Subgroups — D₄₇×#2276 #26374499

Order 12032.

3654 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×D₄₇	C₂	1	3	yes	maximal
2	#20245753	C₂	1	3	yes	maximal, complemented
3	#20245752	C₂	1	9	yes	maximal, complemented
4	#20245751	C₂	1	3	yes	maximal, complemented
5	C₄×C₄⋊C₄×D₄₇	C₂	1	9	yes	maximal
6	#376×D₄₇	C₂	1	3	yes	maximal
7	C₄₇×#2276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×D₄₇	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₄₇	C₂²	1	3	yes	—
10	C₂×C₄×Dic₉₄	C₂²	1	9	yes	—
11	C₂×#13093989	C₂²	1	9	yes	—
12	C₄×Q₈×D₄₇	C₄	1	48	yes	complemented
13	C₂×C₄⋊C₄×D₄₇	C₂²	1	9	yes	—
14	C₂×C₄²×D₄₇	C₂²	1	6	yes	—
15	C₂×C₄²×D₄₇	C₂²	1	9	yes	—
16	C₄⋊C₄×Dic₄₇	C₂²	1	18	yes	—
17	C₄×Dic₄₇⋊C₄	C₂²	1	18	yes	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	9	yes	—
19	#15688974	C₂²	1	18	yes	—
20	#15688969	C₂²	1	9	yes	—
21	#15688968	C₂²	1	3	yes	—
22	#15688968	C₂²	1	9	yes	—
23	#95×D₄₇	C₂²	1	1	yes	characteristic
24	#95×D₄₇	C₂²	1	9	yes	—
25	C₂×C₄×Q₈×C₄₇	C₂²	1	3	yes	—
26	C₄×C₄⋊C₄×C₄₇	C₂²	1	9	yes	—
27	C₄₇×#376	C₂²	1	3	yes	—
28	C₂²×Dic₉₄	C₂³	1	3	yes	—
29	C₂×Q₈×D₄₇	C₂×C₄	1	24	yes	—
30	C₂²×C₄×D₄₇	C₂³	1	3	yes	—
31	C₂²×C₄×D₄₇	C₂³	1	9	yes	—
32	C₂²×C₄×D₄₇	C₂³	1	3	yes	—
33	Q₈×Dic₄₇	C₂×C₄	1	24	yes	complemented
34	C₄×Dic₉₄	C₂×C₄	1	72	yes	complemented
35	#13093989	C₂×C₄	1	72	yes	complemented
36	C₂×C₄⋊Dic₄₇	C₂³	1	9	yes	—
37	C₂×Dic₄₇⋊C₄	C₂³	1	18	yes	—
38	C₄⋊C₄×D₄₇	C₂×C₄	1	72	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	18	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	9	yes	—
41	C₂×C₄×Dic₄₇	C₂³	1	9	yes	—
42	C₂×C₄×Dic₄₇	C₂³	1	9	yes	—
43	C₄²×D₄₇	—	4	12	no	—
44	C₄²×D₄₇	C₂×C₄	1	72	yes	—
45	C₄₇⋊#95	C₂³	1	3	yes	—
46	C₄₇⋊#95	C₂³	1	9	yes	—
47	C₄₇⋊#95	C₂³	1	18	yes	—
48	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₄×Q₈×C₄₇	C₂×C₄	1	24	yes	complemented
50	C₂×C₄⋊C₄×C₄₇	C₂³	1	9	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	9	yes	—
53	#95×C₄₇	C₂³	1	1	yes	characteristic
54	#95×C₄₇	C₂³	1	9	yes	—
55	C₂³×D₄₇	C₂⁴	1	1	yes	characteristic
56	C₂×Q₈×C₄₇	C₂²×C₄	1	12	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	9	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
60	C₂²×Dic₄₇	C₂⁴	1	3	yes	—
61	C₂²×Dic₄₇	C₂⁴	1	9	yes	—
62	C₂²×Dic₄₇	C₂⁴	1	3	yes	—
63	Q₈×D₄₇	C₄²	1	64	yes	complemented
64	C₂×C₄×D₄₇	—	2	18	no	—
65	C₂×C₄×D₄₇	—	2	12	no	—
66	C₂×C₄×D₄₇	—	2	18	no	—
67	C₂×C₄×D₄₇	C₂²×C₄	1	12	yes	—
68	C₂×C₄×D₄₇	C₂²×C₄	1	36	yes	—
69	C₂×C₄×D₄₇	C₂²×C₄	1	36	yes	—
70	C₂×Dic₉₄	C₂²×C₄	1	36	yes	—
71	C₄⋊C₄×C₄₇	C₂²×C₄	1	36	yes	—
72	C₄²×C₄₇	—	4	6	no	—
73	C₄²×C₄₇	C₂²×C₄	1	36	yes	—
74	C₄⋊Dic₄₇	C₂²×C₄	1	36	yes	—
75	Dic₄₇⋊C₄	C₂²×C₄	1	72	yes	—
76	C₄×Dic₄₇	—	4	18	no	—
77	C₄×Dic₄₇	C₂²×C₄	1	36	yes	—
78	C₄×Dic₄₇	C₂²×C₄	1	36	yes	—
79	C₄×Dic₄₇	C₂²×C₄	1	36	yes	—
80	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
81	C₂²×D₄₇	C₂³×C₄	1	6	yes	—
82	C₂²×D₄₇	H(5,𝔽₂)	1	2	yes	—
83	C₂²×D₄₇	2₋¹⁺⁴	1	6	yes	—
84	Q₈×C₄₇	C₂×C₄²	1	16	yes	complemented
85	C₂×C₄×C₄₇	—	2	9	no	—
86	C₂×C₄×C₄₇	—	2	6	no	—
87	C₂×C₄×C₄₇	—	2	9	no	—
88	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
89	C₂×C₄×C₄₇	C₂³×C₄	1	18	yes	—
90	C₂×C₄×C₄₇	C₂³×C₄	1	18	yes	—
91	C₂×Dic₄₇	—	2	9	no	—
92	C₂×Dic₄₇	—	2	6	no	—
93	C₂×Dic₄₇	—	2	9	no	—
94	C₂×Dic₄₇	C₂³×C₄	1	6	yes	—
95	C₂×Dic₄₇	C₂³×C₄	1	18	yes	—
96	C₂×Dic₄₇	C₂³×C₄	1	18	yes	—
97	C₄×D₄₇	—	2	24	no	—
98	C₄×D₄₇	—	2	72	no	—
99	C₄×D₄₇	C₂×C₄²	1	48	yes	—
100	Dic₉₄	C₂×C₄²	1	48	yes	complemented
101	C₂×#2276	—	47	1	no	maximal, Sylow
102	C₂²×C₄₇	C₂⁴×C₄	1	3	yes	—
103	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	3	yes	—
105	C₂×D₄₇	C₂²×C₄²	1	4	yes	—
106	C₂×D₄₇	#519	1	12	yes	—
107	C₂×D₄₇	#520	1	12	yes	—
108	C₄×C₄₇	—	2	6	no	—
109	C₄×C₄₇	—	2	18	no	—
110	C₄×C₄₇	C₂²×C₄²	1	12	yes	—
111	Dic₄₇	—	2	6	no	—
112	Dic₄₇	—	2	18	no	—
113	Dic₄₇	C₂²×C₄²	1	12	yes	—
114	C₂²×C₄×Q₈	—	47	3	no	—
115	#2276	—	47	9	no	—
116	#2276	—	47	3	no	—
117	#2276	—	47	3	no	—
118	#2276	D₄₇	1	1	yes	characteristic, direct product factor
119	C₂×C₄×C₄⋊C₄	—	47	9	no	—
120	C₂×#376	—	47	3	no	—
121	C₂×C₄₇	C₂³×C₄²	1	1	yes	characteristic
122	C₂×C₄₇	C₂×#519	1	3	yes	—
123	C₂×C₄₇	C₂×#520	1	3	yes	—
124	D₄₇	#2276	1	8	yes	direct product factor
125	C₂³×Q₈	—	47	1	no	—
126	C₂×C₄×Q₈	—	47	3	no	—
127	C₂×C₄×Q₈	—	47	48	no	—
128	C₂×C₄×Q₈	—	47	9	no	—
129	C₂×C₄×Q₈	—	47	9	no	—
130	C₂×C₄×Q₈	C₂×D₄₇	1	3	yes	—
131	C₂²×C₄⋊C₄	—	47	9	no	—
132	C₂²×C₄²	—	47	9	no	—
133	C₂²×C₄²	—	47	6	no	—
134	C₄×C₄⋊C₄	—	47	18	no	—
135	C₄×C₄⋊C₄	—	47	9	no	—
136	C₄×C₄⋊C₄	—	47	18	no	—
137	C₄×C₄⋊C₄	—	47	18	no	—
138	C₄×C₄⋊C₄	C₂×D₄₇	1	9	yes	—
139	#376	—	47	9	no	—
140	#376	—	47	9	no	—
141	#376	—	47	3	no	—
142	#376	C₂×D₄₇	1	3	yes	—
143	C₂×#95	—	47	1	no	—
144	C₂×#95	—	47	9	no	—
145	C₄₇	C₂×#2276	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₈	—	47	24	no	—
147	C₂²×Q₈	—	47	3	no	—
148	C₂²×Q₈	C₂²×D₄₇	1	1	yes	characteristic
149	C₂³×C₄	—	47	3	no	—
150	C₂³×C₄	—	47	9	no	—
151	C₂³×C₄	—	47	3	no	—
152	C₄×Q₈	—	47	24	no	—
153	C₄×Q₈	—	47	72	no	—
154	C₄×Q₈	—	47	72	no	—
155	C₄×Q₈	C₄×D₄₇	1	24	yes	complemented
156	C₂×C₄⋊C₄	—	47	18	no	—
157	C₂×C₄⋊C₄	—	47	9	no	—
158	C₂×C₄⋊C₄	—	47	72	no	—
159	C₂×C₄⋊C₄	C₂²×D₄₇	1	9	yes	—
160	C₂×C₄²	—	47	9	no	—
161	C₂×C₄²	—	47	9	no	—
162	C₂×C₄²	—	47	9	no	—
163	C₂×C₄²	—	47	72	no	—
164	C₂×C₄²	—	47	18	no	—
165	C₂×C₄²	—	188	12	no	—
166	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
167	C₂×C₄²	C₂²×D₄₇	1	9	yes	—
168	#95	—	47	3	no	—
169	#95	—	47	18	no	—
170	#95	—	47	9	no	—
171	#95	C₂²×D₄₇	1	1	yes	characteristic
172	#95	C₂²×D₄₇	1	9	yes	—
173	C₂⁴	—	47	1	no	—
174	C₂×Q₈	—	47	64	no	—
175	C₂×Q₈	—	47	36	no	—
176	C₂×Q₈	C₂×C₄×D₄₇	1	12	yes	—
177	C₂²×C₄	—	47	3	no	—
178	C₂²×C₄	—	47	36	no	—
179	C₂²×C₄	—	47	3	no	—
180	C₂²×C₄	—	47	9	no	—
181	C₂²×C₄	—	47	12	no	—
182	C₂²×C₄	—	47	36	no	—
183	C₂²×C₄	—	94	18	no	—
184	C₂²×C₄	—	94	18	no	—
185	C₂²×C₄	—	94	12	no	—
186	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
187	C₂²×C₄	C₂³×D₄₇	1	9	yes	—
188	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
189	C₄⋊C₄	—	47	72	no	—
190	C₄⋊C₄	—	47	36	no	—
191	C₄⋊C₄	C₂×C₄×D₄₇	1	36	yes	—
192	C₄²	—	4	6	no	—
193	C₄²	—	47	36	no	—
194	C₄²	—	47	36	no	—
195	C₄²	—	47	36	no	—
196	C₄²	—	188	18	no	—
197	C₄²	C₂×C₄×D₄₇	1	36	yes	—
198	C₂³	—	47	6	no	—
199	C₂³	—	47	2	no	—
200	C₂³	—	47	6	no	—
201	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
202	Q₈	—	47	48	no	—
203	Q₈	C₄²×D₄₇	1	16	yes	complemented
204	C₂×C₄	—	2	9	no	—
205	C₂×C₄	—	2	6	no	—
206	C₂×C₄	—	2	9	no	—
207	C₂×C₄	—	47	18	no	—
208	C₂×C₄	—	47	48	no	—
209	C₂×C₄	—	47	6	no	—
210	C₂×C₄	—	47	18	no	—
211	C₂×C₄	—	94	24	no	—
212	C₂×C₄	—	94	72	no	—
213	C₂×C₄	—	94	9	no	—
214	C₂×C₄	—	94	6	no	—
215	C₂×C₄	—	94	9	no	—
216	C₂×C₄	C₂²×C₄×D₄₇	1	6	yes	—
217	C₂×C₄	C₂²×C₄×D₄₇	1	18	yes	—
218	C₂×C₄	C₂²×C₄×D₄₇	1	18	yes	—
219	C₂²	—	47	4	no	—
220	C₂²	—	47	12	no	—
221	C₂²	—	47	12	no	—
222	C₂²	C₂³×C₄×D₄₇	1	3	yes	—
223	C₂²	H(5,𝔽₂)×D₄₇	1	1	yes	characteristic
224	C₂²	2₋¹⁺⁴×D₄₇	1	3	yes	—
225	C₄	—	2	6	no	—
226	C₄	—	2	18	no	—
227	C₄	—	47	12	no	—
228	C₄	—	94	6	no	—
229	C₄	—	94	18	no	—
230	C₄	C₂×C₄²×D₄₇	1	12	yes	—
231	C₂	—	47	8	no	—
232	C₂	C₂²×C₄²×D₄₇	1	1	yes	characteristic
233	C₂	#519×D₄₇	1	3	yes	—
234	C₂	#520×D₄₇	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator	C₂×C₄₇	C₂³×C₄²
Fitting	C₄₇×#2276	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂³×D₄₇	C₂⁴
hypercenter	#2276	D₄₇
hypocenter	C₄₇	C₂×#2276
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #2276

Complements

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

N	H ≅ G/N	type
2. #20245753	C₂	semidirect
3. #20245752	C₂	semidirect
4. #20245751	C₂	semidirect
7. C₄₇×#2276	C₂	semidirect
12. C₄×Q₈×D₄₇	C₄	semidirect
33. Q₈×Dic₄₇	C₂×C₄	semidirect
34. C₄×Dic₉₄	C₂×C₄	semidirect
35. #13093989	C₂×C₄	semidirect
49. C₄×Q₈×C₄₇	C₂×C₄	semidirect
63. Q₈×D₄₇	C₄²	semidirect
84. Q₈×C₄₇	C₂×C₄²	semidirect
100. Dic₉₄	C₂×C₄²	semidirect
118. #2276	D₄₇	direct (N × H)
124. D₄₇	#2276	direct (N × H)
145. C₄₇	C₂×#2276	semidirect
155. C₄×Q₈	C₄×D₄₇	semidirect
203. Q₈	C₄²×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×D₄₇	2
#20245753	2
#20245752	2
#20245751	2
C₄×C₄⋊C₄×D₄₇	2
#376×D₄₇	2
C₄₇×#2276	2
C₂×#2276	47

Subgroups — D47×#2276 #26374499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₇×#2276 #26374499