Subgroups — #26214245

Order 12032.

1166 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689078	C₂	1	2	yes	maximal
3	#20245795	C₂	1	1	yes	maximal, characteristic
4	#20243255	C₂	1	4	yes	maximal, complemented
5	C₈×C₄⋊Dic₄₇	C₂	1	4	yes	maximal, complemented
6	#20243084	C₂	1	2	yes	maximal
7	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂×#13093983	C₂²	1	2	yes	—
10	#15689147	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
12	#15689078	—	2	8	no	—
13	C₂×#13093934	C₂²	1	4	yes	—
14	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
15	#15688993	C₄	1	2	yes	—
16	#15688989	C₄	1	2	yes	—
17	#15688983	C₄	1	4	yes	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
19	#15688877	C₂²	1	8	yes	—
20	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₇	C₄	1	8	yes	complemented
24	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
26	C₂×#10625329	—	2	4	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	C₂×C₄	1	8	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	#13093983	—	2	4	no	—
31	#13093983	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
36	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	—
37	#13093934	—	2	8	no	—
38	C₈×Dic₄₇	—	2	8	no	—
39	C₄₇⋊#95	C₂×C₄	1	8	yes	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₄₇	D₄	1	4	yes	—
47	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
56	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	—
58	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	2	8	no	—
60	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
72	C₂×C₈×C₄₇	—	2	8	no	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
76	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
83	C₄²×C₄₇	C₄²	1	4	yes	—
84	C₄²×C₄₇	C₄²	1	4	yes	—
85	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
86	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
87	C₂²⋊Dic₄₇	—	2	8	no	—
88	C₄⋊Dic₄₇	—	2	8	no	—
89	C₄⋊Dic₄₇	—	2	16	no	—
90	C₄×Dic₄₇	—	2	4	no	—
91	C₄×Dic₄₇	—	2	4	no	—
92	#10625329	—	2	16	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
98	C₂³×C₄₇	C₄×D₄	1	2	yes	—
99	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
100	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	16	no	—
103	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₄₇	#117	1	4	yes	—
108	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
109	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
111	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₄₇	#131	1	8	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₄₇	—	2	8	no	—
120	C₂×Dic₄₇	—	2	16	no	—
121	C₈×C₄₇	—	2	4	no	—
122	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
123	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
124	C₄₇⋊C₈	—	2	8	no	—
125	#20437	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
130	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×#131	1	4	yes	—
134	C₄×C₄₇	—	2	4	no	—
135	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
136	C₄×C₄₇	#405	1	8	yes	complemented
137	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
139	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
141	Dic₄₇	—	4	8	no	—
142	C₂×#423	—	47	1	no	—
143	C₂×#405	—	47	2	no	—
144	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
145	#2285	—	47	1	no	—
146	#1770	—	47	4	no	—
147	C₈×C₄⋊C₄	—	47	4	no	—
148	#1729	—	47	2	no	—
149	C₂×C₄₇	—	2	4	no	—
150	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#405	1	2	yes	—
152	C₂×C₄₇	#2931	1	2	yes	—
153	C₂×C₄₇	#2960	1	2	yes	—
154	C₂²×M₁₆	—	47	2	no	—
155	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
156	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
157	C₂×#117	—	47	2	no	—
158	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
159	#423	—	47	8	no	—
160	C₂×C₄⋊C₈	—	47	4	no	—
161	#405	—	94	8	no	—
162	C₂×#97	—	47	4	no	—
163	C₂×C₄×C₈	—	47	4	no	—
164	C₂×C₄×C₈	Dic₄₇	1	8	yes	complemented
165	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
166	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
167	#384	—	47	2	no	—
168	#382	—	47	2	no	—
169	#380	—	47	4	no	—
170	C₄×C₄⋊C₄	—	47	4	no	—
171	#336	—	47	8	no	—
172	C₄₇	#20437	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	C₄×D₄₇	1	2	yes	—
174	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
175	C₂×M₁₆	—	47	8	no	—
176	C₂×M₁₆	—	94	4	no	—
177	C₂×M₁₆	—	94	4	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	47	4	no	—
181	C₂²×C₈	Dic₉₄	1	4	yes	—
182	C₂²×C₈	D₁₈₈	1	4	yes	—
183	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
184	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
185	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
186	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
188	#117	—	94	4	no	—
189	#117	—	94	4	no	—
190	C₂×C₄⋊C₄	—	47	4	no	—
191	C₂×C₄⋊C₄	—	47	8	no	—
192	C₂×C₂²⋊C₄	—	47	4	no	—
193	C₂×C₄²	—	47	4	no	—
194	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₄₇	1	4	yes	—
197	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
198	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
199	C₄⋊C₈	—	47	16	no	—
200	#97	—	94	8	no	—
201	C₄×C₈	—	94	8	no	—
202	C₄×C₈	C₂×Dic₄₇	1	8	yes	—
203	C₄×C₈	C₂×Dic₄₇	1	8	yes	—
204	#95	—	47	8	no	—
205	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	47	8	no	—
208	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
209	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
210	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
211	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
213	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
214	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
215	C₂²×C₄	#10625350	1	2	yes	—
216	C₂²×C₄	#10625350	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
218	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
219	M₁₆	—	94	16	no	—
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	47	8	no	—
224	C₂×C₈	—	94	4	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	C₄⋊Dic₄₇	1	16	yes	complemented
227	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
228	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
229	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
230	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
231	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
232	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
233	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
234	C₄⋊C₄	—	94	8	no	—
235	C₄⋊C₄	—	94	16	no	—
236	C₂²⋊C₄	—	94	8	no	—
237	C₄²	—	94	4	no	—
238	C₄²	—	94	4	no	—
239	C₄²	C₄×Dic₄₇	1	4	yes	—
240	C₄²	C₄×Dic₄₇	1	4	yes	—
241	C₄²	C₂×C₄×D₄₇	1	4	yes	—
242	C₄²	C₂²×Dic₄₇	1	4	yes	—
243	C₂³	—	2	4	no	—
244	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
245	C₂³	C₄×D₁₈₈	1	2	yes	—
246	C₂³	C₄×Dic₉₄	1	2	yes	—
247	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	94	8	no	—
251	C₂×C₄	—	94	16	no	—
252	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
253	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
254	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
256	C₂×C₄	#13093983	1	4	yes	—
257	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
258	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
259	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
260	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
261	C₂×C₄	#13094030	1	8	yes	—
262	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
263	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
265	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
266	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
267	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	94	8	no	—
270	C₈	C₂×C₄⋊Dic₄₇	1	4	yes	—
271	C₈	C₂×C₄⋊Dic₄₇	1	4	yes	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₄×C₄⋊Dic₄₇	1	4	yes	—
276	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
277	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
278	C₂²	C₂×C₄×Dic₉₄	1	1	yes	characteristic
279	C₂²	C₂×#13094030	1	4	yes	—
280	C₄	—	2	4	no	—
281	C₄	—	188	8	no	—
282	C₄	C₄×C₄⋊Dic₄₇	1	4	yes	—
283	C₄	#15689078	1	8	yes	complemented
284	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
285	C₄	C₂×#13093983	1	1	yes	characteristic
286	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
287	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
288	C₂	—	2	4	no	—
289	C₂	C₂×C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
290	C₂	C₂×#15689078	1	2	yes	—
291	C₂	#20252101	1	2	yes	—
292	C₂	#20252322	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	#20437
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #20243255	C₂	semidirect
5. C₈×C₄⋊Dic₄₇	C₂	semidirect
10. #15689147	C₄	semidirect
23. C₂×C₄×C₈×C₄₇	C₄	semidirect
75. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
136. C₄×C₄₇	#405	semidirect
164. C₂×C₄×C₈	Dic₄₇	semidirect
172. C₄₇	#20437	semidirect
226. C₂×C₈	C₄⋊Dic₄₇	semidirect
283. C₄	#15689078	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂×#15689078	2
#20245795	2
#20243255	2
C₈×C₄⋊Dic₄₇	2
#20243084	2
C₂²×C₄×C₈×C₄₇	2
#20437	47