Subgroups — C₂×#20243276 #26214316

Order 12032.

1358 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093955	C₂	1	2	yes	maximal
2	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×Dic₄₇	C₂	1	2	yes	maximal
4	#20243276	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688886	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄₇×#340	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₄₇	—	2	2	no	—
10	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×#13093955	C₄	1	16	yes	complemented
12	C₂×#13093955	C₂²	1	8	yes	—
13	C₂×C₈×Dic₄₇	C₄	1	8	yes	complemented
14	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
15	C₄⋊C₄×Dic₄₇	—	2	4	no	—
16	C₄⋊C₄×Dic₄₇	C₂²	1	8	yes	—
17	C₂×C₄₇⋊#95	—	2	2	no	—
18	#15688886	C₂²	1	4	yes	—
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₄₇×#340	C₂²	1	4	yes	—
22	C₂³×Dic₄₇	—	2	2	no	—
23	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
24	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
27	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	C₂×C₄×Dic₄₇	—	2	8	no	—
33	C₂×C₄×Dic₄₇	—	4	8	no	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
38	#13093955	C₂×C₄	1	32	yes	complemented
39	C₈×Dic₄₇	C₂×C₄	1	16	yes	complemented
40	C₄₇⋊#95	—	2	8	no	—
41	C₂³×C₄×C₄₇	—	2	2	no	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	—	2	4	no	—
44	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
49	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	8	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	—	4	4	no	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₇	#41	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	2	2	no	—
62	C₂²×Dic₄₇	—	2	2	no	—
63	C₂²×Dic₄₇	—	2	8	no	—
64	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	—	4	4	no	—
66	C₂²×Dic₄₇	—	4	4	no	—
67	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
68	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
69	C₂²×Dic₄₇	SD₁₆	1	8	yes	—
70	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
71	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₄₇	—	2	8	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
75	C₄⋊C₄×C₄₇	—	2	8	no	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
78	C₄⋊Dic₄₇	—	2	8	no	—
79	C₄⋊Dic₄₇	—	2	4	no	—
80	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
81	C₄×Dic₄₇	—	4	16	no	—
82	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
83	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
84	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
85	C₂×C₄₇⋊C₈	C₄²	1	16	yes	complemented
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₄₇	C₄×D₄	1	4	yes	—
90	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
91	C₂³×C₄₇	#135	1	2	yes	—
92	C₂³×C₄₇	#135	1	2	yes	—
93	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	—	4	16	no	—
98	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₇	#117	1	4	yes	—
101	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
102	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
107	C₂×Dic₄₇	—	2	2	no	—
108	C₂×Dic₄₇	—	2	2	no	—
109	C₂×Dic₄₇	—	2	8	no	—
110	C₂×Dic₄₇	—	2	8	no	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	4	16	no	—
114	C₂×Dic₄₇	#106	1	16	yes	complemented
115	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×Dic₄₇	C₂×SD₁₆	1	8	yes	—
118	C₈×C₄₇	—	2	4	no	—
119	C₄₇⋊C₈	C₂×C₄²	1	8	yes	complemented
120	C₂×C₄×#106	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
122	C₂²×C₄₇	C₂×#106	1	4	yes	—
123	C₂²×C₄₇	#427	1	4	yes	—
124	C₂²×C₄₇	C₄×SD₁₆	1	8	yes	—
125	C₂²×C₄₇	C₄×SD₁₆	1	8	yes	—
126	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
128	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
129	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
131	C₄×C₄₇	—	4	8	no	—
132	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
133	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
134	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
135	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
136	Dic₄₇	—	2	4	no	—
137	Dic₄₇	—	4	8	no	—
138	Dic₄₇	C₂×#106	1	8	yes	complemented
139	C₂²×#106	—	47	2	no	—
140	C₂²×C₄×C₈	—	47	1	no	—
141	C₂×C₄×C₄⋊C₄	—	47	2	no	—
142	C₄×#106	—	47	8	no	—
143	C₂×#340	—	47	1	no	—
144	C₂×#340	D₄₇	1	1	yes	characteristic
145	C₂×C₄₇	C₄×#106	1	8	yes	complemented
146	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄₇	C₂²×#106	1	1	yes	characteristic
148	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
149	C₂×C₄₇	C₂×C₄×SD₁₆	1	2	yes	—
150	C₂×C₄₇	C₂×C₄×SD₁₆	1	2	yes	—
151	C₂³×C₈	—	47	1	no	—
152	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
153	C₂²×C₄⋊C₄	—	47	2	no	—
154	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
155	C₂²×C₄²	—	47	1	no	—
156	C₂²×C₄²	—	94	2	no	—
157	C₂×#106	—	47	16	no	—
158	C₂×#106	—	47	8	no	—
159	C₂×C₄×C₈	—	47	8	no	—
160	C₂×C₄×C₈	—	47	4	no	—
161	C₄×C₄⋊C₄	—	47	8	no	—
162	C₄×C₄⋊C₄	—	94	4	no	—
163	C₂×#95	—	94	2	no	—
164	#340	—	47	4	no	—
165	#340	C₂×D₄₇	1	4	yes	—
166	C₄₇	C₂×C₄×#106	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	—	2	2	no	—
168	C₂³×C₄	—	47	1	no	—
169	C₂³×C₄	—	47	1	no	—
170	C₂³×C₄	—	94	2	no	—
171	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
172	C₂²×C₈	—	2	4	no	—
173	C₂²×C₈	—	47	4	no	—
174	C₂²×C₈	—	47	8	no	—
175	C₂²×C₈	—	47	2	no	—
176	C₂²×C₈	C₄×D₄₇	1	4	yes	—
177	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	47	8	no	—
181	C₂×C₄⋊C₄	—	47	4	no	—
182	C₂×C₄⋊C₄	—	94	2	no	—
183	C₂×C₄⋊C₄	—	94	4	no	—
184	C₂×C₄⋊C₄	C₄×D₄₇	1	8	yes	—
185	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
186	C₂×C₄²	—	47	4	no	—
187	C₂×C₄²	—	47	2	no	—
188	C₂×C₄²	—	47	4	no	—
189	C₂×C₄²	—	47	2	no	—
190	C₂×C₄²	—	94	8	no	—
191	C₂×C₄²	—	188	8	no	—
192	#106	—	47	32	no	—
193	C₄×C₈	—	47	16	no	—
194	#95	—	94	8	no	—
195	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	—	47	4	no	—
201	C₂²×C₄	—	47	8	no	—
202	C₂²×C₄	—	47	2	no	—
203	C₂²×C₄	—	47	4	no	—
204	C₂²×C₄	—	47	2	no	—
205	C₂²×C₄	—	94	2	no	—
206	C₂²×C₄	—	94	2	no	—
207	C₂²×C₄	—	94	8	no	—
208	C₂²×C₄	—	94	4	no	—
209	C₂²×C₄	—	188	4	no	—
210	C₂²×C₄	—	188	4	no	—
211	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
213	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
214	C₂²×C₄	#10625352	1	4	yes	—
215	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
216	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	47	16	no	—
220	C₂×C₈	—	47	4	no	—
221	C₂×C₈	—	47	8	no	—
222	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	47	8	no	—
226	C₄⋊C₄	—	94	8	no	—
227	C₄⋊C₄	—	94	4	no	—
228	C₄⋊C₄	C₂×C₄×D₄₇	1	8	yes	—
229	C₄²	—	47	8	no	—
230	C₄²	—	47	4	no	—
231	C₄²	—	47	4	no	—
232	C₄²	—	188	16	no	—
233	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
234	C₂³	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
235	C₂³	SD₁₆×D₄₇	1	4	yes	—
236	C₂³	#13094043	1	2	yes	—
237	C₂³	#13094047	1	2	yes	—
238	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	16	no	—
243	C₂×C₄	—	47	4	no	—
244	C₂×C₄	—	47	8	no	—
245	C₂×C₄	—	47	16	no	—
246	C₂×C₄	—	47	4	no	—
247	C₂×C₄	—	94	2	no	—
248	C₂×C₄	—	94	2	no	—
249	C₂×C₄	—	94	8	no	—
250	C₂×C₄	—	94	8	no	—
251	C₂×C₄	—	188	4	no	—
252	C₂×C₄	—	188	4	no	—
253	C₂×C₄	—	188	16	no	—
254	C₂×C₄	C₄²×D₄₇	1	4	yes	—
255	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
256	C₂×C₄	#13093980	1	4	yes	—
257	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
258	C₂×C₄	#13093989	1	8	yes	—
259	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
261	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
262	C₂×C₄	C₂×#10625352	1	2	yes	—
263	C₈	—	2	4	no	—
264	C₈	—	47	8	no	—
265	C₂²	#15688974	1	4	yes	—
266	C₂²	#106×D₄₇	1	4	yes	—
267	C₂²	#15689160	1	4	yes	—
268	C₂²	#15689208	1	8	yes	—
269	C₂²	#15689214	1	8	yes	—
270	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
271	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
272	C₂²	C₂×SD₁₆×D₄₇	1	2	yes	—
273	C₂²	C₂×#13094043	1	1	yes	characteristic
274	C₂²	C₂×#13094047	1	1	yes	characteristic
275	C₄	—	4	8	no	—
276	C₄	—	47	8	no	—
277	C₄	—	94	4	no	—
278	C₄	—	188	8	no	—
279	C₄	#15688974	1	4	yes	—
280	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
281	C₄	C₂×#13093980	1	1	yes	characteristic
282	C₄	C₂×#13093989	1	2	yes	—
283	C₂	#20243276	1	8	yes	direct product factor
284	C₂	C₂×#15688974	1	1	yes	characteristic
285	C₂	C₂×#106×D₄₇	1	1	yes	characteristic
286	C₂	C₂×#15689160	1	1	yes	characteristic
287	C₂	C₂×#15689208	1	2	yes	—
288	C₂	C₂×#15689214	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#340	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#340	D₄₇
hypocenter	C₄₇	C₂×C₄×#106
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
4. #20243276	C₂	direct (N × H)
11. C₂×#13093955	C₄	semidirect
13. C₂×C₈×Dic₄₇	C₄	semidirect
38. #13093955	C₂×C₄	semidirect
39. C₈×Dic₄₇	C₂×C₄	semidirect
85. C₂×C₄₇⋊C₈	C₄²	semidirect
114. C₂×Dic₄₇	#106	semidirect
119. C₄₇⋊C₈	C₂×C₄²	semidirect
138. Dic₄₇	C₂×#106	semidirect
145. C₂×C₄₇	C₄×#106	semidirect
166. C₄₇	C₂×C₄×#106	semidirect
283. C₂	#20243276	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×#13093955	2
C₂²×C₈×Dic₄₇	2
C₂×C₄⋊C₄×Dic₄₇	2
#20243276	2
C₂×#15688886	2
C₂×C₄₇×#340	2
C₂×C₄×#106	47

Subgroups — C2×#20243276 #26214316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243276 #26214316