Subgroups — #26226548

Order 12032.

1266 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	C₂×#15688983	C₂	1	2	yes	maximal
2	#20243578	C₂	1	8	yes	maximal, complemented
3	#20243417	C₂	1	4	yes	maximal, complemented
4	C₂²×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
6	C₂²×C₂²⋊Dic₄₇	—	2	2	no	—
7	C₂×#13093958	C₂²	1	8	yes	complemented
8	#15688983	—	2	2	no	—
9	#15688983	—	2	2	no	—
10	#15688983	C₂²	1	4	yes	—
11	#15688983	C₂²	1	4	yes	—
12	C₂×C₄₇⋊#95	—	2	2	no	—
13	#15688886	C₂²	1	8	yes	complemented
14	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₄₇	C₄	1	8	yes	complemented
17	C₂×C₂²⋊C₈×C₄₇	C₂²	1	4	yes	—
18	C₂³×Dic₄₇	—	2	4	no	—
19	C₂×C₄⋊Dic₄₇	—	2	4	no	—
20	C₂×C₄⋊Dic₄₇	—	4	8	no	—
21	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
22	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
23	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
24	C₂×C₂²⋊Dic₄₇	—	4	8	no	—
25	#13093958	—	4	8	no	—
26	C₄₇⋊#95	—	2	4	no	—
27	C₄₇⋊#95	—	2	4	no	—
28	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	—	4	4	no	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₄₇	D₄	1	2	yes	—
32	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	Q₈	1	4	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₄₇	—	4	4	no	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
38	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
39	C₂²⋊C₈×C₄₇	C₂×C₄	1	16	yes	complemented
40	C₂⁴×C₄₇	—	4	2	no	—
41	C₂⁴×C₄₇	—	4	2	no	—
42	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
43	C₂⁴×C₄₇	D₈	1	4	yes	—
44	C₂⁴×C₄₇	Q₁₆	1	4	yes	—
45	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₄₇	—	2	4	no	—
48	C₂²×C₄×C₄₇	—	2	8	no	—
49	C₂²×C₄×C₄₇	—	4	8	no	—
50	C₂²×C₄×C₄₇	—	4	8	no	—
51	C₂²×C₄×C₄₇	—	4	2	no	—
52	C₂²×C₄×C₄₇	—	4	4	no	—
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₄₇	—	2	8	no	—
63	C₂²×Dic₄₇	—	4	8	no	—
64	C₂²×Dic₄₇	—	4	8	no	—
65	C₂²×Dic₄₇	—	4	8	no	—
66	C₂×C₈×C₄₇	—	2	8	no	—
67	C₂×C₈×C₄₇	—	4	8	no	—
68	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
69	C₂²⋊Dic₄₇	—	4	8	no	—
70	C₂²⋊Dic₄₇	—	4	8	no	—
71	C₄⋊Dic₄₇	—	2	16	no	—
72	C₄⋊Dic₄₇	—	4	8	no	—
73	C₂³×C₄₇	—	2	4	no	—
74	C₂³×C₄₇	—	2	2	no	—
75	C₂³×C₄₇	—	2	4	no	—
76	C₂³×C₄₇	—	2	2	no	—
77	C₂³×C₄₇	—	4	4	no	—
78	C₂³×C₄₇	—	4	4	no	—
79	C₂³×C₄₇	—	4	8	no	—
80	C₂³×C₄₇	—	4	8	no	—
81	C₂³×C₄₇	#107	1	16	yes	complemented
82	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
83	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
84	C₂³×C₄₇	C₂×D₈	1	2	yes	—
85	C₂³×C₄₇	C₂×D₈	1	4	yes	—
86	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
87	C₂³×C₄₇	C₂×Q₁₆	1	4	yes	—
88	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	2	8	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	4	8	no	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
102	C₂×Dic₄₇	—	4	8	no	—
103	C₂×Dic₄₇	—	4	8	no	—
104	C₂×Dic₄₇	—	4	8	no	—
105	C₂×Dic₄₇	—	8	16	no	—
106	C₈×C₄₇	—	2	8	no	—
107	#21940	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	—	2	2	no	—
109	C₂²×C₄₇	—	2	4	no	—
110	C₂²×C₄₇	—	2	4	no	—
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	—	2	8	no	—
113	C₂²×C₄₇	—	2	4	no	—
114	C₂²×C₄₇	—	2	8	no	—
115	C₂²×C₄₇	—	2	4	no	—
116	C₂²×C₄₇	—	4	16	no	—
117	C₂²×C₄₇	C₂×#107	1	4	yes	—
118	C₂²×C₄₇	C₂×#107	1	8	yes	—
119	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₄₇	#534	1	2	yes	—
121	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
122	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
123	C₂²×C₄₇	#575	1	2	yes	—
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	—	2	2	no	—
126	C₄×C₄₇	#518	1	2	yes	—
127	C₄×C₄₇	#543	1	2	yes	—
128	Dic₄₇	—	8	8	no	—
129	C₂²×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
130	C₂×#380	—	47	2	no	—
131	#1852	—	47	8	no	—
132	#1808	—	47	4	no	—
133	C₂×C₄₇	—	2	8	no	—
134	C₂×C₄₇	—	2	4	no	—
135	C₂×C₄₇	#2315	1	1	yes	characteristic
136	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
137	C₂×C₄₇	#2911	1	1	yes	characteristic
138	C₂×C₄₇	#3186	1	2	yes	—
139	C₂×C₄₇	#3189	1	2	yes	—
140	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
141	C₂³×C₈	C₂×D₄₇	1	2	yes	—
142	C₂²×C₄⋊C₄	—	47	4	no	—
143	C₂²×C₂²⋊C₄	—	94	2	no	—
144	C₂×#107	—	47	8	no	—
145	C₂×C₂²⋊C₈	Dic₄₇	1	8	yes	complemented
146	C₂×C₂²⋊C₈	C₂×D₄₇	1	4	yes	—
147	#380	—	47	4	no	—
148	#380	—	47	4	no	—
149	#380	—	94	2	no	—
150	#380	—	94	2	no	—
151	C₂×#95	—	94	2	no	—
152	#340	—	47	8	no	—
153	C₄₇	#21940	1	1	yes	characteristic, complemented, Sylow
154	C₂⁵	D₁₈₈	1	1	yes	characteristic
155	C₂³×C₄	—	4	4	no	—
156	C₂³×C₄	—	94	4	no	—
157	C₂³×C₄	Dic₉₄	1	4	yes	—
158	C₂³×C₄	D₁₈₈	1	2	yes	—
159	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
160	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
161	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
162	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
163	C₂²×C₈	—	4	4	no	—
164	C₂²×C₈	C₂×Dic₄₇	1	8	yes	complemented
165	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
166	C₂×C₄⋊C₄	—	47	8	no	—
167	C₂×C₄⋊C₄	—	94	4	no	—
168	C₂×C₄⋊C₄	—	188	8	no	—
169	C₂×C₂²⋊C₄	—	94	4	no	—
170	C₂×C₂²⋊C₄	—	94	4	no	—
171	C₂×C₂²⋊C₄	—	188	8	no	—
172	#107	—	188	8	no	—
173	C₂²⋊C₈	C₂×Dic₄₇	1	16	yes	complemented
174	#95	—	94	4	no	—
175	#95	—	94	4	no	—
176	C₂⁴	—	4	2	no	—
177	C₂⁴	—	4	2	no	—
178	C₂⁴	D₃₇₆	1	4	yes	—
179	C₂⁴	Dic₁₈₈	1	4	yes	—
180	C₂⁴	C₄⋊Dic₄₇	1	4	yes	—
181	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
182	C₂⁴	C₂×D₁₈₈	1	2	yes	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	4	8	no	—
186	C₂²×C₄	—	4	8	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	4	no	—
189	C₂²×C₄	—	94	8	no	—
190	C₂²×C₄	—	188	8	no	—
191	C₂²×C₄	—	188	8	no	—
192	C₂²×C₄	—	188	8	no	—
193	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
194	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
195	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
196	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
197	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
200	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
201	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	—	4	8	no	—
204	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
205	C₄⋊C₄	—	94	16	no	—
206	C₄⋊C₄	—	188	8	no	—
207	C₂²⋊C₄	—	188	8	no	—
208	C₂²⋊C₄	—	188	8	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	4	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	8	no	—
215	C₂³	—	4	8	no	—
216	C₂³	—	4	4	no	—
217	C₂³	#13093958	1	16	yes	complemented
218	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
219	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
220	C₂³	C₂×D₃₇₆	1	2	yes	—
221	C₂³	C₂×D₃₇₆	1	4	yes	—
222	C₂³	C₂×Dic₁₈₈	1	2	yes	—
223	C₂³	C₂×Dic₁₈₈	1	4	yes	—
224	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	—	376	16	no	—
234	C₂×C₄	—	188	8	no	—
235	C₂×C₄	—	188	8	no	—
236	C₂×C₄	—	188	8	no	—
237	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
238	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
239	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
240	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
241	C₂×C₄	#13094069	1	4	yes	—
242	C₈	—	2	8	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	4	16	no	—
252	C₂²	C₂×#13093958	1	4	yes	—
253	C₂²	C₂×#13093958	1	8	yes	—
254	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
255	C₂²	#15689685	1	2	yes	—
256	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
257	C₂²	C₂²×Dic₁₈₈	1	1	yes	characteristic
258	C₂²	#15689923	1	2	yes	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	376	8	no	—
262	C₄	#15689598	1	2	yes	—
263	C₄	#15689749	1	2	yes	—
264	C₂	—	2	8	no	—
265	C₂	—	2	4	no	—
266	C₂	#20246002	1	1	yes	characteristic
267	C₂	C₂²×#13093958	1	1	yes	characteristic
268	C₂	#20251978	1	1	yes	characteristic
269	C₂	#20254139	1	2	yes	—
270	C₂	#20254163	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₂²⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#21940
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20243578	C₂	semidirect
3. #20243417	C₂	semidirect
7. C₂×#13093958	C₂²	semidirect
13. #15688886	C₂²	semidirect
16. C₂×C₂²⋊C₈×C₄₇	C₄	semidirect
37. C₂²×C₈×C₄₇	C₂×C₄	semidirect
39. C₂²⋊C₈×C₄₇	C₂×C₄	semidirect
81. C₂³×C₄₇	#107	semidirect
145. C₂×C₂²⋊C₈	Dic₄₇	semidirect
153. C₄₇	#21940	semidirect
164. C₂²×C₈	C₂×Dic₄₇	semidirect
173. C₂²⋊C₈	C₂×Dic₄₇	semidirect
217. C₂³	#13093958	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688983	2
#20243578	2
#20243417	2
C₂²×C₂²⋊C₈×C₄₇	2
#21940	47