Subgroups — C₂×#19134074 #25079098

Order 11008.

926 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583276	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583094	C₂	1	2	yes	maximal, complemented
3	#19134074	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582650	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582642	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₄₄	—	2	1	no	—
8	C₂²×D₃₄₄	—	2	1	no	—
9	C₂²×#10621882	—	2	2	no	—
10	C₂×#13063322	C₂²	1	2	yes	—
11	#14583276	—	2	2	no	—
12	#14583276	C₂²	1	4	yes	complemented
13	#14583094	—	2	4	no	—
14	#14583094	C₂²	1	8	yes	complemented
15	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
16	C₂×#13063245	—	2	2	no	—
17	#14582650	C₂²	1	4	yes	complemented
18	#14582642	C₂²	1	8	yes	complemented
19	C₂²×M₁₆×C₄₃	—	2	1	no	—
20	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
21	C₄₃×#436	—	2	2	no	—
22	C₄₃×#436	C₂²	1	4	yes	complemented
23	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₄₃	—	2	1	no	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×D₁₇₂	—	2	2	no	—
27	C₂²×D₁₇₂	—	2	2	no	—
28	C₂²×D₁₇₂	—	2	2	no	—
29	C₂×D₃₄₄	—	2	4	no	—
30	C₂×D₃₄₄	—	2	4	no	—
31	C₂×D₃₄₄	—	4	4	no	—
32	C₂×D₃₄₄	—	4	4	no	—
33	C₂×#10621882	—	2	8	no	—
34	C₂×#10621882	—	4	8	no	—
35	C₂²×C₄₃⋊C₈	—	2	2	no	—
36	#13063322	—	2	2	no	—
37	#13063322	—	2	2	no	—
38	#13063322	—	2	2	no	—
39	#13063322	C₂³	1	4	yes	—
40	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
41	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
42	#13063245	—	2	8	no	—
43	C₂²×D₄×C₄₃	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₃	—	2	1	no	—
45	C₂×M₁₆×C₄₃	—	2	4	no	—
46	C₂×M₁₆×C₄₃	—	4	4	no	—
47	C₂²×C₈×C₄₃	—	2	1	no	—
48	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
50	C₄×D₄×C₄₃	—	2	1	no	—
51	C₄×D₄×C₄₃	—	2	1	no	—
52	C₄×D₄×C₄₃	D₄	1	2	yes	—
53	C₄×D₄×C₄₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
55	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₃	—	2	1	no	—
58	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
59	C₂²⋊C₈×C₄₃	—	2	4	no	—
60	C₄×C₈×C₄₃	D₄	1	2	yes	—
61	C₄×C₈×C₄₃	C₂³	1	2	yes	—
62	C₂⁴×C₄₃	—	2	1	no	—
63	C₂³×D₄₃	—	4	2	no	—
64	C₂×D₄×C₄₃	—	2	1	no	—
65	C₂×D₄×C₄₃	—	2	4	no	—
66	C₂×D₄×C₄₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₃	—	2	2	no	—
68	C₂²×C₄×C₄₃	—	2	2	no	—
69	C₂²×C₄×C₄₃	—	2	1	no	—
70	C₂²×C₄×C₄₃	—	4	2	no	—
71	C₂²×C₄×C₄₃	—	4	2	no	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
75	C₂×D₁₇₂	—	2	4	no	—
76	C₂×D₁₇₂	—	2	4	no	—
77	C₂×D₁₇₂	—	2	4	no	—
78	C₂×D₁₇₂	—	4	2	no	—
79	C₂×D₁₇₂	—	4	2	no	—
80	C₂×D₁₇₂	—	4	2	no	—
81	C₂×D₁₇₂	—	4	4	no	—
82	C₂×D₁₇₂	—	4	4	no	—
83	C₂×D₁₇₂	—	4	4	no	—
84	C₂×D₁₇₂	—	4	4	no	—
85	C₂×D₁₇₂	—	8	4	no	—
86	M₁₆×C₄₃	—	4	8	no	—
87	C₂×C₈×C₄₃	—	2	2	no	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	4	2	no	—
90	C₂×C₈×C₄₃	D₈	1	4	yes	complemented
91	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
93	C₄⋊C₄×C₄₃	—	2	1	no	—
94	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
95	C₂²⋊C₄×C₄₃	—	2	2	no	—
96	C₂²⋊C₄×C₄₃	—	2	2	no	—
97	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₃	#41	1	1	yes	characteristic
100	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
101	#10621882	—	4	16	no	—
102	C₂×C₄₃⋊C₈	—	2	4	no	—
103	C₂×C₄₃⋊C₈	—	4	4	no	—
104	D₃₄₄	—	4	8	no	—
105	D₃₄₄	—	4	8	no	—
106	C₂³×C₄₃	—	2	2	no	—
107	C₂³×C₄₃	—	4	2	no	—
108	C₂³×C₄₃	—	4	2	no	—
109	C₂³×C₄₃	—	4	2	no	—
110	C₂³×C₄₃	#121	1	1	yes	characteristic
111	C₂²×D₄₃	—	4	4	no	—
112	C₂²×D₄₃	—	8	4	no	—
113	C₂²×D₄₃	—	8	4	no	—
114	C₂²×D₄₃	—	8	4	no	—
115	D₄×C₄₃	—	2	4	no	—
116	D₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	4	2	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	—	4	4	no	—
123	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
129	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
132	D₁₇₂	—	4	4	no	—
133	D₁₇₂	—	4	4	no	—
134	D₁₇₂	—	4	4	no	—
135	D₁₇₂	—	4	4	no	—
136	D₁₇₂	—	8	4	no	—
137	D₁₇₂	—	8	4	no	—
138	D₁₇₂	—	8	4	no	—
139	D₁₇₂	—	8	4	no	—
140	C₈×C₄₃	—	2	2	no	—
141	C₈×C₄₃	—	4	2	no	—
142	C₈×C₄₃	C₂×D₈	1	4	yes	complemented
143	C₄₃⋊C₈	—	4	4	no	—
144	C₂×#1687	—	43	1	no	maximal, Sylow
145	C₂²×C₄₃	—	4	2	no	—
146	C₂²×C₄₃	—	4	2	no	—
147	C₂²×C₄₃	—	4	2	no	—
148	C₂²×C₄₃	—	8	4	no	—
149	C₂²×C₄₃	#459	1	2	yes	—
150	C₂²×C₄₃	#469	1	2	yes	—
151	C₂²×C₄₃	#472	1	2	yes	—
152	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
153	C₂×D₄₃	—	8	4	no	—
154	C₂×D₄₃	—	8	4	no	—
155	C₂×D₄₃	—	8	4	no	—
156	C₂×D₄₃	—	16	8	no	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	—	4	2	no	—
160	C₄×C₄₃	#459	1	2	yes	—
161	C₄×C₄₃	#469	1	2	yes	—
162	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
165	C₂×#493	—	43	1	no	—
166	C₂×#459	—	43	2	no	—
167	C₂×#436	D₄₃	1	1	yes	characteristic, complemented
168	#1687	—	43	8	no	—
169	C₂×#334	—	43	1	no	—
170	C₂×#331	—	43	2	no	—
171	C₂×C₄₃	—	8	2	no	—
172	C₂×C₄₃	#1687	1	4	yes	complemented
173	C₂×C₄₃	C₂×#459	1	1	yes	characteristic
174	C₂×C₄₃	C₂×#469	1	1	yes	characteristic
175	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
176	D₄₃	—	16	4	no	—
177	C₂²×D₈	—	86	2	no	—
178	C₂²×D₈	—	86	1	no	—
179	C₂²×D₈	—	86	1	no	—
180	C₂²×M₁₆	—	2	1	no	—
181	C₂×#127	—	43	2	no	—
182	C₂×C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
183	#493	—	43	4	no	—
184	#493	—	86	2	no	—
185	#459	—	43	8	no	—
186	#459	—	86	4	no	—
187	#436	—	2	2	no	—
188	#436	C₂×D₄₃	1	4	yes	complemented
189	C₂×C₄⋊C₈	—	43	2	no	—
190	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
191	C₂×#102	—	86	2	no	—
192	C₂×C₂²⋊C₈	—	2	1	no	—
193	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
194	#334	—	43	4	no	—
195	#331	—	43	8	no	—
196	C₄₃	C₂×#1687	1	1	yes	characteristic, complemented, Sylow
197	C₂²×D₄	—	86	2	no	—
198	C₂²×D₄	—	86	2	no	—
199	C₂²×D₄	—	86	2	no	—
200	C₂²×D₄	D₁₇₂	1	1	yes	characteristic
201	C₂³×C₄	—	2	1	no	—
202	C₂×D₈	—	86	8	no	—
203	C₂×D₈	—	86	4	no	—
204	C₂×D₈	—	86	4	no	—
205	C₂×D₈	—	172	4	no	—
206	C₂×D₈	—	172	8	no	—
207	C₂×D₈	—	172	4	no	—
208	C₂×M₁₆	—	2	4	no	—
209	C₂×M₁₆	—	4	4	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	86	2	no	—
212	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
214	#127	—	43	4	no	—
215	#127	—	86	2	no	—
216	#127	—	86	2	no	—
217	#127	—	86	2	no	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	—	2	1	no	—
220	C₄×D₄	D₁₇₂	1	2	yes	—
221	C₄×D₄	C₂²×D₄₃	1	2	yes	—
222	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	—	2	1	no	—
224	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	43	4	no	—
227	C₄⋊C₈	—	86	2	no	—
228	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
229	#102	—	86	8	no	—
230	C₂²⋊C₈	—	2	4	no	—
231	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
232	C₄×C₈	C₂²×D₄₃	1	2	yes	—
233	C₂⁴	—	2	1	no	—
234	C₂⁴	—	172	2	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	86	4	no	—
238	C₂×D₄	—	86	4	no	—
239	C₂×D₄	—	86	4	no	—
240	C₂×D₄	—	344	4	no	—
241	C₂×D₄	—	172	2	no	—
242	C₂×D₄	—	172	2	no	—
243	C₂×D₄	—	172	2	no	—
244	C₂×D₄	—	172	4	no	—
245	C₂×D₄	—	172	4	no	—
246	C₂×D₄	—	172	4	no	—
247	C₂×D₄	—	172	4	no	—
248	C₂×D₄	C₂×D₁₇₂	1	2	yes	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
255	C₂²×C₄	#10621898	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
257	D₈	—	172	8	no	—
258	D₈	—	172	16	no	—
259	D₈	—	172	8	no	—
260	M₁₆	—	4	8	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	86	4	no	—
265	C₂×C₈	—	172	4	no	—
266	C₂×C₈	#10621882	1	4	yes	complemented
267	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
268	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	C₂×D₁₇₂	1	2	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
274	C₄²	#10621898	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
276	C₄²	C₂³×D₄₃	1	1	yes	characteristic
277	C₂³	—	2	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	344	4	no	—
282	C₂³	—	344	4	no	—
283	C₂³	—	344	4	no	—
284	C₂³	—	172	4	no	—
285	C₂³	#13063299	1	1	yes	characteristic
286	D₄	—	2	4	no	—
287	D₄	—	4	2	no	—
288	D₄	—	344	4	no	—
289	D₄	—	344	4	no	—
290	D₄	—	344	4	no	—
291	D₄	—	344	4	no	—
292	D₄	—	172	4	no	—
293	D₄	—	172	4	no	—
294	D₄	—	172	4	no	—
295	D₄	—	172	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	#13063299	1	1	yes	characteristic
303	C₂×C₄	#13063299	1	1	yes	characteristic
304	C₂×C₄	#13063299	1	1	yes	characteristic
305	C₂×C₄	C₂×#10621882	1	2	yes	—
306	C₂×C₄	#13063350	1	2	yes	—
307	C₂×C₄	#13063353	1	4	yes	—
308	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
310	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	4	2	no	—
313	C₈	—	172	4	no	—
314	C₈	C₂×#10621882	1	4	yes	complemented
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	8	4	no	—
319	C₂²	—	344	4	no	—
320	C₂²	—	344	4	no	—
321	C₂²	—	344	4	no	—
322	C₂²	—	688	8	no	—
323	C₂²	#14583094	1	2	yes	—
324	C₂²	#14583152	1	2	yes	—
325	C₂²	#14583169	1	2	yes	—
326	C₂²	C₂×#13063299	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	4	2	no	—
330	C₄	#14583094	1	2	yes	—
331	C₄	#14583152	1	2	yes	—
332	C₄	C₂²×#10621882	1	1	yes	characteristic
333	C₄	C₂×#13063350	1	1	yes	characteristic
334	C₄	C₂×#13063353	1	2	yes	—
335	C₂	—	8	2	no	—
336	C₂	—	688	4	no	—
337	C₂	#19134074	1	4	yes	direct product factor
338	C₂	C₂×#14583094	1	1	yes	characteristic
339	C₂	C₂×#14583152	1	1	yes	characteristic
340	C₂	C₂×#14583169	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063299
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#436	C₂
socle	C₂³×C₄₃	#121
hypercenter	C₂×#436	D₄₃
hypocenter	C₄₃	C₂×#1687
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. C₂×#14583276	C₂	semidirect
2. C₂×#14583094	C₂	semidirect
3. #19134074	C₂	direct (N × H)
4. C₂×#14582650	C₂	semidirect
5. C₂×#14582642	C₂	semidirect
6. C₂×C₄₃×#436	C₂	semidirect
12. #14583276	C₂²	semidirect
14. #14583094	C₂²	semidirect
17. #14582650	C₂²	semidirect
18. #14582642	C₂²	semidirect
22. C₄₃×#436	C₂²	semidirect
90. C₂×C₈×C₄₃	D₈	semidirect
142. C₈×C₄₃	C₂×D₈	semidirect
167. C₂×#436	D₄₃	semidirect
172. C₂×C₄₃	#1687	semidirect
188. #436	C₂×D₄₃	semidirect
196. C₄₃	C₂×#1687	semidirect
266. C₂×C₈	#10621882	semidirect
314. C₈	C₂×#10621882	semidirect
337. C₂	#19134074	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×#14583276	2
C₂×#14583094	2
#19134074	2
C₂×#14582650	2
C₂×#14582642	2
C₂×C₄₃×#436	2
C₂×#1687	43

Subgroups — C2×#19134074 #25079098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19134074 #25079098