Subgroups — #26215909

Order 12032.

1322 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093960	C₂	1	2	yes	maximal, complemented
2	#20251833	C₂	1	2	yes	maximal
3	#20243430	C₂	1	4	yes	maximal, complemented
4	#20243426	C₂	1	4	yes	maximal, complemented
5	#20243115	C₂	1	2	yes	maximal, complemented
6	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	4	yes	—
8	C₂×#13093960	—	2	8	no	—
9	C₂×#13093960	C₂²	1	4	yes	—
10	C₂×#13093960	C₂²	1	4	yes	—
11	#15689090	—	2	8	no	—
12	#15689087	C₂²	1	8	yes	—
13	#15688888	C₂²	1	8	yes	—
14	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#131×C₄₇	—	2	8	no	—
16	C₂×#131×C₄₇	C₄	1	8	yes	—
17	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
19	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
20	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#10625329	—	2	4	no	—
22	C₂×#10625329	—	2	8	no	—
23	C₂×#10625329	—	2	8	no	—
24	C₂×#10625329	C₂³	1	8	yes	—
25	C₂²×C₄₇⋊C₈	—	2	4	no	—
26	#13093960	—	2	8	no	—
27	#13093960	—	2	8	no	—
28	C₂²⋊(C₄₇⋊C₈)	—	2	16	no	—
29	C₂×#41×C₄₇	—	2	4	no	—
30	C₂×#41×C₄₇	—	2	4	no	—
31	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×#41×C₄₇	D₄	1	4	yes	—
33	C₂²×Q₈×C₄₇	Q₈	1	1	yes	characteristic
34	C₂²×D₄×C₄₇	Q₈	1	2	yes	—
35	C₂²×D₄×C₄₇	Q₈	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	#131×C₄₇	—	2	32	no	—
39	C₂×M₁₆×C₄₇	—	2	4	no	—
40	C₂×M₁₆×C₄₇	—	2	8	no	—
41	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
42	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
43	C₂²×C₈×C₄₇	—	2	8	no	—
44	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₄₇	D₄	1	8	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₂⁴×C₄₇	C₂×Q₈	1	2	yes	—
51	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	#41×C₄₇	—	2	8	no	—
53	#41×C₄₇	—	2	8	no	—
54	#41×C₄₇	—	2	8	no	—
55	#41×C₄₇	—	2	8	no	—
56	C₂×Q₈×C₄₇	—	2	4	no	—
57	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
58	C₂×D₄×C₄₇	—	2	8	no	—
59	C₂×D₄×C₄₇	—	2	4	no	—
60	C₂×D₄×C₄₇	C₄⋊C₄	1	8	yes	—
61	C₂×D₄×C₄₇	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	#41	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₄₇	—	2	8	no	—
74	M₁₆×C₄₇	—	2	16	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	2	16	no	—
77	C₂×C₈×C₄₇	C₂²⋊C₄	1	16	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
83	#10625329	—	2	8	no	—
84	#10625329	—	2	8	no	—
85	#10625329	—	4	8	no	—
86	C₂×C₄₇⋊C₈	—	2	8	no	—
87	C₂×C₄₇⋊C₈	—	4	8	no	—
88	C₂³×C₄₇	—	2	4	no	—
89	C₂³×C₄₇	—	4	2	no	—
90	C₂³×C₄₇	—	4	4	no	—
91	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
92	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
94	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
95	Q₈×C₄₇	—	2	8	no	—
96	D₄×C₄₇	—	2	16	no	—
97	D₄×C₄₇	—	2	8	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₇	#117	1	4	yes	—
106	C₂×C₄×C₄₇	#121	1	4	yes	—
107	C₂×C₄×C₄₇	#121	1	4	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
113	C₈×C₄₇	—	2	4	no	—
114	C₈×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
115	C₈×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
116	C₄₇⋊C₈	—	4	8	no	—
117	#20668	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	2	4	no	—
120	C₂²×C₄₇	—	2	8	no	—
121	C₂²×C₄₇	—	2	4	no	—
122	C₂²×C₄₇	—	4	4	no	—
123	C₂²×C₄₇	—	4	8	no	—
124	C₂²×C₄₇	#380	1	4	yes	—
125	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	#380	1	4	yes	—
129	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
131	C₄×C₄₇	C₂×#121	1	2	yes	—
132	C₂²×#131	D₄₇	1	1	yes	characteristic
133	C₂²×#108	—	47	2	no	—
134	#2879	—	47	2	no	—
135	#1811	—	47	4	no	—
136	#1810	—	47	4	no	—
137	#1738	—	47	2	no	—
138	C₂×C₄₇	—	2	2	no	—
139	C₂×C₄₇	—	2	4	no	—
140	C₂×C₄₇	—	4	4	no	—
141	C₂×C₄₇	C₂×#380	1	1	yes	characteristic
142	C₂×C₄₇	#2916	1	2	yes	—
143	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
144	C₂×#131	—	2	8	no	—
145	C₂×#131	Dic₄₇	1	8	yes	—
146	C₂²×M₁₆	—	47	4	no	—
147	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
148	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
149	C₂³×C₈	C₂×D₄₇	1	2	yes	—
150	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
151	C₂×#108	—	47	4	no	—
152	C₂×#108	—	47	4	no	—
153	C₂×#108	—	94	8	no	—
154	#408	—	94	8	no	—
155	#407	—	47	8	no	—
156	#341	—	47	8	no	—
157	C₄₇	#20668	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	2	4	no	—
159	C₂×#41	—	2	4	no	—
160	C₂×#41	D₁₈₈	1	4	yes	—
161	C₂×#41	C₂×Dic₄₇	1	4	yes	—
162	C₂²×Q₈	Dic₉₄	1	1	yes	characteristic
163	C₂²×D₄	Dic₉₄	1	2	yes	—
164	C₂²×D₄	Dic₉₄	1	1	yes	characteristic
165	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
166	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
167	#131	—	2	32	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	8	no	—
170	C₂×M₁₆	—	47	8	no	—
171	C₂×M₁₆	—	94	4	no	—
172	C₂×M₁₆	—	94	8	no	—
173	C₂×M₁₆	—	94	8	no	—
174	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
175	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
176	C₂²×C₈	—	2	8	no	—
177	C₂²×C₈	—	94	4	no	—
178	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
179	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
180	C₂²×C₈	C₂²⋊D₄₇	1	8	yes	—
181	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
182	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
184	#108	—	94	8	no	—
185	#108	—	94	8	no	—
186	C₂²⋊C₈	—	94	16	no	—
187	C₂⁴	C₂×Dic₉₄	1	2	yes	—
188	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
189	#41	—	2	8	no	—
190	#41	—	2	8	no	—
191	#41	—	2	8	no	—
192	#41	—	2	8	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	C₄⋊Dic₄₇	1	4	yes	—
195	C₂×D₄	—	2	8	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	C₄⋊Dic₄₇	1	8	yes	—
198	C₂×D₄	C₄⋊Dic₄₇	1	4	yes	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
202	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
204	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
205	C₂²×C₄	#10625350	1	4	yes	—
206	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
207	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
209	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
210	M₁₆	—	2	8	no	—
211	M₁₆	—	2	16	no	—
212	M₁₆	—	94	8	no	—
213	M₁₆	—	94	8	no	—
214	M₁₆	—	188	8	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	16	no	—
217	C₂×C₈	—	94	8	no	—
218	C₂×C₈	—	188	8	no	—
219	C₂×C₈	C₂²⋊Dic₄₇	1	16	yes	—
220	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
221	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
222	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
223	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
224	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
225	C₂³	—	2	4	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	4	no	—
228	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
229	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
230	C₂³	#13094003	1	4	yes	—
231	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
232	Q₈	—	2	8	no	—
233	D₄	—	2	16	no	—
234	D₄	—	2	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
240	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
242	C₂×C₄	#13093983	1	4	yes	—
243	C₂×C₄	#13093998	1	4	yes	—
244	C₂×C₄	#13093998	1	4	yes	—
245	C₂×C₄	#13094003	1	4	yes	—
246	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
248	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
249	C₂×C₄	C₂×#10625350	1	2	yes	—
250	C₈	—	2	4	no	—
251	C₈	—	188	8	no	—
252	C₈	C₂×C₂²⋊Dic₄₇	1	4	yes	—
253	C₈	C₂×C₂²⋊Dic₄₇	1	4	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	4	8	no	—
260	C₂²	#15688983	1	4	yes	—
261	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
262	C₂²	C₂×#13094003	1	2	yes	—
263	C₄	—	2	4	no	—
264	C₄	#15688983	1	4	yes	—
265	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
266	C₄	C₂×#13093983	1	1	yes	characteristic
267	C₄	C₂×#13093998	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	2	4	no	—
270	C₂	—	4	4	no	—
271	C₂	C₂×#15688983	1	1	yes	characteristic
272	C₂	#20251999	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#131×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	#20668
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093960	C₂	semidirect
3. #20243430	C₂	semidirect
4. #20243426	C₂	semidirect
5. #20243115	C₂	semidirect
157. C₄₇	#20668	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093960	2
#20251833	2
#20243430	2
#20243426	2
#20243115	2
C₂²×#131×C₄₇	2
#20668	47