Subgroups — #26216922

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 230 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×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
3	#20243559	C₂	1	2	yes	maximal, complemented
4	#20243526	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243429	C₂	1	4	yes	maximal
6	#20243293	C₂	1	4	yes	maximal, complemented
7	C₄₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625324	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×#13093959	C₂²	1	8	yes	—
13	#15689147	C₂²	1	2	yes	—
14	#15689086	C₂²	1	4	yes	—
15	M₁₆×Dic₄₇	C₂²	1	4	yes	—
16	C₄×#10625324	C₄	1	4	yes	—
17	C₄×#10625324	C₄	1	4	yes	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	#15688888	C₂²	1	4	yes	—
21	C₄₇×#423	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#341	C₂²	1	4	yes	—
24	C₂²×C₄×D₄₇	Q₈	1	2	yes	—
25	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
26	C₂×#10625329	C₂³	1	4	yes	—
27	C₂×#10625324	—	2	8	no	—
28	C₂×#10625324	C₂×C₄	1	8	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
31	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
32	C₄²×D₄₇	C₂×C₄	1	2	yes	—
33	C₄²×D₄₇	C₂×C₄	1	2	yes	—
34	C₄²×D₄₇	D₄	1	4	yes	—
35	#13093959	—	2	16	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
37	(C₂×D₄₇)⋊C₈	—	2	8	no	—
38	#13093934	—	2	4	no	—
39	#13093934	C₂×C₄	1	8	yes	—
40	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
41	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
42	C₈×Dic₄₇	—	2	4	no	—
43	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
44	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₄₇	—	2	4	no	—
47	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
49	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
52	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
53	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×D₄₇	—	2	4	no	—
55	C₂×C₄×D₄₇	—	4	4	no	—
56	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
57	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
58	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
60	M₁₆×C₄₇	—	2	8	no	—
61	C₂×C₈×C₄₇	—	2	4	no	—
62	C₂×C₈×C₄₇	—	2	4	no	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
64	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₄₇	C₂×D₄	1	2	yes	—
67	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
68	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
69	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
70	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
71	#10625329	—	2	8	no	—
72	C₂×C₄₇⋊C₈	—	2	4	no	—
73	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
74	C₂×C₄₇⋊C₈	#41	1	8	yes	—
75	#10625324	—	2	16	no	—
76	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
77	C₂²×D₄₇	—	2	2	no	—
78	C₂²×D₄₇	—	4	2	no	—
79	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
83	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
86	C₂×Dic₄₇	—	2	2	no	—
87	C₂×Dic₄₇	—	4	2	no	—
88	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₄×D₄₇	—	2	4	no	—
91	C₄×D₄₇	—	2	4	no	—
92	C₄×D₄₇	—	2	8	no	—
93	C₄×D₄₇	—	4	4	no	—
94	C₈×C₄₇	—	2	4	no	—
95	C₈×C₄₇	—	4	4	no	—
96	C₄₇⋊C₈	—	4	4	no	—
97	C₄₇⋊C₈	#117	1	8	yes	—
98	#20831	—	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	4	no	—
103	C₂×D₄₇	—	2	2	no	—
104	C₂×D₄₇	—	4	4	no	—
105	C₄×C₄₇	—	2	2	no	—
106	C₄×C₄₇	#430	1	4	yes	—
107	C₄×C₄₇	C₂×#117	1	2	yes	—
108	C₄×C₄₇	#532	1	2	yes	—
109	Dic₄₇	—	2	4	no	—
110	Dic₄₇	—	2	2	no	—
111	#2920	—	47	2	no	—
112	C₂×C₄×M₁₆	—	47	1	no	—
113	#1848	—	47	2	no	—
114	#1837	—	47	1	no	—
115	#1837	D₄₇	1	1	yes	characteristic, complemented
116	#1811	—	47	4	no	—
117	#1778	—	47	4	no	—
118	C₂×C₄₇	—	2	2	no	—
119	C₂×C₄₇	#2285	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#430	1	2	yes	—
121	D₄₇	—	4	2	no	—
122	C₂²×M₁₆	—	47	2	no	—
123	C₂²×C₄²	—	47	1	no	—
124	#432	—	94	4	no	—
125	#431	—	94	4	no	—
126	C₂×#108	—	47	8	no	—
127	#423	—	47	2	no	—
128	#423	C₂×D₄₇	1	2	yes	—
129	#407	—	47	4	no	—
130	C₄×M₁₆	—	47	4	no	—
131	C₄×M₁₆	—	47	4	no	—
132	C₄×M₁₆	—	47	4	no	—
133	C₂×#97	—	47	2	no	—
134	C₂×C₄×C₈	—	47	1	no	—
135	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
136	#341	—	47	4	no	—
137	#341	C₂×D₄₇	1	4	yes	—
138	C₄₇	#20831	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	—	47	1	no	—
140	C₂³×C₄	—	47	2	no	—
141	C₂×M₁₆	—	47	8	no	—
142	C₂×M₁₆	—	47	4	no	—
143	C₂×M₁₆	—	94	8	no	—
144	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
145	C₂²×C₈	—	47	2	no	—
146	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
147	C₂×C₄²	—	47	2	no	—
148	C₂×C₄²	—	47	1	no	—
149	C₂×C₄²	—	47	2	no	—
150	C₂×C₄²	—	47	2	no	—
151	C₂×C₄²	—	47	4	no	—
152	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
153	#108	—	94	16	no	—
154	C₄⋊C₈	—	2	4	no	—
155	C₄⋊C₈	—	94	4	no	—
156	C₂²⋊C₈	—	94	8	no	—
157	#97	—	47	8	no	—
158	#97	—	94	4	no	—
159	C₄×C₈	—	47	2	no	—
160	C₄×C₈	—	47	2	no	—
161	C₄×C₈	—	94	4	no	—
162	C₄×C₈	C₄×D₄₇	1	2	yes	—
163	C₄×C₈	C₄×D₄₇	1	2	yes	—
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	2	no	—
169	C₂²×C₄	—	47	2	no	—
170	C₂²×C₄	—	47	4	no	—
171	C₂²×C₄	—	94	4	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	16	no	—
177	M₁₆	—	94	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	47	4	no	—
181	C₂×C₈	—	47	8	no	—
182	C₂×C₈	—	94	4	no	—
183	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
184	C₄²	—	47	4	no	—
185	C₄²	—	47	4	no	—
186	C₄²	—	47	2	no	—
187	C₄²	—	47	2	no	—
188	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
189	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
190	C₄²	D₄×D₄₇	1	2	yes	—
191	C₂³	—	47	2	no	—
192	C₂³	—	94	2	no	—
193	C₂³	—	188	2	no	—
194	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	47	2	no	—
197	C₂×C₄	—	47	4	no	—
198	C₂×C₄	—	94	2	no	—
199	C₂×C₄	—	94	4	no	—
200	C₂×C₄	—	94	4	no	—
201	C₂×C₄	—	94	8	no	—
202	C₂×C₄	—	188	4	no	—
203	C₂×C₄	—	188	2	no	—
204	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
205	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
206	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
207	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
208	C₂×C₄	#41×D₄₇	1	4	yes	—
209	C₈	—	2	4	no	—
210	C₈	—	4	4	no	—
211	C₈	—	47	8	no	—
212	C₈	—	188	4	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	94	2	no	—
215	C₂²	—	94	4	no	—
216	C₂²	—	188	4	no	—
217	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
218	C₂²	#15689679	1	2	yes	—
219	C₄	—	2	2	no	—
220	C₄	—	94	2	no	—
221	C₄	—	94	4	no	—
222	C₄	#15689170	1	4	yes	—
223	C₄	#117×D₄₇	1	2	yes	—
224	C₄	#15689672	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	—	188	2	no	—
227	C₂	#20245794	1	1	yes	characteristic
228	C₂	C₂×#15689170	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×#1837	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1837	D₄₇
hypocenter	C₄₇	#20831
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

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

N	H ≅ G/N	type
3. #20243559	C₂	semidirect
4. #20243526	C₂	semidirect
6. #20243293	C₂	semidirect
7. C₄₇×#1837	C₂	semidirect
115. #1837	D₄₇	semidirect
138. C₄₇	#20831	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×C₄×#10625324	2
#20243559	2
#20243526	2
#20243429	2
#20243293	2
C₄₇×#1837	2
#20831	47