Subgroups — C₂×#19134115 #25079129

Order 11008.

694 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583186	C₂	1	2	yes	maximal
2	C₂×#14582978	C₂	1	1	yes	maximal, characteristic
3	#19134115	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582650	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582648	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10621872	—	2	1	no	—
8	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
9	#14583186	—	2	4	no	—
10	#14583186	C₂²	1	8	yes	—
11	#14582978	—	2	2	no	—
12	#14582978	C₂²	1	4	yes	—
13	C₂×#13063256	—	2	2	no	—
14	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
16	C₂×#13063245	—	2	2	no	—
17	C₂×(C₂×D₄₃)⋊C₈	—	2	1	no	—
18	#14582650	C₂²	1	4	yes	complemented
19	#14582648	C₂²	1	8	yes	complemented
20	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
21	C₄₃×#500	—	2	2	no	—
22	C₄₃×#500	C₂²	1	4	yes	complemented
23	C₂×#107×C₄₃	—	2	1	no	—
24	C₂×#107×C₄₃	—	2	1	no	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₄₃	—	2	1	no	—
28	C₂×#10621872	—	2	4	no	—
29	C₂×#10621872	—	4	4	no	—
30	C₂²×C₄₃⋊C₈	—	2	1	no	—
31	C₂²×C₄₃⋊C₈	—	2	2	no	—
32	C₄×D₁₇₂	—	2	1	no	—
33	C₄×D₁₇₂	—	2	1	no	—
34	C₄×D₁₇₂	D₄	1	2	yes	—
35	C₄×D₁₇₂	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
37	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
38	#13063256	—	2	8	no	—
39	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
40	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
41	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
42	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
43	#13063245	—	2	8	no	—
44	(C₂×D₄₃)⋊C₈	—	2	4	no	—
45	C₂²×Q₈×C₄₃	—	2	2	no	—
46	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
48	C₄⋊Q₈×C₄₃	—	2	2	no	—
49	C₄⋊Q₈×C₄₃	—	2	2	no	—
50	C₄⋊Q₈×C₄₃	—	2	2	no	—
51	C₄⋊Q₈×C₄₃	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	#107×C₄₃	—	2	4	no	—
56	#107×C₄₃	—	2	4	no	—
57	C₄×C₈×C₄₃	D₄	1	2	yes	—
58	C₄×C₈×C₄₃	C₂³	1	2	yes	—
59	C₂³×D₄₃	—	2	1	no	—
60	C₂×Q₈×C₄₃	—	2	4	no	—
61	C₂×Q₈×C₄₃	—	4	2	no	—
62	C₂×Q₈×C₄₃	—	8	4	no	—
63	C₂²×C₄×C₄₃	—	4	2	no	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
67	C₂²×Dic₄₃	—	2	1	no	—
68	C₂×D₁₇₂	—	2	1	no	—
69	C₂×D₁₇₂	—	2	4	no	—
70	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
71	C₂×C₄×D₄₃	—	2	2	no	—
72	C₂×C₄×D₄₃	—	2	2	no	—
73	C₂×C₄×D₄₃	—	4	2	no	—
74	C₂×C₄×D₄₃	—	4	2	no	—
75	C₂×C₈×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
77	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
79	C₄⋊C₄×C₄₃	—	2	4	no	—
80	C₄⋊C₄×C₄₃	—	2	4	no	—
81	C₄⋊C₄×C₄₃	—	4	2	no	—
82	C₄⋊C₄×C₄₃	—	4	2	no	—
83	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₄₃	#41	1	1	yes	characteristic
86	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
87	(C₂×D₄₃)⋊C₄	—	2	2	no	—
88	(C₂×D₄₃)⋊C₄	—	2	2	no	—
89	C₄⋊Dic₄₃	—	2	1	no	—
90	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
91	C₂×C₄₃⋊C₈	—	2	2	no	—
92	C₂×C₄₃⋊C₈	—	2	4	no	—
93	C₂×C₄₃⋊C₈	—	4	2	no	—
94	C₂×C₄₃⋊C₈	—	4	4	no	—
95	#10621872	—	4	8	no	—
96	C₂³×C₄₃	#121	1	1	yes	characteristic
97	C₂²×D₄₃	—	2	2	no	—
98	C₂²×D₄₃	—	4	2	no	—
99	C₂²×D₄₃	—	4	2	no	—
100	C₂²×D₄₃	—	4	2	no	—
101	Q₈×C₄₃	—	8	4	no	—
102	Q₈×C₄₃	—	8	4	no	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	4	4	no	—
105	C₂×C₄×C₄₃	—	8	4	no	—
106	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
110	C₂×C₄×C₄₃	#137	1	2	yes	—
111	C₂×C₄×C₄₃	#137	1	4	yes	—
112	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
115	C₂×Dic₄₃	—	2	2	no	—
116	C₂×Dic₄₃	—	4	2	no	—
117	D₁₇₂	—	2	4	no	—
118	D₁₇₂	—	4	2	no	—
119	C₄×D₄₃	—	4	2	no	—
120	C₄×D₄₃	—	4	2	no	—
121	C₄×D₄₃	—	4	4	no	—
122	C₈×C₄₃	—	2	2	no	—
123	C₈×C₄₃	C₂×SD₁₆	1	4	yes	—
124	C₄₃⋊C₈	—	4	2	no	—
125	C₄₃⋊C₈	—	4	4	no	—
126	C₂×#1695	—	43	1	no	maximal, Sylow
127	C₂²×C₄₃	#461	1	2	yes	—
128	C₂²×C₄₃	#470	1	2	yes	—
129	C₂²×C₄₃	#472	1	2	yes	—
130	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
131	C₂×D₄₃	—	4	2	no	—
132	C₂×D₄₃	—	4	2	no	—
133	C₂×D₄₃	—	4	2	no	—
134	C₂×D₄₃	—	8	4	no	—
135	C₄×C₄₃	—	2	1	no	—
136	C₄×C₄₃	—	2	1	no	—
137	C₄×C₄₃	—	8	4	no	—
138	C₄×C₄₃	#461	1	2	yes	—
139	C₄×C₄₃	#470	1	2	yes	—
140	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
141	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
142	C₄×C₄₃	C₂×#137	1	2	yes	—
143	Dic₄₃	—	4	2	no	—
144	C₂×#500	D₄₃	1	1	yes	characteristic, complemented
145	C₂×#476	—	43	2	no	—
146	C₂×#436	—	43	1	no	—
147	#1695	—	43	8	no	—
148	C₂×#334	—	43	1	no	—
149	C₂×#333	—	43	2	no	—
150	C₂×C₄₃	#1695	1	4	yes	complemented
151	C₂×C₄₃	C₂×#461	1	1	yes	characteristic
152	C₂×C₄₃	C₂×#470	1	1	yes	characteristic
153	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
154	D₄₃	—	8	2	no	—
155	C₂²×M₁₆	—	86	1	no	—
156	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
157	C₂×C₄×D₄	—	43	1	no	—
158	#500	—	2	2	no	—
159	#500	C₂×D₄₃	1	4	yes	complemented
160	#476	—	43	8	no	—
161	#476	—	86	4	no	—
162	#436	—	43	4	no	—
163	#436	—	86	2	no	—
164	C₂×#107	—	2	1	no	—
165	C₂×#107	—	2	1	no	—
166	C₂×#106	—	86	2	no	—
167	C₂×C₄⋊C₈	—	43	1	no	—
168	C₂×C₄⋊C₈	—	43	2	no	—
169	C₂×#102	—	86	2	no	—
170	C₂×C₂²⋊C₈	—	86	1	no	—
171	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
172	#334	—	43	4	no	—
173	#333	—	43	8	no	—
174	C₄₃	C₂×#1695	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₈	—	2	2	no	—
176	C₂²×D₄	—	43	1	no	—
177	C₂³×C₄	—	86	1	no	—
178	C₂×M₁₆	—	86	4	no	—
179	C₂×M₁₆	—	172	4	no	—
180	C₂²×C₈	—	86	2	no	—
181	C₂²×C₈	—	86	1	no	—
182	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	—	2	2	no	—
187	C₄⋊Q₈	C₂²×D₄₃	1	4	yes	—
188	C₄×D₄	—	43	2	no	—
189	C₄×D₄	—	43	2	no	—
190	C₄×D₄	—	86	1	no	—
191	C₄×D₄	—	86	1	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	43	1	no	—
195	C₂×C₂²⋊C₄	—	86	1	no	—
196	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
197	#107	—	2	4	no	—
198	#107	—	2	4	no	—
199	#106	—	86	8	no	—
200	C₄⋊C₈	—	43	2	no	—
201	C₄⋊C₈	—	43	4	no	—
202	C₄⋊C₈	—	86	2	no	—
203	C₄⋊C₈	—	86	1	no	—
204	#102	—	86	8	no	—
205	C₂²⋊C₈	—	86	4	no	—
206	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
207	C₄×C₈	C₂²×D₄₃	1	2	yes	—
208	C₂⁴	—	86	1	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	8	4	no	—
212	C₂×D₄	—	43	2	no	—
213	C₂×D₄	—	86	1	no	—
214	C₂×D₄	—	86	4	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	86	1	no	—
217	C₂²×C₄	—	86	2	no	—
218	C₂²×C₄	—	86	2	no	—
219	C₂²×C₄	—	172	2	no	—
220	C₂²×C₄	—	172	2	no	—
221	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
222	C₂²×C₄	#10621900	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
224	M₁₆	—	172	8	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	86	4	no	—
227	C₂×C₈	—	86	2	no	—
228	C₂×C₈	—	172	2	no	—
229	C₂×C₈	—	172	4	no	—
230	C₂×C₈	#10621884	1	4	yes	—
231	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
232	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	43	2	no	—
238	C₄⋊C₄	—	86	1	no	—
239	C₂²⋊C₄	—	86	2	no	—
240	C₂²⋊C₄	—	86	2	no	—
241	C₄²	D₄×D₄₃	1	1	yes	characteristic
242	C₄²	#10621900	1	1	yes	characteristic
243	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
244	C₄²	C₂³×D₄₃	1	1	yes	characteristic
245	C₂³	—	86	2	no	—
246	C₂³	—	172	2	no	—
247	C₂³	—	172	2	no	—
248	C₂³	—	172	2	no	—
249	C₂³	#13063296	1	1	yes	characteristic
250	Q₈	—	8	4	no	—
251	Q₈	—	8	4	no	—
252	D₄	—	86	4	no	—
253	D₄	—	172	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	86	2	no	—
258	C₂×C₄	—	172	2	no	—
259	C₂×C₄	—	172	4	no	—
260	C₂×C₄	—	172	2	no	—
261	C₂×C₄	—	172	2	no	—
262	C₂×C₄	#13063296	1	1	yes	characteristic
263	C₂×C₄	#13063296	1	1	yes	characteristic
264	C₂×C₄	#13063296	1	1	yes	characteristic
265	C₂×C₄	C₂×#10621884	1	2	yes	—
266	C₂×C₄	#13063356	1	2	yes	—
267	C₂×C₄	#13063357	1	4	yes	—
268	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
270	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	172	2	no	—
273	C₈	—	172	4	no	—
274	C₈	C₂×#10621884	1	4	yes	—
275	C₂²	—	344	4	no	—
276	C₂²	—	172	2	no	—
277	C₂²	—	172	2	no	—
278	C₂²	—	172	2	no	—
279	C₂²	#14583101	1	2	yes	—
280	C₂²	#14583154	1	2	yes	—
281	C₂²	#14583166	1	2	yes	—
282	C₂²	C₂×#13063296	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	8	4	no	—
286	C₄	—	172	2	no	—
287	C₄	#14583101	1	2	yes	—
288	C₄	#14583154	1	2	yes	—
289	C₄	C₂²×#10621884	1	1	yes	characteristic
290	C₄	C₂×#13063356	1	1	yes	characteristic
291	C₄	C₂×#13063357	1	2	yes	—
292	C₂	—	344	2	no	—
293	C₂	#19134115	1	4	yes	direct product factor
294	C₂	C₂×#14583101	1	1	yes	characteristic
295	C₂	C₂×#14583154	1	1	yes	characteristic
296	C₂	C₂×#14583166	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063296
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#500	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂²×D₁₇₂	D₄
hypercenter	C₂×#500	D₄₃
hypocenter	C₄₃	C₂×#1695
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #19134115	C₂	direct (N × H)
4. C₂×#14582650	C₂	semidirect
5. C₂×#14582648	C₂	semidirect
6. C₂×C₄₃×#500	C₂	semidirect
18. #14582650	C₂²	semidirect
19. #14582648	C₂²	semidirect
22. C₄₃×#500	C₂²	semidirect
144. C₂×#500	D₄₃	semidirect
150. C₂×C₄₃	#1695	semidirect
159. #500	C₂×D₄₃	semidirect
174. C₄₃	C₂×#1695	semidirect
293. C₂	#19134115	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×#14583186	2
C₂×#14582978	2
#19134115	2
C₂×#14582650	2
C₂×#14582648	2
C₂×C₄₃×#500	2
C₂×#1695	43

Subgroups — C2×#19134115 #25079129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19134115 #25079129