Subgroups — #26125014

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	#20245872	C₂	1	1	yes	maximal, characteristic
2	#20243121	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688884	C₂	1	2	yes	maximal
4	C₂×#15688880	C₂	1	2	yes	maximal
5	C₄₇×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	#15689022	—	2	2	no	—
10	#15688998	—	2	2	no	—
11	C₄×C₄⋊Dic₄₇	C₄	1	4	yes	complemented
12	#15688890	—	2	2	no	—
13	#15688884	—	2	4	no	—
14	#15688880	—	2	4	no	—
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#343	—	2	2	no	—
18	C₂³×Dic₄₇	—	2	1	no	—
19	#13093983	—	2	2	no	—
20	#13093983	—	2	2	no	—
21	#13093983	—	2	2	no	—
22	#13093983	—	2	2	no	—
23	#13093983	—	2	2	no	—
24	#13093983	D₄	1	4	yes	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	4	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
29	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
36	C₄₇⋊#95	—	2	4	no	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	—	2	2	no	—
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
44	C₂²×C₈×C₄₇	—	2	2	no	—
45	#117×C₄₇	—	2	2	no	—
46	#117×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₄₇	#41	1	2	yes	—
57	C₂²×C₄×C₄₇	#41	1	2	yes	—
58	C₂²×C₄×C₄₇	#41	1	2	yes	—
59	C₂²×C₄×C₄₇	#41	1	2	yes	—
60	C₂²×Dic₄₇	—	2	2	no	—
61	C₂²×Dic₄₇	—	2	2	no	—
62	C₂²×Dic₄₇	—	4	4	no	—
63	C₂²×Dic₄₇	—	4	2	no	—
64	C₂²×Dic₄₇	—	4	2	no	—
65	M₁₆×C₄₇	—	2	4	no	—
66	M₁₆×C₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	4	2	no	—
68	M₁₆×C₄₇	—	4	2	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	4	2	no	—
71	C₂×C₈×C₄₇	—	4	2	no	—
72	C₄⋊C₄×C₄₇	—	2	4	no	—
73	C₂²⋊C₄×C₄₇	—	2	4	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	—	2	2	no	—
76	C₂²⋊Dic₄₇	—	2	4	no	—
77	C₂²⋊Dic₄₇	—	4	2	no	—
78	C₂²⋊Dic₄₇	—	4	2	no	—
79	C₄⋊Dic₄₇	—	2	4	no	—
80	C₄⋊Dic₄₇	—	4	8	no	—
81	C₄⋊Dic₄₇	—	4	1	no	—
82	C₄⋊Dic₄₇	—	4	2	no	—
83	C₄⋊Dic₄₇	—	4	1	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	C₄×Dic₄₇	—	2	4	no	—
86	C₄×Dic₄₇	—	2	4	no	—
87	C₄×Dic₄₇	—	4	1	no	—
88	C₄×Dic₄₇	—	4	2	no	—
89	C₄×Dic₄₇	—	4	1	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	—	4	4	no	—
98	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₄₇	#123	1	2	yes	—
106	C₂×C₄×C₄₇	#123	1	2	yes	—
107	C₂×C₄×C₄₇	#123	1	2	yes	—
108	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#125	1	2	yes	—
110	C₂×C₄×C₄₇	#125	1	2	yes	—
111	C₂×C₄×C₄₇	#126	1	2	yes	—
112	C₂×C₄×C₄₇	#126	1	2	yes	—
113	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
114	C₂×Dic₄₇	—	2	4	no	—
115	C₂×Dic₄₇	—	4	2	no	—
116	C₂×Dic₄₇	—	4	8	no	—
117	C₂×Dic₄₇	—	4	2	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	4	2	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	8	4	no	—
122	C₈×C₄₇	—	4	4	no	—
123	#11347	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	2	4	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	#387	1	2	yes	—
136	C₄×C₄₇	#387	1	2	yes	—
137	C₄×C₄₇	#399	1	2	yes	—
138	C₄×C₄₇	#400	1	2	yes	—
139	Dic₄₇	—	4	4	no	—
140	Dic₄₇	—	8	2	no	—
141	Dic₄₇	—	8	4	no	—
142	#2296	—	47	1	no	—
143	#1742	D₄₇	1	1	yes	characteristic
144	#1740	—	47	1	no	—
145	C₂×#339	—	47	2	no	—
146	C₂×H(3,ℤ/4)	—	47	2	no	—
147	C₂×C₄₇	—	2	1	no	—
148	C₂×C₄₇	—	2	2	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	C₂×D₄₇	1	2	yes	—
154	C₂×#117	—	47	1	no	—
155	C₂×#117	—	47	2	no	—
156	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
157	C₂²×C₄²	—	47	1	no	—
158	#389	—	94	2	no	—
159	#385	—	94	2	no	—
160	C₄×C₄⋊C₄	—	47	4	no	—
161	#343	—	2	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	—	94	1	no	—
167	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	Dic₉₄	1	4	yes	—
174	C₂²×C₈	—	2	2	no	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	47	4	no	—
178	#117	—	94	2	no	—
179	#117	—	94	2	no	—
180	#117	—	94	2	no	—
181	#117	—	94	2	no	—
182	#117	—	94	2	no	—
183	C₂×C₄⋊C₄	—	94	2	no	—
184	C₂×C₄⋊C₄	—	94	4	no	—
185	C₂×C₄⋊C₄	—	94	1	no	—
186	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
187	C₂×C₂²⋊C₄	—	94	2	no	—
188	C₂×C₂²⋊C₄	—	94	1	no	—
189	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
190	C₂×C₄²	—	47	2	no	—
191	C₂×C₄²	—	47	2	no	—
192	C₂×C₄²	—	94	2	no	—
193	C₂×C₄²	—	94	2	no	—
194	C₂×C₄²	—	94	2	no	—
195	C₂×C₄²	—	94	1	no	—
196	C₂×C₄²	C₄×D₄₇	1	2	yes	—
197	#95	—	94	4	no	—
198	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
199	C₂²×C₄	—	94	2	no	—
200	C₂²×C₄	—	94	2	no	—
201	C₂²×C₄	—	188	4	no	—
202	C₂²×C₄	—	188	2	no	—
203	C₂²×C₄	—	188	2	no	—
204	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
205	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
206	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	#10625350	1	2	yes	—
208	C₂²×C₄	D₄×D₄₇	1	2	yes	—
209	C₂²×C₄	#10625352	1	2	yes	—
210	C₂²×C₄	#10625352	1	2	yes	—
211	C₂²×C₄	#10625354	1	2	yes	—
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	94	4	no	—
222	C₄⋊C₄	—	188	8	no	—
223	C₄⋊C₄	—	188	1	no	—
224	C₄⋊C₄	—	188	2	no	—
225	C₄⋊C₄	—	188	1	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	94	4	no	—
228	C₂²⋊C₄	—	188	2	no	—
229	C₂²⋊C₄	—	188	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	94	4	no	—
233	C₄²	—	94	4	no	—
234	C₄²	—	94	4	no	—
235	C₄²	—	188	1	no	—
236	C₄²	—	188	2	no	—
237	C₄²	—	188	1	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
240	C₂³	#13093999	1	2	yes	—
241	C₂³	#13094008	1	2	yes	—
242	C₂³	#13094012	1	1	yes	characteristic
243	C₂³	#13094024	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	94	4	no	—
247	C₂×C₄	—	376	4	no	—
248	C₂×C₄	—	188	4	no	—
249	C₂×C₄	—	188	2	no	—
250	C₂×C₄	—	188	4	no	—
251	C₂×C₄	—	188	2	no	—
252	C₂×C₄	—	188	8	no	—
253	C₂×C₄	—	188	2	no	—
254	C₂×C₄	#13093980	1	1	yes	characteristic
255	C₂×C₄	#13093980	1	1	yes	characteristic
256	C₂×C₄	#13093986	1	2	yes	—
257	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
258	C₂×C₄	#13094001	1	2	yes	—
259	C₂×C₄	#13094001	1	2	yes	—
260	C₂×C₄	#13094003	1	2	yes	—
261	C₂×C₄	#13094005	1	2	yes	—
262	C₂×C₄	#13094006	1	2	yes	—
263	C₂×C₄	#13094006	1	2	yes	—
264	C₂×C₄	#13094013	1	1	yes	characteristic
265	C₂×C₄	#13094015	1	2	yes	—
266	C₂×C₄	#13094015	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	—	4	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	#15688984	1	1	yes	characteristic
277	C₂²	#15689024	1	1	yes	characteristic
278	C₂²	#15689029	1	1	yes	characteristic
279	C₂²	#15689050	1	2	yes	—
280	C₂²	#15689055	1	2	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	376	4	no	—
283	C₄	—	376	2	no	—
284	C₄	—	188	4	no	—
285	C₄	#15689006	1	2	yes	—
286	C₄	#15689006	1	2	yes	—
287	C₄	#15689059	1	2	yes	—
288	C₄	#15689062	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	#20241657	1	1	yes	characteristic
294	C₂	#20243493	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093980
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1742	C₂
socle	C₂²×C₄₇	#381
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1742	D₄₇
hypocenter	C₄₇	#11347
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
11. C₄×C₄⋊Dic₄₇	C₄	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
#20245872	2
#20243121	2
C₂×#15688884	2
C₂×#15688880	2
C₄₇×#1742	2
#11347	47