Subgroups — #26125013

Order 12032.

666 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243127	C₂	1	1	yes	maximal, characteristic
2	#20243121	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688883	C₂	1	2	yes	maximal
4	C₂×#15688879	C₂	1	2	yes	maximal
5	C₄₇×#2296	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	2	yes	—
8	#15688892	—	2	2	no	—
9	#15688890	—	2	2	no	—
10	#15688883	—	2	4	no	—
11	#15688879	—	2	4	no	—
12	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#389	—	2	2	no	—
16	C₄₇×#385	—	2	2	no	—
17	C₄×C₄⋊C₄×C₄₇	C₄	1	4	yes	complemented
18	C₂×#10625329	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	Q₈	1	4	yes	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	#13093983	—	2	2	no	—
26	#13093983	—	2	2	no	—
27	#13093983	—	2	2	no	—
28	#13093983	D₄	1	4	yes	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂³×C₄×C₄₇	—	2	1	no	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	#117×C₄₇	—	2	2	no	—
35	#117×C₄₇	—	2	2	no	—
36	#117×C₄₇	—	2	2	no	—
37	#117×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
42	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
43	C₂×C₄²×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	—	2	1	no	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
49	#95×C₄₇	—	2	4	no	—
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×Dic₄₇	—	4	2	no	—
65	C₄⋊C₄×C₄₇	—	2	4	no	—
66	C₄⋊C₄×C₄₇	—	4	8	no	—
67	C₄⋊C₄×C₄₇	—	4	1	no	—
68	C₄⋊C₄×C₄₇	—	4	1	no	—
69	C₂²⋊C₄×C₄₇	—	2	4	no	—
70	C₂²⋊C₄×C₄₇	—	4	2	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	2	2	no	—
73	C₄²×C₄₇	—	2	4	no	—
74	C₄²×C₄₇	—	2	4	no	—
75	C₄²×C₄₇	—	4	1	no	—
76	C₄²×C₄₇	—	4	1	no	—
77	C₂²⋊Dic₄₇	—	2	4	no	—
78	C₂²⋊Dic₄₇	—	4	2	no	—
79	C₄⋊Dic₄₇	—	2	4	no	—
80	C₄⋊Dic₄₇	—	4	2	no	—
81	C₄×Dic₄₇	—	2	4	no	—
82	C₄×Dic₄₇	—	4	2	no	—
83	#10625329	—	2	4	no	—
84	#10625329	—	2	4	no	—
85	#10625329	—	4	2	no	—
86	#10625329	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂³×C₄₇	—	4	2	no	—
91	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂³×C₄₇	#121	1	2	yes	—
93	C₂³×C₄₇	#123	1	2	yes	—
94	C₂³×C₄₇	#124	1	1	yes	characteristic
95	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	4	2	no	—
99	C₂×C₄×C₄₇	—	4	8	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	4	4	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	#123	1	2	yes	—
112	C₂×C₄×C₄₇	#123	1	2	yes	—
113	C₂×C₄×C₄₇	#123	1	2	yes	—
114	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#125	1	2	yes	—
116	C₂×C₄×C₄₇	#125	1	2	yes	—
117	C₂×C₄×C₄₇	#126	1	2	yes	—
118	C₂×C₄×C₄₇	#126	1	2	yes	—
119	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	8	4	no	—
122	C₄₇⋊C₈	—	4	4	no	—
123	#11347	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	4	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	4	2	no	—
127	C₂²×C₄₇	—	4	1	no	—
128	C₂²×C₄₇	—	4	1	no	—
129	C₂²×C₄₇	#381	1	1	yes	characteristic
130	C₂²×C₄₇	#390	1	1	yes	characteristic
131	C₂²×C₄₇	#391	1	1	yes	characteristic
132	C₂²×C₄₇	#397	1	2	yes	—
133	C₂²×C₄₇	#398	1	2	yes	—
134	C₄×C₄₇	—	4	4	no	—
135	C₄×C₄₇	—	4	4	no	—
136	C₄×C₄₇	—	8	2	no	—
137	C₄×C₄₇	#387	1	2	yes	—
138	C₄×C₄₇	#387	1	2	yes	—
139	C₄×C₄₇	#399	1	2	yes	—
140	C₄×C₄₇	#400	1	2	yes	—
141	Dic₄₇	—	8	4	no	—
142	#2296	D₄₇	1	1	yes	characteristic
143	#1742	—	47	1	no	—
144	#1740	—	47	1	no	—
145	C₂×#339	—	47	2	no	—
146	C₂×H(3,ℤ/4)	—	47	2	no	—
147	C₂×C₄₇	—	2	2	no	—
148	C₂×C₄₇	—	2	1	no	—
149	C₂×C₄₇	—	2	1	no	—
150	C₂×C₄₇	—	2	2	no	—
151	C₂×C₄₇	#1445	1	1	yes	characteristic
152	C₂×C₄₇	#1827	1	2	yes	—
153	C₂²×M₁₆	—	47	2	no	—
154	C₂×#117	—	47	2	no	—
155	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
156	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
157	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
158	#389	—	2	2	no	—
159	#385	—	2	2	no	—
160	C₄×C₄⋊C₄	Dic₄₇	1	4	yes	complemented
161	#343	—	94	2	no	—
162	#342	—	94	2	no	—
163	#339	—	94	4	no	—
164	H(3,ℤ/4)	—	94	4	no	—
165	C₄₇	#11347	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	—	2	1	no	—
167	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
168	C₂×M₁₆	—	47	4	no	—
169	C₂×M₁₆	—	94	2	no	—
170	C₂×M₁₆	—	94	2	no	—
171	C₂×M₁₆	—	94	2	no	—
172	C₂×M₁₆	—	94	2	no	—
173	C₂×M₁₆	—	94	2	no	—
174	C₂²×C₈	—	94	2	no	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	—	47	4	no	—
180	#117	—	94	2	no	—
181	#117	—	94	2	no	—
182	#117	—	94	2	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	94	2	no	—
186	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
187	C₂×C₂²⋊C₄	—	2	1	no	—
188	C₂×C₂²⋊C₄	—	94	2	no	—
189	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	94	2	no	—
194	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
195	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
196	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
197	#95	—	2	4	no	—
198	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	188	2	no	—
204	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
205	C₂²×C₄	#10625350	1	2	yes	—
206	C₂²×C₄	#10625350	1	2	yes	—
207	C₂²×C₄	D₄×D₄₇	1	2	yes	—
208	C₂²×C₄	#10625352	1	2	yes	—
209	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
210	C₂²×C₄	#10625354	1	2	yes	—
211	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
213	M₁₆	—	94	4	no	—
214	M₁₆	—	94	4	no	—
215	M₁₆	—	188	2	no	—
216	M₁₆	—	188	2	no	—
217	C₂×C₈	—	94	4	no	—
218	C₂×C₈	—	188	2	no	—
219	C₂×C₈	—	188	2	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	8	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	94	4	no	—
225	C₄⋊C₄	—	188	2	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	94	4	no	—
229	C₂²⋊C₄	—	188	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	94	4	no	—
237	C₄²	—	188	2	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
240	C₂³	#13093995	1	2	yes	—
241	C₂³	#13094007	1	2	yes	—
242	C₂³	#13094013	1	1	yes	characteristic
243	C₂³	#13094022	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	376	4	no	—
253	C₂×C₄	—	188	4	no	—
254	C₂×C₄	#13093983	1	1	yes	characteristic
255	C₂×C₄	#13093983	1	1	yes	characteristic
256	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
257	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
258	C₂×C₄	#13094000	1	2	yes	—
259	C₂×C₄	#13094002	1	2	yes	—
260	C₂×C₄	#13094002	1	2	yes	—
261	C₂×C₄	#13094007	1	2	yes	—
262	C₂×C₄	#13094009	1	2	yes	—
263	C₂×C₄	#13094009	1	2	yes	—
264	C₂×C₄	#13094011	1	1	yes	characteristic
265	C₂×C₄	#13094014	1	2	yes	—
266	C₂×C₄	#13094014	1	2	yes	—
267	C₂×C₄	#13094019	1	2	yes	—
268	C₂×C₄	#13094019	1	2	yes	—
269	C₂×C₄	#13094020	1	1	yes	characteristic
270	C₈	—	188	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	#15688986	1	1	yes	characteristic
277	C₂²	#15689026	1	1	yes	characteristic
278	C₂²	#15689030	1	1	yes	characteristic
279	C₂²	#15689049	1	2	yes	—
280	C₂²	#15689054	1	2	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	4	4	no	—
283	C₄	—	8	2	no	—
284	C₄	—	376	4	no	—
285	C₄	#15689009	1	2	yes	—
286	C₄	#15689009	1	2	yes	—
287	C₄	#15689058	1	2	yes	—
288	C₄	#15689061	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	#20241656	1	1	yes	characteristic
294	C₂	#20243492	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093983
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2296	C₂
socle	C₂²×C₄₇	#381
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2296	D₄₇
hypocenter	C₄₇	#11347
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ #2296

Complements

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

N	H ≅ G/N	type
17. C₄×C₄⋊C₄×C₄₇	C₄	semidirect
160. C₄×C₄⋊C₄	Dic₄₇	semidirect
165. C₄₇	#11347	semidirect

Maximal subgroups

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

subgroup	index
#20243127	2
#20243121	2
C₂×#15688883	2
C₂×#15688879	2
C₄₇×#2296	2
#11347	47