Subgroups — #26132201

Order 12032.

478 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×#15689078	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#20241465	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₄	1	1	yes	characteristic
7	C₂×#13093983	C₄	1	1	yes	characteristic
8	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	#15689078	—	2	4	no	—
10	C₂×#13093934	C₄	1	1	yes	characteristic
11	C₂×#13093934	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
14	Dic₄₇⋊C₁₆	—	2	4	no	—
15	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
18	C₄₇×C₂²⋊C₁₆	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
24	#13093983	—	2	2	no	—
25	#13093983	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	C₈	1	2	yes	—
27	C₂×C₂²⋊Dic₄₇	C₈	1	2	yes	—
28	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄₇⋊C₁₆	—	2	2	no	—
31	C₂×C₄₇⋊C₁₆	—	2	2	no	—
32	C₂×C₄₇⋊C₁₆	—	2	2	no	—
33	#13093934	—	2	2	no	—
34	#13093934	—	2	2	no	—
35	C₈×Dic₄₇	—	2	2	no	—
36	C₈×Dic₄₇	D₄	1	4	yes	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	D₄	1	2	yes	—
42	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
43	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₁₆×C₄₇	—	2	2	no	—
45	C₂×C₁₆×C₄₇	—	2	2	no	—
46	C₂×C₁₆×C₄₇	—	2	2	no	—
47	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	1	no	—
49	C₂²×C₄×C₄₇	—	2	1	no	—
50	C₂²×C₄×C₄₇	—	2	1	no	—
51	C₂²×C₄×C₄₇	—	2	1	no	—
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
57	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₄₇	—	2	1	no	—
59	C₂×C₈×C₄₇	—	2	1	no	—
60	C₂×C₈×C₄₇	—	2	4	no	—
61	C₂×C₈×C₄₇	—	2	4	no	—
62	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₄₇	#41	1	2	yes	—
67	C₂²⋊Dic₄₇	—	2	4	no	—
68	C₄⋊Dic₄₇	—	2	4	no	—
69	C₄×Dic₄₇	—	2	1	no	—
70	C₄×Dic₄₇	—	2	1	no	—
71	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
72	C₄×Dic₄₇	M₁₆	1	2	yes	—
73	#10625329	—	2	8	no	—
74	C₂×C₄₇⋊C₈	—	2	1	no	—
75	C₂×C₄₇⋊C₈	—	2	1	no	—
76	C₂×C₄₇⋊C₈	—	2	1	no	—
77	C₂×C₄₇⋊C₈	—	2	1	no	—
78	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
79	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
80	C₁₆×C₄₇	—	4	2	no	—
81	C₄₇⋊C₁₆	—	4	2	no	—
82	C₂³×C₄₇	—	2	1	no	—
83	C₂³×C₄₇	—	2	1	no	—
84	C₂³×C₄₇	—	2	2	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
87	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
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	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	—	2	2	no	—
104	C₂×Dic₄₇	—	2	2	no	—
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	C₂²⋊C₈	1	4	yes	—
107	C₈×C₄₇	—	2	2	no	—
108	C₈×C₄₇	#121	1	2	yes	—
109	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
110	C₄₇⋊C₈	—	2	2	no	—
111	C₄₇⋊C₈	—	2	2	no	—
112	#12784	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	4	2	no	—
117	C₂²×C₄₇	—	4	2	no	—
118	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₂²×C₄₇	#431	1	1	yes	characteristic
121	C₂²×C₄₇	#504	1	2	yes	—
122	C₂²×C₄₇	#504	1	2	yes	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	#380	1	1	yes	characteristic
125	C₄×C₄₇	#407	1	1	yes	characteristic
126	C₄×C₄₇	#423	1	1	yes	characteristic
127	C₄×C₄₇	#432	1	1	yes	characteristic
128	Dic₄₇	—	2	4	no	—
129	Dic₄₇	—	4	2	no	—
130	C₂×#405	—	47	1	no	—
131	C₂×C₄⋊C₁₆	—	47	2	no	—
132	C₂×C₂²⋊C₁₆	—	47	1	no	—
133	C₂×C₂²⋊C₁₆	D₄₇	1	1	yes	characteristic
134	#1374	—	47	2	no	—
135	C₂×C₄₇	—	4	2	no	—
136	C₂×C₄₇	#1797	1	1	yes	characteristic
137	C₂×C₄₇	#2113	1	2	yes	—
138	C₂×C₄₇	#2151	1	1	yes	characteristic
139	C₂×C₄₇	#2151	1	1	yes	characteristic
140	C₂×C₄₇	#2164	1	2	yes	—
141	C₂²×M₁₆	—	47	1	no	—
142	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
143	C₂×#117	—	47	1	no	—
144	C₂²×C₁₆	—	47	2	no	—
145	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
146	#405	—	94	4	no	—
147	C₂×#97	—	47	1	no	—
148	C₂×#97	—	47	1	no	—
149	C₂×C₄×C₈	—	47	1	no	—
150	C₂×C₄×C₈	—	47	1	no	—
151	C₄⋊C₁₆	—	94	4	no	—
152	C₂²⋊C₁₆	—	2	2	no	—
153	C₂²⋊C₁₆	—	94	2	no	—
154	C₄₇	#12784	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
156	C₂×M₁₆	—	94	2	no	—
157	C₂×M₁₆	—	94	2	no	—
158	C₂×M₁₆	—	94	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	47	1	no	—
162	C₂²×C₈	—	47	1	no	—
163	C₂²×C₈	Dic₉₄	1	2	yes	—
164	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
166	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
167	#117	—	94	2	no	—
168	#117	—	94	2	no	—
169	C₂×C₄⋊C₄	—	47	2	no	—
170	C₂×C₂²⋊C₄	—	47	2	no	—
171	C₂×C₄²	—	47	1	no	—
172	C₂×C₄²	—	47	1	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	94	2	no	—
177	C₂×C₁₆	—	94	2	no	—
178	C₂×C₁₆	—	94	2	no	—
179	#97	—	94	2	no	—
180	#97	—	94	2	no	—
181	C₄×C₈	—	47	4	no	—
182	C₄×C₈	—	94	2	no	—
183	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	47	2	no	—
189	C₂²×C₄	—	47	2	no	—
190	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
191	C₂²×C₄	#10625324	1	2	yes	—
192	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
193	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
194	M₁₆	—	94	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	47	2	no	—
200	C₂×C₈	—	47	2	no	—
201	C₂×C₈	—	94	1	no	—
202	C₂×C₈	—	94	1	no	—
203	C₂×C₈	—	94	1	no	—
204	C₂×C₈	—	94	1	no	—
205	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
206	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
207	C₂×C₈	D₄×D₄₇	1	2	yes	—
208	C₂×C₈	#10625352	1	2	yes	—
209	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
210	C₄⋊C₄	—	94	4	no	—
211	C₂²⋊C₄	—	94	4	no	—
212	C₄²	—	47	2	no	—
213	C₄²	—	47	2	no	—
214	C₄²	—	94	1	no	—
215	C₄²	—	94	1	no	—
216	C₁₆	—	4	2	no	—
217	C₁₆	—	188	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	Dic₄₇⋊C₈	1	2	yes	—
223	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	47	4	no	—
231	C₂×C₄	—	94	2	no	—
232	C₂×C₄	—	94	2	no	—
233	C₂×C₄	—	94	2	no	—
234	C₂×C₄	—	94	2	no	—
235	C₂×C₄	—	94	2	no	—
236	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
237	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
238	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
239	C₂×C₄	#13093980	1	1	yes	characteristic
240	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
241	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
242	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	94	2	no	—
245	C₈	—	94	2	no	—
246	C₈	#13093996	1	2	yes	—
247	C₈	#13094001	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
254	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
255	C₂²	#15689173	1	1	yes	characteristic
256	C₂²	#15689539	1	2	yes	—
257	C₂²	#15689541	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	94	4	no	—
260	C₄	—	188	2	no	—
261	C₄	#15688981	1	1	yes	characteristic
262	C₄	#15689085	1	1	yes	characteristic
263	C₄	#15689146	1	1	yes	characteristic
264	C₄	#15689177	1	1	yes	characteristic
265	C₂	—	4	2	no	—
266	C₂	#20243366	1	1	yes	characteristic
267	C₂	#20245000	1	2	yes	—
268	C₂	#20245146	1	1	yes	characteristic
269	C₂	#20245148	1	1	yes	characteristic
270	C₂	#20245192	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator, socle	C₂³×C₄₇	C₂²×C₈
Fitting	C₂×C₄₇×C₂²⋊C₁₆	C₂
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₄₇
hypocenter	C₄₇	#12784
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₄₇⋊C₁₆	C₂	semidirect
4. #20241465	C₂	semidirect
154. C₄₇	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689078	2
C₂×Dic₄₇⋊C₁₆	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20241465	2
C₂×C₄₇×C₂²⋊C₁₆	2
#12784	47