Subgroups — #26106746

Order 12032.

678 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
2	C₄×(C₂×D₄₇)⋊C₈	C₂	1	2	yes	maximal
3	C₂×#15688845	C₂	1	2	yes	maximal, complemented
4	C₄₇⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₇	C₄	1	2	yes	—
7	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
8	C₂×(C₂×D₄₇)⋊C₈	C₄	1	4	yes	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
12	C₄×C₈×D₄₇	—	2	4	no	—
13	#15688877	C₄	1	4	yes	—
14	#15688845	—	2	4	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₈×D₄₇	—	2	4	no	—
22	C₂×C₈×D₄₇	—	2	4	no	—
23	C₂×C₈×D₄₇	C₈	1	8	yes	complemented
24	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
26	C₄²×D₄₇	—	2	2	no	—
27	C₄²×D₄₇	—	2	1	no	—
28	C₄²×D₄₇	—	2	1	no	—
29	(C₂×D₄₇)⋊C₈	—	2	8	no	—
30	C₄×C₄₇⋊C₈	—	2	2	no	—
31	C₄×C₄₇⋊C₈	—	2	2	no	—
32	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
33	C₈×Dic₄₇	—	2	4	no	—
34	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	C₄×C₈×C₄₇	—	2	2	no	—
38	C₄×C₈×C₄₇	—	2	2	no	—
39	C₄×C₈×C₄₇	—	2	2	no	—
40	C₂³×D₄₇	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
43	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
44	C₂²×Dic₄₇	C₄²	1	2	yes	—
45	C₂×C₄×D₄₇	—	2	1	no	—
46	C₂×C₄×D₄₇	—	2	1	no	—
47	C₂×C₄×D₄₇	—	2	2	no	—
48	C₂×C₄×D₄₇	—	2	2	no	—
49	C₂×C₄×D₄₇	—	2	2	no	—
50	C₂×C₄×D₄₇	—	2	4	no	—
51	C₂×C₄×D₄₇	C₂×C₈	1	4	yes	—
52	C₂×C₄×D₄₇	M₁₆	1	8	yes	—
53	C₂×C₈×C₄₇	—	2	2	no	—
54	C₂×C₈×C₄₇	—	2	2	no	—
55	C₂×C₈×C₄₇	—	2	4	no	—
56	C₂×C₈×C₄₇	—	2	4	no	—
57	C₂×C₈×C₄₇	—	2	4	no	—
58	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
59	C₄²×C₄₇	C₂×D₄	1	2	yes	—
60	C₄²×C₄₇	#41	1	1	yes	characteristic
61	C₄²×C₄₇	#41	1	1	yes	characteristic
62	C₄×Dic₄₇	—	2	1	no	—
63	C₄×Dic₄₇	—	2	1	no	—
64	C₄×Dic₄₇	—	2	2	no	—
65	C₄×Dic₄₇	—	2	2	no	—
66	C₂×C₄₇⋊C₈	—	2	4	no	—
67	C₂×C₄₇⋊C₈	—	2	4	no	—
68	C₂×C₄₇⋊C₈	—	2	4	no	—
69	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
70	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
71	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
72	C₈×D₄₇	—	2	16	no	—
73	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
74	C₂²×D₄₇	—	2	1	no	—
75	C₂²×D₄₇	—	2	1	no	—
76	C₂²×D₄₇	—	2	2	no	—
77	C₂²×D₄₇	C₄×C₈	1	2	yes	—
78	C₂²×D₄₇	#97	1	4	yes	—
79	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
86	C₂×C₄×C₄₇	#131	1	4	yes	—
87	C₂×Dic₄₇	—	2	1	no	—
88	C₂×Dic₄₇	—	2	2	no	—
89	C₂×Dic₄₇	—	2	1	no	—
90	C₂×Dic₄₇	—	2	2	no	—
91	C₂×Dic₄₇	C₄×C₈	1	2	yes	—
92	C₂×Dic₄₇	#97	1	4	yes	—
93	C₄×D₄₇	—	2	4	no	—
94	C₄×D₄₇	—	2	4	no	—
95	C₄×D₄₇	—	2	8	no	—
96	C₄×D₄₇	—	2	8	no	—
97	C₈×C₄₇	—	2	4	no	—
98	C₈×C₄₇	—	4	4	no	—
99	C₄₇⋊C₈	—	4	4	no	—
100	C₄₇⋊C₈	C₂²⋊C₈	1	8	yes	complemented
101	#8144	—	47	1	no	maximal, Sylow
102	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×#97	1	2	yes	—
105	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
106	C₂²×C₄₇	#405	1	2	yes	—
107	C₂×D₄₇	—	2	2	no	—
108	C₂×D₄₇	—	2	4	no	—
109	C₂×D₄₇	—	2	2	no	—
110	C₂×D₄₇	—	2	4	no	—
111	C₂×D₄₇	#322	1	4	yes	complemented
112	C₄×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
113	C₄×C₄₇	#431	1	2	yes	—
114	C₄×C₄₇	#435	1	4	yes	—
115	C₄×C₄₇	#435	1	4	yes	—
116	Dic₄₇	—	2	2	no	—
117	Dic₄₇	—	2	4	no	—
118	C₂²×C₄×C₈	—	47	1	no	—
119	C₄×C₂²⋊C₈	—	47	2	no	—
120	C₂×#322	—	47	2	no	—
121	#1273	—	47	1	no	—
122	#1273	D₄₇	1	1	yes	characteristic, complemented
123	C₂×C₄₇	C₂×#322	1	1	yes	characteristic
124	C₂×C₄₇	#1450	1	1	yes	characteristic
125	C₂×C₄₇	#1451	1	2	yes	—
126	C₂×C₄₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
127	C₂×C₄₇	#1752	1	2	yes	—
128	D₄₇	—	2	4	no	—
129	C₂³×C₈	—	47	2	no	—
130	C₂²×C₄²	—	47	1	no	—
131	C₂×C₂²⋊C₈	—	47	4	no	—
132	C₂×C₄×C₈	—	47	1	no	—
133	C₂×C₄×C₈	—	47	2	no	—
134	C₂×C₄×C₈	—	47	2	no	—
135	C₂×C₄×C₈	—	94	4	no	—
136	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
138	#336	—	47	4	no	—
139	#322	—	94	4	no	—
140	C₄₇	#8144	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₄	—	47	1	no	—
142	C₂³×C₄	—	47	2	no	—
143	C₂²×C₈	—	47	2	no	—
144	C₂²×C₈	—	47	8	no	—
145	C₂²×C₈	—	47	4	no	—
146	C₂²×C₈	—	94	4	no	—
147	C₂²×C₈	—	94	4	no	—
148	C₂²×C₈	C₄×D₄₇	1	4	yes	—
149	C₂²×C₈	C₄×D₄₇	1	2	yes	—
150	C₂×C₄²	—	47	2	no	—
151	C₂×C₄²	—	47	1	no	—
152	C₂×C₄²	—	94	2	no	—
153	C₂×C₄²	—	94	1	no	—
154	C₂×C₄²	—	94	1	no	—
155	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
156	C₂²⋊C₈	—	94	8	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	47	4	no	—
161	C₄×C₈	—	94	4	no	—
162	C₄×C₈	—	94	2	no	—
163	C₄×C₈	—	94	2	no	—
164	C₂⁴	—	47	1	no	—
165	C₂²×C₄	—	47	1	no	—
166	C₂²×C₄	—	47	2	no	—
167	C₂²×C₄	—	47	4	no	—
168	C₂²×C₄	—	47	8	no	—
169	C₂²×C₄	—	94	1	no	—
170	C₂²×C₄	—	94	1	no	—
171	C₂²×C₄	—	94	2	no	—
172	C₂²×C₄	—	94	2	no	—
173	C₂²×C₄	—	94	2	no	—
174	C₂²×C₄	—	94	4	no	—
175	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	47	4	no	—
183	C₂×C₈	—	47	4	no	—
184	C₂×C₈	—	47	4	no	—
185	C₂×C₈	—	94	16	no	—
186	C₂×C₈	—	94	4	no	—
187	C₂×C₈	—	94	4	no	—
188	C₂×C₈	—	94	4	no	—
189	C₂×C₈	C₈×D₄₇	1	4	yes	—
190	C₄²	—	94	1	no	—
191	C₄²	—	94	1	no	—
192	C₄²	—	94	2	no	—
193	C₄²	—	94	2	no	—
194	C₄²	D₄×D₄₇	1	2	yes	—
195	C₄²	#10625352	1	1	yes	characteristic
196	C₄²	#10625354	1	1	yes	characteristic
197	C₂³	—	47	2	no	—
198	C₂³	—	47	4	no	—
199	C₂³	—	94	1	no	—
200	C₂³	—	94	1	no	—
201	C₂³	—	94	2	no	—
202	C₂³	C₄²×D₄₇	1	1	yes	characteristic
203	C₂×C₄	—	47	2	no	—
204	C₂×C₄	—	47	4	no	—
205	C₂×C₄	—	94	1	no	—
206	C₂×C₄	—	94	2	no	—
207	C₂×C₄	—	94	1	no	—
208	C₂×C₄	—	94	2	no	—
209	C₂×C₄	—	94	4	no	—
210	C₂×C₄	—	94	4	no	—
211	C₂×C₄	—	94	8	no	—
212	C₂×C₄	—	94	8	no	—
213	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
214	C₂×C₄	#13093982	1	1	yes	characteristic
215	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
216	C₂×C₄	#13093986	1	2	yes	—
217	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
218	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
219	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
220	C₂×C₄	#13094030	1	4	yes	—
221	C₈	—	2	4	no	—
222	C₈	—	4	4	no	—
223	C₈	—	47	8	no	—
224	C₈	—	188	4	no	—
225	C₂²	—	47	4	no	—
226	C₂²	—	94	2	no	—
227	C₂²	—	94	4	no	—
228	C₂²	—	94	2	no	—
229	C₂²	—	94	4	no	—
230	C₂²	#15688970	1	1	yes	characteristic
231	C₂²	C₄×C₈×D₄₇	1	1	yes	characteristic
232	C₂²	#97×D₄₇	1	2	yes	—
233	C₂²	#15689075	1	1	yes	characteristic
234	C₂²	#15689077	1	2	yes	—
235	C₄	—	94	2	no	—
236	C₄	—	94	4	no	—
237	C₄	C₂²⋊C₈×D₄₇	1	2	yes	—
238	C₄	#15689175	1	2	yes	—
239	C₄	#15689193	1	4	yes	—
240	C₄	#15689197	1	4	yes	—
241	C₂	—	94	4	no	—
242	C₂	#322×D₄₇	1	1	yes	characteristic
243	C₂	#20241670	1	1	yes	characteristic
244	C₂	#20241676	1	2	yes	—
245	C₂	#20243145	1	1	yes	characteristic
246	C₂	#20243163	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₄₇×#1273	C₂
socle	C₂³×C₄₇	C₂×C₄²
prime radical	C₂³×D₄₇	C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1273	D₄₇
hypocenter	C₄₇	#8144
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
3. C₂×#15688845	C₂	semidirect
4. C₄₇⋊#1273	C₂	semidirect
5. C₄₇×#1273	C₂	semidirect
23. C₂×C₈×D₄₇	C₈	semidirect
100. C₄₇⋊C₈	C₂²⋊C₈	semidirect
111. C₂×D₄₇	#322	semidirect
122. #1273	D₄₇	semidirect
140. C₄₇	#8144	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₄₇	2
C₄×(C₂×D₄₇)⋊C₈	2
C₂×#15688845	2
C₄₇⋊#1273	2
C₄₇×#1273	2
#8144	47