Subgroups — C₂×#19136804 #25251376

Order 11008.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583140	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582937	C₂	1	1	yes	maximal, characteristic
3	#19136804	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582727	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582725	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582704	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582704	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂×#13063304	—	2	1	no	—
11	C₂×#13063284	—	2	1	no	—
12	#14583140	—	2	2	no	—
13	#14583140	C₂²	1	4	yes	—
14	#14582937	—	2	2	no	—
15	#14582937	C₂²	1	4	yes	—
16	C₂×#13063259	C₂²	1	1	yes	characteristic
17	C₂×#13063259	C₂²	1	1	yes	characteristic
18	C₂×#13063259	C₂²	1	1	yes	characteristic
19	C₂×#13063257	—	2	1	no	—
20	C₂×#13063250	—	2	1	no	—
21	#14582727	C₂²	1	4	yes	complemented
22	#14582725	C₂²	1	4	yes	complemented
23	#14582704	C₂²	1	4	yes	complemented
24	#14582704	C₂²	1	4	yes	complemented
25	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
29	C₂²×Dic₈₆	—	2	1	no	—
30	C₂×Dic₁₇₂	—	2	2	no	—
31	C₂×Dic₁₇₂	—	2	2	no	—
32	C₂×Dic₁₇₂	—	2	1	no	—
33	C₂×Dic₁₇₂	C₂³	1	2	yes	—
34	#13063304	—	2	4	no	—
35	#13063304	—	4	2	no	—
36	#13063284	—	2	4	no	—
37	#13063284	—	4	2	no	—
38	C₂×C₄⋊Dic₄₃	—	2	1	no	—
39	C₂×C₄⋊Dic₄₃	—	2	1	no	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×Dic₄₃⋊C₄	—	4	1	no	—
42	C₂×C₂²⋊Dic₄₃	—	4	1	no	—
43	C₂×C₂²⋊Dic₄₃	—	4	1	no	—
44	C₂×C₄×Dic₄₃	—	4	1	no	—
45	#13063259	—	2	1	no	—
46	#13063259	—	2	1	no	—
47	#13063259	—	2	1	no	—
48	#13063259	C₂³	1	2	yes	—
49	#13063259	C₂³	1	2	yes	—
50	#13063259	C₂³	1	2	yes	—
51	#13063257	—	2	4	no	—
52	#13063250	—	2	4	no	—
53	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₃	—	2	2	no	—
55	C₂²×C₈×C₄₃	—	2	2	no	—
56	C₂²×C₈×C₄₃	D₄	1	2	yes	—
57	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
59	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₄₃	—	2	2	no	—
61	C₂×C₁₆×C₄₃	—	2	2	no	—
62	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
64	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	—	2	2	no	—
66	C₂²×C₄×C₄₃	—	2	2	no	—
67	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	D₈	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₄₃	—	4	1	no	—
72	C₂²×Dic₄₃	—	4	1	no	—
73	C₂²×Dic₄₃	—	4	1	no	—
74	C₂²×Dic₄₃	—	4	1	no	—
75	C₂×Dic₈₆	—	2	2	no	—
76	C₂×Dic₈₆	—	4	2	no	—
77	C₂×Dic₈₆	—	4	2	no	—
78	C₂×Dic₈₆	—	4	1	no	—
79	C₂×C₈×C₄₃	—	2	2	no	—
80	C₂×C₈×C₄₃	—	2	4	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	—	2	2	no	—
83	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	#41	1	2	yes	—
86	C₂×C₈×C₄₃	#41	1	2	yes	—
87	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂²⋊Dic₄₃	—	4	4	no	—
89	C₂²⋊Dic₄₃	—	4	4	no	—
90	C₄⋊Dic₄₃	—	2	2	no	—
91	C₄⋊Dic₄₃	—	2	2	no	—
92	C₄⋊Dic₄₃	—	2	2	no	—
93	C₄⋊Dic₄₃	—	4	1	no	—
94	C₄⋊Dic₄₃	—	4	1	no	—
95	C₄⋊Dic₄₃	—	4	1	no	—
96	Dic₄₃⋊C₄	—	4	4	no	—
97	C₄×Dic₄₃	—	4	4	no	—
98	Dic₁₇₂	—	2	2	no	—
99	Dic₁₇₂	—	2	2	no	—
100	Dic₁₇₂	—	4	2	no	—
101	C₁₆×C₄₃	—	2	2	no	—
102	C₁₆×C₄₃	—	2	2	no	—
103	C₂³×C₄₃	—	2	2	no	—
104	C₂³×C₄₃	—	2	2	no	—
105	C₂³×C₄₃	—	2	2	no	—
106	C₂³×C₄₃	C₂×D₈	1	2	yes	—
107	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	#123	1	2	yes	—
113	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#137	1	2	yes	—
116	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₄₃	—	4	2	no	—
118	C₂×Dic₄₃	—	4	2	no	—
119	C₂×Dic₄₃	—	4	2	no	—
120	C₂×Dic₄₃	—	4	2	no	—
121	C₂×Dic₄₃	—	8	2	no	—
122	C₂×Dic₄₃	—	8	2	no	—
123	C₂×Dic₄₃	—	8	2	no	—
124	C₂×Dic₄₃	—	8	2	no	—
125	Dic₈₆	—	4	2	no	—
126	Dic₈₆	—	4	2	no	—
127	Dic₈₆	—	8	2	no	—
128	C₈×C₄₃	—	2	2	no	—
129	C₈×C₄₃	#123	1	2	yes	—
130	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
131	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
132	C₂×#2237	—	43	1	no	maximal, Sylow
133	C₂²×C₄₃	—	2	2	no	—
134	C₂²×C₄₃	—	2	2	no	—
135	C₂²×C₄₃	—	2	2	no	—
136	C₂²×C₄₃	—	4	4	no	—
137	C₂²×C₄₃	#480	1	2	yes	—
138	C₂²×C₄₃	#508	1	2	yes	—
139	C₂²×C₄₃	#510	1	2	yes	—
140	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₄₃	—	2	2	no	—
142	C₄×C₄₃	#480	1	2	yes	—
143	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
144	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
145	Dic₄₃	—	8	2	no	—
146	Dic₄₃	—	8	2	no	—
147	Dic₄₃	—	8	2	no	—
148	Dic₄₃	—	8	2	no	—
149	C₂×#467	—	43	1	no	—
150	C₂×#427	—	43	1	no	—
151	#2237	—	43	8	no	—
152	C₂×#367	—	43	1	no	—
153	C₂×#366	—	43	1	no	—
154	C₂×#358	—	43	1	no	—
155	C₂×#358	—	43	1	no	—
156	C₂×C₂²⋊C₁₆	D₄₃	1	1	yes	characteristic
157	C₂×C₄₃	—	4	2	no	—
158	C₂×C₄₃	#2237	1	4	yes	complemented
159	C₂×C₄₃	C₂×#480	1	1	yes	characteristic
160	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
161	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
162	C₂²×Q₁₆	—	43	1	no	—
163	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
164	C₂×C₂²⋊Q₈	—	86	1	no	—
165	C₂×#117	—	86	1	no	—
166	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
167	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
168	#467	—	43	4	no	—
169	#467	—	86	2	no	—
170	#427	—	43	4	no	—
171	#427	—	86	2	no	—
172	C₂×#107	—	43	1	no	—
173	C₂×#107	—	43	1	no	—
174	C₂×#107	—	43	1	no	—
175	C₂×#106	—	86	1	no	—
176	C₂×#103	—	86	1	no	—
177	#367	—	43	4	no	—
178	#366	—	43	4	no	—
179	#358	—	43	4	no	—
180	#358	—	43	4	no	—
181	C₂²⋊C₁₆	C₂×D₄₃	1	4	yes	—
182	C₄₃	C₂×#2237	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	—	86	1	no	—
184	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
185	C₂×Q₁₆	—	43	2	no	—
186	C₂×Q₁₆	—	86	1	no	—
187	C₂×Q₁₆	—	86	2	no	—
188	C₂×Q₁₆	—	86	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
192	C₂²×C₈	D₁₇₂	1	2	yes	—
193	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
194	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
195	C₂²⋊Q₈	—	86	4	no	—
196	C₂²⋊Q₈	—	172	2	no	—
197	#117	—	86	4	no	—
198	#117	—	172	2	no	—
199	C₂×C₄⋊C₄	—	86	1	no	—
200	C₂×C₄⋊C₄	—	86	1	no	—
201	C₂×C₄⋊C₄	—	86	1	no	—
202	C₂×C₄⋊C₄	—	172	1	no	—
203	C₂×C₂²⋊C₄	—	172	1	no	—
204	C₂×C₂²⋊C₄	—	172	1	no	—
205	C₂×C₄²	—	172	1	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	2	2	no	—
208	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
209	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
210	#107	—	43	2	no	—
211	#107	—	43	2	no	—
212	#107	—	43	2	no	—
213	#107	—	86	1	no	—
214	#107	—	86	1	no	—
215	#107	—	86	1	no	—
216	#106	—	86	4	no	—
217	#103	—	86	4	no	—
218	C₂⁴	D₃₄₄	1	1	yes	characteristic
219	C₂×Q₈	—	86	2	no	—
220	C₂×Q₈	—	172	2	no	—
221	C₂×Q₈	—	172	2	no	—
222	C₂×Q₈	—	172	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	172	1	no	—
226	C₂²×C₄	—	172	1	no	—
227	C₂²×C₄	—	172	1	no	—
228	C₂²×C₄	—	172	1	no	—
229	C₂²×C₄	D₃₄₄	1	2	yes	—
230	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
232	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
233	Q₁₆	—	86	2	no	—
234	Q₁₆	—	86	2	no	—
235	Q₁₆	—	172	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
241	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
242	C₂×C₈	#10621900	1	2	yes	—
243	C₂×C₈	#10621900	1	2	yes	—
244	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
245	C₄⋊C₄	—	86	2	no	—
246	C₄⋊C₄	—	86	2	no	—
247	C₄⋊C₄	—	86	2	no	—
248	C₄⋊C₄	—	172	1	no	—
249	C₄⋊C₄	—	172	1	no	—
250	C₄⋊C₄	—	172	1	no	—
251	C₄⋊C₄	—	172	4	no	—
252	C₂²⋊C₄	—	172	4	no	—
253	C₂²⋊C₄	—	172	4	no	—
254	C₄²	—	172	4	no	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×D₃₄₄	1	2	yes	—
261	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
262	Q₈	—	344	2	no	—
263	Q₈	—	172	2	no	—
264	Q₈	—	172	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	344	2	no	—
270	C₂×C₄	—	344	2	no	—
271	C₂×C₄	—	344	2	no	—
272	C₂×C₄	—	344	2	no	—
273	C₂×C₄	—	172	2	no	—
274	C₂×C₄	—	172	2	no	—
275	C₂×C₄	—	172	2	no	—
276	C₂×C₄	—	172	2	no	—
277	C₂×C₄	#13063309	1	2	yes	—
278	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
279	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
280	C₂×C₄	#13063358	1	2	yes	—
281	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	#13063309	1	2	yes	—
284	C₈	C₂×#10621900	1	1	yes	characteristic
285	C₈	C₂×#10621900	1	1	yes	characteristic
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	#14583213	1	2	yes	—
291	C₂²	#14583331	1	2	yes	—
292	C₂²	#14583342	1	2	yes	—
293	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	344	2	no	—
296	C₄	—	344	2	no	—
297	C₄	—	344	2	no	—
298	C₄	—	344	2	no	—
299	C₄	#14583213	1	2	yes	—
300	C₄	C₂×#13063309	1	1	yes	characteristic
301	C₄	C₂×#13063358	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	#19136804	1	4	yes	direct product factor
304	C₂	C₂×#14583213	1	1	yes	characteristic
305	C₂	C₂×#14583331	1	1	yes	characteristic
306	C₂	C₂×#14583342	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×C₂²⋊C₁₆	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂⁴×C₄₃	D₈
hypercenter	C₂×C₂²⋊C₁₆	D₄₃
hypocenter	C₄₃	C₂×#2237
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #19136804	C₂	direct (N × H)
4. C₂×#14582727	C₂	semidirect
5. C₂×#14582725	C₂	semidirect
6. C₂×#14582704	C₂	semidirect
7. C₂×#14582704	C₂	semidirect
21. #14582727	C₂²	semidirect
22. #14582725	C₂²	semidirect
23. #14582704	C₂²	semidirect
24. #14582704	C₂²	semidirect
158. C₂×C₄₃	#2237	semidirect
182. C₄₃	C₂×#2237	semidirect
303. C₂	#19136804	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×#14583140	2
C₂×#14582937	2
#19136804	2
C₂×#14582727	2
C₂×#14582725	2
C₂×#14582704	2
C₂×C₄₃×C₂²⋊C₁₆	2
C₂×#2237	43

Subgroups — C2×#19136804 #25251376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136804 #25251376