Subgroups — #26226539

Order 12032.

1266 conjugacy classes of subgroups.

The subgroup lattice has 260 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	#20243583	C₂	1	8	yes	maximal, complemented
3	#20243413	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×#13093956	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	#13093956	—	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	SD₁₆	1	8	yes	—
44	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₄₇	—	2	4	no	—
47	C₂²×C₄×C₄₇	—	2	8	no	—
48	C₂²×C₄×C₄₇	—	4	8	no	—
49	C₂²×C₄×C₄₇	—	4	8	no	—
50	C₂²×C₄×C₄₇	—	4	2	no	—
51	C₂²×C₄×C₄₇	—	4	4	no	—
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	2	8	no	—
62	C₂²×Dic₄₇	—	4	8	no	—
63	C₂²×Dic₄₇	—	4	8	no	—
64	C₂²×Dic₄₇	—	4	8	no	—
65	C₂×C₈×C₄₇	—	2	8	no	—
66	C₂×C₈×C₄₇	—	4	8	no	—
67	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
68	C₂²⋊Dic₄₇	—	4	8	no	—
69	C₂²⋊Dic₄₇	—	4	8	no	—
70	C₄⋊Dic₄₇	—	2	16	no	—
71	C₄⋊Dic₄₇	—	4	8	no	—
72	C₂³×C₄₇	—	2	4	no	—
73	C₂³×C₄₇	—	2	2	no	—
74	C₂³×C₄₇	—	2	4	no	—
75	C₂³×C₄₇	—	2	2	no	—
76	C₂³×C₄₇	—	4	4	no	—
77	C₂³×C₄₇	—	4	4	no	—
78	C₂³×C₄₇	—	4	16	no	—
79	C₂³×C₄₇	#106	1	16	yes	complemented
80	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
82	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
83	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
84	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
85	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	2	8	no	—
92	C₂×C₄×C₄₇	—	4	4	no	—
93	C₂×C₄×C₄₇	—	4	8	no	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
99	C₂×Dic₄₇	—	4	8	no	—
100	C₂×Dic₄₇	—	4	8	no	—
101	C₂×Dic₄₇	—	4	8	no	—
102	C₂×Dic₄₇	—	8	16	no	—
103	C₈×C₄₇	—	2	8	no	—
104	#21938	—	47	1	no	maximal, Sylow
105	C₂²×C₄₇	—	2	2	no	—
106	C₂²×C₄₇	—	2	4	no	—
107	C₂²×C₄₇	—	2	4	no	—
108	C₂²×C₄₇	—	2	2	no	—
109	C₂²×C₄₇	—	2	8	no	—
110	C₂²×C₄₇	—	2	8	no	—
111	C₂²×C₄₇	—	2	8	no	—
112	C₂²×C₄₇	—	4	16	no	—
113	C₂²×C₄₇	C₂×#106	1	4	yes	—
114	C₂²×C₄₇	C₂×#106	1	8	yes	—
115	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₄₇	#534	1	2	yes	—
117	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
118	C₂²×C₄₇	#575	1	2	yes	—
119	C₄×C₄₇	—	2	4	no	—
120	C₄×C₄₇	—	2	2	no	—
121	C₄×C₄₇	#518	1	2	yes	—
122	C₄×C₄₇	#543	1	2	yes	—
123	Dic₄₇	—	8	8	no	—
124	C₂²×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
125	C₂×#380	—	47	2	no	—
126	#1853	—	47	8	no	—
127	#1807	—	47	4	no	—
128	C₂×C₄₇	—	2	8	no	—
129	C₂×C₄₇	—	2	4	no	—
130	C₂×C₄₇	#2315	1	1	yes	characteristic
131	C₂×C₄₇	C₂²×#106	1	1	yes	characteristic
132	C₂×C₄₇	#2911	1	1	yes	characteristic
133	C₂×C₄₇	#3187	1	4	yes	—
134	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
135	C₂³×C₈	C₂×D₄₇	1	2	yes	—
136	C₂²×C₄⋊C₄	—	47	4	no	—
137	C₂²×C₂²⋊C₄	—	94	2	no	—
138	C₂×#106	—	47	8	no	—
139	C₂×C₂²⋊C₈	Dic₄₇	1	8	yes	complemented
140	C₂×C₂²⋊C₈	C₂×D₄₇	1	4	yes	—
141	#380	—	47	4	no	—
142	#380	—	47	4	no	—
143	#380	—	94	2	no	—
144	#380	—	94	2	no	—
145	C₂×#95	—	94	2	no	—
146	#340	—	47	8	no	—
147	C₄₇	#21938	1	1	yes	characteristic, complemented, Sylow
148	C₂⁵	D₁₈₈	1	1	yes	characteristic
149	C₂³×C₄	—	4	4	no	—
150	C₂³×C₄	—	94	4	no	—
151	C₂³×C₄	Dic₉₄	1	4	yes	—
152	C₂³×C₄	D₁₈₈	1	2	yes	—
153	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
154	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
155	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
156	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
157	C₂²×C₈	—	4	4	no	—
158	C₂²×C₈	C₂×Dic₄₇	1	8	yes	complemented
159	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
160	C₂×C₄⋊C₄	—	47	8	no	—
161	C₂×C₄⋊C₄	—	94	4	no	—
162	C₂×C₄⋊C₄	—	188	8	no	—
163	C₂×C₂²⋊C₄	—	94	4	no	—
164	C₂×C₂²⋊C₄	—	94	4	no	—
165	C₂×C₂²⋊C₄	—	188	8	no	—
166	#106	—	188	8	no	—
167	C₂²⋊C₈	C₂×Dic₄₇	1	16	yes	complemented
168	#95	—	94	4	no	—
169	#95	—	94	4	no	—
170	C₂⁴	—	4	2	no	—
171	C₂⁴	—	4	2	no	—
172	C₂⁴	#10625325	1	8	yes	—
173	C₂⁴	C₄⋊Dic₄₇	1	4	yes	—
174	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
175	C₂⁴	C₂×D₁₈₈	1	2	yes	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	4	8	no	—
179	C₂²×C₄	—	4	8	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	—	94	8	no	—
183	C₂²×C₄	—	188	8	no	—
184	C₂²×C₄	—	188	8	no	—
185	C₂²×C₄	—	188	8	no	—
186	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
187	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
188	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
189	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
190	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
193	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
194	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
195	C₂×C₈	—	2	8	no	—
196	C₂×C₈	—	4	8	no	—
197	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
198	C₄⋊C₄	—	94	16	no	—
199	C₄⋊C₄	—	188	8	no	—
200	C₂²⋊C₄	—	188	8	no	—
201	C₂²⋊C₄	—	188	8	no	—
202	C₂³	—	2	4	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	4	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	4	4	no	—
207	C₂³	—	4	16	no	—
208	C₂³	—	4	4	no	—
209	C₂³	#13093956	1	16	yes	complemented
210	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
211	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
212	C₂³	C₂×#10625325	1	4	yes	—
213	C₂³	C₂×#10625325	1	4	yes	—
214	C₂³	C₂×#10625325	1	4	yes	—
215	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	376	16	no	—
225	C₂×C₄	—	188	8	no	—
226	C₂×C₄	—	188	8	no	—
227	C₂×C₄	—	188	8	no	—
228	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
230	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
231	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
232	C₂×C₄	#13094069	1	4	yes	—
233	C₈	—	2	8	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	8	no	—
237	C₂²	—	2	8	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	4	16	no	—
242	C₂²	C₂×#13093956	1	4	yes	—
243	C₂²	C₂×#13093956	1	8	yes	—
244	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
245	C₂²	#15689685	1	2	yes	—
246	C₂²	C₂²×#10625325	1	2	yes	—
247	C₂²	#15689923	1	2	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	2	no	—
250	C₄	—	376	8	no	—
251	C₄	#15689598	1	2	yes	—
252	C₄	#15689749	1	2	yes	—
253	C₂	—	2	8	no	—
254	C₂	—	2	4	no	—
255	C₂	#20246002	1	1	yes	characteristic
256	C₂	C₂²×#13093956	1	1	yes	characteristic
257	C₂	#20251978	1	1	yes	characteristic
258	C₂	#20254146	1	4	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	#21938
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. #20243583	C₂	semidirect
3. #20243413	C₂	semidirect
7. C₂×#13093956	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
79. C₂³×C₄₇	#106	semidirect
139. C₂×C₂²⋊C₈	Dic₄₇	semidirect
147. C₄₇	#21938	semidirect
158. C₂²×C₈	C₂×Dic₄₇	semidirect
167. C₂²⋊C₈	C₂×Dic₄₇	semidirect
209. C₂³	#13093956	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
#20243583	2
#20243413	2
C₂²×C₂²⋊C₈×C₄₇	2
#21938	47