Subgroups — C₂×#19136458 #25249885

Order 11008.

734 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	C₂²×#13063270	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582993	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582981	C₂	1	1	yes	maximal, characteristic
4	#19136458	C₂	1	8	yes	maximal, direct product factor
5	C₂×#14582702	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582699	C₂	1	1	yes	maximal, characteristic
7	C₂×#14582674	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
11	C₂×#13063290	—	2	1	no	—
12	C₂×#13063287	—	2	1	no	—
13	C₂×#13063270	C₄	1	8	yes	complemented
14	C₂×#13063270	C₂²	1	4	yes	—
15	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14582993	—	2	2	no	—
17	#14582993	C₂²	1	4	yes	complemented
18	#14582981	—	2	2	no	—
19	#14582981	C₂²	1	4	yes	—
20	C₂×#13063248	—	2	1	no	—
21	C₂×#13063245	—	2	1	no	—
22	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
23	#14582702	C₂²	1	4	yes	complemented
24	#14582699	C₂²	1	4	yes	—
25	#14582674	C₂²	1	4	yes	complemented
26	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#366	C₂²	1	4	yes	complemented
30	C₂²×Dic₈₆	—	2	1	no	—
31	C₂²×D₁₇₂	—	2	1	no	—
32	C₂²×C₄×D₄₃	—	4	1	no	—
33	C₂×Dic₁₇₂	—	2	1	no	—
34	C₂×Dic₁₇₂	—	2	2	no	—
35	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	complemented
36	C₂×Dic₁₇₂	C₂³	1	2	yes	—
37	C₂×D₃₄₄	—	2	1	no	—
38	C₂×D₃₄₄	—	2	2	no	—
39	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
40	C₂×D₃₄₄	C₂³	1	2	yes	—
41	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
42	#13063290	—	2	4	no	—
43	#13063290	—	4	2	no	—
44	#13063287	—	2	4	no	—
45	#13063287	—	4	2	no	—
46	C₂×Dic₄₃⋊C₄	—	4	1	no	—
47	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
48	C₂×C₄×Dic₄₃	—	4	1	no	—
49	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
50	#13063270	C₂×C₄	1	16	yes	complemented
51	C₂×C₄₃⋊C₁₆	—	2	2	no	—
52	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
53	#13063248	—	2	4	no	—
54	#13063245	—	2	4	no	—
55	C₈×Dic₄₃	D₄	1	2	yes	—
56	C₈×Dic₄₃	C₂³	1	2	yes	—
57	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
60	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
61	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
62	#107×C₄₃	—	2	1	no	—
63	#107×C₄₃	—	2	1	no	—
64	#107×C₄₃	C₂³	1	2	yes	—
65	#107×C₄₃	C₂³	1	2	yes	—
66	C₂³×D₄₃	—	4	1	no	—
67	C₂²×C₄×C₄₃	—	4	1	no	—
68	C₂²×C₄×C₄₃	—	4	1	no	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₄₃	—	4	1	no	—
71	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
72	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
73	C₂×D₁₇₂	—	2	4	no	—
74	C₂×D₁₇₂	—	2	2	no	—
75	C₂×D₁₇₂	—	4	1	no	—
76	C₂×D₁₇₂	—	4	2	no	—
77	C₂×C₄×D₄₃	—	4	4	no	—
78	C₂×C₄×D₄₃	—	8	4	no	—
79	C₂×Dic₈₆	—	2	4	no	—
80	C₂×Dic₈₆	—	2	2	no	—
81	C₂×Dic₈₆	—	4	1	no	—
82	C₂×Dic₈₆	—	4	2	no	—
83	C₂×C₈×C₄₃	C₂²×C₄	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	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₄₃	—	2	2	no	—
88	C₄⋊C₄×C₄₃	—	2	2	no	—
89	C₄⋊C₄×C₄₃	—	4	1	no	—
90	C₄⋊C₄×C₄₃	—	4	1	no	—
91	(C₂×D₄₃)⋊C₄	—	4	4	no	—
92	Dic₄₃⋊C₄	—	4	4	no	—
93	C₄×Dic₄₃	—	4	4	no	—
94	C₄×Dic₄₃	D₈	1	2	yes	—
95	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
96	C₂×C₄₃⋊C₈	—	2	2	no	—
97	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
98	Dic₁₇₂	—	2	4	no	—
99	Dic₁₇₂	—	2	2	no	—
100	Dic₁₇₂	C₂²×C₄	1	4	yes	complemented
101	D₃₄₄	—	2	4	no	—
102	D₃₄₄	—	2	2	no	—
103	D₃₄₄	C₂²×C₄	1	4	yes	—
104	C₁₆×C₄₃	C₂²×C₄	1	4	yes	complemented
105	C₄₃⋊C₁₆	—	2	2	no	—
106	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₄₃	—	4	2	no	—
108	C₂²×D₄₃	—	4	4	no	—
109	C₂²×D₄₃	—	8	2	no	—
110	C₂²×D₄₃	—	8	2	no	—
111	C₂×C₄×C₄₃	—	4	2	no	—
112	C₂×C₄×C₄₃	—	4	2	no	—
113	C₂×C₄×C₄₃	—	8	2	no	—
114	C₂×C₄×C₄₃	—	8	2	no	—
115	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#135	1	2	yes	—
118	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₄₃	—	2	2	no	—
120	C₂×Dic₄₃	—	2	2	no	—
121	C₂×Dic₄₃	—	4	2	no	—
122	C₂×Dic₄₃	—	4	4	no	—
123	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
124	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
125	D₁₇₂	—	2	4	no	—
126	D₁₇₂	—	4	2	no	—
127	D₁₇₂	—	4	4	no	—
128	C₄×D₄₃	—	8	8	no	—
129	Dic₈₆	—	2	4	no	—
130	Dic₈₆	—	4	2	no	—
131	Dic₈₆	—	4	4	no	—
132	C₈×C₄₃	C₄×D₄	1	2	yes	—
133	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
134	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
135	C₄₃⋊C₈	—	2	2	no	—
136	C₂×#2175	—	43	1	no	maximal, Sylow
137	C₂²×C₄₃	C₄×D₈	1	2	yes	—
138	C₂²×C₄₃	#509	1	2	yes	—
139	C₂²×C₄₃	#510	1	2	yes	—
140	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₄₃	—	4	4	no	—
142	C₂×D₄₃	—	8	2	no	—
143	C₂×D₄₃	—	8	2	no	—
144	C₂×D₄₃	—	8	8	no	—
145	C₄×C₄₃	—	8	2	no	—
146	C₄×C₄₃	—	8	2	no	—
147	C₄×C₄₃	C₄×D₈	1	2	yes	—
148	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
149	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
150	Dic₄₃	—	2	2	no	—
151	Dic₄₃	—	2	2	no	—
152	Dic₄₃	—	4	4	no	—
153	C₂²×SD₃₂	—	43	1	no	—
154	C₂×C₄×Q₁₆	—	43	1	no	—
155	C₂×C₄×D₈	—	43	1	no	—
156	#2175	—	43	8	no	—
157	C₂×#366	D₄₃	1	1	yes	characteristic, complemented
158	C₂×#358	—	43	1	no	—
159	C₂×#357	—	43	1	no	—
160	C₂×#346	—	43	1	no	—
161	C₂×C₄₃	#2175	1	4	yes	complemented
162	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
163	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
164	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
165	D₄₃	—	8	4	no	—
166	C₂²×Q₁₆	—	43	1	no	—
167	C₂²×D₈	—	43	1	no	—
168	C₂×C₄×Q₈	—	86	1	no	—
169	C₂×C₄×D₄	—	86	1	no	—
170	C₂×SD₃₂	—	43	4	no	—
171	C₂×SD₃₂	—	43	8	no	—
172	C₂²×C₁₆	—	43	1	no	—
173	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
174	C₄×Q₁₆	—	43	4	no	—
175	C₄×Q₁₆	—	86	2	no	—
176	C₄×D₈	—	43	4	no	—
177	C₄×D₈	—	86	2	no	—
178	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
179	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
180	C₂×#103	—	86	1	no	—
181	C₂×#102	—	86	1	no	—
182	C₂×C₄×C₈	—	43	1	no	—
183	#366	C₂×D₄₃	1	4	yes	complemented
184	#358	—	43	4	no	—
185	#357	—	43	4	no	—
186	#346	—	43	4	no	—
187	C₄₃	C₂×#2175	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	—	86	1	no	—
189	C₂²×D₄	—	86	1	no	—
190	C₂³×C₄	—	172	1	no	—
191	C₂×Q₁₆	—	43	4	no	—
192	C₂×Q₁₆	—	43	2	no	—
193	C₂×Q₁₆	—	86	1	no	—
194	C₂×Q₁₆	—	86	2	no	—
195	C₂×D₈	—	43	4	no	—
196	C₂×D₈	—	43	2	no	—
197	C₂×D₈	—	86	1	no	—
198	C₂×D₈	—	86	2	no	—
199	C₂²×C₈	—	43	1	no	—
200	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
201	C₄×Q₈	—	86	4	no	—
202	C₄×Q₈	—	172	2	no	—
203	C₄×D₄	—	86	4	no	—
204	C₄×D₄	—	172	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	172	1	no	—
208	C₂×C₂²⋊C₄	—	172	1	no	—
209	C₂×C₄²	—	43	1	no	—
210	C₂×C₄²	—	172	1	no	—
211	SD₃₂	—	43	16	no	—
212	C₂×C₁₆	—	43	2	no	—
213	C₂×C₁₆	—	86	2	no	—
214	C₂×C₁₆	C₄×D₄₃	1	4	yes	complemented
215	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
216	#107	—	2	1	no	—
217	#107	—	2	1	no	—
218	#107	C₂²×D₄₃	1	2	yes	—
219	#107	C₂²×D₄₃	1	2	yes	—
220	#103	—	86	4	no	—
221	#102	—	86	4	no	—
222	C₄×C₈	—	43	2	no	—
223	C₄×C₈	—	43	2	no	—
224	C₂⁴	—	172	1	no	—
225	C₂×Q₈	—	86	4	no	—
226	C₂×Q₈	—	86	2	no	—
227	C₂×Q₈	—	172	1	no	—
228	C₂×Q₈	—	172	2	no	—
229	C₂×D₄	—	86	4	no	—
230	C₂×D₄	—	86	2	no	—
231	C₂×D₄	—	172	1	no	—
232	C₂×D₄	—	172	2	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	43	1	no	—
236	C₂²×C₄	—	43	1	no	—
237	C₂²×C₄	—	344	4	no	—
238	C₂²×C₄	—	172	4	no	—
239	C₂²×C₄	—	172	1	no	—
240	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
241	Q₁₆	—	43	4	no	—
242	Q₁₆	—	86	4	no	—
243	Q₁₆	—	86	2	no	—
244	D₈	—	43	4	no	—
245	D₈	—	86	4	no	—
246	D₈	—	86	2	no	—
247	C₂×C₈	—	43	2	no	—
248	C₂×C₈	—	86	2	no	—
249	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
250	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
251	C₂×C₈	#10621902	1	2	yes	—
252	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	172	4	no	—
258	C₂²⋊C₄	—	172	4	no	—
259	C₄²	—	43	2	no	—
260	C₄²	—	43	2	no	—
261	C₄²	—	172	4	no	—
262	C₁₆	—	86	2	no	—
263	C₁₆	C₂×C₄×D₄₃	1	4	yes	complemented
264	C₂³	—	344	2	no	—
265	C₂³	—	344	2	no	—
266	C₂³	—	172	2	no	—
267	C₂³	—	172	4	no	—
268	C₂³	D₈×D₄₃	1	1	yes	characteristic
269	Q₈	—	86	4	no	—
270	Q₈	—	172	2	no	—
271	Q₈	—	172	4	no	—
272	D₄	—	86	4	no	—
273	D₄	—	172	2	no	—
274	D₄	—	172	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	43	2	no	—
280	C₂×C₄	—	43	2	no	—
281	C₂×C₄	—	86	2	no	—
282	C₂×C₄	—	86	2	no	—
283	C₂×C₄	—	344	8	no	—
284	C₂×C₄	—	172	2	no	—
285	C₂×C₄	—	172	4	no	—
286	C₂×C₄	#13063287	1	2	yes	—
287	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
288	C₂×C₄	#13063348	1	2	yes	—
289	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
290	C₈	—	86	2	no	—
291	C₈	#13063287	1	2	yes	—
292	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
293	C₈	C₂×#10621902	1	1	yes	characteristic
294	C₂²	—	344	2	no	—
295	C₂²	—	344	8	no	—
296	C₂²	—	344	2	no	—
297	C₂²	—	172	4	no	—
298	C₂²	#14582981	1	2	yes	—
299	C₂²	#14583333	1	2	yes	—
300	C₂²	#14583341	1	2	yes	—
301	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	86	2	no	—
305	C₄	—	86	2	no	—
306	C₄	—	172	4	no	—
307	C₄	#14582981	1	2	yes	—
308	C₄	C₂×#13063287	1	1	yes	characteristic
309	C₄	C₂×#13063348	1	1	yes	characteristic
310	C₂	—	344	4	no	—
311	C₂	#19136458	1	4	yes	direct product factor
312	C₂	C₂×#14582981	1	1	yes	characteristic
313	C₂	C₂×#14583333	1	1	yes	characteristic
314	C₂	C₂×#14583341	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#366	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#366	D₄₃
hypocenter	C₄₃	C₂×#2175
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. C₂×#14582993	C₂	semidirect
4. #19136458	C₂	direct (N × H)
5. C₂×#14582702	C₂	semidirect
7. C₂×#14582674	C₂	semidirect
8. C₂×C₄₃×#366	C₂	semidirect
13. C₂×#13063270	C₄	semidirect
17. #14582993	C₂²	semidirect
23. #14582702	C₂²	semidirect
25. #14582674	C₂²	semidirect
29. C₄₃×#366	C₂²	semidirect
35. C₂×Dic₁₇₂	C₂×C₄	semidirect
50. #13063270	C₂×C₄	semidirect
60. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
100. Dic₁₇₂	C₂²×C₄	semidirect
104. C₁₆×C₄₃	C₂²×C₄	semidirect
157. C₂×#366	D₄₃	semidirect
161. C₂×C₄₃	#2175	semidirect
183. #366	C₂×D₄₃	semidirect
187. C₄₃	C₂×#2175	semidirect
214. C₂×C₁₆	C₄×D₄₃	semidirect
263. C₁₆	C₂×C₄×D₄₃	semidirect
311. C₂	#19136458	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13063270	2
C₂×#14582993	2
C₂×#14582981	2
#19136458	2
C₂×#14582702	2
C₂×#14582699	2
C₂×#14582674	2
C₂×C₄₃×#366	2
C₂×#2175	43

Subgroups — C2×#19136458 #25249885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136458 #25249885