Subgroups — #26109041

Order 12032.

540 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243241	C₂	1	2	yes	maximal, complemented
2	C₂×#15688863	C₂	1	2	yes	maximal
3	#20241269	C₂	1	2	yes	maximal, complemented
4	C₄×#124×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂×#13093935	C₂²	1	4	yes	—
7	#15688878	—	2	2	no	—
8	#15688863	—	2	4	no	—
9	C₂×#124×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₄₇	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₄₇	C₄	1	1	yes	characteristic
12	C₄₇×#391	C₄	1	1	yes	characteristic
13	C₄₇×#390	C₄	1	1	yes	characteristic
14	C₄₇×#388	—	2	2	no	—
15	C₄₇×#382	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊C₄×C₄₇	—	2	2	no	—
17	C₄³×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
20	#13093935	—	2	4	no	—
21	#13093935	—	2	4	no	—
22	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	—	2	1	no	—
25	#124×C₄₇	—	2	2	no	—
26	#124×C₄₇	D₄	1	2	yes	—
27	#124×C₄₇	Q₈	1	2	yes	—
28	C₄×Q₈×C₄₇	—	4	2	no	—
29	C₄×D₄×C₄₇	—	4	2	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₄²×C₄₇	—	2	1	no	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	2	yes	—
40	C₂×C₄²×C₄₇	D₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	#95×C₄₇	—	2	2	no	—
43	#95×C₄₇	—	2	2	no	—
44	C₂⁴×C₄₇	—	2	1	no	—
45	C₂×Q₈×C₄₇	—	2	1	no	—
46	C₂×Q₈×C₄₇	—	4	1	no	—
47	C₂×Q₈×C₄₇	—	4	1	no	—
48	C₂×Q₈×C₄₇	C₄⋊C₄	1	2	yes	—
49	C₂×D₄×C₄₇	—	2	1	no	—
50	C₂×D₄×C₄₇	—	4	1	no	—
51	C₂×D₄×C₄₇	—	4	1	no	—
52	C₂×D₄×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	—	2	1	no	—
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	4	no	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₄⋊C₄×C₄₇	—	4	2	no	—
64	C₄⋊C₄×C₄₇	—	4	1	no	—
65	C₄⋊C₄×C₄₇	—	4	1	no	—
66	C₂²⋊C₄×C₄₇	—	2	4	no	—
67	C₂²⋊C₄×C₄₇	—	2	4	no	—
68	C₂²⋊C₄×C₄₇	—	4	4	no	—
69	C₂²⋊C₄×C₄₇	—	4	2	no	—
70	C₄²×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	—	2	4	no	—
72	C₄²×C₄₇	—	4	1	no	—
73	C₄²×C₄₇	—	4	2	no	—
74	C₄²×C₄₇	—	4	2	no	—
75	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
77	C₄²×C₄₇	#41	1	2	yes	—
78	C₂×C₄₇⋊C₈	—	2	8	no	—
79	C₂×C₄₇⋊C₈	—	4	4	no	—
80	C₂×C₄₇⋊C₈	—	4	4	no	—
81	C₂³×C₄₇	—	2	2	no	—
82	C₂³×C₄₇	—	4	1	no	—
83	C₂³×C₄₇	—	4	1	no	—
84	C₂³×C₄₇	—	4	2	no	—
85	C₂³×C₄₇	—	4	2	no	—
86	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₄₇	—	4	2	no	—
88	Q₈×C₄₇	—	4	2	no	—
89	D₄×C₄₇	—	4	2	no	—
90	D₄×C₄₇	—	4	2	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	4	1	no	—
102	C₂×C₄×C₄₇	—	4	1	no	—
103	C₂×C₄×C₄₇	—	4	1	no	—
104	C₂×C₄×C₄₇	—	4	1	no	—
105	C₂×C₄×C₄₇	—	4	2	no	—
106	C₂×C₄×C₄₇	—	4	2	no	—
107	C₂×C₄×C₄₇	—	8	4	no	—
108	C₂×C₄×C₄₇	#100	1	2	yes	—
109	C₂×C₄×C₄₇	#104	1	4	yes	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#121	1	2	yes	—
113	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
114	C₄₇⋊C₈	—	4	8	no	—
115	#8579	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	4	2	no	—
117	C₂²×C₄₇	—	4	4	no	—
118	C₂²×C₄₇	—	8	1	no	—
119	C₂²×C₄₇	—	8	1	no	—
120	C₂²×C₄₇	—	8	2	no	—
121	C₂²×C₄₇	#380	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
123	C₂²×C₄₇	#413	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×#104	1	2	yes	—
125	C₂²×C₄₇	#421	1	2	yes	—
126	C₄×C₄₇	—	2	1	no	—
127	C₄×C₄₇	—	2	1	no	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	4	1	no	—
131	C₄×C₄₇	—	4	2	no	—
132	C₄×C₄₇	—	4	2	no	—
133	C₄×C₄₇	—	8	1	no	—
134	C₄×C₄₇	—	8	1	no	—
135	C₄×C₄₇	#329	1	4	yes	complemented
136	C₄×#124	D₄₇	1	1	yes	characteristic
137	#1767	—	47	2	no	—
138	C₂×#329	—	47	2	no	—
139	#1295	—	47	2	no	—
140	C₂×C₄₇	—	8	2	no	—
141	C₂×C₄₇	C₂×#329	1	1	yes	characteristic
142	C₂×C₄₇	#1529	1	2	yes	—
143	C₂×C₄₇	#1536	1	1	yes	characteristic
144	C₂×C₄₇	#1800	1	1	yes	characteristic
145	C₂×C₄₇	#1806	1	2	yes	—
146	C₂×#124	C₂×D₄₇	1	1	yes	characteristic
147	C₂×C₄×Q₈	Dic₄₇	1	1	yes	characteristic
148	C₂×C₄×D₄	Dic₄₇	1	1	yes	characteristic
149	C₂×C₄⋊C₈	—	94	2	no	—
150	C₂×#97	—	47	4	no	—
151	#391	Dic₄₇	1	1	yes	characteristic
152	#390	Dic₄₇	1	1	yes	characteristic
153	#388	—	2	2	no	—
154	#382	C₂×D₄₇	1	1	yes	characteristic
155	C₄×C₂²⋊C₄	—	2	2	no	—
156	C₄³	C₂×D₄₇	1	1	yes	characteristic
157	#336	—	94	2	no	—
158	#329	—	94	4	no	—
159	C₄₇	#8579	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
161	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
162	C₂³×C₄	—	2	1	no	—
163	C₂²×C₈	—	94	4	no	—
164	#124	—	2	2	no	—
165	#124	Dic₉₄	1	2	yes	—
166	#124	D₁₈₈	1	2	yes	—
167	C₄×Q₈	—	4	2	no	—
168	C₄×D₄	—	4	2	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	C₂×C₂²⋊C₄	—	2	2	no	—
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
179	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
180	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
181	C₄⋊C₈	—	94	8	no	—
182	#97	—	94	4	no	—
183	#97	—	94	4	no	—
184	#95	—	2	2	no	—
185	#95	—	2	2	no	—
186	C₂⁴	—	2	1	no	—
187	C₂×Q₈	—	2	1	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×Q₈	—	4	1	no	—
190	C₂×Q₈	C₄⋊Dic₄₇	1	2	yes	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	C₄⋊Dic₄₇	1	2	yes	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
201	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
205	C₂×C₈	—	94	8	no	—
206	C₂×C₈	—	188	4	no	—
207	C₂×C₈	—	188	4	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	4	1	no	—
218	C₄²	—	4	2	no	—
219	C₄²	—	4	2	no	—
220	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
221	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
222	C₄²	#10625350	1	2	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
229	Q₈	—	4	2	no	—
230	Q₈	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	#13093941	1	2	yes	—
251	C₂×C₄	#13093951	1	4	yes	—
252	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
253	C₂×C₄	#13093983	1	1	yes	characteristic
254	C₂×C₄	#13093998	1	2	yes	—
255	C₂×C₄	#13094003	1	2	yes	—
256	C₈	—	188	8	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	8	1	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	1	no	—
262	C₂²	#15688983	1	1	yes	characteristic
263	C₂²	C₂×#13093941	1	1	yes	characteristic
264	C₂²	#15689107	1	1	yes	characteristic
265	C₂²	C₂×#13093951	1	2	yes	—
266	C₂²	#15689140	1	2	yes	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	2	no	—
274	C₄	—	8	1	no	—
275	C₄	—	8	1	no	—
276	C₄	#15688863	1	4	yes	complemented
277	C₂	—	8	2	no	—
278	C₂	C₂×#15688863	1	1	yes	characteristic
279	C₂	#20241995	1	2	yes	—
280	C₂	#20242023	1	1	yes	characteristic
281	C₂	#20243380	1	1	yes	characteristic
282	C₂	#20243408	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#124×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#124	D₄₇
hypocenter	C₄₇	#8579
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#124

Complements

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

N	H ≅ G/N	type
1. #20243241	C₂	semidirect
3. #20241269	C₂	semidirect
135. C₄×C₄₇	#329	semidirect
159. C₄₇	#8579	semidirect
276. C₄	#15688863	semidirect

Maximal subgroups

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

subgroup	index
#20243241	2
C₂×#15688863	2
#20241269	2
C₄×#124×C₄₇	2
#8579	47