Subgroups — #26220564

Order 12032.

766 conjugacy classes of subgroups.

The subgroup lattice has 268 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	#20246034	C₂	1	2	yes	maximal
3	#20244257	C₂	1	2	yes	maximal
4	#20244124	C₂	1	2	yes	maximal
5	#20244062	C₂	1	4	yes	maximal
6	#20243221	C₂	1	2	yes	maximal
7	#20243074	C₂	1	1	yes	maximal, characteristic
8	C₄×#106×C₄₇	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	1	yes	characteristic
13	C₂×#13093949	C₂²	1	2	yes	—
14	C₂×#13093947	C₂²	1	4	yes	—
15	C₂×#13093935	C₂²	1	1	yes	characteristic
16	C₂×#13093934	C₂²	1	2	yes	—
17	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
18	#15689028	C₂²	1	4	yes	—
19	#15689019	C₂²	1	4	yes	—
20	#15689007	—	2	4	no	—
21	#15688990	—	2	2	no	—
22	#15688987	—	2	2	no	—
23	#15688969	C₂²	1	1	yes	characteristic
24	#15688885	C₂²	1	4	yes	—
25	C₂×#106×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#340	C₂²	1	2	yes	—
29	C₂²×Dic₉₄	C₂³	1	2	yes	—
30	C₂×Dic₁₈₈	—	2	4	no	—
31	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	C₄×Dic₉₄	—	2	4	no	—
34	C₄×Dic₉₄	C₂×C₄	1	8	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	C₂³	1	2	yes	—
40	#13093958	C₂×C₄	1	4	yes	—
41	#13093949	C₂×C₄	1	8	yes	—
42	#13093947	—	2	8	no	—
43	#13093935	—	2	2	no	—
44	#13093934	—	2	4	no	—
45	C₈×Dic₄₇	—	2	4	no	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₄₇⋊#95	—	2	4	no	—
48	C₄₇⋊#95	—	2	4	no	—
49	C₄₇⋊#95	D₄	1	2	yes	—
50	C₄₇⋊#95	D₄	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
54	C₂×C₄²×C₄₇	—	2	2	no	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	#106×C₄₇	—	2	4	no	—
57	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
58	#95×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	2	2	no	—
64	C₂²×Dic₄₇	—	2	2	no	—
65	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
66	C₂×Dic₉₄	—	2	4	no	—
67	C₂×Dic₉₄	—	2	4	no	—
68	C₂×Dic₉₄	—	2	2	no	—
69	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₇	#41	1	4	yes	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄²×C₄₇	—	8	2	no	—
77	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₄₇	#41	1	2	yes	—
79	C₄⋊Dic₄₇	—	2	2	no	—
80	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
81	Dic₄₇⋊C₄	—	2	8	no	—
82	Dic₄₇⋊C₄	—	8	4	no	—
83	C₄×Dic₄₇	—	2	8	no	—
84	C₄×Dic₄₇	—	2	2	no	—
85	C₄×Dic₄₇	—	2	2	no	—
86	C₂×C₄₇⋊C₈	—	2	2	no	—
87	C₂×C₄₇⋊C₈	—	2	2	no	—
88	C₂×C₄₇⋊C₈	#41	1	4	yes	—
89	Dic₁₈₈	—	2	8	no	—
90	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	4	4	no	—
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₄₇	—	2	4	no	—
103	C₂×Dic₄₇	—	2	4	no	—
104	C₂×Dic₄₇	—	2	4	no	—
105	C₂×Dic₄₇	—	2	4	no	—
106	C₂×Dic₄₇	—	2	4	no	—
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	—	4	2	no	—
109	C₂×Dic₄₇	—	4	2	no	—
110	Dic₉₄	—	2	4	no	—
111	Dic₉₄	—	2	4	no	—
112	Dic₉₄	—	4	4	no	—
113	C₈×C₄₇	—	2	2	no	—
114	C₈×C₄₇	#117	1	4	yes	—
115	C₄₇⋊C₈	—	2	4	no	—
116	#21283	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
118	C₂²×C₄₇	#532	1	2	yes	—
119	C₂²×C₄₇	#577	1	2	yes	—
120	C₂²×C₄₇	#578	1	2	yes	—
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	—	8	4	no	—
124	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
125	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
126	C₄×C₄₇	#529	1	2	yes	—
127	Dic₄₇	—	4	4	no	—
128	Dic₄₇	—	4	4	no	—
129	Dic₄₇	—	4	4	no	—
130	C₂×C₄×Q₁₆	—	47	1	no	—
131	#2320	—	47	2	no	—
132	#1971	—	47	2	no	—
133	#1943	—	47	2	no	—
134	#1933	—	47	4	no	—
135	C₄×#106	D₄₇	1	1	yes	characteristic
136	#1763	—	47	2	no	—
137	#1726	—	47	1	no	—
138	C₂×C₄₇	#2293	1	1	yes	characteristic
139	C₂×C₄₇	#2950	1	1	yes	characteristic
140	C₂×C₄₇	#2952	1	1	yes	characteristic
141	C₂×C₄₇	#3178	1	4	yes	—
142	C₂²×Q₁₆	—	47	1	no	—
143	C₂×C₄×Q₈	—	47	2	no	—
144	C₄×Q₁₆	—	47	8	no	—
145	C₂×#107	—	47	1	no	—
146	C₂×#106	C₂×D₄₇	1	2	yes	—
147	C₂×#103	—	47	2	no	—
148	C₂×#103	—	47	4	no	—
149	C₂×#97	—	47	2	no	—
150	C₂×#97	—	47	1	no	—
151	C₂×C₄×C₈	—	47	2	no	—
152	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
153	#391	—	47	4	no	—
154	#389	—	47	4	no	—
155	#387	—	94	4	no	—
156	#383	—	94	2	no	—
157	#382	—	94	2	no	—
158	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
159	#376	—	47	1	no	—
160	#340	—	47	4	no	—
161	#340	C₂×D₄₇	1	2	yes	—
162	C₄₇	#21283	1	1	yes	characteristic, complemented, Sylow
163	C₂²×Q₈	—	47	2	no	—
164	C₂×Q₁₆	—	47	4	no	—
165	C₂×Q₁₆	—	94	4	no	—
166	C₂²×C₈	—	47	2	no	—
167	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
169	C₄×Q₈	—	47	8	no	—
170	C₄×Q₈	—	94	4	no	—
171	C₂×C₄⋊C₄	—	47	2	no	—
172	C₂×C₄⋊C₄	—	94	2	no	—
173	C₂×C₄⋊C₄	—	94	4	no	—
174	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	—	47	2	no	—
177	C₂×C₄²	—	94	2	no	—
178	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
179	#107	—	47	4	no	—
180	#106	—	2	4	no	—
181	#103	—	47	8	no	—
182	#103	—	94	8	no	—
183	#97	—	94	4	no	—
184	#97	—	94	2	no	—
185	C₄×C₈	—	94	4	no	—
186	C₄×C₈	C₄×D₄₇	1	4	yes	—
187	#95	—	2	2	no	—
188	#95	—	47	2	no	—
189	#95	—	47	2	no	—
190	#95	—	94	4	no	—
191	#95	—	94	4	no	—
192	#95	—	94	4	no	—
193	C₂×Q₈	—	47	4	no	—
194	C₂×Q₈	—	94	2	no	—
195	C₂×Q₈	—	94	4	no	—
196	C₂×Q₈	—	94	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	47	4	no	—
199	C₂²×C₄	—	94	2	no	—
200	C₂²×C₄	—	94	2	no	—
201	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
204	Q₁₆	—	94	8	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	47	4	no	—
207	C₂×C₈	—	94	2	no	—
208	C₂×C₈	—	94	2	no	—
209	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
210	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
211	C₂×C₈	#10625350	1	4	yes	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	47	4	no	—
215	C₄⋊C₄	—	94	8	no	—
216	C₄⋊C₄	—	94	2	no	—
217	C₄⋊C₄	—	376	4	no	—
218	C₄²	—	8	2	no	—
219	C₄²	—	94	8	no	—
220	C₄²	—	94	2	no	—
221	C₄²	—	94	2	no	—
222	C₄²	C₂×C₄×D₄₇	1	2	yes	—
223	C₄²	#10625354	1	2	yes	—
224	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
225	Q₈	—	94	4	no	—
226	Q₈	—	94	4	no	—
227	Q₈	—	188	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	94	4	no	—
234	C₂×C₄	—	94	4	no	—
235	C₂×C₄	—	94	4	no	—
236	C₂×C₄	—	94	4	no	—
237	C₂×C₄	—	94	4	no	—
238	C₂×C₄	—	188	4	no	—
239	C₂×C₄	—	188	2	no	—
240	C₂×C₄	—	188	2	no	—
241	C₂×C₄	#13093986	1	2	yes	—
242	C₂×C₄	#13093986	1	2	yes	—
243	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
244	C₂×C₄	#41×D₄₇	1	2	yes	—
245	C₂×C₄	C₂×#10625350	1	2	yes	—
246	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	94	4	no	—
249	C₈	#13093981	1	4	yes	—
250	C₂²	C₂×#13093986	1	1	yes	characteristic
251	C₂²	#15689670	1	2	yes	—
252	C₂²	#15689929	1	2	yes	—
253	C₂²	#15689937	1	2	yes	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	8	4	no	—
257	C₄	—	188	4	no	—
258	C₄	—	188	4	no	—
259	C₄	—	188	4	no	—
260	C₄	C₂×#13093981	1	1	yes	characteristic
261	C₄	#15689591	1	1	yes	characteristic
262	C₄	#15689660	1	2	yes	—
263	C₂	#20245849	1	1	yes	characteristic
264	C₂	#20252255	1	1	yes	characteristic
265	C₂	#20252271	1	1	yes	characteristic
266	C₂	#20254052	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#106×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#106	D₄₇
hypocenter	C₄₇	#21283
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₁₈₈	C₄	semidirect
162. C₄₇	#21283	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
#20246034	2
#20244257	2
#20244124	2
#20244062	2
#20243221	2
#20243074	2
C₄×#106×C₄₇	2
#21283	47