Subgroups — #26368956

Order 12032.

1162 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093964	C₂	1	2	yes	maximal, complemented
2	#20245002	C₂	1	8	yes	maximal, complemented
3	#20244999	C₂	1	4	yes	maximal, complemented
4	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	—	2	8	no	—
6	C₂×#13093964	C₂²	1	4	yes	—
7	C₂×#13093964	C₂²	1	4	yes	complemented
8	C₂²×C₄₇⋊C₁₆	C₂²	1	4	yes	complemented
9	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	16	yes	complemented
10	C₂²×#41×C₄₇	C₄	1	1	yes	characteristic
11	C₂×#131×C₄₇	—	2	8	no	—
12	C₂×#131×C₄₇	C₄	1	4	yes	—
13	C₂×#131×C₄₇	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₄₇	C₄	1	1	yes	characteristic
15	C₂²×M₁₆×C₄₇	C₄	1	2	yes	—
16	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
17	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	#13093964	—	2	8	no	—
19	#13093964	—	2	8	no	—
20	C₂×C₄₇⋊C₁₆	—	2	8	no	—
21	C₂×C₄₇⋊C₁₆	C₂³	1	8	yes	—
22	C₂×#41×C₄₇	—	2	4	no	—
23	C₂×#41×C₄₇	—	2	4	no	—
24	C₂×#41×C₄₇	C₈	1	4	yes	—
25	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
26	C₂²×Q₈×C₄₇	C₈	1	1	yes	characteristic
27	C₂²×D₄×C₄₇	C₈	1	2	yes	—
28	C₂²×D₄×C₄₇	C₈	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	#131×C₄₇	—	2	16	no	—
32	#131×C₄₇	—	2	8	no	—
33	#131×C₄₇	—	2	8	no	—
34	C₂×M₁₆×C₄₇	—	2	4	no	—
35	C₂×M₁₆×C₄₇	—	2	8	no	—
36	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
38	C₂²×C₈×C₄₇	—	2	8	no	—
39	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₄₇	D₄	1	8	yes	—
44	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
45	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂⁴×C₄₇	C₂×C₈	1	2	yes	—
48	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
49	#41×C₄₇	—	2	16	no	—
50	#41×C₄₇	—	2	8	no	—
51	#41×C₄₇	—	2	8	no	—
52	C₂×Q₈×C₄₇	—	2	4	no	—
53	C₂×Q₈×C₄₇	C₂×C₈	1	4	yes	—
54	C₂×D₄×C₄₇	—	2	8	no	—
55	C₂×D₄×C₄₇	—	2	4	no	—
56	C₂×D₄×C₄₇	C₂×C₈	1	8	yes	—
57	C₂×D₄×C₄₇	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	M₁₆×C₄₇	—	2	8	no	—
70	M₁₆×C₄₇	—	2	16	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	8	no	—
73	C₂×C₈×C₄₇	—	2	8	no	—
74	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
75	C₂×C₈×C₄₇	C₂²⋊C₄	1	8	yes	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₄₇⋊C₁₆	—	2	8	no	—
84	C₂³×C₄₇	—	2	4	no	—
85	C₂³×C₄₇	—	4	2	no	—
86	C₂³×C₄₇	—	4	4	no	—
87	C₂³×C₄₇	C₂²⋊C₈	1	4	yes	—
88	C₂³×C₄₇	C₂²×C₈	1	4	yes	—
89	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
90	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
91	Q₈×C₄₇	—	2	8	no	—
92	D₄×C₄₇	—	2	16	no	—
93	D₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	8	no	—
98	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
99	C₂×C₄×C₄₇	C₂²⋊C₈	1	8	yes	—
100	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
103	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
107	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₄₇	—	2	4	no	—
109	C₈×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
110	C₈×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
111	C₈×C₄₇	C₂²×D₄	1	2	yes	—
112	#33333	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	4	no	—
115	C₂²×C₄₇	—	2	4	no	—
116	C₂²×C₄₇	—	2	8	no	—
117	C₂²×C₄₇	—	4	4	no	—
118	C₂²×C₄₇	—	4	8	no	—
119	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
120	C₂²×C₄₇	C₂×C₂²⋊C₈	1	4	yes	—
121	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
122	C₄×C₄₇	—	2	4	no	—
123	C₄×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
124	C₄×C₄₇	C₂×C₂²⋊C₈	1	4	yes	—
125	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
127	C₂²×#131	D₄₇	1	1	yes	characteristic
128	C₂²×M₃₂	—	47	2	no	—
129	#2113	—	47	8	no	—
130	#2112	—	47	4	no	—
131	C₂×C₄₇	—	2	2	no	—
132	C₂×C₄₇	—	2	4	no	—
133	C₂×C₄₇	—	4	4	no	—
134	C₂×C₄₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₄₇	#3407	1	2	yes	—
136	C₂²×#41	Dic₄₇	1	1	yes	characteristic
137	C₂×#131	—	2	8	no	—
138	C₂×#131	Dic₄₇	1	4	yes	—
139	C₂×#131	C₂×D₄₇	1	4	yes	—
140	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
141	C₂²×M₁₆	Dic₄₇	1	2	yes	—
142	C₂³×C₈	C₂×D₄₇	1	2	yes	—
143	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
144	C₂×M₃₂	—	47	4	no	—
145	C₂×M₃₂	—	47	4	no	—
146	C₂×M₃₂	—	94	8	no	—
147	C₂²×C₁₆	—	47	4	no	—
148	C₂²⋊C₁₆	—	47	16	no	—
149	C₄₇	#33333	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	—	2	4	no	—
151	C₂×#41	—	2	4	no	—
152	C₂×#41	C₄₇⋊C₈	1	4	yes	—
153	C₂×#41	C₂×Dic₄₇	1	4	yes	—
154	C₂²×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
155	C₂²×D₄	C₄₇⋊C₈	1	2	yes	—
156	C₂²×D₄	C₄₇⋊C₈	1	1	yes	characteristic
157	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
158	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
159	#131	—	2	16	no	—
160	#131	—	2	8	no	—
161	#131	—	2	8	no	—
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	—	2	8	no	—
164	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
165	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
166	C₂²×C₈	—	2	8	no	—
167	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
168	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
169	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
170	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
171	C₂²×C₈	C₂²⋊D₄₇	1	8	yes	—
172	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
173	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
175	M₃₂	—	94	8	no	—
176	M₃₂	—	94	8	no	—
177	C₂×C₁₆	—	47	8	no	—
178	C₂×C₁₆	—	94	8	no	—
179	C₂⁴	C₂×C₄₇⋊C₈	1	2	yes	—
180	C₂⁴	C₂×C₄₇⋊C₈	1	1	yes	characteristic
181	#41	—	2	16	no	—
182	#41	—	2	8	no	—
183	#41	—	2	8	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	C₂×C₄₇⋊C₈	1	4	yes	—
186	C₂×D₄	—	2	8	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	C₂×C₄₇⋊C₈	1	8	yes	—
189	C₂×D₄	C₂×C₄₇⋊C₈	1	4	yes	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
193	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
194	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
196	C₂²×C₄	#10625329	1	4	yes	—
197	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
198	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
201	M₁₆	—	2	8	no	—
202	M₁₆	—	2	16	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	8	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
207	C₂×C₈	C₂²⋊Dic₄₇	1	8	yes	—
208	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
209	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
210	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
211	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
212	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
213	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
214	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
215	C₁₆	—	94	8	no	—
216	C₂³	—	2	4	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	4	no	—
219	C₂³	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
220	C₂³	C₂²×C₄₇⋊C₈	1	4	yes	—
221	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
222	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
223	Q₈	—	2	8	no	—
224	D₄	—	2	16	no	—
225	D₄	—	2	8	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
231	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	8	yes	—
232	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
233	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
234	C₂×C₄	C₂²×C₄₇⋊C₈	1	4	yes	—
235	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
237	C₂×C₄	C₂×#10625329	1	2	yes	—
238	C₂×C₄	C₂×#10625329	1	4	yes	—
239	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	C₂×C₂²⋊Dic₄₇	1	2	yes	—
242	C₈	C₂×C₂²⋊Dic₄₇	1	4	yes	—
243	C₈	C₂²×C₂²⋊D₄₇	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	4	4	no	—
249	C₂²	—	4	8	no	—
250	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
251	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
252	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
255	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
256	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
257	C₄	C₂²×#10625329	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	2	4	no	—
260	C₂	—	4	4	no	—
261	C₂	C₂²×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
262	C₂	#20256491	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₂²×#131×C₄₇	C₂
prime radical	C₂²×#41×C₄₇	C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	#33333
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093964	C₂	semidirect
2. #20245002	C₂	semidirect
3. #20244999	C₂	semidirect
7. C₂×#13093964	C₂²	semidirect
8. C₂²×C₄₇⋊C₁₆	C₂²	semidirect
9. C₂²⋊(C₄₇⋊C₁₆)	C₂²	semidirect
149. C₄₇	#33333	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093964	2
#20245002	2
#20244999	2
C₂²×#131×C₄₇	2
#33333	47