Subgroups — #26186844

Order 12032.

734 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252100	C₂	1	2	yes	maximal, complemented
2	#20242302	C₂	1	4	yes	maximal, complemented
3	#20242212	C₂	1	4	yes	maximal, complemented
4	#20242144	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241839	C₂	1	2	yes	maximal, complemented
6	C₂×#15688844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#3169	C₂	1	1	yes	maximal, characteristic
8	C₂×#13094031	—	2	2	no	—
9	#15689201	—	2	2	no	—
10	#15689201	C₂²	1	4	yes	—
11	D₄×C₄₇⋊C₈	—	2	2	no	—
12	D₄×C₄₇⋊C₈	C₂²	1	4	yes	—
13	#15689147	C₂²	1	2	yes	—
14	#15689090	—	2	2	no	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	#15688874	C₂²	1	4	yes	—
18	D₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
19	#15688844	C₂²	1	2	yes	—
20	#15688844	C₂²	1	2	yes	—
21	C₄₇×#532	C₂²	1	2	yes	—
22	C₄₇×#500	C₄	1	1	yes	characteristic
23	C₄₇×#493	C₄	1	1	yes	characteristic
24	C₄₇×#486	C₄	1	2	yes	—
25	C₄₇×#466	C₄	1	4	yes	—
26	C₄×D₈×C₄₇	C₂²	1	4	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	#13094031	—	4	8	no	—
29	C₂×#10625329	—	2	2	no	—
30	C₂×#10625329	—	2	4	no	—
31	C₂²×C₄₇⋊C₈	—	2	1	no	—
32	C₂²×C₄₇⋊C₈	—	2	4	no	—
33	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
34	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
36	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
37	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
38	C₄×C₄₇⋊C₈	—	2	1	no	—
39	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
40	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
41	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂×#41×C₄₇	—	2	2	no	—
44	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
47	#127×C₄₇	C₂×C₄	1	2	yes	—
48	#124×C₄₇	—	2	2	no	—
49	#121×C₄₇	—	2	2	no	—
50	#121×C₄₇	C₂×C₄	1	4	yes	—
51	C₄×D₄×C₄₇	—	2	2	no	—
52	C₄×D₄×C₄₇	C₂³	1	4	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	#107×C₄₇	C₂×C₄	1	4	yes	—
55	#102×C₄₇	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₄₇	C₂³	1	2	yes	—
59	#41×C₄₇	—	4	4	no	—
60	#41×C₄₇	—	4	4	no	—
61	C₂×Q₈×C₄₇	—	2	2	no	—
62	C₂×D₄×C₄₇	—	2	4	no	—
63	C₂×D₄×C₄₇	—	2	2	no	—
64	C₂×D₄×C₄₇	—	2	2	no	—
65	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
70	D₈×C₄₇	—	2	8	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
74	C₄⋊C₄×C₄₇	—	2	2	no	—
75	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
76	C₂²⋊C₄×C₄₇	—	2	4	no	—
77	C₂²⋊C₄×C₄₇	—	2	4	no	—
78	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₄₇	#41	1	1	yes	characteristic
81	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
82	#10625329	—	4	8	no	—
83	#10625329	—	4	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	2	2	no	—
87	C₂×C₄₇⋊C₈	—	4	8	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
90	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
91	C₂³×C₄₇	—	2	4	no	—
92	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
93	Q₈×C₄₇	—	4	4	no	—
94	D₄×C₄₇	—	2	8	no	—
95	D₄×C₄₇	—	4	8	no	—
96	D₄×C₄₇	—	4	4	no	—
97	D₄×C₄₇	—	4	4	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	4	4	no	—
103	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
107	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
110	C₈×C₄₇	—	2	2	no	—
111	C₈×C₄₇	#131	1	4	yes	complemented
112	C₄₇⋊C₈	—	2	2	no	—
113	C₄₇⋊C₈	—	2	2	no	—
114	C₄₇⋊C₈	—	4	2	no	—
115	C₄₇⋊C₈	—	4	4	no	—
116	#17962	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	1	no	—
118	C₂²×C₄₇	—	2	1	no	—
119	C₂²×C₄₇	—	4	4	no	—
120	C₂²×C₄₇	—	4	4	no	—
121	C₂²×C₄₇	—	4	4	no	—
122	C₂²×C₄₇	#442	1	2	yes	—
123	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
124	C₄×C₄₇	—	2	1	no	—
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	—	4	4	no	—
127	C₄×C₄₇	#442	1	2	yes	—
128	C₄×C₄₇	#444	1	2	yes	—
129	C₄×C₄₇	C₂×#131	1	2	yes	—
130	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
131	#3169	D₄₇	1	1	yes	characteristic
132	#2931	—	47	2	no	—
133	#1594	—	47	4	no	—
134	#1580	—	47	4	no	—
135	#1566	—	47	1	no	—
136	#1492	—	47	2	no	—
137	C₂×#322	—	47	1	no	—
138	C₂×C₄₇	—	2	2	no	—
139	C₂×C₄₇	—	8	4	no	—
140	C₂×C₄₇	#2931	1	1	yes	characteristic
141	C₂×C₄₇	C₂×#442	1	1	yes	characteristic
142	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
143	C₂×#131	—	94	2	no	—
144	#532	C₂×D₄₇	1	2	yes	—
145	#500	Dic₄₇	1	1	yes	characteristic
146	#493	Dic₄₇	1	1	yes	characteristic
147	#486	Dic₄₇	1	2	yes	—
148	#466	Dic₄₇	1	4	yes	—
149	C₄×D₈	C₂×D₄₇	1	4	yes	—
150	#436	—	47	4	no	—
151	#436	—	94	2	no	—
152	C₈×D₄	—	47	4	no	—
153	C₈×D₄	—	94	2	no	—
154	#423	—	47	2	no	—
155	#408	—	94	2	no	—
156	C₂×C₄×C₈	—	47	1	no	—
157	C₂×C₄×C₈	—	47	1	no	—
158	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
159	#334	—	47	4	no	—
160	D₄⋊C₈	—	47	8	no	—
161	#322	—	47	2	no	—
162	#322	—	47	2	no	—
163	C₄₇	#17962	1	1	yes	characteristic, complemented, Sylow
164	C₂×#41	—	2	2	no	—
165	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
166	#131	—	188	8	no	—
167	C₂×M₁₆	—	94	4	no	—
168	C₂×M₁₆	—	94	2	no	—
169	C₂²×C₈	—	47	2	no	—
170	C₂²×C₈	—	94	4	no	—
171	C₂²×C₈	—	94	1	no	—
172	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
173	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
174	#127	C₂×Dic₄₇	1	2	yes	—
175	#124	—	2	2	no	—
176	#121	—	2	2	no	—
177	#121	C₂×Dic₄₇	1	4	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂²×D₄₇	1	4	yes	—
180	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
181	#107	C₂×Dic₄₇	1	4	yes	—
182	C₄⋊C₈	—	47	4	no	—
183	C₄⋊C₈	—	94	2	no	—
184	#102	C₂×Dic₄₇	1	8	yes	—
185	C₂²⋊C₈	—	94	4	no	—
186	C₂²⋊C₈	—	94	4	no	—
187	C₄×C₈	—	47	2	no	—
188	C₄×C₈	—	47	1	no	—
189	C₄×C₈	—	47	1	no	—
190	C₄×C₈	—	47	2	no	—
191	C₄×C₈	—	94	1	no	—
192	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
193	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₄₇	1	2	yes	—
195	#41	—	4	4	no	—
196	#41	—	4	4	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
204	C₂²×C₄	#10625352	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
206	D₈	—	2	8	no	—
207	M₁₆	—	188	8	no	—
208	M₁₆	—	188	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	47	2	no	—
211	C₂×C₈	—	47	2	no	—
212	C₂×C₈	—	94	4	no	—
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	—	94	2	no	—
215	C₂×C₈	—	188	8	no	—
216	C₂×C₈	—	188	2	no	—
217	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
218	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₄²	D₄×D₄₇	1	1	yes	characteristic
224	C₄²	#10625352	1	1	yes	characteristic
225	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
226	C₄²	C₂³×D₄₇	1	1	yes	characteristic
227	C₂³	—	2	4	no	—
228	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
240	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
241	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
242	C₂×C₄	#13094048	1	4	yes	—
243	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
245	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	94	2	no	—
248	C₈	—	94	2	no	—
249	C₈	—	188	2	no	—
250	C₈	—	188	4	no	—
251	C₈	#13094031	1	4	yes	complemented
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	#15689235	1	2	yes	—
258	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	4	no	—
262	C₄	#15689235	1	2	yes	—
263	C₄	#15689245	1	2	yes	—
264	C₄	C₂×#13094031	1	2	yes	—
265	C₄	C₂×#13094048	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	8	4	no	—
268	C₂	#20252100	1	1	yes	characteristic
269	C₂	C₂×#15689235	1	1	yes	characteristic
270	C₂	C₂×#15689245	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3169	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#3169	D₄₇
hypocenter	C₄₇	#17962
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20252100	C₂	semidirect
2. #20242302	C₂	semidirect
3. #20242212	C₂	semidirect
4. #20242144	C₂	semidirect
5. #20241839	C₂	semidirect
6. C₂×#15688844	C₂	semidirect
111. C₈×C₄₇	#131	semidirect
163. C₄₇	#17962	semidirect
251. C₈	#13094031	semidirect

Maximal subgroups

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

subgroup	index
#20252100	2
#20242302	2
#20242212	2
#20242144	2
#20241839	2
C₂×#15688844	2
C₄₇×#3169	2
#17962	47