Subgroups — #19147799

Order 5504.

1660 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063370	C₂	1	1	yes	maximal, characteristic
2	#14583379	C₂	1	2	yes	maximal, complemented
3	#14583378	C₂	1	4	yes	maximal, complemented
4	#14583372	C₂	1	4	yes	maximal, complemented
5	#14583371	C₂	1	4	yes	maximal, complemented
6	#14583367	C₂	1	2	yes	maximal, complemented
7	C₄×#10621900	C₂	1	4	yes	maximal, complemented
8	C₂×D₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×(C₂²⋊D₄₃)⋊C₄	C₂	1	4	yes	maximal, complemented
10	C₄×D₄×D₄₃	C₂	1	4	yes	maximal, complemented
11	C₄₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
12	#13063370	C₄	1	16	yes	complemented
13	C₂×#10621898	C₂²	1	2	yes	—
14	C₂×#10621900	C₂²	1	4	yes	—
15	C₂²×C₂²⋊D₄₃	C₂²	1	4	yes	—
16	C₂×D₄×D₄₃	C₂²	1	4	yes	—
17	C₂³×Dic₄₃	C₂²	1	2	yes	complemented
18	C₂²×C₄×D₄₃	C₂²	1	4	yes	complemented
19	C₄×Dic₈₆	C₂²	1	4	yes	complemented
20	#13063290	C₂²	1	2	yes	complemented
21	#13063290	C₂²	1	2	yes	complemented
22	C₄×D₁₇₂	C₂²	1	4	yes	complemented
23	D₄×Dic₄₃	C₂²	1	4	yes	complemented
24	D₄×Dic₄₃	C₂²	1	4	yes	complemented
25	#13063287	C₂²	1	2	yes	complemented
26	#13063287	C₂²	1	2	yes	complemented
27	C₄×C₂²⋊D₄₃	C₂²	1	8	yes	complemented
28	C₄×C₂²⋊D₄₃	C₂²	1	8	yes	complemented
29	(C₂²⋊D₄₃)⋊C₄	C₂²	1	8	yes	complemented
30	(C₂²⋊D₄₃)⋊C₄	C₂²	1	4	yes	complemented
31	(C₂²⋊D₄₃)⋊C₄	C₂²	1	8	yes	complemented
32	(C₂²⋊D₄₃)⋊C₄	C₂²	1	8	yes	complemented
33	(C₂²⋊D₄₃)⋊C₄	C₂²	1	4	yes	complemented
34	#13063283	C₂²	1	4	yes	complemented
35	#13063282	C₂²	1	4	yes	complemented
36	#13063281	C₂²	1	8	yes	complemented
37	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic, complemented
38	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	complemented
39	C₄⋊C₄×D₄₃	C₂²	1	4	yes	complemented
40	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	complemented
41	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	complemented
42	C₂²⋊C₄×D₄₃	C₂²	1	8	yes	complemented
43	C₂×C₄×Dic₄₃	C₂²	1	4	yes	complemented
44	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
45	C₄²×D₄₃	C₂²	1	4	yes	complemented
46	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
47	C₄×D₄×C₄₃	C₂²	1	4	yes	complemented
48	C₄×D₄×C₄₃	C₂²	1	4	yes	complemented
49	#117×C₄₃	C₂²	1	2	yes	complemented
50	C₂×C₂²⋊C₄×C₄₃	C₂²	1	4	yes	complemented
51	C₂⁴×C₄₃	C₂³	1	2	yes	—
52	C₂³×D₄₃	C₂³	1	4	yes	—
53	C₂×D₄×C₄₃	C₂×C₄	1	8	yes	complemented
54	C₂×D₄×C₄₃	C₂³	1	4	yes	—
55	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
56	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
57	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₂²⋊D₄₃	C₂×C₄	1	32	yes	complemented
59	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
60	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
61	C₂²×Dic₄₃	—	4	4	no	—
62	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
63	C₂²×Dic₄₃	C₂³	1	4	yes	—
64	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
65	C₂²×Dic₄₃	C₂³	1	4	yes	—
66	C₂²×Dic₄₃	C₂³	1	2	yes	—
67	C₂²×Dic₄₃	C₂³	1	4	yes	—
68	#10621900	C₂×C₄	1	32	yes	complemented
69	D₄×D₄₃	C₂×C₄	1	32	yes	complemented
70	#10621898	C₂×C₄	1	16	yes	complemented
71	C₂×D₁₇₂	C₂³	1	2	yes	—
72	C₂×C₄×D₄₃	—	4	8	no	—
73	C₂×C₄×D₄₃	C₂³	1	4	yes	—
74	C₂×C₄×D₄₃	C₂³	1	4	yes	—
75	C₂×C₄×D₄₃	C₂³	1	8	yes	—
76	C₂×C₄×D₄₃	C₂³	1	4	yes	—
77	C₂×Dic₈₆	C₂³	1	2	yes	—
78	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
79	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
81	C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
82	C₄²×C₄₃	C₂³	1	4	yes	—
83	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
84	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
85	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
86	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
87	C₄⋊Dic₄₃	C₂³	1	4	yes	—
88	Dic₄₃⋊C₄	C₂³	1	8	yes	—
89	Dic₄₃⋊C₄	C₂³	1	4	yes	—
90	Dic₄₃⋊C₄	C₂³	1	4	yes	—
91	C₄×Dic₄₃	C₂³	1	2	yes	—
92	C₄×Dic₄₃	C₂³	1	4	yes	—
93	C₄×Dic₄₃	C₂³	1	4	yes	—
94	C₄×Dic₄₃	C₂³	1	8	yes	—
95	C₄×Dic₄₃	C₂³	1	2	yes	—
96	C₂³×C₄₃	—	4	2	no	—
97	C₂³×C₄₃	—	4	2	no	—
98	C₂³×C₄₃	C₂²×C₄	1	8	yes	complemented
99	C₂³×C₄₃	C₂⁴	1	4	yes	—
100	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
101	C₂²×D₄₃	—	4	4	no	—
102	C₂²×D₄₃	—	4	4	no	—
103	C₂²×D₄₃	C₂²×C₄	1	16	yes	complemented
104	C₂²×D₄₃	C₂⁴	1	4	yes	—
105	D₄×C₄₃	C₂²×C₄	1	16	yes	complemented
106	C₂×C₄×C₄₃	—	4	4	no	—
107	C₂×C₄×C₄₃	—	4	4	no	—
108	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	complemented
109	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
110	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
111	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
112	C₂²⋊D₄₃	C₂²×C₄	1	16	yes	complemented
113	C₂²⋊D₄₃	C₂²×C₄	1	32	yes	complemented
114	C₂×Dic₄₃	—	2	8	no	—
115	C₂×Dic₄₃	—	2	2	no	—
116	C₂×Dic₄₃	—	4	2	no	—
117	C₂×Dic₄₃	—	4	4	no	—
118	C₂×Dic₄₃	—	4	1	no	—
119	C₂×Dic₄₃	C₂²×C₄	1	16	yes	complemented
120	C₂×Dic₄₃	C₂⁴	1	2	yes	—
121	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₄₃	C₂⁴	1	4	yes	—
123	C₂×Dic₄₃	C₂⁴	1	4	yes	—
124	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
125	D₁₇₂	C₂²×C₄	1	8	yes	complemented
126	C₄×D₄₃	—	2	8	no	—
127	C₄×D₄₃	—	4	8	no	—
128	C₄×D₄₃	—	4	4	no	—
129	C₄×D₄₃	C₂²×C₄	1	16	yes	complemented
130	Dic₈₆	C₂²×C₄	1	8	yes	complemented
131	C₂²×C₄₃	—	2	4	no	—
132	C₂²×C₄₃	—	2	1	no	—
133	C₂²×C₄₃	—	2	4	no	—
134	C₂²×C₄₃	—	2	1	no	—
135	C₂²×C₄₃	—	4	8	no	—
136	C₂²×C₄₃	C₂³×C₄	1	8	yes	—
137	C₂²×C₄₃	C₂³×C₄	1	2	yes	—
138	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
139	C₂×D₄₃	—	2	4	no	—
140	C₂×D₄₃	—	2	4	no	—
141	C₂×D₄₃	—	4	16	no	—
142	C₂×D₄₃	C₂³×C₄	1	8	yes	—
143	C₄×C₄₃	—	2	4	no	—
144	C₄×C₄₃	—	2	4	no	—
145	C₄×C₄₃	C₂³×C₄	1	4	yes	—
146	Dic₄₃	—	2	4	no	—
147	Dic₄₃	—	2	1	no	—
148	Dic₄₃	—	2	2	no	—
149	Dic₄₃	C₂³×C₄	1	8	yes	—
150	Dic₄₃	H(5,𝔽₂)	1	2	yes	complemented
151	C₄×H(5,𝔽₂)	—	43	1	no	maximal, Sylow
152	C₂×C₄₃	—	2	8	no	—
153	C₂×C₄₃	—	2	2	no	—
154	C₂×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
155	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×C₄₃	#585	1	1	yes	characteristic
157	D₄₃	—	2	8	no	—
158	C₂×H(5,𝔽₂)	—	43	1	no	—
159	#520	—	43	4	no	—
160	#520	—	43	2	no	—
161	#518	—	43	4	no	—
162	#518	—	43	4	no	—
163	#518	D₄₃	1	1	yes	characteristic, complemented
164	C₄×#41	—	43	4	no	—
165	C₄×#41	—	43	2	no	—
166	C₂×C₄×D₄	—	43	1	no	—
167	C₂×C₄×D₄	—	43	4	no	—
168	C₂×C₄×D₄	—	43	4	no	—
169	C₄₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
170	H(5,𝔽₂)	—	43	16	no	—
171	C₂×#41	—	43	4	no	—
172	C₂×#41	—	43	2	no	—
173	C₂²×D₄	—	43	4	no	—
174	C₂²×D₄	—	43	4	no	—
175	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
176	C₂³×C₄	—	43	2	no	—
177	C₂³×C₄	—	43	4	no	—
178	C₄×Q₈	—	43	2	no	—
179	C₄×Q₈	—	43	2	no	—
180	C₄×Q₈	—	43	4	no	—
181	C₄×D₄	—	43	4	no	—
182	C₄×D₄	—	43	4	no	—
183	C₄×D₄	—	43	8	no	—
184	C₄×D₄	—	43	8	no	—
185	C₄×D₄	—	43	2	no	—
186	C₄×D₄	—	43	2	no	—
187	C₄×D₄	—	43	8	no	—
188	C₄×D₄	—	43	4	no	—
189	C₄×D₄	—	43	8	no	—
190	C₄×D₄	—	43	8	no	—
191	C₄×D₄	—	43	4	no	—
192	C₄×D₄	—	43	4	no	—
193	C₄×D₄	C₂×D₄₃	1	4	yes	complemented
194	C₄×D₄	C₂×D₄₃	1	4	yes	complemented
195	#117	—	43	4	no	—
196	#117	—	43	8	no	—
197	#117	—	43	4	no	—
198	#117	C₂×D₄₃	1	2	yes	complemented
199	C₂×C₄⋊C₄	—	43	4	no	—
200	C₂×C₄⋊C₄	—	43	1	no	—
201	C₂×C₄⋊C₄	—	43	4	no	—
202	C₂×C₂²⋊C₄	—	43	2	no	—
203	C₂×C₂²⋊C₄	—	43	4	no	—
204	C₂×C₂²⋊C₄	—	43	8	no	—
205	C₂×C₂²⋊C₄	C₂×D₄₃	1	4	yes	complemented
206	C₂×C₄²	—	43	4	no	—
207	C₂×C₄²	—	43	1	no	—
208	C₂×C₄²	—	43	4	no	—
209	C₂⁴	—	43	4	no	—
210	C₂⁴	C₂²×D₄₃	1	2	yes	—
211	#41	—	43	32	no	—
212	#41	—	43	16	no	—
213	C₂×Q₈	—	43	2	no	—
214	C₂×D₄	—	43	32	no	—
215	C₂×D₄	—	43	8	no	—
216	C₂×D₄	—	43	4	no	—
217	C₂×D₄	—	43	32	no	—
218	C₂×D₄	—	43	2	no	—
219	C₂×D₄	C₄×D₄₃	1	8	yes	complemented
220	C₂×D₄	C₂²×D₄₃	1	4	yes	—
221	C₂²×C₄	—	43	4	no	—
222	C₂²×C₄	—	43	4	no	—
223	C₂²×C₄	—	43	4	no	—
224	C₂²×C₄	—	43	1	no	—
225	C₂²×C₄	—	43	1	no	—
226	C₂²×C₄	—	43	4	no	—
227	C₂²×C₄	—	43	2	no	—
228	C₂²×C₄	—	43	4	no	—
229	C₂²×C₄	—	43	4	no	—
230	C₂²×C₄	—	43	8	no	—
231	C₂²×C₄	—	172	8	no	—
232	C₂²×C₄	—	172	4	no	—
233	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
234	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
235	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
236	C₄⋊C₄	—	43	4	no	—
237	C₄⋊C₄	—	43	8	no	—
238	C₄⋊C₄	—	43	4	no	—
239	C₄⋊C₄	—	43	4	no	—
240	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
241	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
242	C₂²⋊C₄	—	43	8	no	—
243	C₂²⋊C₄	—	43	4	no	—
244	C₂²⋊C₄	—	43	4	no	—
245	C₂²⋊C₄	—	43	8	no	—
246	C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
247	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
248	C₄²	—	43	4	no	—
249	C₄²	—	43	2	no	—
250	C₄²	—	43	4	no	—
251	C₄²	—	43	8	no	—
252	C₄²	—	43	2	no	—
253	C₄²	C₂²×D₄₃	1	4	yes	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	43	4	no	—
257	C₂³	—	43	16	no	—
258	C₂³	—	172	4	no	—
259	C₂³	—	172	4	no	—
260	C₂³	C₂×C₄×D₄₃	1	8	yes	complemented
261	C₂³	C₂³×D₄₃	1	4	yes	—
262	C₂³	C₂³×D₄₃	1	1	yes	characteristic
263	Q₈	—	43	8	no	—
264	D₄	—	43	32	no	—
265	D₄	—	43	16	no	—
266	D₄	—	43	8	no	—
267	D₄	C₂×C₄×D₄₃	1	16	yes	complemented
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	43	16	no	—
271	C₂×C₄	—	43	1	no	—
272	C₂×C₄	—	43	4	no	—
273	C₂×C₄	—	43	16	no	—
274	C₂×C₄	—	43	2	no	—
275	C₂×C₄	—	43	4	no	—
276	C₂×C₄	—	43	1	no	—
277	C₂×C₄	—	86	8	no	—
278	C₂×C₄	—	86	8	no	—
279	C₂×C₄	—	86	2	no	—
280	C₂×C₄	—	172	8	no	—
281	C₂×C₄	—	172	4	no	—
282	C₂×C₄	—	172	4	no	—
283	C₂×C₄	—	172	2	no	—
284	C₂×C₄	—	172	1	no	—
285	C₂×C₄	C₂×C₄×D₄₃	1	4	yes	complemented
286	C₂×C₄	C₂³×D₄₃	1	4	yes	—
287	C₂×C₄	C₂³×D₄₃	1	4	yes	—
288	C₂×C₄	C₂³×D₄₃	1	2	yes	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	4	8	no	—
294	C₂²	—	43	8	no	—
295	C₂²	—	86	4	no	—
296	C₂²	—	86	4	no	—
297	C₂²	—	172	16	no	—
298	C₂²	C₂²×C₄×D₄₃	1	8	yes	—
299	C₂²	C₂²×C₄×D₄₃	1	2	yes	—
300	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
301	C₄	—	2	4	no	—
302	C₄	—	2	4	no	—
303	C₄	—	43	8	no	—
304	C₄	—	43	2	no	—
305	C₄	—	86	4	no	—
306	C₄	—	86	1	no	—
307	C₄	—	86	2	no	—
308	C₄	C₂²×C₄×D₄₃	1	4	yes	—
309	C₂	—	2	2	no	—
310	C₂	—	2	8	no	—
311	C₂	—	86	8	no	—
312	C₂	C₂³×C₄×D₄₃	1	1	yes	characteristic
313	C₂	H(5,𝔽₂)×D₄₃	1	1	yes	characteristic
314	C₂	#14583734	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator	C₂×C₄₃	C₂⁴×C₄
Fitting	C₄₃×#518	C₂
socle	C₂²×C₄₃	C₂⁵
hypercenter	#518	D₄₃
hypocenter	C₄₃	C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #518

Complements

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

N	H ≅ G/N	type
2. #14583379	C₂	semidirect
3. #14583378	C₂	semidirect
4. #14583372	C₂	semidirect
5. #14583371	C₂	semidirect
6. #14583367	C₂	semidirect
7. C₄×#10621900	C₂	semidirect
8. C₂×D₄×Dic₄₃	C₂	semidirect
9. C₂×(C₂²⋊D₄₃)⋊C₄	C₂	semidirect
10. C₄×D₄×D₄₃	C₂	semidirect
11. C₄₃×#518	C₂	semidirect
12. #13063370	C₄	semidirect
17. C₂³×Dic₄₃	C₂²	semidirect
18. C₂²×C₄×D₄₃	C₂²	semidirect
19. C₄×Dic₈₆	C₂²	semidirect
20. #13063290	C₂²	semidirect
21. #13063290	C₂²	semidirect
22. C₄×D₁₇₂	C₂²	semidirect
23. D₄×Dic₄₃	C₂²	semidirect
24. D₄×Dic₄₃	C₂²	semidirect
25. #13063287	C₂²	semidirect
26. #13063287	C₂²	semidirect
27. C₄×C₂²⋊D₄₃	C₂²	semidirect
28. C₄×C₂²⋊D₄₃	C₂²	semidirect
29. (C₂²⋊D₄₃)⋊C₄	C₂²	semidirect
30. (C₂²⋊D₄₃)⋊C₄	C₂²	semidirect
31. (C₂²⋊D₄₃)⋊C₄	C₂²	semidirect
32. (C₂²⋊D₄₃)⋊C₄	C₂²	semidirect
33. (C₂²⋊D₄₃)⋊C₄	C₂²	semidirect
34. #13063283	C₂²	semidirect
35. #13063282	C₂²	semidirect
36. #13063281	C₂²	semidirect
37. C₂×C₄⋊Dic₄₃	C₂²	semidirect
38. C₂×Dic₄₃⋊C₄	C₂²	semidirect
39. C₄⋊C₄×D₄₃	C₂²	semidirect
40. C₂×C₂²⋊Dic₄₃	C₂²	semidirect
41. C₂×(C₂×D₄₃)⋊C₄	C₂²	semidirect
42. C₂²⋊C₄×D₄₃	C₂²	semidirect
43. C₂×C₄×Dic₄₃	C₂²	semidirect
44. C₂×C₄×Dic₄₃	C₂²	semidirect
45. C₄²×D₄₃	C₂²	semidirect
47. C₄×D₄×C₄₃	C₂²	semidirect
48. C₄×D₄×C₄₃	C₂²	semidirect
49. #117×C₄₃	C₂²	semidirect
50. C₂×C₂²⋊C₄×C₄₃	C₂²	semidirect
53. C₂×D₄×C₄₃	C₂×C₄	semidirect
58. C₂×C₂²⋊D₄₃	C₂×C₄	semidirect
68. #10621900	C₂×C₄	semidirect
69. D₄×D₄₃	C₂×C₄	semidirect
70. #10621898	C₂×C₄	semidirect
98. C₂³×C₄₃	C₂²×C₄	semidirect
103. C₂²×D₄₃	C₂²×C₄	semidirect
105. D₄×C₄₃	C₂²×C₄	semidirect
108. C₂×C₄×C₄₃	C₂²×C₄	semidirect
112. C₂²⋊D₄₃	C₂²×C₄	semidirect
113. C₂²⋊D₄₃	C₂²×C₄	semidirect
119. C₂×Dic₄₃	C₂²×C₄	semidirect
125. D₁₇₂	C₂²×C₄	semidirect
129. C₄×D₄₃	C₂²×C₄	semidirect
130. Dic₈₆	C₂²×C₄	semidirect
150. Dic₄₃	H(5,𝔽₂)	semidirect
163. #518	D₄₃	semidirect
169. C₄₃	C₄×H(5,𝔽₂)	semidirect
193. C₄×D₄	C₂×D₄₃	semidirect
194. C₄×D₄	C₂×D₄₃	semidirect
198. #117	C₂×D₄₃	semidirect
205. C₂×C₂²⋊C₄	C₂×D₄₃	semidirect
219. C₂×D₄	C₄×D₄₃	semidirect
260. C₂³	C₂×C₄×D₄₃	semidirect
267. D₄	C₂×C₄×D₄₃	semidirect
285. C₂×C₄	C₂×C₄×D₄₃	semidirect