Subgroups — #26226565

Order 12032.

1090 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	#20245771	C₂	1	2	yes	maximal
2	#20243589	C₂	1	8	yes	maximal, complemented
3	#20243421	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#407	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	C₂²	1	4	yes	—
6	C₂×#13093958	C₂²	1	4	yes	complemented
7	C₂×#13093956	C₂²	1	4	yes	complemented
8	#15688986	—	2	2	no	—
9	#15688983	C₂²	1	4	yes	—
10	#15688983	C₂²	1	4	yes	—
11	#15688979	—	2	2	no	—
12	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
13	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
14	#15688886	C₂²	1	8	yes	complemented
15	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
17	C₄₇×#407	C₄	1	8	yes	complemented
18	C₂×C₂²⋊C₈×C₄₇	C₂²	1	4	yes	—
19	#13093983	—	4	4	no	—
20	#13093983	—	4	4	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	—	2	4	no	—
25	C₂×C₄×Dic₄₇	—	2	4	no	—
26	#13093958	—	4	4	no	—
27	#13093956	—	4	4	no	—
28	C₄₇⋊#95	—	2	4	no	—
29	C₄₇⋊#95	—	2	4	no	—
30	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	—	4	2	no	—
32	C₂³×C₄×C₄₇	—	4	2	no	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
34	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	D₄	1	2	yes	—
36	C₂³×C₄×C₄₇	Q₈	1	4	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
39	C₂×M₁₆×C₄₇	—	4	2	no	—
40	C₂×M₁₆×C₄₇	—	4	2	no	—
41	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
42	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
43	C₂²⋊C₈×C₄₇	C₂×C₄	1	16	yes	—
44	C₂⁴×C₄₇	—	4	2	no	—
45	C₂⁴×C₄₇	—	4	4	no	—
46	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
47	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
48	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	4	8	no	—
54	C₂²×C₄×C₄₇	—	4	4	no	—
55	C₂²×C₄×C₄₇	—	4	4	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	4	no	—
58	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	2	8	no	—
68	M₁₆×C₄₇	—	4	8	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
72	C₂²⋊Dic₄₇	—	4	8	no	—
73	C₂²⋊Dic₄₇	—	4	8	no	—
74	C₂²⋊Dic₄₇	—	4	8	no	—
75	C₄⋊Dic₄₇	—	2	8	no	—
76	C₄⋊Dic₄₇	—	2	8	no	—
77	C₄×Dic₄₇	—	4	8	no	—
78	C₂³×C₄₇	—	2	4	no	—
79	C₂³×C₄₇	—	2	2	no	—
80	C₂³×C₄₇	—	2	4	no	—
81	C₂³×C₄₇	—	2	4	no	—
82	C₂³×C₄₇	—	2	4	no	—
83	C₂³×C₄₇	—	4	4	no	—
84	C₂³×C₄₇	—	4	8	no	—
85	C₂³×C₄₇	—	4	16	no	—
86	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³×C₄₇	C₂×C₄⋊C₄	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	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	4	4	no	—
98	C₂×C₄×C₄₇	—	4	8	no	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
101	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
104	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
105	C₂×Dic₄₇	—	4	8	no	—
106	C₂×Dic₄₇	—	4	8	no	—
107	C₂×Dic₄₇	—	4	8	no	—
108	C₈×C₄₇	—	2	8	no	—
109	#21943	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	2	no	—
111	C₂²×C₄₇	—	2	4	no	—
112	C₂²×C₄₇	—	2	4	no	—
113	C₂²×C₄₇	—	2	4	no	—
114	C₂²×C₄₇	—	2	4	no	—
115	C₂²×C₄₇	—	4	16	no	—
116	C₂²×C₄₇	—	4	8	no	—
117	C₂²×C₄₇	—	4	4	no	—
118	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₄₇	#534	1	2	yes	—
120	C₂²×C₄₇	#575	1	4	yes	—
121	C₄×C₄₇	—	2	4	no	—
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	#518	1	2	yes	—
124	C₄×C₄₇	#543	1	1	yes	characteristic
125	C₄×C₄₇	#543	1	1	yes	characteristic
126	Dic₄₇	—	8	8	no	—
127	C₂×#407	D₄₇	1	1	yes	characteristic
128	#2280	—	47	2	no	—
129	#1854	—	47	8	no	—
130	#1809	—	47	4	no	—
131	C₂×C₄₇	—	4	2	no	—
132	C₂×C₄₇	—	4	4	no	—
133	C₂×C₄₇	#2315	1	1	yes	characteristic
134	C₂×C₄₇	#2911	1	2	yes	—
135	C₂×C₄₇	#3197	1	4	yes	—
136	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
137	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
138	C₂×#117	—	47	4	no	—
139	C₂×#107	—	47	4	no	—
140	C₂×#106	—	47	4	no	—
141	#407	Dic₄₇	1	8	yes	complemented
142	C₂×C₂²⋊C₈	C₂×D₄₇	1	4	yes	—
143	#381	—	94	2	no	—
144	#380	—	47	4	no	—
145	#380	—	47	4	no	—
146	#379	—	94	2	no	—
147	C₄×C₂²⋊C₄	—	94	2	no	—
148	C₄×C₂²⋊C₄	—	94	2	no	—
149	#340	—	47	8	no	—
150	C₄₇	#21943	1	1	yes	characteristic, complemented, Sylow
151	C₂⁵	D₁₈₈	1	1	yes	characteristic
152	C₂³×C₄	—	4	2	no	—
153	C₂³×C₄	—	4	2	no	—
154	C₂³×C₄	Dic₉₄	1	4	yes	—
155	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
156	C₂³×C₄	D₁₈₈	1	2	yes	—
157	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
158	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
159	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
160	C₂×M₁₆	—	4	2	no	—
161	C₂×M₁₆	—	4	2	no	—
162	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
163	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
164	#117	—	188	4	no	—
165	#117	—	188	4	no	—
166	C₂×C₄⋊C₄	—	47	8	no	—
167	C₂×C₂²⋊C₄	—	94	4	no	—
168	C₂×C₂²⋊C₄	—	94	4	no	—
169	C₂×C₂²⋊C₄	—	94	4	no	—
170	C₂×C₄²	—	94	4	no	—
171	#107	—	188	4	no	—
172	#106	—	188	4	no	—
173	C₂²⋊C₈	C₂×Dic₄₇	1	16	yes	—
174	#95	—	94	4	no	—
175	#95	—	94	4	no	—
176	C₂⁴	—	4	2	no	—
177	C₂⁴	—	4	4	no	—
178	C₂⁴	C₄⋊Dic₄₇	1	4	yes	—
179	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
180	C₂⁴	C₂×D₁₈₈	1	2	yes	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	4	8	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	—	94	8	no	—
191	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
192	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
193	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
194	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
195	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
197	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
198	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
199	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
200	M₁₆	—	4	8	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
204	C₄⋊C₄	—	94	8	no	—
205	C₄⋊C₄	—	94	8	no	—
206	C₂²⋊C₄	—	188	8	no	—
207	C₂²⋊C₄	—	188	8	no	—
208	C₂²⋊C₄	—	188	8	no	—
209	C₄²	—	188	8	no	—
210	C₂³	—	2	4	no	—
211	C₂³	—	2	4	no	—
212	C₂³	—	2	4	no	—
213	C₂³	—	2	4	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	4	no	—
216	C₂³	—	4	8	no	—
217	C₂³	—	4	16	no	—
218	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
219	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
220	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	188	8	no	—
232	C₂×C₄	—	188	8	no	—
233	C₂×C₄	—	188	8	no	—
234	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
235	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
236	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
237	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
238	C₂×C₄	#13094069	1	2	yes	—
239	C₂×C₄	#13094069	1	2	yes	—
240	C₈	—	2	8	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	16	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	4	8	no	—
249	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
250	C₂²	#15689685	1	2	yes	—
251	C₂²	#15689923	1	4	yes	—
252	C₄	—	2	4	no	—
253	C₄	—	2	2	no	—
254	C₄	—	376	8	no	—
255	C₄	#15689598	1	2	yes	—
256	C₄	#15689749	1	1	yes	characteristic
257	C₄	#15689749	1	1	yes	characteristic
258	C₂	—	4	2	no	—
259	C₂	—	4	4	no	—
260	C₂	#20246002	1	1	yes	characteristic
261	C₂	#20251978	1	2	yes	—
262	C₂	#20254230	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#407	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#407	D₄₇
hypocenter	C₄₇	#21943
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂×#407

Complements

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

N	H ≅ G/N	type
2. #20243589	C₂	semidirect
3. #20243421	C₂	semidirect
6. C₂×#13093958	C₂²	semidirect
7. C₂×#13093956	C₂²	semidirect
14. #15688886	C₂²	semidirect
17. C₄₇×#407	C₄	semidirect
141. #407	Dic₄₇	semidirect
150. C₄₇	#21943	semidirect

Maximal subgroups

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

subgroup	index
#20245771	2
#20243589	2
#20243421	2
C₂×C₄₇×#407	2
#21943	47