Subgroups — #26216915

Order 12032.

770 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252020	C₂	1	2	yes	maximal
2	C₂×#97×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243563	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243528	C₂	1	2	yes	maximal, complemented
5	#20243423	C₂	1	4	yes	maximal
6	#20243296	C₂	1	4	yes	maximal, complemented
7	C₄₇×#1849	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	C₂²	1	2	yes	—
9	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
10	#15689179	—	2	4	no	—
11	#15689175	—	2	4	no	—
12	C₂×#13093960	C₂²	1	4	yes	—
13	#15689147	C₂²	1	2	yes	—
14	#15689086	C₂²	1	4	yes	—
15	M₁₆×Dic₄₇	C₂²	1	4	yes	—
16	C₂×#13093935	C₂²	1	1	yes	characteristic
17	C₂×#13093934	C₂²	1	2	yes	—
18	#97×D₄₇	C₄	1	8	yes	—
19	#15688887	C₂²	1	8	yes	—
20	C₂×#108×C₄₇	C₂²	1	4	yes	—
21	C₄₇×#423	C₂²	1	2	yes	—
22	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄×D₄₇	Q₈	1	2	yes	—
24	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
25	C₂×#10625329	C₂³	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
27	C₂×C₈×D₄₇	—	2	4	no	—
28	C₂×C₈×D₄₇	—	2	4	no	—
29	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
30	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
31	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
32	C₄²×D₄₇	C₂×C₄	1	4	yes	—
33	C₄²×D₄₇	D₄	1	4	yes	—
34	#13093960	—	2	8	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
36	(C₂×D₄₇)⋊C₈	—	2	8	no	—
37	#13093935	C₂×C₄	1	4	yes	—
38	#13093934	—	2	4	no	—
39	#13093934	C₂×C₄	1	8	yes	—
40	C₈×Dic₄₇	—	2	4	no	—
41	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
42	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	#108×C₄₇	—	2	8	no	—
45	C₄⋊C₈×C₄₇	—	2	4	no	—
46	#97×C₄₇	C₂×C₄	1	4	yes	—
47	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
52	C₂×C₄×D₄₇	—	2	2	no	—
53	C₂×C₄×D₄₇	—	2	2	no	—
54	C₂×C₄×D₄₇	—	4	4	no	—
55	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
57	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
59	M₁₆×C₄₇	—	2	8	no	—
60	C₂×C₈×C₄₇	—	2	4	no	—
61	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₄₇	#41	1	4	yes	—
63	C₂×C₈×C₄₇	#41	1	4	yes	—
64	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
65	C₄²×C₄₇	C₂×D₄	1	2	yes	—
66	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
67	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
68	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
69	#10625329	—	2	8	no	—
70	C₂×C₄₇⋊C₈	—	2	2	no	—
71	C₂×C₄₇⋊C₈	—	2	2	no	—
72	C₂×C₄₇⋊C₈	—	2	4	no	—
73	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
74	C₈×D₄₇	—	2	16	no	—
75	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×D₄₇	—	2	2	no	—
77	C₂²×D₄₇	—	2	4	no	—
78	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
84	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
85	C₂×Dic₄₇	—	2	2	no	—
86	C₂×Dic₄₇	—	4	2	no	—
87	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
89	C₄×D₄₇	—	2	8	no	—
90	C₄×D₄₇	—	2	4	no	—
91	C₄×D₄₇	—	2	2	no	—
92	C₄×D₄₇	—	2	2	no	—
93	C₄×D₄₇	—	4	4	no	—
94	C₈×C₄₇	—	4	4	no	—
95	C₈×C₄₇	#117	1	8	yes	—
96	C₄₇⋊C₈	—	2	4	no	—
97	C₄₇⋊C₈	—	4	4	no	—
98	#20830	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	2	2	no	—
100	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₄₇	#533	1	2	yes	—
102	C₂×D₄₇	—	2	8	no	—
103	C₂×D₄₇	—	2	2	no	—
104	C₂×D₄₇	—	2	2	no	—
105	C₂×D₄₇	#430	1	4	yes	—
106	C₄×C₄₇	—	2	4	no	—
107	C₄×C₄₇	C₂×#117	1	2	yes	—
108	C₄×C₄₇	#532	1	1	yes	characteristic
109	C₄×C₄₇	#532	1	1	yes	characteristic
110	Dic₄₇	—	2	4	no	—
111	Dic₄₇	—	2	2	no	—
112	#2920	—	47	2	no	—
113	C₂²×#97	—	47	1	no	—
114	#1849	—	47	1	no	—
115	#1849	D₄₇	1	1	yes	characteristic, complemented
116	#1838	—	47	2	no	—
117	#1810	—	47	4	no	—
118	#1778	—	47	4	no	—
119	C₂×C₄₇	—	2	2	no	—
120	C₂×C₄₇	#2285	1	1	yes	characteristic
121	C₂×C₄₇	C₂×#430	1	2	yes	—
122	D₄₇	—	2	4	no	—
123	C₂³×C₈	—	47	2	no	—
124	C₂²×C₄²	—	47	1	no	—
125	#432	—	94	4	no	—
126	#431	—	94	4	no	—
127	C₂×#108	—	47	4	no	—
128	C₂×#108	C₂×D₄₇	1	4	yes	—
129	#423	—	47	2	no	—
130	#423	C₂×D₄₇	1	2	yes	—
131	#407	—	47	4	no	—
132	C₄×M₁₆	—	47	4	no	—
133	C₂×#97	—	47	2	no	—
134	C₂×#97	—	47	1	no	—
135	C₂×#97	—	47	8	no	—
136	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
137	#341	—	47	8	no	—
138	C₄₇	#20830	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	—	47	1	no	—
140	C₂³×C₄	—	47	2	no	—
141	C₂×M₁₆	—	47	4	no	—
142	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
143	C₂²×C₈	—	47	2	no	—
144	C₂²×C₈	—	47	8	no	—
145	C₂²×C₈	—	94	4	no	—
146	C₂²×C₈	—	94	4	no	—
147	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
148	C₂×C₄²	—	47	2	no	—
149	C₂×C₄²	—	47	1	no	—
150	C₂×C₄²	—	47	4	no	—
151	C₂×C₄²	—	47	4	no	—
152	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
153	#108	—	2	8	no	—
154	#108	—	94	8	no	—
155	C₄⋊C₈	—	2	4	no	—
156	C₄⋊C₈	—	94	4	no	—
157	C₂²⋊C₈	—	94	8	no	—
158	#97	—	47	8	no	—
159	#97	—	47	4	no	—
160	#97	—	94	4	no	—
161	#97	C₄×D₄₇	1	4	yes	—
162	C₄×C₈	—	94	4	no	—
163	C₂⁴	—	47	1	no	—
164	C₂²×C₄	—	47	1	no	—
165	C₂²×C₄	—	47	2	no	—
166	C₂²×C₄	—	47	4	no	—
167	C₂²×C₄	—	47	2	no	—
168	C₂²×C₄	—	47	2	no	—
169	C₂²×C₄	—	47	4	no	—
170	C₂²×C₄	—	94	2	no	—
171	C₂²×C₄	—	94	2	no	—
172	C₂²×C₄	—	188	4	no	—
173	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
174	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
175	M₁₆	—	2	8	no	—
176	M₁₆	—	94	8	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	47	4	no	—
179	C₂×C₈	—	94	2	no	—
180	C₂×C₈	—	94	2	no	—
181	C₂×C₈	—	94	16	no	—
182	C₂×C₈	—	94	4	no	—
183	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
184	C₂×C₈	#10625352	1	4	yes	—
185	C₂×C₈	#10625354	1	4	yes	—
186	C₄²	—	47	8	no	—
187	C₄²	—	47	2	no	—
188	C₄²	—	47	2	no	—
189	C₄²	C₂×C₄×D₄₇	1	2	yes	—
190	C₄²	D₄×D₄₇	1	2	yes	—
191	C₂³	—	47	2	no	—
192	C₂³	—	94	4	no	—
193	C₂³	—	94	2	no	—
194	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	47	2	no	—
197	C₂×C₄	—	47	4	no	—
198	C₂×C₄	—	94	8	no	—
199	C₂×C₄	—	94	2	no	—
200	C₂×C₄	—	94	4	no	—
201	C₂×C₄	—	94	2	no	—
202	C₂×C₄	—	94	2	no	—
203	C₂×C₄	—	188	4	no	—
204	C₂×C₄	—	188	2	no	—
205	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
206	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
207	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂×#10625352	1	2	yes	—
209	C₂×C₄	C₂×#10625354	1	2	yes	—
210	C₈	—	4	4	no	—
211	C₈	—	94	4	no	—
212	C₈	—	188	4	no	—
213	C₈	#13093982	1	8	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	—	47	4	no	—
216	C₂²	—	94	8	no	—
217	C₂²	—	94	2	no	—
218	C₂²	—	94	2	no	—
219	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
220	C₂²	#15689678	1	2	yes	—
221	C₄	—	2	4	no	—
222	C₄	—	94	2	no	—
223	C₄	—	94	4	no	—
224	C₄	C₂×#13093982	1	2	yes	—
225	C₄	#15689671	1	1	yes	characteristic
226	C₄	#15689674	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	—	94	4	no	—
229	C₂	#20245793	1	1	yes	characteristic
230	C₂	#430×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1849	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1849	D₄₇
hypocenter	C₄₇	#20830
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

Complements

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

N	H ≅ G/N	type
3. #20243563	C₂	semidirect
4. #20243528	C₂	semidirect
6. #20243296	C₂	semidirect
7. C₄₇×#1849	C₂	semidirect
115. #1849	D₄₇	semidirect
138. C₄₇	#20830	semidirect

Maximal subgroups

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

subgroup	index
#20252020	2
C₂×#97×D₄₇	2
#20243563	2
#20243528	2
#20243423	2
#20243296	2
C₄₇×#1849	2
#20830	47