Subgroups — #26220551

Order 12032.

766 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#20246040	C₂	1	2	yes	maximal
3	#20244249	C₂	1	2	yes	maximal
4	#20244121	C₂	1	2	yes	maximal
5	#20244078	C₂	1	4	yes	maximal
6	C₄×#13093958	C₂	1	1	yes	maximal, characteristic
7	#20243219	C₂	1	2	yes	maximal
8	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
11	C₄×Dic₁₈₈	C₄	1	8	yes	complemented
12	C₂×#13093958	C₂²	1	2	yes	—
13	C₂×#13093958	C₂²	1	1	yes	characteristic
14	C₂×#13093949	C₂²	1	4	yes	—
15	C₂×#13093949	C₂²	1	2	yes	—
16	#15689029	C₂²	1	4	yes	—
17	#15689021	C₂²	1	4	yes	—
18	#15689005	—	2	4	no	—
19	#15688990	—	2	2	no	—
20	#15688987	—	2	2	no	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	#15688886	C₂²	1	4	yes	—
23	#15688886	C₂²	1	2	yes	—
24	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#376	C₂²	1	1	yes	characteristic
29	C₂²×Dic₉₄	C₂³	1	2	yes	—
30	C₂×Dic₁₈₈	—	2	4	no	—
31	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
32	C₄×Dic₉₄	—	2	4	no	—
33	C₄×Dic₉₄	C₂×C₄	1	8	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
36	C₂×Dic₄₇⋊C₄	—	2	2	no	—
37	C₂×Dic₄₇⋊C₄	—	2	4	no	—
38	C₂×C₄×Dic₄₇	—	2	2	no	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	#13093958	—	2	4	no	—
41	#13093958	C₂×C₄	1	4	yes	—
42	#13093949	—	2	8	no	—
43	#13093949	C₂×C₄	1	8	yes	—
44	C₄₇⋊#95	—	2	2	no	—
45	C₄₇⋊#95	—	2	4	no	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₄₇⋊#95	—	2	4	no	—
48	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	#97×C₄₇	—	2	4	no	—
54	#97×C₄₇	—	2	2	no	—
55	C₄×C₈×C₄₇	—	2	4	no	—
56	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
57	#95×C₄₇	D₄	1	1	yes	characteristic
58	#95×C₄₇	D₄	1	1	yes	characteristic
59	#95×C₄₇	D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₄₇	—	2	2	no	—
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂×Dic₉₄	—	2	4	no	—
69	C₂×Dic₉₄	—	2	4	no	—
70	C₂×Dic₉₄	—	2	2	no	—
71	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₇	#41	1	4	yes	—
78	C₂×C₈×C₄₇	#41	1	4	yes	—
79	C₄²×C₄₇	—	2	2	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₇	#41	1	2	yes	—
83	C₄⋊Dic₄₇	—	2	2	no	—
84	C₄⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
87	Dic₄₇⋊C₄	—	2	8	no	—
88	Dic₄₇⋊C₄	—	8	4	no	—
89	C₄×Dic₄₇	—	2	8	no	—
90	C₄×Dic₄₇	—	8	2	no	—
91	Dic₁₈₈	—	2	8	no	—
92	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	1	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	2	no	—
102	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
108	C₂×Dic₄₇	—	2	4	no	—
109	C₂×Dic₄₇	—	2	4	no	—
110	C₂×Dic₄₇	—	4	4	no	—
111	C₂×Dic₄₇	—	4	2	no	—
112	C₂×Dic₄₇	—	4	2	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₂×Dic₄₇	—	4	4	no	—
116	Dic₉₄	—	2	4	no	—
117	Dic₉₄	—	2	4	no	—
118	Dic₉₄	—	4	4	no	—
119	C₈×C₄₇	—	2	2	no	—
120	C₈×C₄₇	—	2	4	no	—
121	C₈×C₄₇	#117	1	4	yes	—
122	#21281	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂²×C₄₇	#532	1	2	yes	—
125	C₂²×C₄₇	#576	1	1	yes	characteristic
126	C₂²×C₄₇	#578	1	2	yes	—
127	C₂²×C₄₇	#578	1	1	yes	characteristic
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	4	2	no	—
131	C₄×C₄₇	—	4	2	no	—
132	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
133	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
134	C₄×C₄₇	#529	1	2	yes	—
135	Dic₄₇	—	4	4	no	—
136	Dic₄₇	—	4	4	no	—
137	Dic₄₇	—	8	4	no	—
138	C₂×C₄×Q₁₆	—	47	1	no	—
139	#2320	—	47	2	no	—
140	#1969	—	47	2	no	—
141	#1942	—	47	2	no	—
142	#1935	—	47	4	no	—
143	C₄×#107	—	47	1	no	—
144	#1763	—	47	2	no	—
145	#1726	D₄₇	1	1	yes	characteristic
146	C₂×C₄₇	#2293	1	1	yes	characteristic
147	C₂×C₄₇	#2950	1	1	yes	characteristic
148	C₂×C₄₇	#2951	1	1	yes	characteristic
149	C₂×C₄₇	#3175	1	2	yes	—
150	C₂×C₄₇	#3177	1	2	yes	—
151	C₂²×Q₁₆	—	47	1	no	—
152	C₂×C₄×Q₈	—	47	2	no	—
153	C₄×Q₁₆	—	47	8	no	—
154	C₂×#107	—	47	1	no	—
155	C₂×#107	—	47	2	no	—
156	C₂×#103	—	47	2	no	—
157	C₂×#103	—	47	4	no	—
158	C₂×#97	C₂×D₄₇	1	2	yes	—
159	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
161	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
162	#391	—	47	4	no	—
163	#389	—	47	4	no	—
164	#387	—	94	4	no	—
165	#383	—	94	2	no	—
166	#382	—	94	2	no	—
167	C₄×C₄⋊C₄	—	47	2	no	—
168	#376	C₂×D₄₇	1	1	yes	characteristic
169	#340	—	47	2	no	—
170	#340	—	47	4	no	—
171	C₄₇	#21281	1	1	yes	characteristic, complemented, Sylow
172	C₂²×Q₈	—	47	2	no	—
173	C₂×Q₁₆	—	47	4	no	—
174	C₂×Q₁₆	—	94	4	no	—
175	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
177	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
178	C₄×Q₈	—	47	8	no	—
179	C₄×Q₈	—	94	4	no	—
180	C₂×C₄⋊C₄	—	47	2	no	—
181	C₂×C₄⋊C₄	—	47	4	no	—
182	C₂×C₄⋊C₄	—	94	2	no	—
183	C₂×C₄⋊C₄	—	94	4	no	—
184	C₂×C₄²	—	94	2	no	—
185	C₂×C₄²	—	94	2	no	—
186	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
187	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
188	#107	—	47	4	no	—
189	#107	—	94	4	no	—
190	#103	—	47	8	no	—
191	#103	—	94	8	no	—
192	#97	—	2	4	no	—
193	#97	—	2	2	no	—
194	C₄×C₈	—	2	4	no	—
195	C₄×C₈	C₄×D₄₇	1	4	yes	—
196	#95	—	94	2	no	—
197	#95	—	94	4	no	—
198	#95	—	94	4	no	—
199	#95	—	94	4	no	—
200	#95	D₁₈₈	1	1	yes	characteristic
201	#95	D₁₈₈	1	1	yes	characteristic
202	#95	D₁₈₈	1	2	yes	—
203	C₂×Q₈	—	47	4	no	—
204	C₂×Q₈	—	94	2	no	—
205	C₂×Q₈	—	94	4	no	—
206	C₂×Q₈	—	94	4	no	—
207	C₂²×C₄	—	94	2	no	—
208	C₂²×C₄	—	94	2	no	—
209	C₂²×C₄	—	94	4	no	—
210	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
211	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
212	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
214	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
215	Q₁₆	—	94	8	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
221	C₂×C₈	#10625350	1	4	yes	—
222	C₂×C₈	#10625350	1	4	yes	—
223	C₄⋊C₄	—	47	4	no	—
224	C₄⋊C₄	—	94	8	no	—
225	C₄⋊C₄	—	94	2	no	—
226	C₄⋊C₄	—	94	4	no	—
227	C₄⋊C₄	—	94	4	no	—
228	C₄⋊C₄	—	376	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	94	8	no	—
232	C₄²	—	376	2	no	—
233	C₄²	C₂×C₄×D₄₇	1	2	yes	—
234	C₄²	#10625350	1	2	yes	—
235	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
236	Q₈	—	94	4	no	—
237	Q₈	—	94	4	no	—
238	Q₈	—	188	4	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	94	4	no	—
249	C₂×C₄	—	94	4	no	—
250	C₂×C₄	—	188	4	no	—
251	C₂×C₄	—	188	4	no	—
252	C₂×C₄	—	188	4	no	—
253	C₂×C₄	—	188	2	no	—
254	C₂×C₄	—	188	4	no	—
255	C₂×C₄	—	188	2	no	—
256	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
257	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
258	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
259	C₂×C₄	C₂×#10625350	1	2	yes	—
260	C₂×C₄	C₂×#10625350	1	2	yes	—
261	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	—	2	4	no	—
264	C₈	#13093981	1	4	yes	—
265	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
266	C₂²	#15689673	1	2	yes	—
267	C₂²	#15689928	1	1	yes	characteristic
268	C₂²	#15689940	1	2	yes	—
269	C₂²	#15689940	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	376	4	no	—
275	C₄	—	188	4	no	—
276	C₄	—	188	4	no	—
277	C₄	C₂×#13093981	1	1	yes	characteristic
278	C₄	C₄×#10625350	1	1	yes	characteristic
279	C₄	#15689657	1	2	yes	—
280	C₂	#20245852	1	1	yes	characteristic
281	C₂	#20252257	1	1	yes	characteristic
282	C₂	#20252265	1	1	yes	characteristic
283	C₂	#20254027	1	2	yes	—
284	C₂	#20254047	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1726	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#21281
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₁₈₈	C₄	semidirect
171. C₄₇	#21281	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₈₈	2
#20246040	2
#20244249	2
#20244121	2
#20244078	2
C₄×#13093958	2
#20243219	2
C₄₇×#1726	2
#21281	47