Subgroups — #26369654

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093970	C₂	1	1	yes	maximal, characteristic
2	#20252257	C₂	1	1	yes	maximal, characteristic
3	#20245257	C₂	1	4	yes	maximal, complemented
4	C₄×#13093970	C₂	1	4	yes	maximal, complemented
5	#20245170	C₂	1	1	yes	maximal, characteristic
6	#20245108	C₂	1	1	yes	maximal, characteristic
7	#20245107	C₂	1	1	yes	maximal, characteristic
8	#20244985	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#2950	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
11	#15689600	—	2	1	no	—
12	C₂×#13093970	C₄	1	8	yes	complemented
13	C₂×#13093970	C₂²	1	2	yes	—
14	C₂×#13093970	C₂²	1	2	yes	—
15	C₂×#13093964	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	#15689231	—	2	2	no	—
18	C₄×Dic₁₈₈	C₂²	1	4	yes	—
19	#15689161	C₂²	1	1	yes	characteristic
20	#15689133	—	2	1	no	—
21	#15688950	C₂²	1	2	yes	—
22	#15688948	C₂²	1	2	yes	—
23	#15688927	C₂²	1	4	yes	—
24	#15688926	C₂²	1	4	yes	—
25	#15688899	C₂²	1	2	yes	—
26	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
27	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#519	—	2	1	no	—
29	C₄₇×#441	—	2	2	no	—
30	C₄×Q₁₆×C₄₇	C₂²	1	4	yes	—
31	C₄₇×#427	C₂²	1	1	yes	characteristic
32	C₄₇×#419	—	2	1	no	—
33	C₄₇×#405	C₂²	1	1	yes	characteristic
34	C₂²×Dic₉₄	—	2	1	no	—
35	C₂×Dic₁₈₈	—	2	1	no	—
36	C₂×Dic₁₈₈	—	2	2	no	—
37	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
38	C₂×Dic₁₈₈	C₂³	1	2	yes	—
39	C₄×Dic₉₄	—	2	4	no	—
40	#13093989	—	4	2	no	—
41	#13093983	—	2	1	no	—
42	#13093980	—	4	2	no	—
43	#13093970	C₂×C₄	1	8	yes	—
44	#13093970	C₂×C₄	1	8	yes	—
45	#13093964	—	2	2	no	—
46	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
47	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
50	#13093958	C₂³	1	2	yes	—
51	#13093956	—	2	1	no	—
52	#13093949	—	2	4	no	—
53	C₂²×Q₈×C₄₇	—	2	1	no	—
54	C₂×Q₁₆×C₄₇	—	2	1	no	—
55	C₂×Q₁₆×C₄₇	—	2	2	no	—
56	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	—
57	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
58	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₄×Q₈×C₄₇	—	2	4	no	—
61	C₄×Q₈×C₄₇	—	4	2	no	—
62	#117×C₄₇	—	2	1	no	—
63	#117×C₄₇	—	4	2	no	—
64	#117×C₄₇	D₄	1	1	yes	characteristic
65	#107×C₄₇	C₂³	1	2	yes	—
66	#106×C₄₇	—	2	1	no	—
67	#103×C₄₇	—	2	4	no	—
68	#97×C₄₇	D₄	1	2	yes	—
69	C₄×C₈×C₄₇	C₂³	1	2	yes	—
70	C₂×Q₈×C₄₇	—	2	4	no	—
71	C₂×Q₈×C₄₇	—	2	2	no	—
72	C₂×Q₈×C₄₇	—	4	1	no	—
73	C₂×Q₈×C₄₇	—	4	2	no	—
74	C₂²×C₄×C₄₇	—	4	1	no	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	4	1	no	—
77	C₂×Dic₉₄	—	2	4	no	—
78	C₂×Dic₉₄	—	2	2	no	—
79	C₂×Dic₉₄	—	4	1	no	—
80	C₂×Dic₉₄	—	4	2	no	—
81	Q₁₆×C₄₇	—	2	4	no	—
82	Q₁₆×C₄₇	—	2	2	no	—
83	Q₁₆×C₄₇	C₂²×C₄	1	4	yes	—
84	M₁₆×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	#41	1	2	yes	—
89	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₄⋊C₄×C₄₇	—	4	4	no	—
92	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
93	C₂²⋊C₄×C₄₇	—	4	1	no	—
94	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
95	C₄²×C₄₇	—	4	1	no	—
96	C₄²×C₄₇	—	4	4	no	—
97	C₄²×C₄₇	C₂×D₄	1	2	yes	—
98	C₂²⋊Dic₄₇	—	4	1	no	—
99	C₄⋊Dic₄₇	—	2	2	no	—
100	Dic₄₇⋊C₄	—	4	4	no	—
101	C₄×Dic₄₇	—	4	4	no	—
102	C₄×Dic₄₇	—	4	1	no	—
103	Dic₁₈₈	—	2	4	no	—
104	Dic₁₈₈	—	2	2	no	—
105	Dic₁₈₈	C₂²×C₄	1	4	yes	—
106	C₄₇⋊C₁₆	—	2	2	no	—
107	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
109	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
110	Q₈×C₄₇	—	2	4	no	—
111	Q₈×C₄₇	—	4	2	no	—
112	Q₈×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
118	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#135	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	Dic₉₄	—	2	4	no	—
125	Dic₉₄	—	4	2	no	—
126	Dic₉₄	—	4	4	no	—
127	C₈×C₄₇	—	2	2	no	—
128	C₈×C₄₇	C₄×D₄	1	2	yes	—
129	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
131	#33458	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	C₄×D₈	1	2	yes	—
134	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₄₇	—	2	4	no	—
136	C₄×C₄₇	—	4	4	no	—
137	C₄×C₄₇	—	8	2	no	—
138	C₄×C₄₇	C₄×D₈	1	2	yes	—
139	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
141	Dic₄₇	—	4	4	no	—
142	Dic₄₇	—	8	2	no	—
143	C₂²×Q₃₂	—	47	1	no	—
144	#2950	—	47	1	no	—
145	#2950	D₄₇	1	1	yes	characteristic
146	#2176	—	47	4	no	—
147	C₄×Q₃₂	—	47	4	no	—
148	#2158	—	47	1	no	—
149	#2141	—	47	1	no	—
150	#2141	—	47	1	no	—
151	#2108	—	47	1	no	—
152	C₂×C₄₇	—	2	2	no	—
153	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
154	C₂×C₄₇	#3417	1	2	yes	—
155	C₂²×Q₁₆	—	47	1	no	—
156	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
157	#519	—	2	1	no	—
158	#519	—	94	1	no	—
159	C₂×Q₃₂	—	47	8	no	—
160	C₂×Q₃₂	—	47	2	no	—
161	C₂×Q₃₂	—	47	2	no	—
162	C₂×M₃₂	—	47	1	no	—
163	C₂²×C₁₆	—	47	1	no	—
164	#441	—	2	2	no	—
165	#441	—	94	2	no	—
166	C₄×Q₁₆	—	47	4	no	—
167	C₄×Q₁₆	C₂×D₄₇	1	4	yes	—
168	#427	—	47	1	no	—
169	#427	C₂×D₄₇	1	1	yes	characteristic
170	#419	—	2	1	no	—
171	#419	—	94	1	no	—
172	#405	C₂×D₄₇	1	1	yes	characteristic
173	#367	—	47	2	no	—
174	#366	—	47	2	no	—
175	#358	—	47	4	no	—
176	#358	—	47	4	no	—
177	#346	—	47	2	no	—
178	C₄×C₁₆	—	47	2	no	—
179	C₄₇	#33458	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×Q₈	—	94	1	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	1	no	—
184	C₂×Q₁₆	—	47	4	no	—
185	C₂×Q₁₆	—	47	2	no	—
186	C₂×Q₁₆	—	94	2	no	—
187	C₂×Q₁₆	—	94	1	no	—
188	C₂×Q₁₆	C₄×D₄₇	1	4	yes	—
189	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
190	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
192	C₄×Q₈	—	2	4	no	—
193	C₄×Q₈	—	4	2	no	—
194	C₄×Q₈	—	94	4	no	—
195	C₄×Q₈	—	188	2	no	—
196	#117	—	2	1	no	—
197	#117	—	4	2	no	—
198	#117	—	94	1	no	—
199	#117	—	188	2	no	—
200	#117	C₂²⋊D₄₇	1	1	yes	characteristic
201	Q₃₂	—	47	8	no	—
202	Q₃₂	—	47	8	no	—
203	M₃₂	—	94	2	no	—
204	C₂×C₁₆	—	47	4	no	—
205	C₂×C₁₆	—	47	1	no	—
206	C₂×C₁₆	—	47	1	no	—
207	C₂×C₁₆	—	47	2	no	—
208	#107	—	47	2	no	—
209	#107	C₂²×D₄₇	1	2	yes	—
210	#106	—	2	1	no	—
211	#106	—	94	1	no	—
212	#103	—	2	4	no	—
213	#103	—	94	4	no	—
214	#97	C₂²⋊D₄₇	1	2	yes	—
215	C₄×C₈	C₂²×D₄₇	1	2	yes	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	4	1	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	94	4	no	—
221	C₂×Q₈	—	94	2	no	—
222	C₂×Q₈	—	188	1	no	—
223	C₂×Q₈	—	188	2	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	188	1	no	—
226	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
227	Q₁₆	—	2	4	no	—
228	Q₁₆	—	2	2	no	—
229	Q₁₆	—	47	4	no	—
230	Q₁₆	—	94	4	no	—
231	Q₁₆	—	94	2	no	—
232	Q₁₆	C₂×C₄×D₄₇	1	4	yes	—
233	M₁₆	—	2	2	no	—
234	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
235	C₂×C₈	#10625350	1	2	yes	—
236	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	94	2	no	—
242	C₄⋊C₄	—	188	4	no	—
243	C₄⋊C₄	#10625334	1	2	yes	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	188	1	no	—
246	C₂²⋊C₄	#10625334	1	2	yes	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	4	no	—
249	C₄²	—	188	1	no	—
250	C₄²	—	188	4	no	—
251	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
252	C₁₆	—	47	2	no	—
253	C₁₆	—	47	2	no	—
254	C₁₆	—	94	2	no	—
255	C₂³	C₂×#10625334	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	4	2	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	94	4	no	—
260	Q₈	—	188	2	no	—
261	Q₈	—	188	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	188	2	no	—
266	C₂×C₄	—	188	2	no	—
267	C₂×C₄	—	188	4	no	—
268	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
269	C₂×C₄	C₂×#10625334	1	4	yes	—
270	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
271	C₂×C₄	#13094044	1	2	yes	—
272	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	C₄×C₂²⋊D₄₇	1	2	yes	—
275	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
276	C₈	C₂×#10625350	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	C₄×#10625334	1	2	yes	—
279	C₂²	C₂²×#10625334	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	4	4	no	—
282	C₄	—	8	2	no	—
283	C₄	—	376	2	no	—
284	C₄	—	188	4	no	—
285	C₄	C₄×#10625334	1	2	yes	—
286	C₄	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
287	C₄	C₂×#13094044	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	C₂×C₄×#10625334	1	1	yes	characteristic
290	C₂	#20256539	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2950	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2950	D₄₇
hypocenter	C₄₇	#33458
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
3. #20245257	C₂	semidirect
4. C₄×#13093970	C₂	semidirect
12. C₂×#13093970	C₄	semidirect
179. C₄₇	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093970	2
#20252257	2
#20245257	2
C₄×#13093970	2
#20245170	2
#20245108	2
#20245107	2
#20244985	2
C₄₇×#2950	2
#33458	47