Subgroups — #26134412

Order 12032.

306 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688947	C₂	1	2	yes	maximal
2	#20245144	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093956	C₂	1	1	yes	maximal, characteristic
4	#20241488	C₂	1	2	yes	maximal
5	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	C₂×#13093956	C₂²	1	1	yes	characteristic
8	C₂×#13093956	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊Dic₄₇	—	2	1	no	—
10	#15688947	—	2	4	no	—
11	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	complemented
12	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	complemented
13	#15688886	—	2	1	no	—
14	C₂×#108×C₄₇	—	2	1	no	—
15	C₂×#108×C₄₇	C₂²	1	2	yes	—
16	C₄₇×#423	—	2	1	no	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₄₇	—	2	1	no	—
19	C₂×C₄⋊Dic₄₇	—	2	1	no	—
20	C₂×C₄×Dic₄₇	—	4	1	no	—
21	#13093964	—	2	2	no	—
22	#13093964	—	2	2	no	—
23	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
24	#13093956	—	2	1	no	—
25	#13093956	—	2	1	no	—
26	#13093956	D₄	1	2	yes	—
27	#13093956	Q₈	1	2	yes	—
28	C₄₇⋊#95	—	4	1	no	—
29	C₂×M₁₆×C₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
33	#108×C₄₇	—	2	2	no	—
34	#108×C₄₇	—	2	2	no	—
35	#108×C₄₇	—	4	2	no	—
36	C₄⋊C₈×C₄₇	—	4	1	no	—
37	C₄⋊C₈×C₄₇	—	4	1	no	—
38	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₄₇	—	4	1	no	—
46	C₂²×Dic₄₇	—	4	1	no	—
47	M₁₆×C₄₇	—	4	2	no	—
48	M₁₆×C₄₇	—	4	2	no	—
49	C₂×C₈×C₄₇	—	2	1	no	—
50	C₂×C₈×C₄₇	—	2	1	no	—
51	C₂×C₈×C₄₇	—	4	2	no	—
52	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
59	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
60	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
61	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
62	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
63	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
64	C₄⋊Dic₄₇	—	2	2	no	—
65	C₄⋊Dic₄₇	—	2	2	no	—
66	C₄⋊Dic₄₇	—	4	1	no	—
67	C₄⋊Dic₄₇	—	4	1	no	—
68	C₄×Dic₄₇	—	8	2	no	—
69	C₄₇⋊C₁₆	—	2	4	no	—
70	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	4	1	no	—
73	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
83	C₂×Dic₄₇	—	4	2	no	—
84	C₂×Dic₄₇	—	4	2	no	—
85	C₂×Dic₄₇	—	8	1	no	—
86	C₂×Dic₄₇	—	8	1	no	—
87	C₂×Dic₄₇	—	8	1	no	—
88	C₂×Dic₄₇	—	8	1	no	—
89	C₈×C₄₇	—	2	1	no	—
90	C₈×C₄₇	—	2	1	no	—
91	C₈×C₄₇	—	8	2	no	—
92	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
93	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
94	C₈×C₄₇	#125	1	1	yes	characteristic
95	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
96	#13185	—	47	1	no	maximal, Sylow
97	C₂²×C₄₇	—	2	1	no	—
98	C₂²×C₄₇	—	2	1	no	—
99	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
100	C₂²×C₄₇	#461	1	1	yes	characteristic
101	C₂²×C₄₇	#476	1	1	yes	characteristic
102	C₄×C₄₇	—	2	1	no	—
103	C₄×C₄₇	—	2	1	no	—
104	C₄×C₄₇	—	2	1	no	—
105	C₄×C₄₇	#365	1	2	yes	complemented
106	C₄×C₄₇	#389	1	1	yes	characteristic
107	C₄×C₄₇	#428	1	1	yes	characteristic
108	C₄×C₄₇	#460	1	1	yes	characteristic
109	C₄×C₄₇	#475	1	1	yes	characteristic
110	Dic₄₇	—	8	2	no	—
111	Dic₄₇	—	8	2	no	—
112	C₂×#365	—	47	2	no	—
113	#2150	—	47	1	no	—
114	#1848	D₄₇	1	1	yes	characteristic
115	C₄×#106	—	47	1	no	—
116	#1383	—	47	2	no	—
117	C₂×C₄₇	—	2	1	no	—
118	C₂×C₄₇	—	2	1	no	—
119	C₂×C₄₇	#1948	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#365	1	1	yes	characteristic
121	C₂×C₄₇	#2155	1	1	yes	characteristic
122	C₂×M₃₂	—	47	2	no	—
123	C₂×#108	—	2	1	no	—
124	C₂×#108	C₂×D₄₇	1	2	yes	—
125	C₂×#106	—	47	1	no	—
126	C₂×#106	—	47	1	no	—
127	#423	—	2	1	no	—
128	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
129	C₄×C₄⋊C₄	—	94	1	no	—
130	#365	—	94	4	no	—
131	C₄⋊C₁₆	—	47	2	no	—
132	C₄⋊C₁₆	—	47	2	no	—
133	#340	—	94	1	no	—
134	C₄₇	#13185	1	1	yes	characteristic, complemented, Sylow
135	C₂×M₁₆	—	2	2	no	—
136	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	94	1	no	—
139	C₂×C₄⋊C₄	—	94	1	no	—
140	C₂×C₄²	—	188	1	no	—
141	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
142	M₃₂	—	94	2	no	—
143	M₃₂	—	94	2	no	—
144	C₂×C₁₆	—	47	4	no	—
145	#108	—	2	2	no	—
146	#108	—	2	2	no	—
147	#108	—	4	2	no	—
148	#106	—	47	2	no	—
149	#106	—	47	2	no	—
150	#106	—	94	1	no	—
151	#106	—	94	1	no	—
152	C₄⋊C₈	—	4	1	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄×C₈	Dic₉₄	1	1	yes	characteristic
155	C₄×C₈	Dic₉₄	1	1	yes	characteristic
156	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
157	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
158	#95	—	188	1	no	—
159	C₂²×C₄	—	188	1	no	—
160	C₂²×C₄	—	188	1	no	—
161	C₂²×C₄	#10625336	1	1	yes	characteristic
162	C₂²×C₄	#10625336	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
164	M₁₆	—	4	2	no	—
165	M₁₆	—	4	2	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
170	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
171	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
172	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
173	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
174	C₂×C₈	#10625352	1	1	yes	characteristic
175	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
176	C₂×C₈	#10625354	1	1	yes	characteristic
177	C₄⋊C₄	—	94	2	no	—
178	C₄⋊C₄	—	94	2	no	—
179	C₄⋊C₄	—	188	1	no	—
180	C₄⋊C₄	—	188	1	no	—
181	C₄²	—	376	2	no	—
182	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
183	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
184	C₄²	#10625335	1	1	yes	characteristic
185	C₄²	#10625335	1	1	yes	characteristic
186	C₁₆	—	94	4	no	—
187	C₂³	C₂×#10625336	1	1	yes	characteristic
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	376	1	no	—
191	C₂×C₄	—	376	1	no	—
192	C₂×C₄	—	376	1	no	—
193	C₂×C₄	—	376	1	no	—
194	C₂×C₄	—	188	2	no	—
195	C₂×C₄	—	188	2	no	—
196	C₂×C₄	#13093955	1	1	yes	characteristic
197	C₂×C₄	#13093955	1	1	yes	characteristic
198	C₂×C₄	#13093955	1	1	yes	characteristic
199	C₂×C₄	#13093955	1	1	yes	characteristic
200	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
201	C₂×C₄	#13093995	1	1	yes	characteristic
202	C₂×C₄	#13094000	1	1	yes	characteristic
203	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
204	C₂×C₄	#13094050	1	1	yes	characteristic
205	C₂×C₄	#13094055	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	8	2	no	—
209	C₈	#13093986	1	1	yes	characteristic
210	C₈	#13093989	1	1	yes	characteristic
211	C₈	#13094015	1	1	yes	characteristic
212	C₈	#13094023	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂×#13093955	1	1	yes	characteristic
216	C₂²	#15689325	1	1	yes	characteristic
217	C₂²	#15689410	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	376	2	no	—
222	C₄	—	376	2	no	—
223	C₄	#15688947	1	2	yes	complemented
224	C₄	#15689019	1	1	yes	characteristic
225	C₄	#15689163	1	1	yes	characteristic
226	C₄	#15689320	1	1	yes	characteristic
227	C₄	#15689405	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	#20244149	1	1	yes	characteristic
231	C₂	C₂×#15688947	1	1	yes	characteristic
232	C₂	#20245162	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093955
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1848	C₂
socle	C₂²×C₄₇	C₂×#106
prime radical	C₂³×C₄₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#13185
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
11. C₄⋊(C₄₇⋊C₁₆)	C₄	semidirect
12. C₄⋊(C₄₇⋊C₁₆)	C₄	semidirect
105. C₄×C₄₇	#365	semidirect
134. C₄₇	#13185	semidirect
223. C₄	#15688947	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688947	2
#20245144	2
C₄×#13093956	2
#20241488	2
C₄₇×#1848	2
#13185	47