Subgroups — C₂×#19134080 #25079198

Order 11008.

850 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	C₂×#14583276	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583100	C₂	1	2	yes	maximal, complemented
3	C₂×#14583029	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134080	C₂	1	8	yes	maximal, direct product factor
5	C₂×#14582642	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10621873	—	2	2	no	—
8	C₂²×D₃₄₄	—	2	1	no	—
9	C₂²×D₃₄₄	—	2	1	no	—
10	C₂×#13063322	C₂²	1	2	yes	—
11	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
12	#14583276	—	2	2	no	—
13	#14583276	C₂²	1	4	yes	—
14	#14583100	—	2	4	no	—
15	#14583100	C₂²	1	8	yes	complemented
16	#14583029	—	2	2	no	—
17	#14583029	C₂²	1	4	yes	complemented
18	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₃⋊C₈	—	2	1	no	—
20	C₂×#13063247	—	2	2	no	—
21	C₂×#13063235	—	2	1	no	—
22	#14582642	C₂²	1	8	yes	complemented
23	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#334	C₂²	1	4	yes	complemented
26	C₂²×Dic₈₆	D₄	1	1	yes	characteristic
27	C₂²×D₁₇₂	—	2	2	no	—
28	C₂²×D₁₇₂	—	2	2	no	—
29	C₂²×D₁₇₂	—	2	2	no	—
30	C₂×#10621873	—	2	8	no	—
31	C₂×#10621873	—	4	8	no	—
32	C₂×D₃₄₄	—	2	4	no	—
33	C₂×D₃₄₄	—	2	4	no	—
34	C₂×D₃₄₄	—	4	4	no	—
35	C₂×D₃₄₄	—	4	4	no	—
36	C₂²×C₄₃⋊C₈	—	2	1	no	—
37	#13063322	—	2	2	no	—
38	#13063322	—	2	2	no	—
39	#13063322	—	2	2	no	—
40	#13063322	C₂³	1	4	yes	—
41	C₄×Dic₈₆	—	2	1	no	—
42	C₄×Dic₈₆	—	2	1	no	—
43	C₄×Dic₈₆	D₄	1	2	yes	—
44	C₄×Dic₈₆	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
46	C₂×Dic₄₃⋊C₄	—	2	1	no	—
47	C₂×C₄×Dic₄₃	—	2	1	no	—
48	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
49	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
50	Dic₄₃⋊C₈	—	2	4	no	—
51	#13063247	—	2	8	no	—
52	#13063235	—	2	4	no	—
53	C₂²×C₈×C₄₃	—	2	2	no	—
54	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₃	—	2	2	no	—
58	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
59	C₄×C₈×C₄₃	D₄	1	2	yes	—
60	C₄×C₈×C₄₃	C₂³	1	2	yes	—
61	C₂³×D₄₃	—	4	2	no	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
65	C₂²×Dic₄₃	—	2	1	no	—
66	C₂²×Dic₄₃	—	2	1	no	—
67	C₂×D₁₇₂	—	2	4	no	—
68	C₂×D₁₇₂	—	2	4	no	—
69	C₂×D₁₇₂	—	2	4	no	—
70	C₂×D₁₇₂	—	4	2	no	—
71	C₂×D₁₇₂	—	4	2	no	—
72	C₂×D₁₇₂	—	4	2	no	—
73	C₂×D₁₇₂	—	4	4	no	—
74	C₂×D₁₇₂	—	4	4	no	—
75	C₂×D₁₇₂	—	4	4	no	—
76	C₂×D₁₇₂	—	4	4	no	—
77	C₂×D₁₇₂	—	8	4	no	—
78	C₂×Dic₈₆	—	2	4	no	—
79	C₂×Dic₈₆	—	2	1	no	—
80	C₂×Dic₈₆	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	—	2	4	no	—
83	C₂×C₈×C₄₃	—	4	4	no	—
84	C₂×C₈×C₄₃	SD₁₆	1	4	yes	complemented
85	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
87	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₃	#41	1	1	yes	characteristic
90	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
91	C₄⋊Dic₄₃	—	2	1	no	—
92	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
93	Dic₄₃⋊C₄	—	2	2	no	—
94	Dic₄₃⋊C₄	—	2	2	no	—
95	C₄×Dic₄₃	—	2	2	no	—
96	C₄×Dic₄₃	—	2	2	no	—
97	C₂×C₄₃⋊C₈	—	2	2	no	—
98	C₂×C₄₃⋊C₈	—	4	2	no	—
99	D₃₄₄	—	4	8	no	—
100	D₃₄₄	—	4	8	no	—
101	#10621873	—	4	16	no	—
102	C₂³×C₄₃	#121	1	1	yes	characteristic
103	C₂²×D₄₃	—	4	4	no	—
104	C₂²×D₄₃	—	8	4	no	—
105	C₂²×D₄₃	—	8	4	no	—
106	C₂²×D₄₃	—	8	4	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
112	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
113	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₄₃	—	2	2	no	—
118	C₂×Dic₄₃	—	2	2	no	—
119	C₂×Dic₄₃	—	4	2	no	—
120	C₂×Dic₄₃	—	4	2	no	—
121	D₁₇₂	—	4	4	no	—
122	D₁₇₂	—	4	4	no	—
123	D₁₇₂	—	4	4	no	—
124	D₁₇₂	—	4	4	no	—
125	D₁₇₂	—	8	4	no	—
126	D₁₇₂	—	8	4	no	—
127	D₁₇₂	—	8	4	no	—
128	D₁₇₂	—	8	4	no	—
129	Dic₈₆	—	2	4	no	—
130	Dic₈₆	—	4	2	no	—
131	C₈×C₄₃	—	2	2	no	—
132	C₈×C₄₃	—	4	4	no	—
133	C₈×C₄₃	C₂×SD₁₆	1	4	yes	complemented
134	C₄₃⋊C₈	—	4	2	no	—
135	C₂×#1688	—	43	1	no	maximal, Sylow
136	C₂²×C₄₃	#460	1	2	yes	—
137	C₂²×C₄₃	#469	1	2	yes	—
138	C₂²×C₄₃	#472	1	2	yes	—
139	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
140	C₂×D₄₃	—	8	4	no	—
141	C₂×D₄₃	—	8	4	no	—
142	C₂×D₄₃	—	8	4	no	—
143	C₂×D₄₃	—	16	8	no	—
144	C₄×C₄₃	—	2	1	no	—
145	C₄×C₄₃	—	2	1	no	—
146	C₄×C₄₃	#460	1	2	yes	—
147	C₄×C₄₃	#469	1	2	yes	—
148	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
149	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
150	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	Dic₄₃	—	4	2	no	—
152	Dic₄₃	—	4	2	no	—
153	C₂×#493	—	43	1	no	—
154	C₂×#460	—	43	2	no	—
155	C₂×#446	—	43	1	no	—
156	#1688	—	43	8	no	—
157	C₂×#334	D₄₃	1	1	yes	characteristic, complemented
158	C₂×#331	—	43	2	no	—
159	C₂×C₄₃	#1688	1	4	yes	complemented
160	C₂×C₄₃	C₂×#460	1	1	yes	characteristic
161	C₂×C₄₃	C₂×#469	1	1	yes	characteristic
162	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
163	D₄₃	—	16	4	no	—
164	C₂²×SD₁₆	—	86	2	no	—
165	C₂²×D₈	—	86	1	no	—
166	C₂²×D₈	—	86	1	no	—
167	C₂×#127	—	43	2	no	—
168	C₂×C₄×Q₈	—	43	1	no	—
169	#493	—	43	4	no	—
170	#493	—	86	2	no	—
171	#460	—	43	8	no	—
172	#460	—	86	4	no	—
173	#446	—	43	4	no	—
174	#446	—	86	2	no	—
175	C₂×C₄⋊C₈	—	43	1	no	—
176	C₂×C₄⋊C₈	—	86	1	no	—
177	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
178	C₂×#102	—	86	2	no	—
179	C₂×#97	—	86	1	no	—
180	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
181	#334	C₂×D₄₃	1	4	yes	complemented
182	#331	—	43	8	no	—
183	C₄₃	C₂×#1688	1	1	yes	characteristic, complemented, Sylow
184	C₂²×Q₈	—	43	1	no	—
185	C₂²×D₄	—	86	2	no	—
186	C₂²×D₄	—	86	2	no	—
187	C₂²×D₄	—	86	2	no	—
188	C₂×SD₁₆	—	86	8	no	—
189	C₂×SD₁₆	—	172	8	no	—
190	C₂×D₈	—	86	4	no	—
191	C₂×D₈	—	86	4	no	—
192	C₂×D₈	—	172	4	no	—
193	C₂×D₈	—	172	4	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	86	1	no	—
196	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
197	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
198	#127	—	43	4	no	—
199	#127	—	86	2	no	—
200	#127	—	86	2	no	—
201	#127	—	86	2	no	—
202	C₄×Q₈	—	43	2	no	—
203	C₄×Q₈	—	43	2	no	—
204	C₄×Q₈	—	86	1	no	—
205	C₄×Q₈	—	86	1	no	—
206	C₂×C₄⋊C₄	—	43	1	no	—
207	C₂×C₄⋊C₄	—	86	1	no	—
208	C₂×C₄²	—	86	1	no	—
209	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	43	2	no	—
212	C₄⋊C₈	—	86	4	no	—
213	C₄⋊C₈	—	86	1	no	—
214	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
215	#102	—	86	8	no	—
216	#97	—	86	4	no	—
217	C₄×C₈	D₁₇₂	1	2	yes	—
218	C₄×C₈	C₂²×D₄₃	1	2	yes	—
219	C₂⁴	—	172	2	no	—
220	C₂×Q₈	—	43	2	no	—
221	C₂×Q₈	—	86	1	no	—
222	C₂×Q₈	—	86	4	no	—
223	C₂×D₄	—	86	4	no	—
224	C₂×D₄	—	86	4	no	—
225	C₂×D₄	—	86	4	no	—
226	C₂×D₄	—	344	4	no	—
227	C₂×D₄	—	172	2	no	—
228	C₂×D₄	—	172	2	no	—
229	C₂×D₄	—	172	2	no	—
230	C₂×D₄	—	172	4	no	—
231	C₂×D₄	—	172	4	no	—
232	C₂×D₄	—	172	4	no	—
233	C₂×D₄	—	172	4	no	—
234	C₂²×C₄	—	86	1	no	—
235	C₂²×C₄	—	86	1	no	—
236	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
238	C₂²×C₄	#10621902	1	1	yes	characteristic
239	SD₁₆	—	172	16	no	—
240	D₈	—	172	8	no	—
241	D₈	—	172	8	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	4	no	—
245	C₂×C₈	—	86	2	no	—
246	C₂×C₈	—	172	2	no	—
247	C₂×C₈	#10621873	1	4	yes	complemented
248	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
249	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
250	C₄⋊C₄	—	43	2	no	—
251	C₄⋊C₄	—	86	1	no	—
252	C₄⋊C₄	—	86	2	no	—
253	C₄⋊C₄	—	86	2	no	—
254	C₄²	—	86	2	no	—
255	C₄²	—	86	2	no	—
256	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
257	C₄²	D₄×D₄₃	1	1	yes	characteristic
258	C₄²	#10621902	1	1	yes	characteristic
259	C₄²	C₂³×D₄₃	1	1	yes	characteristic
260	C₂³	—	344	4	no	—
261	C₂³	—	344	4	no	—
262	C₂³	—	344	4	no	—
263	C₂³	—	172	4	no	—
264	C₂³	#13063300	1	1	yes	characteristic
265	Q₈	—	86	4	no	—
266	Q₈	—	172	2	no	—
267	D₄	—	344	4	no	—
268	D₄	—	344	4	no	—
269	D₄	—	344	4	no	—
270	D₄	—	344	4	no	—
271	D₄	—	172	4	no	—
272	D₄	—	172	4	no	—
273	D₄	—	172	4	no	—
274	D₄	—	172	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	86	2	no	—
277	C₂×C₄	—	86	2	no	—
278	C₂×C₄	—	172	2	no	—
279	C₂×C₄	—	172	2	no	—
280	C₂×C₄	#13063300	1	1	yes	characteristic
281	C₂×C₄	#13063300	1	1	yes	characteristic
282	C₂×C₄	#13063300	1	1	yes	characteristic
283	C₂×C₄	C₂×#10621873	1	2	yes	—
284	C₂×C₄	#13063352	1	4	yes	—
285	C₂×C₄	#13063353	1	2	yes	—
286	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
288	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	4	4	no	—
291	C₈	—	172	2	no	—
292	C₈	C₂×#10621873	1	4	yes	complemented
293	C₂²	—	344	4	no	—
294	C₂²	—	344	4	no	—
295	C₂²	—	344	4	no	—
296	C₂²	—	688	8	no	—
297	C₂²	#14583100	1	2	yes	—
298	C₂²	#14583153	1	2	yes	—
299	C₂²	#14583170	1	2	yes	—
300	C₂²	C₂×#13063300	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	172	2	no	—
304	C₄	—	172	2	no	—
305	C₄	#14583100	1	2	yes	—
306	C₄	#14583153	1	2	yes	—
307	C₄	C₂²×#10621873	1	1	yes	characteristic
308	C₄	C₂×#13063352	1	2	yes	—
309	C₄	C₂×#13063353	1	1	yes	characteristic
310	C₂	—	688	4	no	—
311	C₂	#19134080	1	4	yes	direct product factor
312	C₂	C₂×#14583100	1	1	yes	characteristic
313	C₂	C₂×#14583153	1	1	yes	characteristic
314	C₂	C₂×#14583170	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063300
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#334	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂×#14583276	C₂
hypercenter	C₂×#334	D₄₃
hypocenter	C₄₃	C₂×#1688
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
2. C₂×#14583100	C₂	semidirect
3. C₂×#14583029	C₂	semidirect
4. #19134080	C₂	direct (N × H)
5. C₂×#14582642	C₂	semidirect
6. C₂×C₄₃×#334	C₂	semidirect
15. #14583100	C₂²	semidirect
17. #14583029	C₂²	semidirect
22. #14582642	C₂²	semidirect
25. C₄₃×#334	C₂²	semidirect
84. C₂×C₈×C₄₃	SD₁₆	semidirect
133. C₈×C₄₃	C₂×SD₁₆	semidirect
157. C₂×#334	D₄₃	semidirect
159. C₂×C₄₃	#1688	semidirect
181. #334	C₂×D₄₃	semidirect
183. C₄₃	C₂×#1688	semidirect
247. C₂×C₈	#10621873	semidirect
292. C₈	C₂×#10621873	semidirect
311. C₂	#19134080	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×#14583100	2
C₂×#14583029	2
#19134080	2
C₂×#14582642	2
C₂×C₄₃×#334	2
C₂×#1688	43

Subgroups — C2×#19134080 #25079198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19134080 #25079198