Subgroups — #26106961

Order 12032.

498 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094032	C₂	1	1	yes	maximal, characteristic
2	#20241771	C₂	1	1	yes	maximal, characteristic
3	#20241768	C₂	1	1	yes	maximal, characteristic
4	#20241681	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094032	C₄	1	2	yes	—
9	C₄×#10625350	C₂²	1	1	yes	characteristic
10	#15689176	C₂²	1	1	yes	characteristic
11	#15689081	C₄	1	2	yes	—
12	#15689080	—	2	2	no	—
13	C₄×#10625324	—	2	1	no	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₄₇	—	2	1	no	—
17	#15688855	C₄	1	2	yes	—
18	#15688853	C₄	1	2	yes	—
19	#15688852	C₄	1	2	yes	—
20	#15688850	C₄	1	2	yes	—
21	#15688845	—	2	2	no	—
22	#15688845	—	2	1	no	—
23	#15688844	—	2	1	no	—
24	C₄₇×#431	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
28	#13094032	—	2	4	no	—
29	#13094032	—	2	4	no	—
30	C₂×#10625324	—	2	2	no	—
31	C₂²×C₄₇⋊C₈	—	2	1	no	—
32	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₄₇	—	2	2	no	—
34	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
36	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
37	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
38	C₄×C₂²⋊D₄₇	—	2	1	no	—
39	C₄×C₂²⋊D₄₇	—	2	1	no	—
40	#13093983	C₂×C₄	1	1	yes	characteristic
41	#13093981	—	2	1	no	—
42	C₄²×D₄₇	—	2	1	no	—
43	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
44	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
45	#13093934	—	2	2	no	—
46	C₄×C₄₇⋊C₈	—	2	1	no	—
47	C₄×C₄₇⋊C₈	—	2	1	no	—
48	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
49	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
50	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
51	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
52	C₈×Dic₄₇	—	2	2	no	—
53	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₄₇	—	2	1	no	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₄₇	—	2	1	no	—
58	#97×C₄₇	—	2	1	no	—
59	C₄×C₈×C₄₇	—	2	1	no	—
60	C₄×C₈×C₄₇	—	2	1	no	—
61	C₄×C₈×C₄₇	—	2	1	no	—
62	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
66	C₂×C₂²⋊D₄₇	—	2	1	no	—
67	#10625350	—	2	2	no	—
68	#10625350	M₁₆	1	4	yes	—
69	C₂×D₁₈₈	C₄²	1	1	yes	characteristic
70	C₂×C₄×D₄₇	—	2	1	no	—
71	C₂×C₄×D₄₇	—	2	1	no	—
72	C₂×C₄×D₄₇	—	2	1	no	—
73	C₂×Dic₉₄	C₄²	1	1	yes	characteristic
74	M₁₆×C₄₇	—	2	2	no	—
75	M₁₆×C₄₇	M₁₆	1	4	yes	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	4	2	no	—
80	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²⋊Dic₄₇	—	2	1	no	—
85	(C₂×D₄₇)⋊C₄	—	2	1	no	—
86	(C₂×D₄₇)⋊C₄	—	2	1	no	—
87	C₄⋊Dic₄₇	C₄²	1	1	yes	characteristic
88	C₄⋊Dic₄₇	C₄²	1	1	yes	characteristic
89	Dic₄₇⋊C₄	—	2	1	no	—
90	Dic₄₇⋊C₄	—	2	1	no	—
91	C₄×Dic₄₇	—	2	1	no	—
92	C₄×Dic₄₇	—	2	1	no	—
93	C₄×Dic₄₇	—	2	1	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
100	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
101	#10625324	—	2	8	no	—
102	C₈×D₄₇	—	2	8	no	—
103	C₂³×C₄₇	#117	1	1	yes	characteristic
104	C₂²×D₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
115	C₂²⋊D₄₇	—	2	4	no	—
116	C₂×Dic₄₇	—	2	1	no	—
117	C₂×Dic₄₇	—	2	1	no	—
118	C₂×Dic₄₇	—	2	1	no	—
119	D₁₈₈	—	2	1	no	—
120	D₁₈₈	#97	1	2	yes	—
121	C₄×D₄₇	—	2	4	no	—
122	C₄×D₄₇	—	4	2	no	—
123	C₄×D₄₇	—	4	2	no	—
124	Dic₉₄	—	2	1	no	—
125	Dic₉₄	#97	1	2	yes	—
126	C₈×C₄₇	—	2	4	no	—
127	C₈×C₄₇	—	4	2	no	—
128	C₈×C₄₇	—	4	2	no	—
129	C₄₇⋊C₈	—	2	4	no	—
130	C₄₇⋊C₈	—	2	2	no	—
131	C₄₇⋊C₈	—	2	2	no	—
132	C₄₇⋊C₈	—	4	2	no	—
133	#8186	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	—	2	1	no	—
136	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₇	#432	1	2	yes	—
138	C₂×D₄₇	—	2	2	no	—
139	C₂×D₄₇	—	4	1	no	—
140	C₂×D₄₇	—	4	1	no	—
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
144	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
145	C₄×C₄₇	#407	1	2	yes	—
146	C₄×C₄₇	#443	1	2	yes	—
147	C₄×C₄₇	#444	1	2	yes	—
148	Dic₄₇	—	2	2	no	—
149	Dic₄₇	—	4	1	no	—
150	Dic₄₇	—	4	1	no	—
151	C₄×#131	—	47	1	no	—
152	#1476	—	47	1	no	—
153	#1476	—	47	1	no	—
154	#1452	—	47	1	no	—
155	C₂×#322	—	47	1	no	—
156	#1278	—	47	1	no	—
157	#1278	D₄₇	1	1	yes	characteristic, complemented
158	C₂×C₄₇	—	2	2	no	—
159	C₂×C₄₇	#1749	1	1	yes	characteristic
160	C₂×C₄₇	#1764	1	1	yes	characteristic
161	C₂×C₄₇	#1765	1	1	yes	characteristic
162	D₄₇	—	4	2	no	—
163	C₂×#131	—	47	2	no	—
164	C₄×#41	—	47	1	no	—
165	#431	—	47	1	no	—
166	#431	C₂×D₄₇	1	1	yes	characteristic
167	#405	—	47	2	no	—
168	#405	—	94	2	no	—
169	C₄×M₁₆	—	94	1	no	—
170	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
171	C₂×C₄×C₈	—	47	1	no	—
172	C₂×C₄×C₈	—	47	1	no	—
173	C₂×C₄×C₈	—	94	1	no	—
174	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
175	Q₈⋊C₈	—	47	2	no	—
176	Q₈⋊C₈	—	47	2	no	—
177	D₄⋊C₈	—	47	2	no	—
178	D₄⋊C₈	—	47	2	no	—
179	#322	—	94	1	no	—
180	#322	—	94	2	no	—
181	#322	—	94	1	no	—
182	C₄₇	#8186	1	1	yes	characteristic, complemented, Sylow
183	C₂×#41	—	47	1	no	—
184	#131	—	94	4	no	—
185	#131	—	94	4	no	—
186	C₂×M₁₆	—	94	2	no	—
187	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	47	2	no	—
190	C₂²×C₈	—	94	2	no	—
191	C₂²×C₈	—	94	1	no	—
192	C₄×Q₈	—	47	1	no	—
193	C₄×Q₈	—	47	1	no	—
194	C₄×D₄	—	47	1	no	—
195	C₄×D₄	—	47	1	no	—
196	C₄×D₄	—	94	1	no	—
197	C₄×D₄	—	94	1	no	—
198	#117	—	47	1	no	—
199	#117	—	94	1	no	—
200	C₂×C₄²	—	94	1	no	—
201	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
202	C₄⋊C₈	—	47	2	no	—
203	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	94	1	no	—
206	#97	—	2	1	no	—
207	#97	—	94	2	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	47	2	no	—
212	C₄×C₈	—	47	2	no	—
213	C₄×C₈	—	47	1	no	—
214	C₄×C₈	—	47	1	no	—
215	C₄×C₈	—	94	2	no	—
216	C₄×C₈	—	94	1	no	—
217	C₄×C₈	—	94	1	no	—
218	C₄×C₈	C₄×D₄₇	1	2	yes	—
219	#41	—	47	4	no	—
220	#41	—	94	2	no	—
221	C₂×Q₈	—	47	1	no	—
222	C₂×D₄	—	47	1	no	—
223	C₂×D₄	—	94	1	no	—
224	C₂²×C₄	—	94	1	no	—
225	C₂²×C₄	—	94	1	no	—
226	C₂²×C₄	—	94	1	no	—
227	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	#10625352	1	1	yes	characteristic
229	C₂²×C₄	#10625354	1	1	yes	characteristic
230	M₁₆	—	2	2	no	—
231	M₁₆	—	94	8	no	—
232	M₁₆	#10625324	1	4	yes	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	47	2	no	—
238	C₂×C₈	—	47	2	no	—
239	C₂×C₈	—	94	8	no	—
240	C₂×C₈	—	94	4	no	—
241	C₂×C₈	—	94	2	no	—
242	C₂×C₈	—	94	2	no	—
243	C₂×C₈	—	94	2	no	—
244	C₂×C₈	—	94	2	no	—
245	C₄⋊C₄	—	47	1	no	—
246	C₄⋊C₄	—	47	1	no	—
247	C₄⋊C₄	—	94	1	no	—
248	C₄⋊C₄	—	94	1	no	—
249	C₂²⋊C₄	—	94	1	no	—
250	C₂²⋊C₄	—	94	1	no	—
251	C₂²⋊C₄	—	94	1	no	—
252	C₄²	—	94	1	no	—
253	C₄²	—	94	1	no	—
254	C₄²	—	94	1	no	—
255	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
256	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
257	C₄²	D₄×D₄₇	1	1	yes	characteristic
258	C₄²	D₄×D₄₇	1	1	yes	characteristic
259	C₂³	—	94	1	no	—
260	C₂³	#13093982	1	1	yes	characteristic
261	Q₈	—	47	2	no	—
262	Q₈	—	94	1	no	—
263	D₄	—	47	2	no	—
264	D₄	—	94	4	no	—
265	D₄	—	94	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	94	1	no	—
269	C₂×C₄	—	94	4	no	—
270	C₂×C₄	—	94	1	no	—
271	C₂×C₄	—	94	1	no	—
272	C₂×C₄	—	188	2	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
276	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	#13093986	1	1	yes	characteristic
279	C₂×C₄	#13093986	1	1	yes	characteristic
280	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
281	C₂×C₄	C₂×#10625324	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	4	2	no	—
285	C₈	—	94	4	no	—
286	C₈	—	94	2	no	—
287	C₈	—	94	2	no	—
288	C₈	—	188	2	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	94	2	no	—
292	C₂²	—	188	1	no	—
293	C₂²	—	188	1	no	—
294	C₂²	#15688970	1	1	yes	characteristic
295	C₂²	#15689179	1	2	yes	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	94	2	no	—
299	C₄	—	188	1	no	—
300	C₄	—	188	1	no	—
301	C₄	#97×D₄₇	1	1	yes	characteristic
302	C₄	C₄×#10625324	1	1	yes	characteristic
303	C₄	#15689085	1	2	yes	—
304	C₄	#15689239	1	2	yes	—
305	C₄	#15689242	1	2	yes	—
306	C₂	—	2	2	no	—
307	C₂	—	188	2	no	—
308	C₂	#20243149	1	1	yes	characteristic
309	C₂	#20243224	1	1	yes	characteristic
310	C₂	#20243229	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1278	C₂
socle	C₂²×C₄₇	C₄×C₂²⋊C₄
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1278	D₄₇
hypocenter	C₄₇	#8186
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
4. #20241681	C₂	semidirect
5. C₂×#15688845	C₂	semidirect
6. #20241226	C₂	semidirect
7. C₄₇×#1278	C₂	semidirect
157. #1278	D₄₇	semidirect
182. C₄₇	#8186	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094032	2
#20241771	2
#20241768	2
#20241681	2
C₂×#15688845	2
#20241226	2
C₄₇×#1278	2
#8186	47