Subgroups — #26106333

Order 12032.

746 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
2	#20245786	C₂	1	1	yes	maximal, characteristic
3	#20241270	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	—	2	1	no	—
6	C₂×#13093935	—	2	1	no	—
7	C₂×#13093935	—	2	2	no	—
8	C₂×#13093935	C₄	1	4	yes	complemented
9	C₂×#13093935	C₂²	1	2	yes	—
10	#15688991	—	2	1	no	—
11	#15688989	C₂²	1	2	yes	—
12	#15688986	—	2	1	no	—
13	#15688983	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
15	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
16	#15688969	—	2	1	no	—
17	C₂²×M₁₆×C₄₇	—	2	1	no	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₄₇	—	2	2	no	—
20	C₄×M₁₆×C₄₇	C₄	1	4	yes	complemented
21	C₂×#97×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₇	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	—	2	8	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄×Dic₄₇	—	2	2	no	—
29	#13093935	—	2	4	no	—
30	#13093935	—	2	2	no	—
31	#13093935	C₂×C₄	1	4	yes	—
32	C₄₇⋊#95	—	2	2	no	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂×M₁₆×C₄₇	—	2	8	no	—
40	C₂×M₁₆×C₄₇	—	2	4	no	—
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	#97×C₄₇	—	2	2	no	—
51	#97×C₄₇	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₄₇	—	2	4	no	—
53	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	2	4	no	—
70	M₁₆×C₄₇	—	2	16	no	—
71	C₂×C₈×C₄₇	—	2	8	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₄²×C₄₇	C₄²	1	2	yes	—
74	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
77	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²⋊Dic₄₇	—	4	4	no	—
84	C₄⋊Dic₄₇	—	4	4	no	—
85	C₄×Dic₄₇	—	4	4	no	—
86	C₂×C₄₇⋊C₈	—	2	8	no	—
87	C₂×C₄₇⋊C₈	—	2	16	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	#95	1	2	yes	—
92	C₂³×C₄₇	#104	1	4	yes	—
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	2	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	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	#95	1	2	yes	—
106	C₂×C₄×C₄₇	#95	1	2	yes	—
107	C₂×C₄×C₄₇	#95	1	2	yes	—
108	C₂×C₄×C₄₇	#104	1	4	yes	—
109	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×Dic₄₇	—	4	4	no	—
116	C₂×Dic₄₇	—	4	4	no	—
117	C₂×Dic₄₇	—	4	4	no	—
118	C₈×C₄₇	—	2	8	no	—
119	C₄₇⋊C₈	—	2	8	no	—
120	#8029	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	2	8	no	—
124	C₂²×C₄₇	#339	1	4	yes	—
125	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
126	C₂²×C₄₇	#413	1	2	yes	—
127	C₂²×C₄₇	C₂×#104	1	2	yes	—
128	C₂²×C₄₇	C₂×#104	1	2	yes	—
129	C₄×C₄₇	—	2	1	no	—
130	C₄×C₄₇	—	2	1	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	1	no	—
133	C₄×C₄₇	—	2	1	no	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	#339	1	4	yes	—
138	Dic₄₇	—	8	4	no	—
139	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
140	C₂²×#97	—	47	1	no	—
141	#2283	—	47	1	no	—
142	#1295	—	47	4	no	—
143	C₂×C₄₇	—	2	4	no	—
144	C₂×C₄₇	#1524	1	4	yes	—
145	C₂×C₄₇	C₂×#339	1	1	yes	characteristic
146	C₂×C₄₇	C₂×#339	1	1	yes	characteristic
147	C₂×C₄₇	#1742	1	1	yes	characteristic
148	C₂²×M₁₆	—	2	1	no	—
149	C₂³×C₈	—	94	1	no	—
150	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
151	C₄×M₁₆	—	2	2	no	—
152	C₄×M₁₆	Dic₄₇	1	4	yes	complemented
153	C₂×#97	—	47	4	no	—
154	C₂×#97	—	47	2	no	—
155	C₂×#97	—	94	1	no	—
156	C₂×#97	—	94	2	no	—
157	C₂×#97	C₂×D₄₇	1	2	yes	—
158	C₂×C₄×C₈	—	2	1	no	—
159	#383	—	94	1	no	—
160	#382	—	47	2	no	—
161	#381	—	94	1	no	—
162	#380	—	47	2	no	—
163	C₄×C₄⋊C₄	—	47	2	no	—
164	C₄×C₂²⋊C₄	—	94	1	no	—
165	#376	—	94	1	no	—
166	C₄₇	#8029	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
168	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
169	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂×M₁₆	—	2	8	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	94	8	no	—
174	C₂²×C₈	—	94	4	no	—
175	C₂²×C₈	—	94	2	no	—
176	C₂×C₄⋊C₄	—	47	4	no	—
177	C₂×C₂²⋊C₄	—	94	2	no	—
178	C₂×C₄²	—	94	2	no	—
179	C₂×C₄²	Dic₉₄	1	2	yes	—
180	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₄₇	1	2	yes	—
182	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
183	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
184	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
186	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	#97	—	2	2	no	—
188	#97	—	47	4	no	—
189	#97	—	94	4	no	—
190	#97	—	94	2	no	—
191	#97	C₂×Dic₄₇	1	4	yes	—
192	C₄×C₈	—	2	4	no	—
193	#95	—	94	2	no	—
194	#95	—	94	2	no	—
195	#95	—	94	2	no	—
196	#95	—	94	2	no	—
197	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	94	4	no	—
202	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
203	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
204	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
205	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
206	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
209	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
211	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
214	M₁₆	—	2	16	no	—
215	C₂×C₈	—	2	8	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	94	8	no	—
218	C₂×C₈	—	94	16	no	—
219	C₂×C₈	—	94	4	no	—
220	C₄⋊C₄	—	188	4	no	—
221	C₂²⋊C₄	—	188	4	no	—
222	C₄²	—	188	4	no	—
223	C₄²	C₄×Dic₄₇	1	2	yes	—
224	C₄²	Dic₄₇⋊C₄	1	2	yes	—
225	C₄²	Dic₄₇⋊C₄	1	2	yes	—
226	C₄²	C₄⋊Dic₄₇	1	2	yes	—
227	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
228	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
229	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
230	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	C₄₇⋊#95	1	2	yes	—
236	C₂³	#13093952	1	4	yes	—
237	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
238	C₂×C₄	—	2	2	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	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	188	4	no	—
250	C₂×C₄	—	188	4	no	—
251	C₂×C₄	—	188	4	no	—
252	C₂×C₄	C₄₇⋊#95	1	2	yes	—
253	C₂×C₄	C₄₇⋊#95	1	2	yes	—
254	C₂×C₄	C₄₇⋊#95	1	2	yes	—
255	C₂×C₄	#13093950	1	4	yes	—
256	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
257	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
259	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
262	C₈	—	2	8	no	—
263	C₈	—	94	8	no	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	#15688883	1	4	yes	—
268	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
269	C₂²	#15689105	1	2	yes	—
270	C₂²	C₂×#13093950	1	2	yes	—
271	C₂²	C₂×#13093952	1	2	yes	—
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	376	4	no	—
281	C₄	#15688884	1	4	yes	—
282	C₂	—	2	4	no	—
283	C₂	#20241977	1	4	yes	—
284	C₂	C₂×#15688883	1	1	yes	characteristic
285	C₂	C₂×#15688884	1	1	yes	characteristic
286	C₂	#20243126	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#8029
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20241270	C₂	semidirect
8. C₂×#13093935	C₄	semidirect
20. C₄×M₁₆×C₄₇	C₄	semidirect
152. C₄×M₁₆	Dic₄₇	semidirect
166. C₄₇	#8029	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093935	2
#20245786	2
#20241270	2
C₂×C₄×M₁₆×C₄₇	2
#8029	47