Subgroups — #26216516

Order 12032.

818 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689146	C₂	1	2	yes	maximal
2	C₂×M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243527	C₂	1	4	yes	maximal, complemented
4	#20243524	C₂	1	4	yes	maximal, complemented
5	#20243115	C₂	1	1	yes	maximal, characteristic
6	#20243114	C₂	1	2	yes	maximal
7	C₄₇×#1738	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	#15689146	—	2	4	no	—
12	#15689146	C₂²	1	4	yes	—
13	#15689146	C₂²	1	4	yes	—
14	C₂×Dic₄₇⋊C₈	—	2	4	no	—
15	M₁₆×Dic₄₇	C₄	1	4	yes	—
16	M₁₆×Dic₄₇	C₄	1	4	yes	—
17	C₂×#13093934	C₂²	1	2	yes	—
18	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
19	#15688888	C₂²	1	4	yes	—
20	#15688887	C₂²	1	8	yes	—
21	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
22	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#341	C₂²	1	4	yes	—
24	C₂³×Dic₄₇	Q₈	1	2	yes	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	—	2	4	no	—
29	C₂×#10625329	C₂×C₄	1	4	yes	—
30	C₂×#10625329	C₂³	1	4	yes	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
36	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
39	Dic₄₇⋊C₈	—	2	8	no	—
40	Dic₄₇⋊C₈	—	2	8	no	—
41	#13093934	C₂×C₄	1	8	yes	—
42	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
43	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₄₇	—	2	4	no	—
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	—	2	4	no	—
48	C₂×M₁₆×C₄₇	—	2	4	no	—
49	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
50	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
51	C₂²×C₈×C₄₇	—	2	2	no	—
52	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	#41	1	4	yes	—
58	C₂²×C₄×C₄₇	#41	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₇	—	4	4	no	—
61	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
62	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
63	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
64	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
65	M₁₆×C₄₇	—	2	8	no	—
66	M₁₆×C₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	2	4	no	—
68	M₁₆×C₄₇	—	4	4	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	4	4	no	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
73	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
74	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
75	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
78	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
79	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
80	#10625329	—	2	8	no	—
81	#10625329	—	2	4	no	—
82	#10625329	—	2	4	no	—
83	#10625329	—	4	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	4	4	no	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
88	C₂³×C₄₇	—	4	2	no	—
89	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₄₇	#125	1	4	yes	—
91	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₄₇	#117	1	4	yes	—
94	C₂×C₄×C₄₇	#117	1	4	yes	—
95	C₂×C₄×C₄₇	#124	1	4	yes	—
96	C₂×C₄×C₄₇	#124	1	4	yes	—
97	C₂×C₄×C₄₇	#125	1	4	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
102	C₂×Dic₄₇	—	2	8	no	—
103	C₂×Dic₄₇	—	2	4	no	—
104	C₂×Dic₄₇	—	4	4	no	—
105	C₂×Dic₄₇	—	4	2	no	—
106	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₈×C₄₇	—	2	4	no	—
110	C₈×C₄₇	—	4	4	no	—
111	C₄₇⋊C₈	—	2	4	no	—
112	C₄₇⋊C₈	—	4	4	no	—
113	#20769	—	47	1	no	maximal, Sylow
114	C₂²×C₄₇	—	2	4	no	—
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	4	4	no	—
117	C₂²×C₄₇	#382	1	4	yes	—
118	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₄₇	C₂×#125	1	2	yes	—
120	C₄×C₄₇	#382	1	4	yes	—
121	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
122	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
123	C₄×C₄₇	C₂×#124	1	2	yes	—
124	Dic₄₇	—	2	4	no	—
125	Dic₄₇	—	2	2	no	—
126	Dic₄₇	#430	1	4	yes	—
127	C₂×#423	—	47	2	no	—
128	C₂×C₄×M₁₆	—	47	1	no	—
129	#1838	—	47	4	no	—
130	#1837	—	47	4	no	—
131	#1738	—	47	2	no	—
132	#1738	—	47	1	no	—
133	#1738	D₄₇	1	1	yes	characteristic
134	C₂×C₄₇	—	2	2	no	—
135	C₂×C₄₇	—	2	4	no	—
136	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
137	C₂×C₄₇	C₂×#430	1	2	yes	—
138	C₂²×M₁₆	—	47	1	no	—
139	C₂²×M₁₆	—	47	2	no	—
140	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
141	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
142	C₂²×C₄²	—	47	1	no	—
143	#423	—	47	4	no	—
144	#423	—	47	4	no	—
145	#423	—	94	4	no	—
146	C₂×C₄⋊C₈	—	94	4	no	—
147	C₄×M₁₆	—	47	4	no	—
148	C₄×M₁₆	—	47	4	no	—
149	C₂×#97	—	47	2	no	—
150	C₂×C₄×C₈	—	47	2	no	—
151	#341	—	47	8	no	—
152	#341	—	47	4	no	—
153	#341	C₂×D₄₇	1	4	yes	—
154	C₄₇	#20769	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₄	—	47	2	no	—
156	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	4	no	—
160	C₂×M₁₆	—	2	4	no	—
161	C₂×M₁₆	—	47	4	no	—
162	C₂×M₁₆	—	47	4	no	—
163	C₂×M₁₆	—	94	4	no	—
164	C₂×M₁₆	—	94	2	no	—
165	C₂×M₁₆	—	94	4	no	—
166	C₂×M₁₆	—	94	4	no	—
167	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
168	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	47	2	no	—
171	C₂²×C₈	—	94	2	no	—
172	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
173	C₂×C₄²	—	47	2	no	—
174	C₂×C₄²	—	47	2	no	—
175	C₂×C₄²	—	47	4	no	—
176	C₂×C₄²	—	47	2	no	—
177	C₂×C₄²	—	47	1	no	—
178	C₂×C₄²	—	47	1	no	—
179	C₄⋊C₈	—	94	8	no	—
180	C₄⋊C₈	—	94	8	no	—
181	#97	—	47	8	no	—
182	C₄×C₈	—	47	4	no	—
183	C₄×C₈	—	47	4	no	—
184	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
185	C₂²×C₄	—	47	4	no	—
186	C₂²×C₄	—	47	2	no	—
187	C₂²×C₄	—	47	4	no	—
188	C₂²×C₄	—	47	2	no	—
189	C₂²×C₄	—	188	4	no	—
190	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
191	C₂²×C₄	#10625350	1	4	yes	—
192	C₂²×C₄	D₄×D₄₇	1	2	yes	—
193	C₂²×C₄	#10625352	1	4	yes	—
194	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
195	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
196	M₁₆	—	2	8	no	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	2	4	no	—
199	M₁₆	—	4	4	no	—
200	M₁₆	—	94	8	no	—
201	M₁₆	—	94	4	no	—
202	M₁₆	—	94	4	no	—
203	M₁₆	—	188	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	47	4	no	—
208	C₂×C₈	—	94	4	no	—
209	C₂×C₈	—	94	4	no	—
210	C₂×C₈	—	188	4	no	—
211	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
212	C₄²	—	47	4	no	—
213	C₄²	—	47	4	no	—
214	C₄²	—	47	1	no	—
215	C₄²	—	47	1	no	—
216	C₄²	—	47	2	no	—
217	C₄²	—	47	2	no	—
218	C₄²	—	47	2	no	—
219	C₂³	—	4	2	no	—
220	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
221	C₂³	#13094014	1	4	yes	—
222	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
223	C₂×C₄	—	47	2	no	—
224	C₂×C₄	—	47	2	no	—
225	C₂×C₄	—	47	4	no	—
226	C₂×C₄	—	94	8	no	—
227	C₂×C₄	—	94	4	no	—
228	C₂×C₄	—	188	4	no	—
229	C₂×C₄	—	188	2	no	—
230	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
231	C₂×C₄	#13093980	1	4	yes	—
232	C₂×C₄	#13093981	1	4	yes	—
233	C₂×C₄	#13094010	1	4	yes	—
234	C₂×C₄	#13094010	1	4	yes	—
235	C₂×C₄	#13094014	1	4	yes	—
236	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
238	C₂×C₄	C₂×#10625352	1	2	yes	—
239	C₂×C₄	C₂×#10625350	1	2	yes	—
240	C₈	—	2	4	no	—
241	C₈	—	4	4	no	—
242	C₈	—	94	4	no	—
243	C₈	—	188	4	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	4	no	—
247	C₂²	#15688987	1	4	yes	—
248	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
249	C₂²	C₂×#13094014	1	2	yes	—
250	C₄	—	47	4	no	—
251	C₄	—	94	2	no	—
252	C₄	—	94	4	no	—
253	C₄	#15688987	1	4	yes	—
254	C₄	C₂×#13093980	1	1	yes	characteristic
255	C₄	C₂×#13093981	1	1	yes	characteristic
256	C₄	C₂×#13094010	1	2	yes	—
257	C₂	—	2	4	no	—
258	C₂	—	2	2	no	—
259	C₂	C₂×#15688987	1	1	yes	characteristic
260	C₂	#430×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1738	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1738	D₄₇
hypocenter	C₄₇	#20769
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1738

Complements

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

N	H ≅ G/N	type
3. #20243527	C₂	semidirect
4. #20243524	C₂	semidirect
154. C₄₇	#20769	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689146	2
C₂×M₁₆×Dic₄₇	2
#20243527	2
#20243524	2
#20243115	2
#20243114	2
C₄₇×#1738	2
#20769	47