Subgroups — #26215068

Order 12032.

1090 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20243253	C₂	1	4	yes	maximal, complemented
4	C₄⋊C₈×Dic₄₇	C₂	1	4	yes	maximal, complemented
5	C₈×C₂²⋊Dic₄₇	C₂	1	4	yes	maximal
6	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093983	C₄	1	2	yes	—
8	#15689176	C₄	1	8	yes	complemented
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₂×C₈×Dic₄₇	C₂²	1	8	yes	—
13	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
16	C₄×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
17	#15688969	C₄	1	2	yes	—
18	C₄²×Dic₄₇	C₂²	1	2	yes	—
19	#15688877	C₂²	1	8	yes	—
20	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₄₇	C₄	1	8	yes	complemented
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
25	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
26	#13093983	C₈	1	16	yes	complemented
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
32	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
33	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
34	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
35	C₈×Dic₄₇	—	4	8	no	—
36	C₄₇⋊#95	C₂×C₄	1	4	yes	—
37	C₄₇⋊#95	C₂×C₄	1	4	yes	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	—	4	4	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
43	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
51	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
52	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
57	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂×C₈	1	8	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
68	C₂×C₈×C₄₇	—	2	8	no	—
69	C₂×C₈×C₄₇	—	4	8	no	—
70	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
71	C₄²×C₄₇	C₄²	1	4	yes	—
72	C₄²×C₄₇	C₄²	1	4	yes	—
73	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²⋊Dic₄₇	C₂×C₈	1	16	yes	—
78	C₄⋊Dic₄₇	C₂×C₈	1	16	yes	complemented
79	C₄×Dic₄₇	—	2	4	no	—
80	C₄×Dic₄₇	—	2	4	no	—
81	C₄×Dic₄₇	—	4	4	no	—
82	C₄×Dic₄₇	C₂×C₈	1	16	yes	complemented
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	—	2	2	no	—
88	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
89	C₂³×C₄₇	C₂²×C₈	1	4	yes	—
90	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	4	4	no	—
98	C₂×C₄×C₄₇	C₄×C₈	1	16	yes	complemented
99	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
105	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
109	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
110	C₂×Dic₄₇	—	2	8	no	—
111	C₂×Dic₄₇	—	2	8	no	—
112	C₂×Dic₄₇	C₂²×C₈	1	16	yes	—
113	C₈×C₄₇	—	2	8	no	—
114	C₄₇⋊C₈	—	2	8	no	—
115	#20543	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	8	no	—
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	C₂×C₄×C₈	1	4	yes	—
120	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
121	C₂²×C₄₇	#520	1	2	yes	—
122	C₂²×C₄₇	C₂³×C₈	1	2	yes	—
123	C₂²×C₄₇	#568	1	2	yes	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	2	2	no	—
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	C₂×C₄×C₈	1	4	yes	—
128	C₄×C₄₇	C₂×C₄×C₈	1	4	yes	—
129	C₄×C₄₇	#518	1	1	yes	characteristic
130	C₄×C₄₇	#519	1	1	yes	characteristic
131	C₄×C₄₇	#520	1	2	yes	—
132	Dic₄₇	—	2	16	no	—
133	C₂×#431	—	47	1	no	—
134	C₂²×C₄⋊C₈	D₄₇	1	1	yes	characteristic
135	C₄×#117	—	47	1	no	—
136	C₈×C₄⋊C₄	—	47	4	no	—
137	C₄×C₄⋊C₈	—	47	4	no	—
138	C₈×C₂²⋊C₄	—	47	4	no	—
139	C₂×C₄₇	—	2	4	no	—
140	C₂×C₄₇	#2274	1	1	yes	characteristic
141	C₂×C₄₇	C₂²×C₄×C₈	1	1	yes	characteristic
142	C₂×C₄₇	#2873	1	1	yes	characteristic
143	C₂×C₄₇	#2988	1	4	yes	—
144	C₂³×C₈	C₂×D₄₇	1	2	yes	—
145	C₂×#117	—	47	2	no	—
146	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
147	#431	—	47	8	no	—
148	C₂×C₄⋊C₈	—	47	2	no	—
149	C₂×C₄⋊C₈	Dic₄₇	1	8	yes	complemented
150	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
151	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
152	C₂×C₂²⋊C₈	—	47	2	no	—
153	C₂×C₄×C₈	—	47	2	no	—
154	C₂×C₄×C₈	—	47	8	no	—
155	C₄×C₄⋊C₄	—	47	2	no	—
156	C₄×C₄⋊C₄	—	47	2	no	—
157	C₄×C₂²⋊C₄	—	47	2	no	—
158	C₄×C₂²⋊C₄	—	47	2	no	—
159	#376	—	47	2	no	—
160	C₄³	—	47	2	no	—
161	#336	—	47	8	no	—
162	C₄₇	#20543	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
164	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
165	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
166	C₂²×C₈	—	4	4	no	—
167	C₂²×C₈	—	47	4	no	—
168	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
169	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
170	#117	—	47	16	no	—
171	C₂×C₄⋊C₄	—	47	4	no	—
172	C₂×C₂²⋊C₄	—	47	4	no	—
173	C₂×C₄²	—	47	4	no	—
174	C₂×C₄²	—	47	4	no	—
175	C₂×C₄²	—	47	4	no	—
176	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₄₇	1	4	yes	—
179	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
180	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
182	C₄⋊C₈	—	47	8	no	—
183	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
184	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
185	C₂²⋊C₈	—	47	8	no	—
186	C₄×C₈	—	47	8	no	—
187	C₄×C₈	—	188	8	no	—
188	#95	—	47	4	no	—
189	#95	—	47	4	no	—
190	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	47	8	no	—
195	C₂²×C₄	C₈×D₄₇	1	8	yes	—
196	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
197	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
198	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
204	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
205	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	—	4	8	no	—
208	C₂×C₈	—	47	8	no	—
209	C₂×C₈	—	94	4	no	—
210	C₂×C₈	—	94	4	no	—
211	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
212	C₄⋊C₄	—	47	16	no	—
213	C₂²⋊C₄	—	47	16	no	—
214	C₄²	—	47	16	no	—
215	C₄²	—	94	4	no	—
216	C₄²	—	94	4	no	—
217	C₄²	—	188	4	no	—
218	C₄²	C₄×Dic₄₇	1	4	yes	—
219	C₄²	C₄×Dic₄₇	1	4	yes	—
220	C₄²	C₂×C₄×D₄₇	1	2	yes	—
221	C₄²	C₂×C₄×D₄₇	1	2	yes	—
222	C₄²	C₂²×Dic₄₇	1	2	yes	—
223	C₄²	C₂²×Dic₄₇	1	2	yes	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
227	C₂³	C₂×C₈×D₄₇	1	4	yes	—
228	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	47	16	no	—
237	C₂×C₄	—	94	8	no	—
238	C₂×C₄	—	94	8	no	—
239	C₂×C₄	C₈×Dic₄₇	1	16	yes	complemented
240	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
241	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
242	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
244	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
245	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
246	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
247	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
248	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
249	C₂×C₄	#13094069	1	2	yes	—
250	C₂×C₄	#13094072	1	2	yes	—
251	C₈	—	2	8	no	—
252	C₈	—	94	8	no	—
253	C₂²	—	2	8	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	C₂×C₈×Dic₄₇	1	4	yes	—
257	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
258	C₂²	#15689604	1	2	yes	—
259	C₂²	C₂²×C₈×D₄₇	1	2	yes	—
260	C₂²	#15689889	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	94	16	no	—
265	C₄	C₂×C₈×Dic₄₇	1	4	yes	—
266	C₄	C₂×C₈×Dic₄₇	1	4	yes	—
267	C₄	#15689598	1	1	yes	characteristic
268	C₄	#15689601	1	1	yes	characteristic
269	C₄	#15689604	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	#20245745	1	1	yes	characteristic
272	C₂	C₂²×C₈×Dic₄₇	1	1	yes	characteristic
273	C₂	#20251806	1	1	yes	characteristic
274	C₂	#20252497	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂²×C₄×C₈
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#20543
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #20243253	C₂	semidirect
4. C₄⋊C₈×Dic₄₇	C₂	semidirect
8. #15689176	C₄	semidirect
22. C₂×C₄⋊C₈×C₄₇	C₄	semidirect
26. #13093983	C₈	semidirect
78. C₄⋊Dic₄₇	C₂×C₈	semidirect
82. C₄×Dic₄₇	C₂×C₈	semidirect
98. C₂×C₄×C₄₇	C₄×C₈	semidirect
149. C₂×C₄⋊C₈	Dic₄₇	semidirect
162. C₄₇	#20543	semidirect
239. C₂×C₄	C₈×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
C₄×#13093983	2
#20243253	2
C₄⋊C₈×Dic₄₇	2
C₈×C₂²⋊Dic₄₇	2
C₂²×C₄⋊C₈×C₄₇	2
#20543	47