Subgroups — #26216740

Order 12032.

994 conjugacy classes of subgroups.

The subgroup lattice has 302 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	C₂×#15689171	C₂	1	2	yes	maximal
3	C₂×#15689146	C₂	1	2	yes	maximal
4	C₂×M₁₆×Dic₄₇	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	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	#15689172	—	2	4	no	—
12	#15689171	—	2	8	no	—
13	C₂×#13093960	C₂²	1	4	yes	—
14	C₂×#13093959	C₂²	1	8	yes	—
15	#15689146	—	2	4	no	—
16	#15689146	C₂²	1	4	yes	—
17	#15689146	C₂²	1	4	yes	—
18	C₂×Dic₄₇⋊C₈	—	2	4	no	—
19	M₁₆×Dic₄₇	C₄	1	4	yes	—
20	M₁₆×Dic₄₇	C₄	1	4	yes	—
21	C₂×#13093934	C₂²	1	2	yes	—
22	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
23	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
24	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#430	—	2	4	no	—
26	C₂×#108×C₄₇	C₂²	1	4	yes	—
27	C₂³×Dic₄₇	Q₈	1	2	yes	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	4	no	—
30	C₂×#10625329	—	2	4	no	—
31	C₂×#10625329	C₂×C₄	1	4	yes	—
32	C₂×#10625329	D₄	1	4	yes	—
33	C₂×#10625329	Q₈	1	4	yes	—
34	C₂×#10625329	C₂³	1	4	yes	—
35	C₂²×C₄₇⋊C₈	—	2	2	no	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
40	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
41	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
43	#13093960	—	2	8	no	—
44	#13093959	—	2	16	no	—
45	Dic₄₇⋊C₈	—	2	8	no	—
46	Dic₄₇⋊C₈	—	2	8	no	—
47	#13093934	C₂×C₄	1	8	yes	—
48	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
49	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₄₇	—	2	2	no	—
52	C₂×M₁₆×C₄₇	—	2	4	no	—
53	C₂×M₁₆×C₄₇	—	2	4	no	—
54	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
55	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
56	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
57	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
58	C₂²×C₈×C₄₇	—	2	2	no	—
59	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
60	#108×C₄₇	—	2	8	no	—
61	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	4	4	no	—
71	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
72	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
73	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
74	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
75	M₁₆×C₄₇	—	2	4	no	—
76	M₁₆×C₄₇	—	2	4	no	—
77	M₁₆×C₄₇	—	4	4	no	—
78	M₁₆×C₄₇	C₄⋊C₄	1	16	yes	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	4	4	no	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
84	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
85	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
86	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
89	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
90	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
91	#10625329	—	2	4	no	—
92	#10625329	—	2	4	no	—
93	#10625329	—	4	4	no	—
94	#10625329	C₄⋊C₄	1	16	yes	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	4	4	no	—
97	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
98	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
99	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
100	C₂³×C₄₇	—	4	2	no	—
101	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
103	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
104	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×C₄×C₄₇	#127	1	4	yes	—
109	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
110	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
111	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×Dic₄₇	—	2	8	no	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
124	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
125	C₈×C₄₇	—	4	4	no	—
126	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
127	C₄₇⋊C₈	—	4	4	no	—
128	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
129	#20806	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	4	2	no	—
133	C₂²×C₄₇	—	4	2	no	—
134	C₂²×C₄₇	#384	1	4	yes	—
135	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	C₄×C₄₇	#384	1	4	yes	—
139	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
142	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	Dic₄₇	—	2	4	no	—
144	Dic₄₇	—	2	2	no	—
145	Dic₄₇	#430	1	4	yes	complemented
146	C₂×#430	—	47	2	no	—
147	C₂×#430	—	47	1	no	—
148	C₂×#430	D₄₇	1	1	yes	characteristic
149	C₂×#423	—	47	2	no	—
150	C₂×C₄×M₁₆	—	47	1	no	—
151	#1849	—	47	4	no	—
152	#1848	—	47	4	no	—
153	C₂×C₄₇	—	2	1	no	—
154	C₂×C₄₇	—	2	1	no	—
155	C₂×C₄₇	—	2	4	no	—
156	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#430	1	2	yes	—
158	C₂²×M₁₆	—	47	1	no	—
159	C₂²×M₁₆	—	47	2	no	—
160	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
161	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
162	C₂²×C₄²	—	47	1	no	—
163	#430	—	2	4	no	—
164	#430	—	94	8	no	—
165	#430	—	94	4	no	—
166	C₂×#108	—	47	8	no	—
167	C₂×#108	—	47	4	no	—
168	C₂×#108	C₂×D₄₇	1	4	yes	—
169	#423	—	47	4	no	—
170	#423	—	47	4	no	—
171	#423	—	94	4	no	—
172	C₂×C₄⋊C₈	—	94	4	no	—
173	C₄×M₁₆	—	47	4	no	—
174	C₄×M₁₆	—	47	4	no	—
175	C₂×#97	—	47	2	no	—
176	C₂×C₄×C₈	—	47	2	no	—
177	C₄₇	#20806	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	47	2	no	—
179	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	—	47	4	no	—
184	C₂×M₁₆	—	47	4	no	—
185	C₂×M₁₆	—	47	4	no	—
186	C₂×M₁₆	—	47	4	no	—
187	C₂×M₁₆	—	94	2	no	—
188	C₂×M₁₆	—	94	4	no	—
189	C₂×M₁₆	—	94	4	no	—
190	C₂×M₁₆	Dic₉₄	1	4	yes	—
191	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
192	C₂×M₁₆	C₂²⋊D₄₇	1	4	yes	—
193	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	47	2	no	—
196	C₂²×C₈	—	94	2	no	—
197	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
198	C₂×C₄²	—	47	2	no	—
199	C₂×C₄²	—	47	2	no	—
200	C₂×C₄²	—	47	4	no	—
201	C₂×C₄²	—	47	2	no	—
202	C₂×C₄²	—	47	1	no	—
203	C₂×C₄²	—	47	1	no	—
204	#108	—	2	8	no	—
205	#108	—	94	16	no	—
206	#108	—	94	8	no	—
207	C₄⋊C₈	—	94	8	no	—
208	C₄⋊C₈	—	94	8	no	—
209	#97	—	47	8	no	—
210	C₄×C₈	—	47	4	no	—
211	C₄×C₈	—	47	4	no	—
212	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
213	C₂²×C₄	—	47	4	no	—
214	C₂²×C₄	—	47	2	no	—
215	C₂²×C₄	—	47	4	no	—
216	C₂²×C₄	—	47	2	no	—
217	C₂²×C₄	—	188	4	no	—
218	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
219	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
220	C₂²×C₄	D₄×D₄₇	1	2	yes	—
221	C₂²×C₄	D₄×D₄₇	1	2	yes	—
222	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
223	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
224	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
225	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	4	no	—
229	M₁₆	—	47	16	no	—
230	M₁₆	—	94	4	no	—
231	M₁₆	—	94	4	no	—
232	M₁₆	—	188	4	no	—
233	M₁₆	Dic₄₇⋊C₄	1	16	yes	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	4	no	—
236	C₂×C₈	—	47	4	no	—
237	C₂×C₈	—	47	4	no	—
238	C₂×C₈	—	47	4	no	—
239	C₂×C₈	—	94	4	no	—
240	C₂×C₈	—	188	4	no	—
241	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
242	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
243	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
244	C₄²	—	47	4	no	—
245	C₄²	—	47	4	no	—
246	C₄²	—	47	1	no	—
247	C₄²	—	47	1	no	—
248	C₄²	—	47	2	no	—
249	C₄²	—	47	2	no	—
250	C₄²	—	47	2	no	—
251	C₂³	—	4	2	no	—
252	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
253	C₂³	#13094022	1	2	yes	—
254	C₂³	#13094023	1	2	yes	—
255	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
256	C₂×C₄	—	47	2	no	—
257	C₂×C₄	—	47	2	no	—
258	C₂×C₄	—	47	4	no	—
259	C₂×C₄	—	94	8	no	—
260	C₂×C₄	—	94	4	no	—
261	C₂×C₄	—	188	4	no	—
262	C₂×C₄	—	188	2	no	—
263	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
264	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
265	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
266	C₂×C₄	#13094020	1	4	yes	—
267	C₂×C₄	#13094022	1	2	yes	—
268	C₂×C₄	#13094023	1	2	yes	—
269	C₂×C₄	#13094023	1	4	yes	—
270	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
271	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
272	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
276	C₈	—	4	4	no	—
277	C₈	—	47	8	no	—
278	C₈	—	188	4	no	—
279	C₈	C₂×Dic₄₇⋊C₄	1	8	yes	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	#15688992	1	4	yes	—
285	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
286	C₂²	C₂×#13094022	1	1	yes	characteristic
287	C₂²	C₂×#13094023	1	1	yes	characteristic
288	C₄	—	47	4	no	—
289	C₄	—	94	2	no	—
290	C₄	—	94	4	no	—
291	C₄	#15688992	1	4	yes	—
292	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
293	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
294	C₄	C₂×#13094020	1	1	yes	characteristic
295	C₄	C₂×#13094023	1	1	yes	characteristic
296	C₂	—	2	4	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	C₂×#15688992	1	1	yes	characteristic
300	C₂	#430×D₄₇	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	#20806
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
145. Dic₄₇	#430	semidirect
177. C₄₇	#20806	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
C₂×#15689171	2
C₂×#15689146	2
C₂×M₁₆×Dic₄₇	2
#20243561	2
#20243558	2
C₂×C₄₇×#430	2
#20806	47