Subgroups — C₂×#20242902 #26198808

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689528	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689329	C₂	1	2	yes	maximal
3	C₂×#15689202	C₂	1	1	yes	maximal, characteristic
4	#20242902	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688870	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#334	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10625325	—	2	2	no	—
8	C₂²×#10625324	—	2	1	no	—
9	C₂×#13094024	C₂²	1	2	yes	—
10	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	#15689528	—	2	2	no	—
12	#15689528	C₂²	1	4	yes	complemented
13	#15689329	—	2	4	no	—
14	#15689329	C₂²	1	8	yes	—
15	#15689202	—	2	2	no	—
16	#15689202	C₂²	1	4	yes	—
17	C₂×#13093958	—	2	1	no	—
18	C₂×#13093958	—	2	1	no	—
19	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#13093949	—	2	2	no	—
21	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
22	#15688870	C₂²	1	8	yes	complemented
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#334	C₂²	1	4	yes	complemented
26	C₂²×Dic₉₄	—	2	2	no	—
27	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
28	C₂²×C₄×D₄₇	—	2	1	no	—
29	C₂×#10625325	—	2	8	no	—
30	C₂×#10625325	—	4	8	no	—
31	C₂×#10625324	—	2	4	no	—
32	C₂×#10625324	—	4	4	no	—
33	C₂²×C₄₇⋊C₈	—	2	1	no	—
34	#13094024	—	2	2	no	—
35	#13094024	—	2	2	no	—
36	#13094024	—	2	2	no	—
37	#13094024	C₂³	1	4	yes	—
38	C₄×D₁₈₈	—	2	1	no	—
39	C₄×D₁₈₈	—	2	1	no	—
40	C₄×D₁₈₈	D₄	1	2	yes	—
41	C₄×D₁₈₈	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	—	2	2	no	—
43	C₂×C₄⋊Dic₄₇	—	2	2	no	—
44	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
45	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
46	#13093958	—	2	4	no	—
47	#13093958	—	2	4	no	—
48	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
49	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
50	#13093949	—	2	8	no	—
51	(C₂×D₄₇)⋊C₈	—	2	4	no	—
52	C₂²×C₈×C₄₇	—	2	2	no	—
53	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₇	—	2	2	no	—
57	C₄⋊C₈×C₄₇	C₂³	1	4	yes	—
58	C₄×C₈×C₄₇	D₄	1	2	yes	—
59	C₄×C₈×C₄₇	C₂³	1	2	yes	—
60	C₂³×D₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
64	C₂²×Dic₄₇	—	2	1	no	—
65	C₂²×Dic₄₇	—	4	2	no	—
66	C₂×D₁₈₈	—	2	4	no	—
67	C₂×D₁₈₈	—	2	1	no	—
68	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
69	C₂×C₄×D₄₇	—	2	2	no	—
70	C₂×C₄×D₄₇	—	2	2	no	—
71	C₂×C₄×D₄₇	—	4	2	no	—
72	C₂×C₄×D₄₇	—	4	2	no	—
73	C₂×Dic₉₄	—	2	4	no	—
74	C₂×Dic₉₄	—	4	4	no	—
75	C₂×Dic₉₄	—	4	4	no	—
76	C₂×Dic₉₄	—	4	2	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	4	4	no	—
80	C₂×C₈×C₄₇	SD₁₆	1	4	yes	complemented
81	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₄₇	#41	1	1	yes	characteristic
86	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
87	(C₂×D₄₇)⋊C₄	—	2	2	no	—
88	(C₂×D₄₇)⋊C₄	—	2	2	no	—
89	C₄⋊Dic₄₇	—	2	1	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	4	2	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	#10625325	—	4	16	no	—
98	#10625324	—	4	8	no	—
99	C₂³×C₄₇	#121	1	1	yes	characteristic
100	C₂²×D₄₇	—	2	2	no	—
101	C₂²×D₄₇	—	4	2	no	—
102	C₂²×D₄₇	—	4	2	no	—
103	C₂²×D₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
110	C₂×C₄×C₄₇	#137	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
114	C₂×Dic₄₇	—	2	2	no	—
115	C₂×Dic₄₇	—	4	2	no	—
116	C₂×Dic₄₇	—	4	4	no	—
117	C₂×Dic₄₇	—	8	4	no	—
118	D₁₈₈	—	2	4	no	—
119	D₁₈₈	—	4	2	no	—
120	C₄×D₄₇	—	4	2	no	—
121	C₄×D₄₇	—	4	2	no	—
122	C₄×D₄₇	—	4	4	no	—
123	Dic₉₄	—	4	4	no	—
124	Dic₉₄	—	4	4	no	—
125	Dic₉₄	—	8	4	no	—
126	C₈×C₄₇	—	2	2	no	—
127	C₈×C₄₇	—	4	4	no	—
128	C₈×C₄₇	C₂×SD₁₆	1	4	yes	complemented
129	C₄₇⋊C₈	—	4	2	no	—
130	C₂×#1693	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	#461	1	2	yes	—
132	C₂²×C₄₇	#469	1	2	yes	—
133	C₂²×C₄₇	#473	1	2	yes	—
134	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
135	C₂×D₄₇	—	4	2	no	—
136	C₂×D₄₇	—	4	2	no	—
137	C₂×D₄₇	—	4	2	no	—
138	C₂×D₄₇	—	8	4	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	2	1	no	—
141	C₄×C₄₇	#461	1	2	yes	—
142	C₄×C₄₇	#469	1	2	yes	—
143	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
144	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
145	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
146	Dic₄₇	—	4	2	no	—
147	Dic₄₇	—	8	4	no	—
148	C₂×#500	—	47	1	no	—
149	C₂×#461	—	47	2	no	—
150	C₂×#436	—	47	1	no	—
151	#1693	—	47	8	no	—
152	C₂×#334	D₄₇	1	1	yes	characteristic, complemented
153	C₂×#332	—	47	2	no	—
154	C₂×C₄₇	#1693	1	4	yes	complemented
155	C₂×C₄₇	C₂×#461	1	1	yes	characteristic
156	C₂×C₄₇	C₂×#469	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
158	D₄₇	—	8	2	no	—
159	C₂²×SD₁₆	—	94	2	no	—
160	C₂²×M₁₆	—	94	1	no	—
161	C₂×C₄⋊Q₈	—	47	2	no	—
162	C₂×C₄×D₄	—	47	1	no	—
163	#500	—	47	4	no	—
164	#500	—	94	2	no	—
165	#461	—	47	8	no	—
166	#461	—	94	4	no	—
167	#436	—	47	4	no	—
168	#436	—	94	2	no	—
169	C₂×#107	—	94	1	no	—
170	C₂×#107	—	94	1	no	—
171	C₂×C₄⋊C₈	—	47	1	no	—
172	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
173	C₂×#103	—	94	2	no	—
174	C₂×C₂²⋊C₈	—	94	1	no	—
175	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
176	#334	C₂×D₄₇	1	4	yes	complemented
177	#332	—	47	8	no	—
178	C₄₇	C₂×#1693	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₈	—	94	2	no	—
180	C₂²×D₄	—	47	1	no	—
181	C₂³×C₄	—	94	1	no	—
182	C₂×SD₁₆	—	94	8	no	—
183	C₂×SD₁₆	—	188	8	no	—
184	C₂×M₁₆	—	94	4	no	—
185	C₂×M₁₆	—	188	4	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	94	1	no	—
188	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
189	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
190	C₄⋊Q₈	—	47	4	no	—
191	C₄⋊Q₈	—	94	2	no	—
192	C₄⋊Q₈	—	94	2	no	—
193	C₄⋊Q₈	—	94	2	no	—
194	C₄×D₄	—	47	2	no	—
195	C₄×D₄	—	47	2	no	—
196	C₄×D₄	—	94	1	no	—
197	C₄×D₄	—	94	1	no	—
198	C₂×C₄⋊C₄	—	47	1	no	—
199	C₂×C₄⋊C₄	—	94	2	no	—
200	C₂×C₄⋊C₄	—	94	2	no	—
201	C₂×C₂²⋊C₄	—	94	1	no	—
202	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
203	#107	—	94	4	no	—
204	#107	—	94	4	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	47	2	no	—
207	C₄⋊C₈	—	94	1	no	—
208	C₄⋊C₈	C₂²×D₄₇	1	4	yes	—
209	#103	—	94	8	no	—
210	C₂²⋊C₈	—	94	4	no	—
211	C₄×C₈	D₁₈₈	1	2	yes	—
212	C₄×C₈	C₂²×D₄₇	1	2	yes	—
213	C₂⁴	—	94	1	no	—
214	C₂×Q₈	—	94	4	no	—
215	C₂×Q₈	—	188	4	no	—
216	C₂×Q₈	—	188	4	no	—
217	C₂×Q₈	—	188	2	no	—
218	C₂×D₄	—	47	2	no	—
219	C₂×D₄	—	94	1	no	—
220	C₂×D₄	—	94	4	no	—
221	C₂²×C₄	—	94	1	no	—
222	C₂²×C₄	—	94	2	no	—
223	C₂²×C₄	—	94	2	no	—
224	C₂²×C₄	—	188	2	no	—
225	C₂²×C₄	—	188	2	no	—
226	C₂²×C₄	—	188	2	no	—
227	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
228	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	#10625354	1	1	yes	characteristic
230	SD₁₆	—	188	16	no	—
231	M₁₆	—	188	8	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	4	no	—
235	C₂×C₈	—	94	2	no	—
236	C₂×C₈	—	188	2	no	—
237	C₂×C₈	#10625325	1	4	yes	complemented
238	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
239	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
240	C₄⋊C₄	—	47	2	no	—
241	C₄⋊C₄	—	94	1	no	—
242	C₄⋊C₄	—	94	4	no	—
243	C₄⋊C₄	—	94	4	no	—
244	C₄⋊C₄	—	188	2	no	—
245	C₄⋊C₄	—	188	2	no	—
246	C₂²⋊C₄	—	94	2	no	—
247	C₂²⋊C₄	—	94	2	no	—
248	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
249	C₄²	D₄×D₄₇	1	1	yes	characteristic
250	C₄²	#10625354	1	1	yes	characteristic
251	C₄²	C₂³×D₄₇	1	1	yes	characteristic
252	C₂³	—	94	2	no	—
253	C₂³	—	188	2	no	—
254	C₂³	—	188	2	no	—
255	C₂³	—	188	2	no	—
256	C₂³	#13093999	1	1	yes	characteristic
257	Q₈	—	376	4	no	—
258	Q₈	—	188	4	no	—
259	Q₈	—	188	4	no	—
260	D₄	—	94	4	no	—
261	D₄	—	188	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	94	2	no	—
264	C₂×C₄	—	376	4	no	—
265	C₂×C₄	—	188	4	no	—
266	C₂×C₄	—	188	2	no	—
267	C₂×C₄	—	188	4	no	—
268	C₂×C₄	—	188	2	no	—
269	C₂×C₄	—	188	2	no	—
270	C₂×C₄	#13093999	1	1	yes	characteristic
271	C₂×C₄	#13093999	1	1	yes	characteristic
272	C₂×C₄	#13093999	1	1	yes	characteristic
273	C₂×C₄	C₂×#10625325	1	2	yes	—
274	C₂×C₄	#13094051	1	4	yes	—
275	C₂×C₄	#13094057	1	2	yes	—
276	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
278	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	4	4	no	—
281	C₈	—	188	2	no	—
282	C₈	C₂×#10625325	1	4	yes	complemented
283	C₂²	—	376	4	no	—
284	C₂²	—	188	2	no	—
285	C₂²	—	188	2	no	—
286	C₂²	—	188	2	no	—
287	C₂²	#15689329	1	2	yes	—
288	C₂²	#15689377	1	2	yes	—
289	C₂²	#15689399	1	2	yes	—
290	C₂²	C₂×#13093999	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	376	4	no	—
294	C₄	—	188	2	no	—
295	C₄	#15689329	1	2	yes	—
296	C₄	#15689377	1	2	yes	—
297	C₄	C₂²×#10625325	1	1	yes	characteristic
298	C₄	C₂×#13094051	1	2	yes	—
299	C₄	C₂×#13094057	1	1	yes	characteristic
300	C₂	—	376	2	no	—
301	C₂	#20242902	1	4	yes	direct product factor
302	C₂	C₂×#15689329	1	1	yes	characteristic
303	C₂	C₂×#15689377	1	1	yes	characteristic
304	C₂	C₂×#15689399	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093999
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#334	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	C₂×#334	D₄₇
hypocenter	C₄₇	C₂×#1693
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
1. C₂×#15689528	C₂	semidirect
4. #20242902	C₂	direct (N × H)
5. C₂×#15688870	C₂	semidirect
6. C₂×C₄₇×#334	C₂	semidirect
12. #15689528	C₂²	semidirect
22. #15688870	C₂²	semidirect
25. C₄₇×#334	C₂²	semidirect
80. C₂×C₈×C₄₇	SD₁₆	semidirect
128. C₈×C₄₇	C₂×SD₁₆	semidirect
152. C₂×#334	D₄₇	semidirect
154. C₂×C₄₇	#1693	semidirect
176. #334	C₂×D₄₇	semidirect
178. C₄₇	C₂×#1693	semidirect
237. C₂×C₈	#10625325	semidirect
282. C₈	C₂×#10625325	semidirect
301. C₂	#20242902	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×#15689528	2
C₂×#15689329	2
C₂×#15689202	2
#20242902	2
C₂×#15688870	2
C₂×C₄₇×#334	2
C₂×#1693	47

Subgroups — C2×#20242902 #26198808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242902 #26198808