Subgroups — #26106979

Order 12032.

420 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252030	C₂	1	1	yes	maximal, characteristic
2	#20241771	C₂	1	1	yes	maximal, characteristic
3	#20241771	C₂	1	1	yes	maximal, characteristic
4	#20241671	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241225	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241225	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10625350	C₂²	1	1	yes	characteristic
9	#15689182	—	2	2	no	—
10	#15689176	C₂²	1	1	yes	characteristic
11	#15689176	C₂²	1	1	yes	characteristic
12	#15689174	—	2	1	no	—
13	#15689174	—	2	1	no	—
14	#15689149	C₄	1	2	yes	—
15	#15689089	C₄	1	2	yes	—
16	C₄×#10625329	C₂²	1	1	yes	characteristic
17	#15688855	C₄	1	2	yes	—
18	#15688855	C₄	1	2	yes	—
19	#15688852	C₄	1	2	yes	—
20	#15688852	C₄	1	2	yes	—
21	#15688844	—	2	1	no	—
22	C₈×C₄₇⋊C₈	—	2	1	no	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#322	—	2	2	no	—
27	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
28	C₂×#10625329	C₂×C₄	1	2	yes	—
29	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
31	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
32	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₄₇	—	2	1	no	—
34	C₄×C₂²⋊D₄₇	—	2	1	no	—
35	#13093983	C₂×C₄	1	1	yes	characteristic
36	#13093981	—	2	1	no	—
37	C₄²×D₄₇	—	2	1	no	—
38	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
39	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
40	Dic₄₇⋊C₈	—	2	2	no	—
41	Dic₄₇⋊C₈	—	2	2	no	—
42	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
43	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
44	(C₂×D₄₇)⋊C₈	—	2	2	no	—
45	(C₂×D₄₇)⋊C₈	—	2	2	no	—
46	#13093935	—	2	1	no	—
47	C₄×C₄₇⋊C₈	—	2	1	no	—
48	C₄×C₄₇⋊C₈	—	2	1	no	—
49	C₄×C₄₇⋊C₈	—	2	1	no	—
50	C₂²×C₈×C₄₇	—	2	1	no	—
51	C₂²×C₈×C₄₇	—	2	1	no	—
52	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₄₇	—	2	1	no	—
55	C₄×C₈×C₄₇	—	2	1	no	—
56	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
59	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
60	C₄×C₈×C₄₇	D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₇	—	2	1	no	—
65	#10625350	—	2	4	no	—
66	C₂×D₁₈₈	C₄²	1	1	yes	characteristic
67	C₂×C₄×D₄₇	—	2	1	no	—
68	C₂×C₄×D₄₇	—	2	1	no	—
69	C₂×C₄×D₄₇	—	2	1	no	—
70	C₂×Dic₉₄	C₄²	1	1	yes	characteristic
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	4	2	no	—
76	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²⋊Dic₄₇	—	2	1	no	—
83	(C₂×D₄₇)⋊C₄	—	2	1	no	—
84	(C₂×D₄₇)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₄₇	C₄²	1	1	yes	characteristic
86	C₄⋊Dic₄₇	C₄²	1	1	yes	characteristic
87	Dic₄₇⋊C₄	—	2	1	no	—
88	Dic₄₇⋊C₄	—	2	1	no	—
89	C₄×Dic₄₇	—	2	1	no	—
90	C₄×Dic₄₇	—	2	1	no	—
91	C₄×Dic₄₇	—	2	1	no	—
92	#10625329	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂³×C₄₇	#117	1	1	yes	characteristic
97	C₂²×D₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#131	1	4	yes	—
107	C₂²⋊D₄₇	—	4	2	no	—
108	C₂×Dic₄₇	—	2	1	no	—
109	C₂×Dic₄₇	—	2	1	no	—
110	C₂×Dic₄₇	—	2	1	no	—
111	D₁₈₈	—	2	2	no	—
112	C₄×D₄₇	—	4	2	no	—
113	C₄×D₄₇	—	4	2	no	—
114	C₄×D₄₇	—	4	2	no	—
115	Dic₉₄	—	2	2	no	—
116	C₈×C₄₇	—	2	4	no	—
117	C₈×C₄₇	—	2	2	no	—
118	C₈×C₄₇	—	2	2	no	—
119	C₈×C₄₇	—	4	2	no	—
120	C₄₇⋊C₈	—	4	2	no	—
121	C₄₇⋊C₈	—	4	2	no	—
122	C₄₇⋊C₈	—	4	2	no	—
123	#8189	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	1	no	—
125	C₂²×C₄₇	—	2	1	no	—
126	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₄₇	#433	1	2	yes	—
128	C₂×D₄₇	—	4	1	no	—
129	C₂×D₄₇	—	4	1	no	—
130	C₂×D₄₇	—	4	1	no	—
131	C₄×C₄₇	—	2	1	no	—
132	C₄×C₄₇	—	2	1	no	—
133	C₄×C₄₇	#405	1	2	yes	—
134	C₄×C₄₇	#408	1	2	yes	—
135	C₄×C₄₇	#443	1	2	yes	—
136	C₄×C₄₇	#444	1	2	yes	—
137	Dic₄₇	—	4	1	no	—
138	Dic₄₇	—	4	1	no	—
139	Dic₄₇	—	4	1	no	—
140	#2922	—	47	1	no	—
141	#1476	—	47	1	no	—
142	#1476	—	47	1	no	—
143	#1450	—	47	1	no	—
144	C₂×#322	D₄₇	1	1	yes	characteristic, complemented
145	#1278	—	47	1	no	—
146	#1278	—	47	1	no	—
147	C₂×C₄₇	—	2	2	no	—
148	C₂×C₄₇	#1750	1	1	yes	characteristic
149	C₂×C₄₇	#1764	1	1	yes	characteristic
150	C₂×C₄₇	#1765	1	1	yes	characteristic
151	D₄₇	—	8	1	no	—
152	C₄×#41	—	47	1	no	—
153	#433	—	94	2	no	—
154	#431	—	47	1	no	—
155	#431	—	47	1	no	—
156	#431	—	94	1	no	—
157	#431	—	94	1	no	—
158	#424	—	47	2	no	—
159	#408	—	47	2	no	—
160	C₄×M₁₆	—	47	1	no	—
161	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
164	Q₈⋊C₈	—	47	2	no	—
165	Q₈⋊C₈	—	47	2	no	—
166	D₄⋊C₈	—	47	2	no	—
167	D₄⋊C₈	—	47	2	no	—
168	#322	—	2	2	no	—
169	#322	—	94	1	no	—
170	C₈²	—	94	1	no	—
171	C₄₇	#8189	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	47	1	no	—
173	C₂×M₁₆	—	47	2	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	C₄×D₄₇	1	2	yes	—
177	C₄×Q₈	—	47	1	no	—
178	C₄×Q₈	—	47	1	no	—
179	C₄×D₄	—	47	1	no	—
180	C₄×D₄	—	47	1	no	—
181	C₄×D₄	—	94	1	no	—
182	C₄×D₄	—	94	1	no	—
183	#117	—	47	1	no	—
184	#117	—	94	1	no	—
185	C₂×C₄²	—	94	1	no	—
186	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	C₄⋊C₈	—	47	2	no	—
188	C₄⋊C₈	—	47	2	no	—
189	C₄⋊C₈	—	94	2	no	—
190	C₄⋊C₈	—	94	2	no	—
191	C₂²⋊C₈	—	94	1	no	—
192	C₂²⋊C₈	—	94	1	no	—
193	C₂²⋊C₈	—	94	2	no	—
194	C₂²⋊C₈	—	94	2	no	—
195	#97	—	94	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	94	1	no	—
199	C₄×C₈	—	94	1	no	—
200	C₄×C₈	—	94	1	no	—
201	C₄×C₈	C₄×D₄₇	1	2	yes	—
202	C₄×C₈	C₄×D₄₇	1	2	yes	—
203	C₄×C₈	D₁₈₈	1	1	yes	characteristic
204	C₄×C₈	D₁₈₈	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
206	#41	—	94	4	no	—
207	C₂×Q₈	—	47	1	no	—
208	C₂×D₄	—	47	1	no	—
209	C₂×D₄	—	94	1	no	—
210	C₂²×C₄	—	94	1	no	—
211	C₂²×C₄	—	94	1	no	—
212	C₂²×C₄	—	94	1	no	—
213	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
214	C₂²×C₄	#10625350	1	1	yes	characteristic
215	C₂²×C₄	#10625350	1	1	yes	characteristic
216	M₁₆	—	94	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	94	2	no	—
223	C₂×C₈	—	94	2	no	—
224	C₂×C₈	—	94	2	no	—
225	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
226	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
227	C₄⋊C₄	—	47	1	no	—
228	C₄⋊C₄	—	47	1	no	—
229	C₄⋊C₄	—	94	1	no	—
230	C₄⋊C₄	—	94	1	no	—
231	C₂²⋊C₄	—	94	1	no	—
232	C₂²⋊C₄	—	94	1	no	—
233	C₂²⋊C₄	—	94	1	no	—
234	C₄²	—	94	1	no	—
235	C₄²	—	94	1	no	—
236	C₄²	—	94	1	no	—
237	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
240	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
241	C₂³	—	94	1	no	—
242	C₂³	#13093981	1	1	yes	characteristic
243	Q₈	—	94	2	no	—
244	D₄	—	94	2	no	—
245	D₄	—	188	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	94	1	no	—
249	C₂×C₄	—	94	1	no	—
250	C₂×C₄	—	94	1	no	—
251	C₂×C₄	—	188	2	no	—
252	C₂×C₄	—	188	2	no	—
253	C₂×C₄	—	188	2	no	—
254	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
255	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
257	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
258	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
259	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
260	C₂×C₄	#13094032	1	4	yes	—
261	C₈	—	2	4	no	—
262	C₈	—	2	2	no	—
263	C₈	—	2	2	no	—
264	C₈	—	4	2	no	—
265	C₈	—	188	2	no	—
266	C₈	—	188	2	no	—
267	C₈	—	188	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	188	1	no	—
271	C₂²	—	188	1	no	—
272	C₂²	—	188	1	no	—
273	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
274	C₂²	#15689185	1	2	yes	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	188	1	no	—
278	C₄	—	188	1	no	—
279	C₄	—	188	1	no	—
280	C₄	#15689080	1	2	yes	—
281	C₄	#15689091	1	2	yes	—
282	C₄	#15689241	1	2	yes	—
283	C₄	#15689246	1	2	yes	—
284	C₂	—	2	2	no	—
285	C₂	—	376	1	no	—
286	C₂	#20243157	1	1	yes	characteristic
287	C₂	#20243225	1	1	yes	characteristic
288	C₂	#20243232	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂²×C₄₇	C₄×C₂²⋊C₄
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#8189
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. #20241671	C₂	semidirect
5. #20241225	C₂	semidirect
6. #20241225	C₂	semidirect
7. C₂×C₄₇×#322	C₂	semidirect
144. C₂×#322	D₄₇	semidirect
171. C₄₇	#8189	semidirect

Maximal subgroups

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

subgroup	index
#20252030	2
#20241771	2
#20241671	2
#20241225	2
C₂×C₄₇×#322	2
#8189	47