Subgroups — C₂×#19145340 #25966569

Order 11008.

1954 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583456	C₂	1	2	yes	maximal
2	#19145340	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583297	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14583279	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14583250	C₂	1	2	yes	maximal, complemented
6	C₂×#14583157	C₂	1	4	yes	maximal
7	C₂×#14583007	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₂²	1	4	yes	—
10	#14583456	—	2	8	no	—
11	#14583456	C₂²	1	16	yes	—
12	C₂×#13063325	C₂²	1	2	yes	—
13	C₂×#13063325	C₂²	1	2	yes	—
14	C₂×#13063317	—	2	2	no	—
15	C₂×#13063304	—	2	2	no	—
16	C₂×#13063304	C₂²	1	4	yes	—
17	C₂×C₄×Dic₈₆	—	2	2	no	—
18	C₂×C₄×Dic₈₆	C₂²	1	4	yes	—
19	C₂×#13063284	—	2	2	no	—
20	#14583297	C₂²	1	8	yes	complemented
21	#14583279	C₂²	1	8	yes	complemented
22	#14583250	C₂²	1	16	yes	complemented
23	#14583157	C₂²	1	32	yes	—
24	#14583007	C₂²	1	32	yes	complemented
25	C₂×#13063257	C₂²	1	4	yes	—
26	C₂×#13063250	C₂²	1	8	yes	—
27	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
28	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₄₃	C₂²	1	8	yes	—
30	C₂×#97×C₄₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂²×Dic₈₆	—	2	2	no	—
34	C₂²×Dic₈₆	C₂³	1	4	yes	—
35	C₂×Dic₁₇₂	—	2	16	no	—
36	C₂×Dic₁₇₂	C₂³	1	16	yes	—
37	#13063325	—	2	4	no	—
38	#13063325	—	2	4	no	—
39	#13063325	C₂³	1	8	yes	—
40	#13063325	C₂³	1	8	yes	—
41	#13063317	—	2	16	no	—
42	#13063304	—	2	16	no	—
43	#13063304	—	2	8	no	—
44	#13063304	C₂³	1	16	yes	—
45	C₄×Dic₈₆	—	2	16	no	—
46	C₄×Dic₈₆	—	2	8	no	—
47	C₄×Dic₈₆	C₂³	1	16	yes	—
48	#13063284	—	2	16	no	—
49	C₂×C₄⋊Dic₄₃	—	2	2	no	—
50	C₂×C₄⋊Dic₄₃	—	2	2	no	—
51	C₂×C₄⋊Dic₄₃	—	2	2	no	—
52	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
53	C₂×Dic₄₃⋊C₄	—	2	4	no	—
54	C₂×Dic₄₃⋊C₄	—	2	4	no	—
55	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
56	C₂×C₄×Dic₄₃	—	2	4	no	—
57	#13063257	C₂³	1	16	yes	—
58	#13063250	C₂³	1	32	yes	—
59	C₂³×C₄×C₄₃	D₄	1	2	yes	—
60	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₄₃	—	2	8	no	—
62	C₂×M₁₆×C₄₃	C₂³	1	8	yes	—
63	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
65	C₂×C₄²×C₄₃	D₄	1	2	yes	—
66	C₂×C₄²×C₄₃	D₄	1	4	yes	—
67	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
70	#97×C₄₃	C₂³	1	8	yes	—
71	C₄×C₈×C₄₃	C₂³	1	4	yes	—
72	C₄×C₈×C₄₃	C₂³	1	4	yes	—
73	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	—	2	8	no	—
75	C₂²×C₄×C₄₃	—	2	4	no	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₃	C₂×D₄	1	8	yes	—
80	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
82	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₄₃	—	2	4	no	—
84	C₂²×Dic₄₃	—	2	4	no	—
85	C₂×Dic₈₆	—	2	16	no	—
86	C₂×Dic₈₆	—	2	4	no	—
87	C₂×Dic₈₆	—	2	8	no	—
88	C₂×Dic₈₆	—	4	8	no	—
89	C₂×Dic₈₆	C₂⁴	1	8	yes	—
90	M₁₆×C₄₃	—	2	16	no	—
91	C₂×C₈×C₄₃	—	2	4	no	—
92	C₂×C₈×C₄₃	#41	1	8	yes	—
93	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
94	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
95	C₄²×C₄₃	C₂×D₄	1	2	yes	—
96	C₄²×C₄₃	C₂×D₄	1	8	yes	—
97	C₄²×C₄₃	C₂×D₄	1	2	yes	—
98	C₄²×C₄₃	C₂⁴	1	2	yes	—
99	C₄²×C₄₃	C₂⁴	1	2	yes	—
100	C₂²⋊Dic₄₃	—	2	16	no	—
101	C₄⋊Dic₄₃	—	2	8	no	—
102	C₄⋊Dic₄₃	—	2	8	no	—
103	C₄⋊Dic₄₃	—	2	8	no	—
104	C₄⋊Dic₄₃	—	2	4	no	—
105	C₄⋊Dic₄₃	C₂⁴	1	8	yes	—
106	Dic₄₃⋊C₄	—	2	16	no	—
107	Dic₄₃⋊C₄	—	2	16	no	—
108	C₄×Dic₄₃	—	2	16	no	—
109	Dic₁₇₂	—	2	32	no	—
110	C₂³×C₄₃	—	2	2	no	—
111	C₂³×C₄₃	—	2	4	no	—
112	C₂³×C₄₃	C₂²×D₄	1	2	yes	—
113	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	4	no	—
116	C₂×C₄×C₄₃	—	2	4	no	—
117	C₂×C₄×C₄₃	—	2	16	no	—
118	C₂×C₄×C₄₃	—	2	4	no	—
119	C₂×C₄×C₄₃	#137	1	8	yes	—
120	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
121	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
124	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
125	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
126	C₂×Dic₄₃	—	2	8	no	—
127	C₂×Dic₄₃	—	2	8	no	—
128	C₂×Dic₄₃	—	4	8	no	—
129	C₂×Dic₄₃	—	4	8	no	—
130	Dic₈₆	—	2	16	no	—
131	Dic₈₆	—	4	8	no	—
132	Dic₈₆	—	4	16	no	—
133	C₈×C₄₃	—	2	4	no	—
134	C₈×C₄₃	C₂×#41	1	8	yes	—
135	C₂×#3185	—	43	1	no	maximal, Sylow
136	C₂²×C₄₃	—	2	2	no	—
137	C₂²×C₄₃	—	2	4	no	—
138	C₂²×C₄₃	—	4	4	no	—
139	C₂²×C₄₃	#532	1	2	yes	—
140	C₂²×C₄₃	C₂×#137	1	4	yes	—
141	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
142	C₄×C₄₃	—	2	2	no	—
143	C₄×C₄₃	—	2	4	no	—
144	C₄×C₄₃	#532	1	2	yes	—
145	C₄×C₄₃	C₂×#137	1	8	yes	—
146	C₄×C₄₃	C₂²×#41	1	2	yes	—
147	Dic₄₃	—	4	8	no	—
148	Dic₄₃	—	4	8	no	—
149	C₂×#533	—	43	2	no	—
150	#3185	—	43	16	no	—
151	C₂×#498	—	43	1	no	—
152	C₂×#494	—	43	1	no	—
153	C₂×#487	—	43	2	no	—
154	C₂×#470	—	43	4	no	—
155	C₂×#441	—	43	4	no	—
156	C₂×C₄×M₁₆	D₄₃	1	1	yes	characteristic
157	C₂×C₄₃	—	4	2	no	—
158	C₂×C₄₃	#3185	1	4	yes	complemented
159	C₂×C₄₃	C₂×#532	1	1	yes	characteristic
160	C₂×C₄₃	C₂²×#137	1	2	yes	—
161	C₂²×Q₁₆	—	43	4	no	—
162	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
163	#533	—	43	16	no	—
164	#533	—	86	8	no	—
165	C₂×C₄⋊Q₈	—	43	2	no	—
166	C₂×C₄⋊Q₈	—	43	2	no	—
167	C₂×#125	—	86	2	no	—
168	C₂×C₂²⋊Q₈	—	43	4	no	—
169	C₂×C₂²⋊Q₈	—	86	2	no	—
170	C₂×C₄×Q₈	—	43	4	no	—
171	C₂×C₄×Q₈	—	86	2	no	—
172	C₂×#117	—	86	2	no	—
173	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
174	#498	—	43	8	no	—
175	#494	—	43	8	no	—
176	#487	—	43	16	no	—
177	#470	—	43	32	no	—
178	#441	—	43	32	no	—
179	C₂×#106	—	43	4	no	—
180	C₂×#103	—	43	8	no	—
181	C₄×M₁₆	C₂×D₄₃	1	8	yes	—
182	C₂×#97	C₂×D₄₃	1	2	yes	—
183	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
185	C₄₃	C₂×#3185	1	1	yes	characteristic, complemented, Sylow
186	C₂²×Q₈	—	43	4	no	—
187	C₂²×Q₈	—	86	2	no	—
188	C₂³×C₄	D₁₇₂	1	2	yes	—
189	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
190	C₂×Q₁₆	—	43	16	no	—
191	C₂×Q₁₆	—	86	16	no	—
192	C₂×M₁₆	—	2	8	no	—
193	C₂×M₁₆	C₂²×D₄₃	1	8	yes	—
194	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
195	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
196	C₄⋊Q₈	—	43	8	no	—
197	C₄⋊Q₈	—	43	8	no	—
198	C₄⋊Q₈	—	86	4	no	—
199	C₄⋊Q₈	—	86	4	no	—
200	#125	—	86	16	no	—
201	C₂²⋊Q₈	—	43	16	no	—
202	C₂²⋊Q₈	—	86	8	no	—
203	C₂²⋊Q₈	—	86	16	no	—
204	C₄×Q₈	—	43	16	no	—
205	C₄×Q₈	—	86	8	no	—
206	C₄×Q₈	—	86	16	no	—
207	#117	—	86	16	no	—
208	C₂×C₄⋊C₄	—	43	4	no	—
209	C₂×C₄⋊C₄	—	86	2	no	—
210	C₂×C₄⋊C₄	—	86	2	no	—
211	C₂×C₄⋊C₄	—	86	4	no	—
212	C₂×C₄⋊C₄	—	86	4	no	—
213	C₂×C₄⋊C₄	—	86	2	no	—
214	C₂×C₂²⋊C₄	—	86	4	no	—
215	C₂×C₄²	—	86	4	no	—
216	C₂×C₄²	D₁₇₂	1	2	yes	—
217	C₂×C₄²	D₁₇₂	1	4	yes	—
218	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
219	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
221	#106	—	43	16	no	—
222	#103	—	43	32	no	—
223	#97	C₂²×D₄₃	1	8	yes	—
224	C₄×C₈	C₂²×D₄₃	1	4	yes	—
225	C₄×C₈	C₂²×D₄₃	1	4	yes	—
226	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
227	C₂×Q₈	—	43	8	no	—
228	C₂×Q₈	—	86	8	no	—
229	C₂×Q₈	—	86	4	no	—
230	C₂×Q₈	—	86	16	no	—
231	C₂×Q₈	—	172	8	no	—
232	C₂²×C₄	—	2	8	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	86	4	no	—
235	C₂²×C₄	—	86	4	no	—
236	C₂²×C₄	C₂×D₁₇₂	1	8	yes	—
237	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
238	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
239	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
241	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
242	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
243	Q₁₆	—	86	32	no	—
244	M₁₆	—	2	16	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	#10621898	1	8	yes	—
247	C₂×C₈	C₂³×D₄₃	1	4	yes	—
248	C₂×C₈	C₂³×D₄₃	1	4	yes	—
249	C₄⋊C₄	—	43	8	no	—
250	C₄⋊C₄	—	86	4	no	—
251	C₄⋊C₄	—	86	8	no	—
252	C₄⋊C₄	—	86	8	no	—
253	C₄⋊C₄	—	86	16	no	—
254	C₄⋊C₄	—	86	16	no	—
255	C₄⋊C₄	—	86	8	no	—
256	C₂²⋊C₄	—	86	16	no	—
257	C₄²	—	86	16	no	—
258	C₄²	C₂×D₁₇₂	1	2	yes	—
259	C₄²	C₂×D₁₇₂	1	8	yes	—
260	C₄²	C₂×D₁₇₂	1	2	yes	—
261	C₄²	C₂³×D₄₃	1	2	yes	—
262	C₄²	C₂³×D₄₃	1	2	yes	—
263	C₂³	—	2	4	no	—
264	C₂³	—	2	2	no	—
265	C₂³	C₂²×D₁₇₂	1	2	yes	—
266	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
267	Q₈	—	86	16	no	—
268	Q₈	—	172	16	no	—
269	Q₈	—	172	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	16	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	86	8	no	—
276	C₂×C₄	—	86	8	no	—
277	C₂×C₄	—	172	8	no	—
278	C₂×C₄	—	172	8	no	—
279	C₂×C₄	#13063358	1	8	yes	—
280	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
281	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
282	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
283	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
284	C₂×C₄	C₂×#10621898	1	4	yes	—
285	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
286	C₈	—	2	4	no	—
287	C₈	C₂×#10621898	1	8	yes	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	4	no	—
291	C₂²	#14583449	1	2	yes	—
292	C₂²	C₂×#13063358	1	4	yes	—
293	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
294	C₄	—	2	4	no	—
295	C₄	—	2	2	no	—
296	C₄	—	172	8	no	—
297	C₄	—	172	8	no	—
298	C₄	#14583449	1	2	yes	—
299	C₄	C₂×#13063358	1	8	yes	—
300	C₄	C₂²×#10621898	1	2	yes	—
301	C₂	—	4	2	no	—
302	C₂	#19145340	1	4	yes	direct product factor
303	C₂	C₂×#14583449	1	1	yes	characteristic
304	C₂	C₂²×#13063358	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₄×M₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
hypercenter	C₂×C₄×M₁₆	D₄₃
hypocenter	C₄₃	C₂×#3185
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #19145340	C₂	direct (N × H)
3. C₂×#14583297	C₂	semidirect
4. C₂×#14583279	C₂	semidirect
5. C₂×#14583250	C₂	semidirect
7. C₂×#14583007	C₂	semidirect
20. #14583297	C₂²	semidirect
21. #14583279	C₂²	semidirect
22. #14583250	C₂²	semidirect
24. #14583007	C₂²	semidirect
158. C₂×C₄₃	#3185	semidirect
185. C₄₃	C₂×#3185	semidirect
302. C₂	#19145340	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#14583456	2
#19145340	2
C₂×#14583297	2
C₂×#14583279	2
C₂×#14583250	2
C₂×#14583157	2
C₂×#14583007	2
C₂×C₄×M₁₆×C₄₃	2
C₂×#3185	43

Subgroups — C2×#19145340 #25966569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19145340 #25966569