Subgroups — C₂×#20244213 #26219933

Order 12032.

2286 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	C₂²×#13093945	C₂	1	4	yes	maximal, complemented
2	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688989	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244213	C₂	1	8	yes	maximal, direct product factor
5	C₂²×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂×#13093945	—	2	16	no	—
9	C₂×#13093945	C₂²	1	16	yes	complemented
10	C₂×#13093935	—	2	4	no	—
11	C₂×#13093935	C₂²	1	4	yes	complemented
12	#15688989	—	2	2	no	—
13	#15688989	C₂²	1	4	yes	complemented
14	C₂×C₄₇⋊#95	—	2	2	no	—
15	C₂³×D₄×C₄₇	—	2	2	no	—
16	C₂³×D₄×C₄₇	C₂²	1	2	yes	—
17	C₂×#127×C₄₇	—	2	2	no	—
18	C₂×#127×C₄₇	—	2	4	no	—
19	C₂×#127×C₄₇	C₄	1	8	yes	complemented
20	C₂×#127×C₄₇	C₂²	1	4	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂³×Dic₄₇	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	D₄	1	8	yes	complemented
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	C₂×C₄⋊Dic₄₇	—	2	8	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
30	#13093945	—	2	32	no	—
31	#13093935	—	2	8	no	—
32	C₄₇⋊#95	—	2	8	no	—
33	C₂⁵×C₄₇	—	2	2	no	—
34	C₂²×D₄×C₄₇	—	2	16	no	—
35	C₂²×D₄×C₄₇	—	2	8	no	—
36	C₂²×D₄×C₄₇	—	2	2	no	—
37	C₂²×D₄×C₄₇	—	2	4	no	—
38	C₂²×D₄×C₄₇	—	2	8	no	—
39	C₂²×D₄×C₄₇	—	2	8	no	—
40	C₂²×D₄×C₄₇	—	4	16	no	—
41	C₂²×D₄×C₄₇	C₂×C₄	1	8	yes	—
42	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
43	C₂³×C₄×C₄₇	D₄	1	2	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	#127×C₄₇	—	2	16	no	—
46	#127×C₄₇	—	2	8	no	—
47	#127×C₄₇	C₂×C₄	1	16	yes	complemented
48	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₇	D₄	1	2	yes	—
50	C₂×C₄²×C₄₇	D₄	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
52	C₂⁴×C₄₇	—	2	8	no	—
53	C₂⁴×C₄₇	—	2	4	no	—
54	C₂⁴×C₄₇	—	4	4	no	—
55	C₂⁴×C₄₇	—	4	4	no	—
56	C₂⁴×C₄₇	—	4	16	no	—
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×D₄×C₄₇	—	2	8	no	—
59	C₂×D₄×C₄₇	—	2	32	no	—
60	C₂×D₄×C₄₇	—	2	32	no	—
61	C₂×D₄×C₄₇	—	2	8	no	—
62	C₂×D₄×C₄₇	—	2	4	no	—
63	C₂×D₄×C₄₇	—	2	8	no	—
64	C₂×D₄×C₄₇	—	4	32	no	—
65	C₂×D₄×C₄₇	—	4	64	no	—
66	C₂×D₄×C₄₇	—	4	16	no	—
67	C₂×D₄×C₄₇	—	4	4	no	—
68	C₂×D₄×C₄₇	—	4	4	no	—
69	C₂×D₄×C₄₇	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₄₇	—	2	8	no	—
71	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₄₇	#41	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₂²×Dic₄₇	—	4	4	no	—
81	C₂²×Dic₄₇	—	4	8	no	—
82	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
83	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
84	C₄²×C₄₇	C₂×D₄	1	4	yes	—
85	C₄⋊Dic₄₇	—	2	8	no	—
86	C₄⋊Dic₄₇	—	4	4	no	—
87	C₂×C₄₇⋊C₈	—	2	16	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	C₂×D₄	1	8	yes	complemented
91	C₂³×C₄₇	—	2	8	no	—
92	C₂³×C₄₇	—	4	4	no	—
93	C₂³×C₄₇	—	4	4	no	—
94	C₂³×C₄₇	—	4	16	no	—
95	C₂³×C₄₇	—	4	16	no	—
96	C₂³×C₄₇	—	4	32	no	—
97	C₂³×C₄₇	—	8	16	no	—
98	C₂³×C₄₇	—	8	16	no	—
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂³×C₄₇	#124	1	2	yes	—
101	C₂³×C₄₇	#127	1	2	yes	—
102	C₂³×C₄₇	AGL(1,ℤ/8)	1	8	yes	—
103	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	D₄×C₄₇	—	2	16	no	—
105	D₄×C₄₇	—	4	64	no	—
106	D₄×C₄₇	—	4	8	no	—
107	D₄×C₄₇	—	4	16	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	—	2	16	no	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
113	C₂×C₄×C₄₇	#121	1	8	yes	—
114	C₂×C₄×C₄₇	#124	1	2	yes	—
115	C₂×C₄×C₄₇	#127	1	2	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	4	8	no	—
122	C₂×Dic₄₇	—	8	8	no	—
123	C₄₇⋊C₈	—	2	8	no	—
124	C₂×#1961	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	4	16	no	—
126	C₂²×C₄₇	—	4	8	no	—
127	C₂²×C₄₇	—	8	8	no	—
128	C₂²×C₄₇	—	8	8	no	—
129	C₂²×C₄₇	—	8	32	no	—
130	C₂²×C₄₇	#390	1	4	yes	—
131	C₂²×C₄₇	#418	1	8	yes	—
132	C₂²×C₄₇	#490	1	8	yes	—
133	C₂²×C₄₇	#496	1	8	yes	—
134	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
138	C₄×C₄₇	—	2	8	no	—
139	C₄×C₄₇	#390	1	4	yes	—
140	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
141	C₄×C₄₇	C₂×#121	1	2	yes	—
142	Dic₄₇	—	8	4	no	—
143	C₂²×#127	D₄₇	1	1	yes	characteristic
144	C₂²×#102	—	47	4	no	—
145	C₂²×#97	—	47	1	no	—
146	C₂×#382	—	47	1	no	—
147	#1961	—	47	8	no	—
148	C₂×C₄₇	—	8	8	no	—
149	C₂×C₄₇	#1961	1	8	yes	complemented
150	C₂×C₄₇	C₂×#390	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#418	1	2	yes	—
152	C₂×C₄₇	C₂×#490	1	2	yes	—
153	C₂×C₄₇	C₂×#496	1	2	yes	—
154	C₂³×D₄	—	2	2	no	—
155	C₂³×D₄	C₂×D₄₇	1	2	yes	—
156	C₂³×C₈	—	47	2	no	—
157	C₂×#127	—	2	2	no	—
158	C₂×#127	—	2	4	no	—
159	C₂×#127	Dic₄₇	1	8	yes	complemented
160	C₂×#127	C₂×D₄₇	1	4	yes	—
161	C₂²×C₄⋊C₄	—	47	2	no	—
162	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
163	C₂×#102	—	47	16	no	—
164	C₂×#102	—	94	16	no	—
165	C₂×#97	—	47	4	no	—
166	C₂×#97	—	94	4	no	—
167	#382	—	47	4	no	—
168	#382	—	94	2	no	—
169	C₂×#95	—	94	2	no	—
170	C₄₇	C₂×#1961	1	1	yes	characteristic, complemented, Sylow
171	C₂⁵	—	2	2	no	—
172	C₂²×D₄	—	2	16	no	—
173	C₂²×D₄	—	2	8	no	—
174	C₂²×D₄	—	2	8	no	—
175	C₂²×D₄	—	2	2	no	—
176	C₂²×D₄	—	2	4	no	—
177	C₂²×D₄	—	2	8	no	—
178	C₂²×D₄	—	4	16	no	—
179	C₂²×D₄	C₂×Dic₄₇	1	8	yes	—
180	C₂²×D₄	C₂²×D₄₇	1	4	yes	—
181	C₂³×C₄	—	94	2	no	—
182	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
183	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
184	C₂²×C₈	—	47	8	no	—
185	C₂²×C₈	—	47	4	no	—
186	C₂²×C₈	—	94	4	no	—
187	C₂²×C₈	—	94	4	no	—
188	#127	—	2	16	no	—
189	#127	—	2	8	no	—
190	#127	C₂×Dic₄₇	1	16	yes	complemented
191	C₂×C₄⋊C₄	—	47	4	no	—
192	C₂×C₄⋊C₄	—	94	8	no	—
193	C₂×C₄⋊C₄	—	94	2	no	—
194	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
195	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
196	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
197	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
198	#102	—	94	32	no	—
199	#97	—	94	8	no	—
200	#95	—	94	8	no	—
201	C₂⁴	—	2	8	no	—
202	C₂⁴	—	2	4	no	—
203	C₂⁴	—	4	4	no	—
204	C₂⁴	—	4	4	no	—
205	C₂⁴	—	4	16	no	—
206	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
207	C₂×D₄	—	2	8	no	—
208	C₂×D₄	—	2	32	no	—
209	C₂×D₄	—	2	32	no	—
210	C₂×D₄	—	2	8	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	8	no	—
213	C₂×D₄	—	4	64	no	—
214	C₂×D₄	—	4	32	no	—
215	C₂×D₄	—	4	16	no	—
216	C₂×D₄	—	4	4	no	—
217	C₂×D₄	—	4	4	no	—
218	C₂×D₄	C₂²×Dic₄₇	1	8	yes	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	94	4	no	—
221	C₂²×C₄	—	188	4	no	—
222	C₂²×C₄	—	188	8	no	—
223	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
224	C₂²×C₄	D₄×D₄₇	1	4	yes	—
225	C₂²×C₄	#10625352	1	4	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
228	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
230	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
231	C₂×C₈	—	47	8	no	—
232	C₂×C₈	—	94	16	no	—
233	C₂×C₈	—	94	4	no	—
234	C₂×C₈	—	94	4	no	—
235	C₄⋊C₄	—	94	8	no	—
236	C₄⋊C₄	—	188	4	no	—
237	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
238	C₄²	C₂²×Dic₄₇	1	4	yes	—
239	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
240	C₂³	—	2	8	no	—
241	C₂³	—	4	16	no	—
242	C₂³	—	4	4	no	—
243	C₂³	—	4	32	no	—
244	C₂³	—	4	16	no	—
245	C₂³	—	4	4	no	—
246	C₂³	—	8	16	no	—
247	C₂³	—	8	16	no	—
248	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
249	C₂³	#13094011	1	2	yes	—
250	C₂³	#13094020	1	2	yes	—
251	C₂³	#13094050	1	8	yes	—
252	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
253	D₄	—	2	16	no	—
254	D₄	—	4	64	no	—
255	D₄	—	4	16	no	—
256	D₄	—	4	8	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	376	8	no	—
260	C₂×C₄	—	188	8	no	—
261	C₂×C₄	—	188	4	no	—
262	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
263	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
264	C₂×C₄	D₄×Dic₄₇	1	8	yes	—
265	C₂×C₄	#13093995	1	8	yes	—
266	C₂×C₄	#13094011	1	2	yes	—
267	C₂×C₄	#13094020	1	2	yes	—
268	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
270	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
271	C₂×C₄	C₂×#10625352	1	2	yes	—
272	C₈	—	94	8	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	4	16	no	—
275	C₂²	—	8	8	no	—
276	C₂²	—	8	8	no	—
277	C₂²	—	8	32	no	—
278	C₂²	#15689026	1	4	yes	—
279	C₂²	#15689128	1	8	yes	—
280	C₂²	#15689487	1	8	yes	—
281	C₂²	#15689510	1	8	yes	—
282	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
283	C₂²	C₂×#13094011	1	1	yes	characteristic
284	C₂²	C₂×#13094020	1	1	yes	characteristic
285	C₂²	C₂×#13094050	1	4	yes	—
286	C₄	—	2	8	no	—
287	C₄	—	376	4	no	—
288	C₄	#15689026	1	4	yes	—
289	C₄	C₂×D₄×Dic₄₇	1	2	yes	—
290	C₄	C₂×#13093995	1	2	yes	—
291	C₂	—	8	8	no	—
292	C₂	#20244213	1	8	yes	direct product factor
293	C₂	C₂×#15689026	1	1	yes	characteristic
294	C₂	C₂×#15689128	1	2	yes	—
295	C₂	C₂×#15689487	1	2	yes	—
296	C₂	C₂×#15689510	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting, prime radical	C₂²×#127×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#127	D₄₇
hypocenter	C₄₇	C₂×#1961
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093945	C₂	semidirect
2. C₂²×#13093935	C₂	semidirect
3. C₂×#15688989	C₂	semidirect
4. #20244213	C₂	direct (N × H)
9. C₂×#13093945	C₂²	semidirect
11. C₂×#13093935	C₂²	semidirect
13. #15688989	C₂²	semidirect
19. C₂×#127×C₄₇	C₄	semidirect
25. C₂²×C₄₇⋊C₈	D₄	semidirect
47. #127×C₄₇	C₂×C₄	semidirect
90. C₂×C₄₇⋊C₈	C₂×D₄	semidirect
149. C₂×C₄₇	#1961	semidirect
159. C₂×#127	Dic₄₇	semidirect
170. C₄₇	C₂×#1961	semidirect
190. #127	C₂×Dic₄₇	semidirect
292. C₂	#20244213	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13093945	2
C₂²×#13093935	2
C₂×#15688989	2
#20244213	2
C₂²×#127×C₄₇	2
C₂×#1961	47

Subgroups — C2×#20244213 #26219933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244213 #26219933