Subgroups — #26214450

Order 12032.

930 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20243247	C₂	1	4	yes	maximal, complemented
4	#20243156	C₂	1	4	yes	maximal
5	#20243136	C₂	1	2	yes	maximal, complemented
6	#97×Dic₄₇	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093983	C₂²	1	2	yes	—
9	#15689180	C₄	1	8	yes	complemented
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×#13093935	C₂²	1	2	yes	—
13	C₂×#13093934	C₂²	1	4	yes	—
14	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
15	C₄×C₄⋊Dic₄₇	C₄	1	4	yes	—
16	C₄×C₂²⋊Dic₄₇	C₄	1	4	yes	—
17	#15688969	C₂²	1	2	yes	—
18	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
19	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
20	#15688877	C₂²	1	8	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#432	C₄	1	8	yes	complemented
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
25	C₂×#97×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	#13093983	—	2	8	no	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
33	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
34	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
35	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
36	#13093935	C₂×C₄	1	8	yes	—
37	#13093934	—	2	8	no	—
38	C₈×Dic₄₇	—	2	8	no	—
39	C₄₇⋊#95	C₂×C₄	1	8	yes	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
49	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
50	#97×C₄₇	C₂×C₄	1	8	yes	—
51	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
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	2	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
62	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₄₇	—	2	4	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
66	C₄²×C₄₇	C₄²	1	8	yes	—
67	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²⋊Dic₄₇	—	2	8	no	—
72	C₄⋊Dic₄₇	—	2	8	no	—
73	C₄×Dic₄₇	—	2	4	no	—
74	C₄×Dic₄₇	—	4	4	no	—
75	C₄×Dic₄₇	—	4	4	no	—
76	C₄×Dic₄₇	#41	1	8	yes	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
80	C₂³×C₄₇	—	2	2	no	—
81	C₂³×C₄₇	—	2	4	no	—
82	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
83	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	4	8	no	—
90	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
92	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
93	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
96	C₂×Dic₄₇	—	2	8	no	—
97	C₂×Dic₄₇	—	2	4	no	—
98	C₂×Dic₄₇	—	2	8	no	—
99	C₂×Dic₄₇	#117	1	8	yes	—
100	C₈×C₄₇	—	2	8	no	—
101	C₄₇⋊C₈	—	2	8	no	—
102	#20467	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	—	2	4	no	—
104	C₂²×C₄₇	—	2	2	no	—
105	C₂²×C₄₇	—	4	4	no	—
106	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
107	C₂²×C₄₇	C₂×#117	1	2	yes	—
108	C₂²×C₄₇	#568	1	4	yes	—
109	C₄×C₄₇	—	2	4	no	—
110	C₄×C₄₇	—	2	4	no	—
111	C₄×C₄₇	—	2	2	no	—
112	C₄×C₄₇	C₄×#41	1	2	yes	—
113	C₄×C₄₇	C₄×#41	1	2	yes	—
114	Dic₄₇	—	2	8	no	—
115	Dic₄₇	—	4	4	no	—
116	C₂×#432	—	47	1	no	—
117	C₂×#432	D₄₇	1	1	yes	characteristic
118	C₄×#117	—	47	1	no	—
119	#1768	—	47	4	no	—
120	#1750	—	47	4	no	—
121	#1745	—	47	2	no	—
122	C₄×#97	—	47	2	no	—
123	C₂×C₄₇	—	4	2	no	—
124	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
125	C₂×C₄₇	#2873	1	2	yes	—
126	C₂×C₄₇	#2923	1	4	yes	—
127	C₂×#117	—	47	2	no	—
128	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
129	#432	—	47	8	no	—
130	#432	Dic₄₇	1	8	yes	complemented
131	C₂×C₄⋊C₈	—	47	2	no	—
132	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
133	C₂×C₂²⋊C₈	—	47	2	no	—
134	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
135	C₂×#97	—	47	2	no	—
136	C₂×#97	—	47	4	no	—
137	C₂×#97	C₂×D₄₇	1	2	yes	—
138	C₂×C₄×C₈	—	47	4	no	—
139	C₄×C₄⋊C₄	—	47	4	no	—
140	C₄×C₂²⋊C₄	—	47	4	no	—
141	#376	—	47	2	no	—
142	C₄³	—	47	1	no	—
143	C₄³	—	47	1	no	—
144	#336	—	47	8	no	—
145	C₄₇	#20467	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₄	C₄×D₄₇	1	2	yes	—
147	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
148	C₂²×C₈	—	47	4	no	—
149	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
150	#117	—	94	8	no	—
151	C₂×C₄⋊C₄	—	47	4	no	—
152	C₂×C₂²⋊C₄	—	47	4	no	—
153	C₂×C₄²	—	47	2	no	—
154	C₂×C₄²	—	47	2	no	—
155	C₂×C₄²	—	47	4	no	—
156	C₂×C₄²	—	47	4	no	—
157	C₂×C₄²	C₄×D₄₇	1	2	yes	—
158	C₂×C₄²	C₄×D₄₇	1	4	yes	—
159	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
160	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
162	C₄⋊C₈	—	47	8	no	—
163	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
164	C₂²⋊C₈	—	47	8	no	—
165	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
166	#97	—	47	8	no	—
167	#97	—	94	8	no	—
168	#97	C₂×Dic₄₇	1	8	yes	—
169	C₄×C₈	—	94	8	no	—
170	#95	—	47	8	no	—
171	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	4	4	no	—
174	C₂²×C₄	—	47	4	no	—
175	C₂²×C₄	—	47	4	no	—
176	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
177	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
179	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
182	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	47	8	no	—
186	C₂×C₈	—	94	4	no	—
187	C₂×C₈	—	94	4	no	—
188	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
189	C₄⋊C₄	—	94	8	no	—
190	C₂²⋊C₄	—	94	8	no	—
191	C₄²	—	47	8	no	—
192	C₄²	—	94	4	no	—
193	C₄²	—	188	4	no	—
194	C₄²	—	188	4	no	—
195	C₄²	C₄×Dic₄₇	1	8	yes	—
196	C₄²	C₂×C₄×D₄₇	1	2	yes	—
197	C₄²	C₂×C₄×D₄₇	1	2	yes	—
198	C₄²	C₂²×Dic₄₇	1	2	yes	—
199	C₄²	C₂²×Dic₄₇	1	2	yes	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
203	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	8	no	—
210	C₂×C₄	—	47	8	no	—
211	C₂×C₄	—	94	8	no	—
212	C₂×C₄	—	94	4	no	—
213	C₂×C₄	—	94	8	no	—
214	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
215	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
216	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
217	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
218	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
219	C₂×C₄	#41×D₄₇	1	4	yes	—
220	C₈	—	2	8	no	—
221	C₈	—	94	8	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
226	C₂²	#117×D₄₇	1	2	yes	—
227	C₂²	#15689889	1	4	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	4	no	—
230	C₄	—	2	2	no	—
231	C₄	—	94	8	no	—
232	C₄	—	188	4	no	—
233	C₄	#15689591	1	2	yes	—
234	C₄	#41×Dic₄₇	1	2	yes	—
235	C₂	—	4	2	no	—
236	C₂	#117×Dic₄₇	1	1	yes	characteristic
237	C₂	#20251806	1	2	yes	—
238	C₂	#20252034	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#20467
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20243247	C₂	semidirect
5. #20243136	C₂	semidirect
6. #97×Dic₄₇	C₂	semidirect
9. #15689180	C₄	semidirect
22. C₄₇×#432	C₄	semidirect
130. #432	Dic₄₇	semidirect
145. C₄₇	#20467	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
C₄×#13093983	2
#20243247	2
#20243156	2
#20243136	2
#97×Dic₄₇	2
C₂×C₄₇×#432	2
#20467	47