Subgroups — C₂×#19132603 #25006815

Order 11008.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	#19132603	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₄₄	C₄	1	1	yes	characteristic, complemented
6	C₂×#13063299	—	2	1	no	—
7	C₂²×C₄₃⋊C₁₆	—	2	1	no	—
8	#14583135	—	2	1	no	—
9	#14583135	—	2	1	no	—
10	#14583135	C₄	1	2	yes	complemented
11	#14583135	C₂²	1	2	yes	—
12	C₂×#13063259	C₄	1	1	yes	characteristic, complemented
13	C₂×#13063247	—	2	1	no	—
14	C₂²⋊(C₄₃⋊C₁₆)	—	2	1	no	—
15	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	complemented
16	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₄₃	—	2	1	no	—
18	C₄₃×#340	—	2	1	no	—
19	C₄₃×#340	C₂²	1	2	yes	complemented
20	C₂²×D₁₇₂	—	2	1	no	—
21	C₂²×C₂²⋊D₄₃	—	4	1	no	—
22	C₂×D₃₄₄	—	2	1	no	—
23	C₂×D₃₄₄	—	4	2	no	—
24	C₂×D₃₄₄	C₂×C₄	1	2	yes	complemented
25	#13063299	—	2	2	no	—
26	#13063299	—	2	2	no	—
27	#13063299	—	4	1	no	—
28	#13063299	—	4	1	no	—
29	C₂×C₄⋊Dic₄₃	—	2	1	no	—
30	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
31	C₂×C₄₃⋊C₁₆	—	2	2	no	—
32	C₂×C₄₃⋊C₁₆	—	4	2	no	—
33	#13063259	—	2	1	no	—
34	#13063259	C₂×C₄	1	2	yes	complemented
35	#13063247	—	2	2	no	—
36	#13063247	—	2	2	no	—
37	C₂³×C₄×C₄₃	—	4	1	no	—
38	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₃	—	4	2	no	—
40	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
49	C₂×C₄⋊C₄×C₄₃	—	4	1	no	—
50	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
51	C₂³×D₄₃	—	4	1	no	—
52	C₂²×C₄×C₄₃	—	2	2	no	—
53	C₂²×C₄×C₄₃	—	2	2	no	—
54	C₂²×C₄×C₄₃	—	4	4	no	—
55	C₂²×C₄×C₄₃	—	4	2	no	—
56	C₂²×C₄×C₄₃	—	8	2	no	—
57	C₂²×C₄×C₄₃	—	8	2	no	—
58	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₃	—	4	2	no	—
65	C₂×C₂²⋊D₄₃	—	4	2	no	—
66	C₂×C₂²⋊D₄₃	—	8	2	no	—
67	C₂×C₂²⋊D₄₃	—	8	2	no	—
68	C₂²×Dic₄₃	—	4	1	no	—
69	C₂×D₁₇₂	—	2	2	no	—
70	C₂×D₁₇₂	—	4	1	no	—
71	C₂×D₁₇₂	—	4	2	no	—
72	C₂×D₁₇₂	—	4	2	no	—
73	C₂×C₈×C₄₃	—	2	2	no	—
74	C₂×C₈×C₄₃	—	2	2	no	—
75	C₂×C₈×C₄₃	—	4	1	no	—
76	C₂×C₈×C₄₃	—	4	1	no	—
77	C₂×C₈×C₄₃	—	4	1	no	—
78	C₂×C₈×C₄₃	—	4	1	no	—
79	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₄⋊C₄×C₄₃	—	2	4	no	—
84	C₄⋊C₄×C₄₃	—	4	4	no	—
85	C₄⋊C₄×C₄₃	—	4	2	no	—
86	(C₂×D₄₃)⋊C₄	—	4	2	no	—
87	(C₂×D₄₃)⋊C₄	—	4	2	no	—
88	C₄⋊Dic₄₃	—	2	2	no	—
89	C₄⋊Dic₄₃	—	4	1	no	—
90	D₃₄₄	—	4	2	no	—
91	D₃₄₄	—	4	2	no	—
92	C₄₃⋊C₁₆	—	4	2	no	—
93	C₂³×C₄₃	—	2	2	no	—
94	C₂³×C₄₃	—	2	2	no	—
95	C₂³×C₄₃	#102	1	1	yes	characteristic
96	C₂³×C₄₃	#102	1	1	yes	characteristic
97	C₂³×C₄₃	D₁₆	1	2	yes	—
98	C₂³×C₄₃	SD₃₂	1	2	yes	—
99	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₄₃	—	4	2	no	—
101	C₂²×D₄₃	—	8	2	no	—
102	C₂²×D₄₃	—	8	2	no	—
103	C₂²×D₄₃	—	8	2	no	—
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	4	4	no	—
112	C₂×C₄×C₄₃	—	8	8	no	—
113	C₂×C₄×C₄₃	—	8	2	no	—
114	C₂×C₄×C₄₃	—	8	2	no	—
115	C₂×C₄×C₄₃	#99	1	2	yes	—
116	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#104	1	2	yes	—
119	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂²⋊D₄₃	—	8	4	no	—
122	C₂²⋊D₄₃	—	8	2	no	—
123	C₂²⋊D₄₃	—	8	2	no	—
124	C₂×Dic₄₃	—	4	2	no	—
125	C₂×Dic₄₃	—	8	2	no	—
126	D₁₇₂	—	4	2	no	—
127	D₁₇₂	—	4	2	no	—
128	D₁₇₂	—	8	2	no	—
129	C₈×C₄₃	—	2	1	no	—
130	C₈×C₄₃	—	2	1	no	—
131	C₈×C₄₃	—	2	1	no	—
132	C₈×C₄₃	—	2	1	no	—
133	C₂×#1347	—	43	1	no	maximal, Sylow
134	C₂²×C₄₃	—	2	4	no	—
135	C₂²×C₄₃	—	2	4	no	—
136	C₂²×C₄₃	—	2	1	no	—
137	C₂²×C₄₃	—	2	1	no	—
138	C₂²×C₄₃	—	2	1	no	—
139	C₂²×C₄₃	—	2	1	no	—
140	C₂²×C₄₃	#327	1	2	yes	—
141	C₂²×C₄₃	#357	1	2	yes	—
142	C₂²×C₄₃	#357	1	2	yes	—
143	C₂²×C₄₃	#360	1	2	yes	—
144	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
145	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
146	C₂²×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
147	C₂×D₄₃	—	8	2	no	—
148	C₂×D₄₃	—	8	2	no	—
149	C₂×D₄₃	—	8	2	no	—
150	C₂×D₄₃	—	16	4	no	—
151	C₄×C₄₃	—	2	1	no	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	—	8	4	no	—
154	C₄×C₄₃	#327	1	2	yes	—
155	C₄×C₄₃	C₂×#99	1	1	yes	characteristic
156	C₄×C₄₃	C₂×#104	1	1	yes	characteristic
157	Dic₄₃	—	8	2	no	—
158	C₂×#466	—	43	1	no	—
159	C₂×C₂²⋊C₁₆	—	43	1	no	—
160	C₂×#340	D₄₃	1	1	yes	characteristic, complemented
161	#1347	—	43	4	no	—
162	C₂×C₄₃	—	2	1	no	—
163	C₂×C₄₃	—	2	2	no	—
164	C₂×C₄₃	—	2	1	no	—
165	C₂×C₄₃	#1347	1	4	yes	complemented
166	C₂×C₄₃	C₂×#327	1	1	yes	characteristic
167	C₂×C₄₃	C₂×#357	1	1	yes	characteristic
168	C₂×C₄₃	C₂×#360	1	1	yes	characteristic
169	D₄₃	—	16	2	no	—
170	C₂²×D₈	—	43	1	no	—
171	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
172	C₂×#121	—	86	1	no	—
173	C₂²×C₄⋊C₄	—	2	1	no	—
174	C₂²×C₁₆	—	86	1	no	—
175	#466	—	43	2	no	—
176	#466	—	43	2	no	—
177	#466	—	86	1	no	—
178	#466	—	86	1	no	—
179	C₂×#107	—	43	1	no	—
180	C₂×#102	—	86	1	no	—
181	C₂²⋊C₁₆	—	43	2	no	—
182	C₂²⋊C₁₆	—	86	1	no	—
183	#340	—	2	1	no	—
184	#340	C₂×D₄₃	1	2	yes	complemented
185	C₄₃	C₂×#1347	1	1	yes	characteristic, complemented, Sylow
186	C₂²×D₄	—	86	1	no	—
187	C₂²×D₄	—	172	1	no	—
188	C₂³×C₄	—	4	1	no	—
189	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
190	C₂×D₈	—	43	2	no	—
191	C₂×D₈	—	86	1	no	—
192	C₂×D₈	—	172	2	no	—
193	C₂²×C₈	—	4	2	no	—
194	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
195	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
196	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
197	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
200	#121	—	86	2	no	—
201	#121	—	86	2	no	—
202	#121	—	172	1	no	—
203	#121	—	172	1	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	4	2	no	—
207	C₂×C₄⋊C₄	—	4	1	no	—
208	C₂×C₄⋊C₄	—	86	1	no	—
209	C₂×C₂²⋊C₄	—	172	1	no	—
210	C₂×C₁₆	—	86	2	no	—
211	C₂×C₁₆	—	172	2	no	—
212	#107	—	43	2	no	—
213	#107	—	86	1	no	—
214	#102	—	86	2	no	—
215	#102	—	86	2	no	—
216	C₂⁴	—	172	1	no	—
217	C₂⁴	#10621882	1	1	yes	characteristic
218	C₂×D₄	—	86	2	no	—
219	C₂×D₄	—	344	2	no	—
220	C₂×D₄	—	344	2	no	—
221	C₂×D₄	—	172	2	no	—
222	C₂×D₄	—	172	2	no	—
223	C₂×D₄	—	172	2	no	—
224	C₂×D₄	—	172	1	no	—
225	C₂×D₄	—	172	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	8	2	no	—
231	C₂²×C₄	—	8	2	no	—
232	C₂²×C₄	—	172	1	no	—
233	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
234	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
235	C₂²×C₄	#10621882	1	1	yes	characteristic
236	C₂²×C₄	#10621884	1	1	yes	characteristic
237	C₂²×C₄	#10621884	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
239	D₈	—	172	2	no	—
240	D₈	—	172	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
248	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
250	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	86	2	no	—
255	C₄⋊C₄	—	172	1	no	—
256	C₂²⋊C₄	—	172	2	no	—
257	C₂²⋊C₄	—	172	2	no	—
258	C₁₆	—	172	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	344	2	no	—
262	C₂³	—	344	2	no	—
263	C₂³	—	344	2	no	—
264	C₂³	—	172	2	no	—
265	C₂³	#13063245	1	1	yes	characteristic
266	C₂³	#13063245	1	1	yes	characteristic
267	C₂³	#13063266	1	2	yes	—
268	C₂³	#13063268	1	2	yes	—
269	C₂³	C₂×#10621882	1	1	yes	characteristic
270	D₄	—	344	2	no	—
271	D₄	—	344	2	no	—
272	D₄	—	344	4	no	—
273	D₄	—	344	2	no	—
274	D₄	—	172	2	no	—
275	D₄	—	172	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	8	8	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	344	2	no	—
288	C₂×C₄	—	172	2	no	—
289	C₂×C₄	#13063239	1	2	yes	—
290	C₂×C₄	#13063245	1	1	yes	characteristic
291	C₂×C₄	#13063245	1	1	yes	characteristic
292	C₂×C₄	#13063253	1	2	yes	—
293	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	344	2	no	—
306	C₂²	—	344	2	no	—
307	C₂²	—	344	2	no	—
308	C₂²	—	688	4	no	—
309	C₂²	#14582632	1	2	yes	—
310	C₂²	#14582699	1	2	yes	—
311	C₂²	#14582699	1	2	yes	—
312	C₂²	#14582708	1	2	yes	—
313	C₂²	C₂×#13063245	1	1	yes	characteristic
314	C₂²	C₂×#13063266	1	1	yes	characteristic
315	C₂²	C₂×#13063268	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	4	no	—
319	C₄	—	344	2	no	—
320	C₄	#14582632	1	2	yes	—
321	C₄	C₂×#13063239	1	1	yes	characteristic
322	C₄	C₂×#13063253	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	2	2	no	—
326	C₂	—	688	2	no	—
327	C₂	#19132603	1	4	yes	direct product factor
328	C₂	C₂×#14582632	1	1	yes	characteristic
329	C₂	C₂×#14582699	1	1	yes	characteristic
330	C₂	C₂×#14582708	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063245
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#340	C₂
socle	C₂³×C₄₃	#102
prime radical	C₂×#14583135	C₂
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#340	D₄₃
hypocenter	C₄₃	C₂×#1347
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
2. C₂×C₂²⋊(C₄₃⋊C₁₆)	C₂	semidirect
3. #19132603	C₂	direct (N × H)
4. C₂×C₄₃×#340	C₂	semidirect
5. C₂²×D₃₄₄	C₄	semidirect
10. #14583135	C₄	semidirect
12. C₂×#13063259	C₄	semidirect
15. C₂²⋊(C₄₃⋊C₁₆)	C₂²	semidirect
19. C₄₃×#340	C₂²	semidirect
24. C₂×D₃₄₄	C₂×C₄	semidirect
34. #13063259	C₂×C₄	semidirect
160. C₂×#340	D₄₃	semidirect
165. C₂×C₄₃	#1347	semidirect
184. #340	C₂×D₄₃	semidirect
185. C₄₃	C₂×#1347	semidirect
327. C₂	#19132603	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×#14583135	2
C₂×C₂²⋊(C₄₃⋊C₁₆)	2
#19132603	2
C₂×C₄₃×#340	2
C₂×#1347	43

Subgroups — C2×#19132603 #25006815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19132603 #25006815