Subgroups — #26105831

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	#20241229	C₂	1	4	yes	maximal, complemented
4	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	#15689176	—	2	2	no	—
7	#15689176	C₄	1	4	yes	complemented
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₄×#10625329	—	2	2	no	—
11	C₄×#10625329	C₄	1	4	yes	complemented
12	C₂×#13093935	—	2	1	no	—
13	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×#117×C₄₇	—	2	1	no	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#384	C₄	1	2	yes	—
18	C₄₇×#382	—	2	1	no	—
19	C₄₇×#380	—	2	1	no	—
20	C₄₇×#380	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
22	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂×#10625329	—	2	8	no	—
24	C₂×#10625329	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
29	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
30	#13093935	—	2	4	no	—
31	C₄×C₄₇⋊C₈	—	2	4	no	—
32	C₄×C₄₇⋊C₈	—	2	2	no	—
33	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
37	#117×C₄₇	—	4	4	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	#95×C₄₇	—	2	2	no	—
52	#95×C₄₇	—	2	2	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	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	C₄²	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	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₄₇	—	2	8	no	—
72	C₄⋊C₄×C₄₇	—	4	4	no	—
73	C₄⋊C₄×C₄₇	—	4	4	no	—
74	C₂²⋊C₄×C₄₇	—	4	4	no	—
75	C₄²×C₄₇	—	4	4	no	—
76	C₄²×C₄₇	C₄²	1	2	yes	—
77	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
80	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
82	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
86	#10625329	—	2	16	no	—
87	C₂×C₄₇⋊C₈	—	2	8	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	#95	1	2	yes	—
97	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	#95	1	2	yes	—
112	C₂×C₄×C₄₇	#95	1	2	yes	—
113	C₂×C₄×C₄₇	#95	1	2	yes	—
114	C₂×C₄×C₄₇	#104	1	4	yes	—
115	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#131	1	4	yes	—
122	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₄₇	#137	1	2	yes	—
124	C₄₇⋊C₈	—	2	8	no	—
125	C₄₇⋊C₈	—	4	4	no	—
126	#7896	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	4	2	no	—
131	C₂²×C₄₇	—	4	2	no	—
132	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
133	C₂²×C₄₇	#408	1	2	yes	—
134	C₂²×C₄₇	#418	1	1	yes	characteristic
135	C₂²×C₄₇	#419	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×#104	1	2	yes	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	8	4	no	—
142	C₄×C₄₇	#339	1	4	yes	—
143	C₄×C₄₇	#405	1	2	yes	—
144	C₄×C₄₇	#417	1	2	yes	—
145	C₄×C₄₇	#424	1	2	yes	—
146	C₄×C₄₇	#428	1	2	yes	—
147	C₂×#431	—	47	1	no	—
148	C₂×C₄×M₁₆	—	47	1	no	—
149	#2285	D₄₇	1	1	yes	characteristic
150	#1279	—	47	4	no	—
151	C₂×C₄₇	—	4	2	no	—
152	C₂×C₄₇	#1484	1	2	yes	—
153	C₂×C₄₇	#1485	1	2	yes	—
154	C₂×C₄₇	#1729	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#339	1	1	yes	characteristic
156	C₂×C₄₇	#1736	1	1	yes	characteristic
157	C₂²×M₁₆	—	94	1	no	—
158	C₂×#117	—	2	1	no	—
159	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
160	#431	—	47	4	no	—
161	#431	—	94	2	no	—
162	C₂×C₄⋊C₈	—	47	2	no	—
163	C₂×C₂²⋊C₈	—	94	1	no	—
164	C₄×M₁₆	—	47	4	no	—
165	C₄×M₁₆	—	94	2	no	—
166	C₂×#97	—	94	1	no	—
167	C₂×C₄×C₈	—	47	2	no	—
168	C₂×C₄×C₈	—	94	1	no	—
169	#384	Dic₄₇	1	2	yes	—
170	#382	—	2	1	no	—
171	#380	—	2	1	no	—
172	#380	Dic₄₇	1	2	yes	—
173	C₄×C₄⋊C₄	—	2	1	no	—
174	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
175	C₄₇	#7896	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
177	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
178	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂×M₁₆	—	94	8	no	—
180	C₂×M₁₆	—	94	4	no	—
181	C₂²×C₈	—	94	2	no	—
182	C₂²×C₈	—	94	2	no	—
183	#117	—	4	4	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	Dic₉₄	1	2	yes	—
190	C₂×C₄²	C₄×D₄₇	1	2	yes	—
191	C₂×C₄²	C₄×D₄₇	1	2	yes	—
192	C₂×C₄²	D₁₈₈	1	2	yes	—
193	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
195	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
197	C₄⋊C₈	—	47	4	no	—
198	C₄⋊C₈	—	94	2	no	—
199	C₂²⋊C₈	—	94	4	no	—
200	#97	—	94	4	no	—
201	C₄×C₈	—	47	4	no	—
202	C₄×C₈	—	94	4	no	—
203	C₄×C₈	—	94	2	no	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
213	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
214	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
215	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
216	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
217	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
218	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	M₁₆	—	94	16	no	—
225	C₂×C₈	—	94	8	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	94	4	no	—
228	C₂×C₈	—	188	2	no	—
229	C₂×C₈	—	188	2	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	4	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	4	4	no	—
235	C₄²	C₄×Dic₄₇	1	2	yes	—
236	C₄²	Dic₄₇⋊C₄	1	2	yes	—
237	C₄²	Dic₄₇⋊C₄	1	2	yes	—
238	C₄²	C₄⋊Dic₄₇	1	2	yes	—
239	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
240	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
241	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
242	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
245	C₂³	—	2	2	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	C₄₇⋊#95	1	2	yes	—
250	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	C₄₇⋊#95	1	2	yes	—
265	C₂×C₄	C₄₇⋊#95	1	2	yes	—
266	C₂×C₄	C₄₇⋊#95	1	2	yes	—
267	C₂×C₄	#13093951	1	4	yes	—
268	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
274	C₂×C₄	#13094031	1	4	yes	—
275	C₂×C₄	#13094050	1	2	yes	—
276	C₂×C₄	#13094055	1	2	yes	—
277	C₈	—	94	8	no	—
278	C₈	—	188	4	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
285	C₂²	#15689090	1	2	yes	—
286	C₂²	#15689128	1	1	yes	characteristic
287	C₂²	#15689132	1	1	yes	characteristic
288	C₂²	C₂×#13093951	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	8	4	no	—
294	C₄	#15688882	1	4	yes	—
295	C₄	#15689079	1	2	yes	—
296	C₄	#15689122	1	2	yes	—
297	C₄	#15689150	1	2	yes	—
298	C₄	#15689163	1	2	yes	—
299	C₂	—	4	2	no	—
300	C₂	#20241805	1	2	yes	—
301	C₂	#20241809	1	2	yes	—
302	C₂	#20243085	1	1	yes	characteristic
303	C₂	C₂×#15688882	1	1	yes	characteristic
304	C₂	#20243107	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#7896
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #20241229	C₂	semidirect
7. #15689176	C₄	semidirect
11. C₄×#10625329	C₄	semidirect
175. C₄₇	#7896	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
C₂×C₄×#10625329	2
#20241229	2
C₄₇×#2285	2
#7896	47