Subgroups — #26188057

Order 12032.

586 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	#20253974	C₂	1	1	yes	maximal, characteristic
2	#20252335	C₂	1	2	yes	maximal
3	#20242286	C₂	1	2	yes	maximal, complemented
4	#20242240	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₁₈₈	C₂	1	2	yes	maximal, complemented
6	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
7	#20241792	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
9	#15689680	C₂²	1	2	yes	—
10	#15689528	C₄	1	1	yes	characteristic
11	#15689503	C₄	1	1	yes	characteristic
12	#15689474	C₄	1	2	yes	—
13	#15689364	C₄	1	2	yes	—
14	#15689364	C₄	1	2	yes	—
15	#15689253	C₂²	1	4	yes	—
16	#15689251	—	2	4	no	—
17	C₈×Dic₉₄	C₂²	1	4	yes	—
18	C₄×Dic₁₈₈	C₂²	1	2	yes	—
19	C₄×Dic₁₈₈	C₂²	1	2	yes	—
20	#15689151	—	2	2	no	—
21	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
22	#15689081	—	2	2	no	—
23	#15688876	C₂²	1	1	yes	characteristic
24	#15688876	C₂²	1	2	yes	—
25	#15688876	C₂²	1	1	yes	characteristic
26	#15688855	C₂²	1	4	yes	—
27	#15688855	C₂²	1	4	yes	—
28	C₄₇×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#322	C₂²	1	2	yes	—
32	C₈²×C₄₇	C₂²	1	2	yes	—
33	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
35	C₂²×C₄₇⋊C₈	—	2	2	no	—
36	#13094024	C₂×C₄	1	2	yes	—
37	#13094024	C₂×C₄	1	2	yes	—
38	#13094016	—	2	2	no	—
39	#13094003	—	2	2	no	—
40	#13094003	C₂×C₄	1	4	yes	—
41	C₄×Dic₉₄	—	2	2	no	—
42	C₄×Dic₉₄	C₂³	1	4	yes	—
43	#13093983	—	2	2	no	—
44	#13093958	C₂×C₄	1	2	yes	—
45	#13093958	C₂×C₄	1	2	yes	—
46	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
48	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
49	Dic₄₇⋊C₈	—	2	4	no	—
50	Dic₄₇⋊C₈	—	2	4	no	—
51	#13093949	C₂×C₄	1	4	yes	—
52	#13093949	C₂×C₄	1	4	yes	—
53	#13093934	—	2	4	no	—
54	C₈×Dic₄₇	—	2	4	no	—
55	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₄₇	—	2	1	no	—
61	C₂²⋊C₈×C₄₇	—	2	1	no	—
62	#97×C₄₇	D₄	1	2	yes	—
63	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₄₇	C₂³	1	2	yes	—
66	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₄₇	C₂³	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
72	C₂×Dic₉₄	—	2	2	no	—
73	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
74	M₁₆×C₄₇	—	2	2	no	—
75	M₁₆×C₄₇	Q₁₆	1	4	yes	complemented
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	4	1	no	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₄₇	#41	1	1	yes	characteristic
88	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₂²⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	2	2	no	—
92	C₄⋊Dic₄₇	—	2	2	no	—
93	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
94	Dic₄₇⋊C₄	—	2	4	no	—
95	Dic₄₇⋊C₄	—	2	4	no	—
96	C₄×Dic₄₇	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	—	4	4	no	—
100	Dic₁₈₈	—	2	4	no	—
101	Dic₁₈₈	—	4	2	no	—
102	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	4	1	no	—
106	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₄₇	#135	1	2	yes	—
111	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
114	C₂×Dic₄₇	—	2	4	no	—
115	C₂×Dic₄₇	—	2	4	no	—
116	Dic₉₄	—	2	4	no	—
117	Dic₉₄	—	2	4	no	—
118	Dic₉₄	—	8	2	no	—
119	C₈×C₄₇	—	2	1	no	—
120	C₈×C₄₇	—	2	1	no	—
121	C₈×C₄₇	—	2	1	no	—
122	C₈×C₄₇	—	2	2	no	—
123	C₈×C₄₇	—	4	2	no	—
124	C₈×C₄₇	#131	1	2	yes	—
125	C₈×C₄₇	C₂×Q₁₆	1	4	yes	—
126	C₄₇⋊C₈	—	4	2	no	—
127	C₄₇⋊C₈	—	4	2	no	—
128	#18068	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	C₄×Q₁₆	1	2	yes	—
132	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₄×C₄₇	—	2	1	no	—
134	C₄×C₄₇	—	2	1	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	C₄×Q₁₆	1	2	yes	—
137	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
138	C₄×C₄₇	#568	1	1	yes	characteristic
139	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
140	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
141	Dic₄₇	—	4	4	no	—
142	Dic₄₇	—	4	4	no	—
143	#3170	—	47	1	no	—
144	#2962	—	47	2	no	—
145	#1591	—	47	2	no	—
146	#1584	—	47	4	no	—
147	C₈×Q₁₆	—	47	2	no	—
148	C₂×#335	—	47	1	no	—
149	#1481	—	47	2	no	—
150	C₈×M₁₆	D₄₇	1	1	yes	characteristic
151	C₂×C₄₇	—	2	2	no	—
152	C₂×C₄₇	#2935	1	1	yes	characteristic
153	C₂×C₄₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
154	C₂×C₄₇	#2957	1	1	yes	characteristic
155	#533	—	47	2	no	—
156	#500	—	47	1	no	—
157	#494	—	47	1	no	—
158	#487	—	47	2	no	—
159	#467	—	47	2	no	—
160	#467	—	47	2	no	—
161	#446	—	47	4	no	—
162	#446	—	94	4	no	—
163	C₈×Q₈	—	47	4	no	—
164	C₄×Q₁₆	—	47	2	no	—
165	C₄×Q₁₆	—	47	2	no	—
166	#431	C₂×D₄₇	1	1	yes	characteristic
167	#424	—	94	2	no	—
168	C₂×C₄⋊C₈	—	47	2	no	—
169	#405	—	94	2	no	—
170	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
172	#335	—	47	1	no	—
173	#335	—	47	2	no	—
174	#335	—	47	1	no	—
175	Q₈⋊C₈	—	47	4	no	—
176	Q₈⋊C₈	—	47	4	no	—
177	#322	C₂×D₄₇	1	2	yes	—
178	C₈²	C₂×D₄₇	1	2	yes	—
179	C₄₇	#18068	1	1	yes	characteristic, complemented, Sylow
180	C₂×Q₁₆	—	47	2	no	—
181	C₂×Q₁₆	—	47	2	no	—
182	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
183	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
184	C₂²×C₈	—	94	2	no	—
185	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
187	C₄⋊Q₈	—	47	2	no	—
188	C₄⋊Q₈	—	47	2	no	—
189	#125	—	94	2	no	—
190	C₂²⋊Q₈	—	47	4	no	—
191	C₂²⋊Q₈	—	94	2	no	—
192	C₄×Q₈	—	47	4	no	—
193	C₄×Q₈	—	94	2	no	—
194	#117	—	94	2	no	—
195	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
196	#107	—	47	2	no	—
197	#107	—	47	2	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	47	2	no	—
200	C₄⋊C₈	—	47	2	no	—
201	C₄⋊C₈	—	94	4	no	—
202	C₄⋊C₈	—	94	4	no	—
203	C₄⋊C₈	—	94	2	no	—
204	#103	—	47	4	no	—
205	#103	—	47	4	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	#97	—	94	4	no	—
208	#97	D₁₈₈	1	2	yes	—
209	C₄×C₈	—	94	4	no	—
210	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
211	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₄₇	1	2	yes	—
213	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₄₇	1	2	yes	—
216	C₂×Q₈	—	47	4	no	—
217	C₂×Q₈	—	94	2	no	—
218	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
220	C₂²×C₄	#10625350	1	1	yes	characteristic
221	Q₁₆	—	94	4	no	—
222	Q₁₆	—	188	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	Dic₁₈₈	1	4	yes	complemented
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	94	2	no	—
229	C₂×C₈	—	94	2	no	—
230	C₂×C₈	—	188	4	no	—
231	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
235	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
236	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
237	C₄⋊C₄	—	47	4	no	—
238	C₄⋊C₄	—	94	4	no	—
239	C₄⋊C₄	—	94	2	no	—
240	C₄⋊C₄	—	94	2	no	—
241	C₄⋊C₄	—	94	2	no	—
242	C₄⋊C₄	—	94	4	no	—
243	C₂²⋊C₄	—	94	4	no	—
244	C₄²	—	94	4	no	—
245	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
246	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
247	C₄²	#10625350	1	1	yes	characteristic
248	C₄²	C₂³×D₄₇	1	1	yes	characteristic
249	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
250	Q₈	—	94	4	no	—
251	Q₈	—	94	4	no	—
252	Q₈	—	376	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	94	4	no	—
257	C₂×C₄	—	94	4	no	—
258	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
259	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
260	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
261	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
262	C₂×C₄	#13094046	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
265	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	188	2	no	—
272	C₈	—	188	2	no	—
273	C₈	#13094032	1	2	yes	—
274	C₈	C₂×Dic₁₈₈	1	4	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₄×Dic₁₈₈	1	2	yes	—
278	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	188	4	no	—
283	C₄	—	188	4	no	—
284	C₄	C₄×Dic₁₈₈	1	2	yes	—
285	C₄	C₂×#13094032	1	1	yes	characteristic
286	C₄	#15689889	1	1	yes	characteristic
287	C₄	C₂²×Dic₁₈₈	1	1	yes	characteristic
288	C₄	C₂×#13094046	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#20252134	1	1	yes	characteristic
291	C₂	C₂×C₄×Dic₁₈₈	1	1	yes	characteristic
292	C₂	#20252306	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₈×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₈×M₁₆	D₄₇
hypocenter	C₄₇	#18068
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20242286	C₂	semidirect
4. #20242240	C₂	semidirect
5. C₈×Dic₁₈₈	C₂	semidirect
75. M₁₆×C₄₇	Q₁₆	semidirect
179. C₄₇	#18068	semidirect
224. M₁₆	Dic₁₈₈	semidirect

Maximal subgroups

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

subgroup	index
#20253974	2
#20252335	2
#20242286	2
#20242240	2
C₈×Dic₁₈₈	2
C₂×#15688876	2
#20241792	2
C₈×M₁₆×C₄₇	2
#18068	47