Subgroups — #26186958

Order 12032.

560 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252483	C₂	1	2	yes	maximal
2	#20242297	C₂	1	2	yes	maximal, complemented
3	#20242285	C₂	1	2	yes	maximal, complemented
4	#20242207	C₂	1	2	yes	maximal, complemented
5	Q₁₆×C₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
6	#20242151	C₂	1	1	yes	maximal, characteristic
7	#20241861	C₂	1	2	yes	maximal
8	#20241671	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#3170	C₂	1	1	yes	maximal, characteristic
10	#15689254	—	2	4	no	—
11	#15689254	C₂²	1	4	yes	—
12	Q₈×C₄₇⋊C₈	C₂²	1	4	yes	—
13	#15689184	—	2	4	no	—
14	#15689180	C₂²	1	2	yes	—
15	#15689176	C₂²	1	1	yes	characteristic
16	#15689176	C₂²	1	1	yes	characteristic
17	#15688876	C₂²	1	2	yes	—
18	#15688874	C₂²	1	2	yes	—
19	Q₈⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
20	Q₈⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
21	#15688844	C₂²	1	1	yes	characteristic
22	#15688844	C₂²	1	1	yes	characteristic
23	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
24	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
25	C₄₇×#533	C₂²	1	2	yes	—
26	C₄₇×#500	C₄	1	1	yes	characteristic
27	C₄₇×#494	C₄	1	1	yes	characteristic
28	C₄₇×#487	C₄	1	2	yes	—
29	C₄₇×#467	C₄	1	2	yes	—
30	C₄₇×#467	C₄	1	2	yes	—
31	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
32	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
37	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
38	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
39	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
40	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
41	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
42	#13093935	—	2	2	no	—
43	#13093935	—	2	4	no	—
44	C₄×C₄₇⋊C₈	—	2	4	no	—
45	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
54	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
55	#125×C₄₇	—	2	2	no	—
56	C₂²⋊Q₈×C₄₇	—	2	2	no	—
57	C₂²⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
58	C₄×Q₈×C₄₇	—	2	2	no	—
59	C₄×Q₈×C₄₇	C₂³	1	4	yes	—
60	#117×C₄₇	—	2	2	no	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	#107×C₄₇	C₂×C₄	1	2	yes	—
63	#107×C₄₇	C₂×C₄	1	2	yes	—
64	#103×C₄₇	C₂×C₄	1	4	yes	—
65	#103×C₄₇	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	C₂×Q₈×C₄₇	—	2	2	no	—
71	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
75	Q₁₆×C₄₇	—	2	4	no	—
76	Q₁₆×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₄₇	—	2	2	no	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₄⋊C₄×C₄₇	—	2	4	no	—
88	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
89	C₂²⋊C₄×C₄₇	—	2	4	no	—
90	C₄²×C₄₇	—	2	4	no	—
91	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₇	#41	1	1	yes	characteristic
94	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	1	no	—
97	C₂×C₄₇⋊C₈	—	2	1	no	—
98	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
99	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
100	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
101	Q₈×C₄₇	—	2	4	no	—
102	Q₈×C₄₇	—	2	4	no	—
103	Q₈×C₄₇	—	8	2	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
116	C₈×C₄₇	—	2	1	no	—
117	C₈×C₄₇	—	2	1	no	—
118	C₈×C₄₇	#131	1	2	yes	—
119	C₈×C₄₇	#131	1	2	yes	—
120	C₄₇⋊C₈	—	2	2	no	—
121	C₄₇⋊C₈	—	2	2	no	—
122	C₄₇⋊C₈	—	4	1	no	—
123	C₄₇⋊C₈	—	4	1	no	—
124	C₄₇⋊C₈	—	4	4	no	—
125	#17972	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	1	no	—
127	C₂²×C₄₇	—	2	1	no	—
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₄₇	—	2	1	no	—
131	C₄×C₄₇	—	2	1	no	—
132	C₄×C₄₇	—	2	1	no	—
133	C₄×C₄₇	—	2	1	no	—
134	C₄×C₄₇	—	4	4	no	—
135	C₄×C₄₇	—	4	4	no	—
136	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
137	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
138	C₄×C₄₇	#578	1	2	yes	—
139	#3170	D₄₇	1	1	yes	characteristic
140	#2985	—	47	2	no	—
141	#1593	—	47	2	no	—
142	#1591	—	47	2	no	—
143	#1579	—	47	2	no	—
144	C₈×Q₁₆	—	47	2	no	—
145	#1568	—	47	1	no	—
146	#1497	—	47	2	no	—
147	#1450	—	47	1	no	—
148	C₂×C₄₇	—	4	1	no	—
149	C₂×C₄₇	#2931	1	1	yes	characteristic
150	C₂×C₄₇	#2950	1	1	yes	characteristic
151	C₂×C₄₇	#2957	1	1	yes	characteristic
152	#533	C₂×D₄₇	1	2	yes	—
153	#500	Dic₄₇	1	1	yes	characteristic
154	#494	Dic₄₇	1	1	yes	characteristic
155	#487	Dic₄₇	1	2	yes	—
156	#467	Dic₄₇	1	2	yes	—
157	#467	Dic₄₇	1	2	yes	—
158	#446	—	47	4	no	—
159	#446	—	94	4	no	—
160	C₈×Q₈	—	47	4	no	—
161	C₄×Q₁₆	C₂×D₄₇	1	2	yes	—
162	C₄×Q₁₆	C₂×D₄₇	1	2	yes	—
163	#433	—	94	4	no	—
164	#432	—	47	2	no	—
165	#431	—	47	1	no	—
166	#431	—	47	1	no	—
167	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
168	#335	—	47	2	no	—
169	#334	—	47	2	no	—
170	Q₈⋊C₈	—	47	4	no	—
171	Q₈⋊C₈	—	47	4	no	—
172	#322	—	47	1	no	—
173	#322	—	47	1	no	—
174	C₈²	—	47	1	no	—
175	C₈²	—	47	1	no	—
176	C₄₇	#17972	1	1	yes	characteristic, complemented, Sylow
177	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
178	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
179	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
181	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
182	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
183	#125	—	2	2	no	—
184	C₂²⋊Q₈	—	2	2	no	—
185	C₂²⋊Q₈	C₂×Dic₄₇	1	4	yes	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	C₂²×D₄₇	1	4	yes	—
188	#117	—	2	2	no	—
189	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
190	#107	C₂×Dic₄₇	1	2	yes	—
191	#107	C₂×Dic₄₇	1	2	yes	—
192	C₄⋊C₈	—	47	2	no	—
193	C₄⋊C₈	—	47	4	no	—
194	C₄⋊C₈	—	94	4	no	—
195	C₄⋊C₈	—	94	4	no	—
196	C₄⋊C₈	—	94	1	no	—
197	#103	C₂×Dic₄₇	1	4	yes	—
198	#103	C₂×Dic₄₇	1	4	yes	—
199	C₂²⋊C₈	—	47	2	no	—
200	C₂²⋊C₈	—	94	2	no	—
201	C₂²⋊C₈	—	94	1	no	—
202	#97	—	94	4	no	—
203	#97	—	94	2	no	—
204	C₄×C₈	—	47	1	no	—
205	C₄×C₈	—	47	1	no	—
206	C₄×C₈	—	47	1	no	—
207	C₄×C₈	—	47	1	no	—
208	C₄×C₈	—	94	4	no	—
209	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
210	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
215	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
216	C₂²×C₄	#10625352	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	47	2	no	—
223	C₂×C₈	—	47	2	no	—
224	C₂×C₈	—	94	4	no	—
225	C₂×C₈	—	94	1	no	—
226	C₂×C₈	—	94	1	no	—
227	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
228	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
229	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
230	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	D₄×D₄₇	1	1	yes	characteristic
240	C₄²	#10625352	1	1	yes	characteristic
241	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
242	C₄²	C₂³×D₄₇	1	1	yes	characteristic
243	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	Q₈	—	8	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
254	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
255	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
256	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
258	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	94	2	no	—
262	C₈	—	94	2	no	—
263	C₈	—	188	1	no	—
264	C₈	—	188	1	no	—
265	C₈	—	188	4	no	—
266	C₈	#13094031	1	2	yes	—
267	C₈	#13094031	1	2	yes	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	4	no	—
278	C₄	C₂×#13094031	1	1	yes	characteristic
279	C₄	C₂×#13094031	1	1	yes	characteristic
280	C₄	#15689936	1	2	yes	—
281	C₂	—	4	1	no	—
282	C₂	#20252100	1	1	yes	characteristic
283	C₂	#20252256	1	1	yes	characteristic
284	C₂	#20252305	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3170	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#3170	D₄₇
hypocenter	C₄₇	#17972
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #20242297	C₂	semidirect
3. #20242285	C₂	semidirect
4. #20242207	C₂	semidirect
5. Q₁₆×C₄₇⋊C₈	C₂	semidirect
176. C₄₇	#17972	semidirect

Maximal subgroups

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

subgroup	index
#20252483	2
#20242297	2
#20242285	2
#20242207	2
Q₁₆×C₄₇⋊C₈	2
#20242151	2
#20241861	2
#20241671	2
C₄₇×#3170	2
#17972	47