Subgroups — #26215314

Order 12032.

990 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₉₄	C₂	1	2	yes	maximal
2	C₄²×Dic₉₄	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₄₇	C₂	1	2	yes	maximal
4	#20243249	C₂	1	4	yes	maximal
5	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#20243236	C₂	1	2	yes	maximal
7	#20243132	C₂	1	2	yes	maximal
8	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₉₄	C₄	1	2	yes	—
10	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
11	C₈×Dic₉₄	C₄	1	16	yes	complemented
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₈	C₂²	1	4	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
17	C₄×Dic₄₇⋊C₄	C₄	1	4	yes	—
18	C₄×Dic₄₇⋊C₄	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₄²×Dic₄₇	C₂²	1	2	yes	—
22	#15688878	C₂²	1	2	yes	—
23	#15688877	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
25	C₄³×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#336	C₂²	1	1	yes	characteristic
27	C₄₇×#336	C₂²	1	2	yes	—
28	C₄₇×#336	C₂²	1	1	yes	characteristic
29	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
31	C₄×Dic₉₄	C₈	1	16	yes	complemented
32	C₄×Dic₉₄	C₂×C₄	1	8	yes	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
41	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
42	Dic₄₇⋊C₈	C₂×C₄	1	16	yes	—
43	C₄×C₄₇⋊C₈	—	4	2	no	—
44	C₈×Dic₄₇	—	4	4	no	—
45	C₈×Dic₄₇	C₂×C₄	1	16	yes	—
46	C₄₇⋊#95	C₂×C₄	1	4	yes	—
47	C₄₇⋊#95	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
63	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
64	C₂×Dic₉₄	C₄²	1	4	yes	—
65	C₂×Dic₉₄	C₂×C₈	1	8	yes	—
66	C₂×C₈×C₄₇	—	2	2	no	—
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
72	C₄²×C₄₇	—	2	1	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	—	2	1	no	—
75	C₄²×C₄₇	—	4	2	no	—
76	C₄²×C₄₇	C₂×C₈	1	8	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
78	C₄⋊Dic₄₇	C₄²	1	4	yes	—
79	C₄⋊Dic₄₇	C₂×C₈	1	8	yes	—
80	Dic₄₇⋊C₄	C₄²	1	8	yes	—
81	Dic₄₇⋊C₄	C₂×C₈	1	16	yes	—
82	C₄×Dic₄₇	—	2	2	no	—
83	C₄×Dic₄₇	—	2	2	no	—
84	C₄×Dic₄₇	—	4	4	no	—
85	C₄×Dic₄₇	C₂×C₈	1	16	yes	—
86	C₄×Dic₄₇	C₂²×C₄	1	8	yes	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
90	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
99	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
102	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
104	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
105	C₂×Dic₄₇	—	2	4	no	—
106	C₂×Dic₄₇	—	2	4	no	—
107	C₂×Dic₄₇	—	2	4	no	—
108	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
109	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
110	C₂×Dic₄₇	C₂²×C₈	1	8	yes	—
111	C₂×Dic₄₇	C₂²×C₈	1	8	yes	—
112	Dic₉₄	C₄×C₈	1	16	yes	complemented
113	C₈×C₄₇	—	2	4	no	—
114	C₈×C₄₇	—	2	4	no	—
115	C₄₇⋊C₈	—	2	8	no	—
116	#20569	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
118	C₂²×C₄₇	#519	1	1	yes	characteristic
119	C₂²×C₄₇	#520	1	1	yes	characteristic
120	C₂²×C₄₇	C₂³×C₈	1	2	yes	—
121	C₂²×C₄₇	#568	1	2	yes	—
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	—	2	4	no	—
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	C₂×C₄×C₈	1	4	yes	—
126	C₄×C₄₇	#519	1	2	yes	—
127	C₄×C₄₇	#520	1	2	yes	—
128	Dic₄₇	—	2	4	no	—
129	Dic₄₇	—	2	8	no	—
130	Dic₄₇	C₂×C₄×C₈	1	8	yes	—
131	C₂×C₈×Q₈	—	47	2	no	—
132	C₄²×Q₈	—	47	1	no	—
133	C₈×C₄⋊C₄	—	47	2	no	—
134	C₈×C₄⋊C₄	—	47	4	no	—
135	C₄×C₄⋊C₈	—	47	1	no	—
136	C₄×C₄⋊C₈	—	47	2	no	—
137	#1744	—	47	2	no	—
138	#1744	D₄₇	1	1	yes	characteristic
139	C₂×C₄₇	#2276	1	1	yes	characteristic
140	C₂×C₄₇	C₂²×C₄×C₈	1	1	yes	characteristic
141	C₂×C₄₇	#2875	1	1	yes	characteristic
142	C₂×C₄₇	#2982	1	2	yes	—
143	C₂×C₄₇	#2988	1	2	yes	—
144	C₂×C₄×Q₈	—	47	2	no	—
145	C₂×C₄×Q₈	—	47	1	no	—
146	C₈×Q₈	—	47	16	no	—
147	C₂×C₄⋊C₈	—	47	4	no	—
148	C₂×C₄⋊C₈	—	47	2	no	—
149	C₂×C₄×C₈	—	47	4	no	—
150	C₂×C₄×C₈	—	47	4	no	—
151	C₂×C₄×C₈	—	47	2	no	—
152	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
153	C₄×C₄⋊C₄	—	47	2	no	—
154	C₄×C₄⋊C₄	—	47	4	no	—
155	C₄×C₄⋊C₄	—	47	1	no	—
156	C₄×C₄⋊C₄	—	47	2	no	—
157	C₄³	—	47	2	no	—
158	C₄³	C₂×D₄₇	1	1	yes	characteristic
159	#336	—	47	4	no	—
160	#336	—	47	2	no	—
161	#336	C₂×D₄₇	1	1	yes	characteristic
162	#336	C₂×D₄₇	1	1	yes	characteristic
163	#336	C₂×D₄₇	1	2	yes	—
164	C₄₇	#20569	1	1	yes	characteristic, complemented, Sylow
165	C₂²×Q₈	—	47	1	no	—
166	C₂²×C₈	—	47	4	no	—
167	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
168	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
169	C₄×Q₈	—	47	16	no	—
170	C₄×Q₈	—	47	8	no	—
171	C₂×C₄⋊C₄	—	47	2	no	—
172	C₂×C₄⋊C₄	—	47	4	no	—
173	C₂×C₄⋊C₄	—	47	1	no	—
174	C₂×C₄⋊C₄	—	47	2	no	—
175	C₂×C₄²	—	47	4	no	—
176	C₂×C₄²	—	47	2	no	—
177	C₂×C₄²	—	47	2	no	—
178	C₂×C₄²	—	47	4	no	—
179	C₂×C₄²	C₄×D₄₇	1	2	yes	—
180	C₂×C₄²	C₄×D₄₇	1	2	yes	—
181	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
182	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
184	C₄⋊C₈	—	47	16	no	—
185	C₄⋊C₈	—	47	8	no	—
186	C₄×C₈	—	47	16	no	—
187	C₄×C₈	—	188	2	no	—
188	C₄×C₈	—	188	4	no	—
189	C₄×C₈	C₄×D₄₇	1	8	yes	—
190	#95	—	47	2	no	—
191	#95	—	47	4	no	—
192	C₂×Q₈	—	47	4	no	—
193	C₂×Q₈	—	47	8	no	—
194	C₂²×C₄	—	47	2	no	—
195	C₂²×C₄	—	47	2	no	—
196	C₂²×C₄	—	47	4	no	—
197	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
199	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
200	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	47	8	no	—
207	C₂×C₈	—	94	4	no	—
208	C₂×C₈	—	94	4	no	—
209	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
210	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
211	C₄⋊C₄	—	47	8	no	—
212	C₄⋊C₄	—	47	4	no	—
213	C₄⋊C₄	—	47	16	no	—
214	C₄⋊C₄	—	47	8	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	1	no	—
218	C₄²	—	4	2	no	—
219	C₄²	—	47	16	no	—
220	C₄²	—	47	8	no	—
221	C₄²	—	94	2	no	—
222	C₄²	—	94	2	no	—
223	C₄²	—	188	4	no	—
224	C₄²	C₈×D₄₇	1	8	yes	—
225	C₄²	C₂×C₄×D₄₇	1	4	yes	—
226	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
227	Q₈	—	47	16	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	47	4	no	—
234	C₂×C₄	—	47	8	no	—
235	C₂×C₄	—	47	4	no	—
236	C₂×C₄	—	47	8	no	—
237	C₂×C₄	—	94	4	no	—
238	C₂×C₄	—	94	4	no	—
239	C₂×C₄	—	94	4	no	—
240	C₂×C₄	C₄²×D₄₇	1	2	yes	—
241	C₂×C₄	C₄²×D₄₇	1	2	yes	—
242	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
243	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
244	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
245	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
246	C₂×C₄	#13094070	1	1	yes	characteristic
247	C₂×C₄	#13094071	1	1	yes	characteristic
248	C₂×C₄	#13094071	1	2	yes	—
249	C₈	—	2	4	no	—
250	C₈	—	2	4	no	—
251	C₈	—	94	8	no	—
252	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
253	C₂²	#15689600	1	1	yes	characteristic
254	C₂²	#15689603	1	1	yes	characteristic
255	C₂²	C₂²×C₈×D₄₇	1	2	yes	—
256	C₂²	#15689889	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	47	8	no	—
261	C₄	—	94	8	no	—
262	C₄	—	94	4	no	—
263	C₄	C₄×C₈×D₄₇	1	4	yes	—
264	C₄	#15689600	1	2	yes	—
265	C₄	#15689603	1	2	yes	—
266	C₂	#20245751	1	1	yes	characteristic
267	C₂	C₂×C₄×C₈×D₄₇	1	1	yes	characteristic
268	C₂	#20251813	1	1	yes	characteristic
269	C₂	#20252468	1	2	yes	—
270	C₂	#20252496	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂²×C₄×C₈
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	C₂³×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#20569
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1744

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₉₄	C₄	semidirect
31. C₄×Dic₉₄	C₈	semidirect
112. Dic₉₄	C₄×C₈	semidirect
164. C₄₇	#20569	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₉₄	2
C₄²×Dic₉₄	2
C₈×C₄⋊Dic₄₇	2
#20243249	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243236	2
#20243132	2
C₄₇×#1744	2
#20569	47