Subgroups — #26216748

Order 12032.

954 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689172	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093959	C₂	1	2	yes	maximal
3	#20251951	C₂	1	2	yes	maximal
4	C₂×#15689081	C₂	1	1	yes	maximal, characteristic
5	#20243561	C₂	1	4	yes	maximal, complemented
6	#20243558	C₂	1	4	yes	maximal, complemented
7	C₂×C₄₇×#430	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×#13093983	C₂²	1	1	yes	characteristic
11	#15689172	—	2	4	no	—
12	C₂×#13093960	C₂²	1	4	yes	—
13	C₂×#13093959	—	2	8	no	—
14	C₂×#13093959	C₂²	1	4	yes	—
15	C₂×#13093959	C₂²	1	4	yes	—
16	#15689151	—	2	4	no	—
17	#15689149	—	2	4	no	—
18	#15689146	C₂²	1	8	yes	—
19	#15689081	C₄	1	8	yes	—
20	C₂×#13093934	C₂²	1	2	yes	—
21	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
22	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
23	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#430	—	2	4	no	—
25	C₂×#108×C₄₇	C₂²	1	4	yes	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	—	2	4	no	—
29	C₂×#10625329	C₂³	1	4	yes	—
30	C₂²×C₄₇⋊C₈	—	2	2	no	—
31	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
33	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
36	#13093983	C₂×C₄	1	4	yes	—
37	#13093983	D₄	1	4	yes	—
38	C₂×C₄⋊Dic₄₇	Q₈	1	2	yes	—
39	C₂×C₂²⋊Dic₄₇	Q₈	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
41	#13093960	—	2	8	no	—
42	#13093959	—	2	8	no	—
43	#13093959	—	2	8	no	—
44	Dic₄₇⋊C₈	—	2	16	no	—
45	#13093934	C₂×C₄	1	8	yes	—
46	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₄₇	—	2	2	no	—
49	C₂×M₁₆×C₄₇	—	2	4	no	—
50	C₂×M₁₆×C₄₇	—	2	4	no	—
51	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
53	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
54	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
55	C₂²×C₈×C₄₇	—	2	2	no	—
56	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
57	#108×C₄₇	—	2	8	no	—
58	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	C₂×Q₈	1	4	yes	—
68	M₁₆×C₄₇	—	2	4	no	—
69	M₁₆×C₄₇	—	2	4	no	—
70	M₁₆×C₄₇	—	4	4	no	—
71	M₁₆×C₄₇	C₄⋊C₄	1	16	yes	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	4	4	no	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
77	C₂²⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
78	C₄⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
79	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
80	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
81	#10625329	—	2	4	no	—
82	#10625329	—	2	4	no	—
83	#10625329	—	4	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	4	4	no	—
86	C₂×C₄₇⋊C₈	C₄⋊C₄	1	16	yes	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
90	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
91	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
92	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
93	C₂³×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
96	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
97	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
101	C₂×C₄×C₄₇	#127	1	4	yes	—
102	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
103	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
104	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
113	C₈×C₄₇	—	4	4	no	—
114	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
115	C₄₇⋊C₈	—	4	4	no	—
116	C₄₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
117	C₄₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
118	#20807	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	4	no	—
121	C₂²×C₄₇	—	4	2	no	—
122	C₂²×C₄₇	—	4	2	no	—
123	C₂²×C₄₇	#384	1	4	yes	—
124	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₄×C₄₇	#384	1	4	yes	—
128	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
131	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	Dic₄₇	—	2	8	no	—
133	C₂×#430	—	47	1	no	—
134	C₂×#430	D₄₇	1	1	yes	characteristic
135	C₂²×#108	—	47	2	no	—
136	#2905	—	47	2	no	—
137	C₂×#405	—	47	1	no	—
138	#1849	—	47	4	no	—
139	#1848	—	47	4	no	—
140	C₂×C₄₇	—	2	1	no	—
141	C₂×C₄₇	—	2	1	no	—
142	C₂×C₄₇	—	2	4	no	—
143	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
144	C₂×C₄₇	#2916	1	2	yes	—
145	C₂²×M₁₆	—	47	2	no	—
146	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
147	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
148	C₂³×C₈	—	47	1	no	—
149	C₂×#117	—	47	1	no	—
150	#430	—	2	4	no	—
151	#430	—	94	4	no	—
152	C₂×#108	—	47	4	no	—
153	C₂×#108	—	47	4	no	—
154	C₂×#108	—	47	4	no	—
155	C₂×#108	—	94	8	no	—
156	C₂×#108	C₂×D₄₇	1	4	yes	—
157	#424	—	94	4	no	—
158	#424	—	94	4	no	—
159	#423	—	47	8	no	—
160	#405	—	47	8	no	—
161	C₂×#97	—	47	2	no	—
162	C₂×C₄×C₈	—	47	2	no	—
163	C₄₇	#20807	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	—	47	4	no	—
169	C₂×M₁₆	—	94	2	no	—
170	C₂×M₁₆	—	94	4	no	—
171	C₂×M₁₆	—	94	4	no	—
172	C₂×M₁₆	Dic₉₄	1	4	yes	—
173	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
174	C₂×M₁₆	C₂²⋊D₄₇	1	4	yes	—
175	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	47	4	no	—
178	C₂²×C₈	—	47	4	no	—
179	C₂²×C₈	—	47	4	no	—
180	C₂²×C₈	—	47	1	no	—
181	C₂²×C₈	—	47	1	no	—
182	C₂²×C₈	—	94	2	no	—
183	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
184	#117	—	47	4	no	—
185	#117	—	47	4	no	—
186	C₂×C₄⋊C₄	—	47	2	no	—
187	C₂×C₂²⋊C₄	—	47	2	no	—
188	C₂×C₄²	—	47	2	no	—
189	#108	—	2	8	no	—
190	#108	—	94	8	no	—
191	#108	—	94	8	no	—
192	#108	—	94	8	no	—
193	C₄⋊C₈	—	94	16	no	—
194	#97	—	47	8	no	—
195	C₄×C₈	—	47	8	no	—
196	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
197	C₂²×C₄	—	47	4	no	—
198	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
199	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
200	C₂²×C₄	D₄×D₄₇	1	2	yes	—
201	C₂²×C₄	D₄×D₄₇	1	2	yes	—
202	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
203	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
204	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
205	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	4	4	no	—
209	M₁₆	—	94	4	no	—
210	M₁₆	—	94	4	no	—
211	M₁₆	—	188	4	no	—
212	M₁₆	Dic₄₇⋊C₄	1	16	yes	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	—	47	16	no	—
216	C₂×C₈	—	47	2	no	—
217	C₂×C₈	—	47	2	no	—
218	C₂×C₈	—	47	2	no	—
219	C₂×C₈	—	47	2	no	—
220	C₂×C₈	—	47	2	no	—
221	C₂×C₈	—	47	2	no	—
222	C₂×C₈	—	94	4	no	—
223	C₂×C₈	—	188	4	no	—
224	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
225	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
226	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
227	C₄⋊C₄	—	47	8	no	—
228	C₂²⋊C₄	—	47	8	no	—
229	C₄²	—	47	4	no	—
230	C₄²	—	47	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
233	C₂³	#13094022	1	2	yes	—
234	C₂³	#13094023	1	2	yes	—
235	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
236	C₂×C₄	—	47	8	no	—
237	C₂×C₄	—	188	4	no	—
238	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
239	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
240	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
241	C₂×C₄	#13094020	1	4	yes	—
242	C₂×C₄	#13094022	1	2	yes	—
243	C₂×C₄	#13094023	1	2	yes	—
244	C₂×C₄	#13094023	1	4	yes	—
245	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
250	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
251	C₈	—	4	4	no	—
252	C₈	—	47	4	no	—
253	C₈	—	47	4	no	—
254	C₈	—	188	4	no	—
255	C₈	C₂×Dic₄₇⋊C₄	1	8	yes	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	#15688992	1	4	yes	—
261	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
262	C₂²	C₂×#13094022	1	1	yes	characteristic
263	C₂²	C₂×#13094023	1	1	yes	characteristic
264	C₄	—	94	8	no	—
265	C₄	#15688992	1	4	yes	—
266	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
267	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
268	C₄	C₂×#13094020	1	1	yes	characteristic
269	C₄	C₂×#13094023	1	1	yes	characteristic
270	C₂	—	2	4	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	C₂×#15688992	1	1	yes	characteristic
274	C₂	#20251998	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#430	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#430	D₄₇
hypocenter	C₄₇	#20807
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
5. #20243561	C₂	semidirect
6. #20243558	C₂	semidirect
163. C₄₇	#20807	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689172	2
C₂²×#13093959	2
#20251951	2
C₂×#15689081	2
#20243561	2
#20243558	2
C₂×C₄₇×#430	2
#20807	47