2	4	no	—
89	#10621877	D₄	1	4	yes	complemented
90	C₂×C₄₃⋊C₈	—	2	4	no	—
91	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
92	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
93	C₂³×C₄₃	—	2	2	no	—
94	C₂³×C₄₃	—	2	2	no	—
95	C₂³×C₄₃	—	2	2	no	—
96	C₂³×C₄₃	—	4	1	no	—
97	C₂³×C₄₃	—	4	2	no	—
98	C₂³×C₄₃	—	4	2	no	—
99	C₂³×C₄₃	—	4	2	no	—
100	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×D₄₃	—	2	1	no	—
104	D₄×C₄₃	—	2	2	no	—
105	D₄×C₄₃	—	2	2	no	—
106	D₄×C₄₃	—	4	4	no	—
107	D₄×C₄₃	—	4	4	no	—
108	D₄×C₄₃	—	4	2	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#41	1	2	yes	—
123	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₄₃	—	4	2	no	—
125	C₂²⋊D₄₃	—	4	2	no	—
126	C₂²⋊D₄₃	—	4	2	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	—	2	1	no	—
130	D₁₇₂	—	2	1	no	—
131	D₁₇₂	—	2	1	no	—
132	C₄×D₄₃	—	4	2	no	—
133	Dic₈₆	—	2	1	no	—
134	Dic₈₆	—	2	1	no	—
135	C₄₃⋊C₈	—	2	2	no	—
136	C₄₃⋊C₈	C₂×D₄	1	4	yes	complemented
137	C₂²×C₄₃	—	2	1	no	—
138	C₂²×C₄₃	—	2	1	no	—
139	C₂²×C₄₃	—	2	1	no	—
140	C₂²×C₄₃	—	2	1	no	—
141	C₂²×C₄₃	—	2	1	no	—
142	C₂²×C₄₃	—	2	1	no	—
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	4	4	no	—
145	C₂²×C₄₃	—	4	2	no	—
146	C₂²×C₄₃	—	4	2	no	—
147	C₂²×C₄₃	—	4	2	no	—
148	C₂²×C₄₃	—	8	4	no	—
149	C₂²×C₄₃	#121	1	2	yes	—
150	C₂²×C₄₃	AGL(1,ℤ/8)	1	2	yes	complemented
151	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₄₃	—	4	1	no	—
153	C₂×D₄₃	—	4	1	no	—
154	C₂×D₄₃	—	4	1	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	—	4	2	no	—
158	C₄×C₄₃	#121	1	2	yes	—
159	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
160	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
161	Dic₄₃	—	4	1	no	—
162	Dic₄₃	—	4	1	no	—
163	Dic₄₃	—	4	1	no	—
164	#3055	—	43	1	no	maximal, Sylow
165	C₂×C₄₃	—	2	2	no	—
166	C₂×C₄₃	—	2	2	no	—
167	C₂×C₄₃	—	4	1	no	—
168	C₂×C₄₃	—	8	2	no	—
169	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
170	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₂×C₄₃	#575	1	1	yes	characteristic
172	D₄₃	—	8	1	no	—
173	C₂×AGL(1,ℤ/8)	—	43	1	no	—
174	C₂²×M₁₆	—	43	1	no	—
175	#525	—	43	1	no	—
176	C₂×#121	D₄₃	1	1	yes	characteristic, complemented
177	#469	—	43	2	no	—
178	#468	—	43	2	no	—
179	#466	—	43	2	no	—
180	#465	—	43	2	no	—
181	#428	—	43	1	no	—
182	#418	—	43	1	no	—
183	#417	—	43	1	no	—
184	C₄₃	#3055	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	43	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
188	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
189	AGL(1,ℤ/8)	—	86	4	no	—
190	C₂×SD₁₆	—	43	2	no	—
191	C₂×D₈	—	43	2	no	—
192	C₂×M₁₆	—	43	1	no	—
193	C₂×M₁₆	—	43	1	no	—
194	C₂×M₁₆	—	43	2	no	—
195	C₂×M₁₆	—	86	4	no	—
196	C₂²×C₈	—	43	2	no	—
197	#123	—	86	1	no	—
198	C₂²⋊Q₈	—	43	1	no	—
199	C₂²⋊Q₈	—	43	1	no	—
200	#121	—	2	1	no	—
201	#121	—	2	1	no	—
202	#121	—	43	1	no	—
203	#121	—	43	1	no	—
204	#121	C₂×D₄₃	1	2	yes	—
205	#121	C₂×D₄₃	1	2	yes	—
206	#120	—	86	1	no	—
207	C₄×D₄	—	86	1	no	—
208	C₄×D₄	—	86	1	no	—
209	#117	—	43	1	no	—
210	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	—	2	1	no	—
212	#107	—	43	2	no	—
213	#106	—	43	2	no	—
214	#103	—	43	2	no	—
215	#102	—	43	2	no	—
216	#102	—	43	2	no	—
217	#102	—	43	2	no	—
218	C₂⁴	—	2	1	no	—
219	C₂⁴	C₂²⋊D₄₃	1	1	yes	characteristic
220	#41	—	86	2	no	—
221	#41	—	86	2	no	—
222	C₂×Q₈	—	43	1	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	43	1	no	—
230	C₂×D₄	—	86	1	no	—
231	C₂×D₄	—	86	1	no	—
232	C₂×D₄	—	86	1	no	—
233	C₂×D₄	C₂²×D₄₃	1	2	yes	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	86	1	no	—
238	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
244	SD₁₆	—	86	4	no	—
245	D₈	—	86	4	no	—
246	M₁₆	—	43	4	no	—
247	M₁₆	—	86	4	no	—
248	M₁₆	—	86	2	no	—
249	C₂×C₈	—	43	2	no	—
250	C₂×C₈	—	43	2	no	—
251	C₂×C₈	—	86	4	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	43	1	no	—
254	C₄⋊C₄	—	43	1	no	—
255	C₄⋊C₄	—	86	1	no	—
256	C₄⋊C₄	—	86	1	no	—
257	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	86	1	no	—
261	C₂²⋊C₄	—	86	1	no	—
262	C₂²⋊C₄	—	86	1	no	—
263	C₂²⋊C₄	—	86	1	no	—
264	C₂²⋊C₄	—	86	1	no	—
265	C₄²	—	86	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	4	1	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	86	1	no	—
274	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
275	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
276	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
277	Q₈	—	86	1	no	—
278	Q₈	—	86	1	no	—
279	D₄	—	2	2	no	—
280	D₄	—	2	2	no	—
281	D₄	—	4	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	86	1	no	—
285	D₄	—	86	1	no	—
286	D₄	—	172	2	no	—
287	D₄	—	172	2	no	—
288	D₄	—	172	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	86	1	no	—
299	C₂×C₄	—	86	1	no	—
300	C₂×C₄	—	86	1	no	—
301	C₂×C₄	—	172	2	no	—
302	C₂×C₄	D₄×D₄₃	1	2	yes	—
303	C₂×C₄	#10621900	1	2	yes	—
304	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
308	C₈	—	43	4	no	—
309	C₈	—	86	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	8	4	no	—
322	C₂²	—	172	1	no	—
323	C₂²	—	172	1	no	—
324	C₂²	—	172	1	no	—
325	C₂²	#13063297	1	2	yes	—
326	C₂²	#13063351	1	2	yes	complemented
327	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	172	1	no	—
332	C₄	—	172	1	no	—
333	C₄	—	172	1	no	—
334	C₄	#13063297	1	2	yes	—
335	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
336	C₄	C₂×#10621900	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	—	2	2	no	—
339	C₂	—	4	1	no	—
340	C₂	—	8	2	no	—
341	C₂	—	344	1	no	—
342	C₂	C₂×#13063297	1	1	yes	characteristic
343	C₂	C₂×#13063351	1	1	yes	characteristic
344	C₂	#14583697	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×#121×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₂×#121	D₄₃
hypocenter	C₄₃	#3055
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. C₂×#13063351	C₂	semidirect
2. C₂²×#10621877	C₂	semidirect
3. #14583410	C₂	semidirect
4. #14583150	C₂	semidirect
5. #14583143	C₂	semidirect
6. #14583133	C₂	semidirect
7. #14583126	C₂	semidirect
8. #14582939	C₂	semidirect
9. #14582904	C₂	semidirect
10. #14582898	C₂	semidirect
11. C₂×#121×C₄₃	C₂	semidirect
14. C₂×#10621883	C₂²	semidirect
15. C₂×#10621882	C₂²	semidirect
19. C₂×#10621877	C₂²	semidirect
20. C₂²×C₄₃⋊C₈	C₂²	semidirect
22. #13063304	C₂²	semidirect
23. #13063304	C₂²	semidirect
24. #13063299	C₂²	semidirect
25. #13063299	C₂²	semidirect
30. #13063258	C₂²	semidirect
31. #13063256	C₂²	semidirect
32. #13063248	C₂²	semidirect
33. #13063246	C₂²	semidirect
34. #13063246	C₂²	semidirect
35. #13063245	C₂²	semidirect
89. #10621877	D₄	semidirect
136. C₄₃⋊C₈	C₂×D₄	semidirect
150. C₂²×C₄₃	AGL(1,ℤ/8)	semidirect
176. C₂×#121	D₄₃	semidirect
184. C₄₃	#3055	semidirect
326. C₂²	#13063351	semidirect