Subgroups — #26216494

Order 12032.

1014 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
3	#20243521	C₂	1	4	yes	maximal, complemented
4	#20243515	C₂	1	4	yes	maximal, complemented
5	#20243109	C₂	1	4	yes	maximal
6	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	—	2	2	no	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₄×#10625329	C₄	1	8	yes	complemented
11	C₂×#13093935	C₂²	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
13	#15688993	—	2	2	no	—
14	#15688993	C₂²	1	2	yes	—
15	#15688993	C₂²	1	2	yes	—
16	#15688885	C₂²	1	16	yes	—
17	C₂×#117×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#384	C₂²	1	2	yes	—
20	C₄₇×#382	C₂²	1	2	yes	—
21	C₄₇×#380	—	2	2	no	—
22	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₂³×Dic₄₇	—	2	2	no	—
24	C₂×#10625329	—	2	4	no	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	C₂×C₄	1	8	yes	—
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	C₂×C₄⋊Dic₄₇	—	2	8	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	4	8	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	#13093935	C₂×C₄	1	8	yes	complemented
35	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
36	C₂³×C₄×C₄₇	D₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	#117×C₄₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₄₇	D₄	1	2	yes	—
46	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	#95×C₄₇	—	2	4	no	—
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	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	#41	1	4	yes	—
61	C₂²×C₄×C₄₇	#41	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	2	4	no	—
64	C₂²×Dic₄₇	—	4	8	no	—
65	C₂²×Dic₄₇	—	4	4	no	—
66	C₄⋊C₄×C₄₇	—	2	8	no	—
67	C₄⋊C₄×C₄₇	—	4	4	no	—
68	C₄⋊C₄×C₄₇	—	4	4	no	—
69	C₂²⋊C₄×C₄₇	—	4	4	no	—
70	C₄²×C₄₇	—	4	4	no	—
71	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
72	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
73	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
75	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
76	C₄⋊Dic₄₇	—	2	8	no	—
77	C₄⋊Dic₄₇	—	4	8	no	—
78	C₄⋊Dic₄₇	—	4	4	no	—
79	C₄⋊Dic₄₇	—	4	8	no	—
80	#10625329	—	2	16	no	—
81	C₂×C₄₇⋊C₈	—	2	4	no	—
82	C₂×C₄₇⋊C₈	—	2	4	no	—
83	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
84	C₂³×C₄₇	—	2	2	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³×C₄₇	#124	1	2	yes	—
88	C₂³×C₄₇	#124	1	2	yes	—
89	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	16	no	—
93	C₂×C₄×C₄₇	—	4	4	no	—
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₄₇	#117	1	8	yes	—
100	C₂×C₄×C₄₇	#124	1	2	yes	—
101	C₂×C₄×C₄₇	#124	1	2	yes	—
102	C₂×C₄×C₄₇	#125	1	8	yes	—
103	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
104	C₂×C₄×C₄₇	#137	1	4	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
109	C₂×Dic₄₇	—	4	4	no	—
110	C₂×Dic₄₇	—	4	8	no	—
111	C₂×Dic₄₇	—	8	8	no	—
112	C₄₇⋊C₈	—	2	8	no	—
113	#20764	—	47	1	no	maximal, Sylow
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	4	no	—
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	#382	1	4	yes	—
119	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
123	C₂²×C₄₇	C₂×#137	1	2	yes	—
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	—	8	4	no	—
126	C₄×C₄₇	#382	1	4	yes	—
127	C₄×C₄₇	#428	1	8	yes	—
128	C₄×C₄₇	C₂×#117	1	2	yes	—
129	C₄×C₄₇	C₂×#125	1	2	yes	—
130	Dic₄₇	—	8	4	no	—
131	C₂×C₄×M₁₆	—	47	1	no	—
132	#2285	D₄₇	1	1	yes	characteristic
133	C₂×#384	—	47	1	no	—
134	#1836	—	47	4	no	—
135	#1834	—	47	4	no	—
136	#1736	—	47	4	no	—
137	C₂×C₄₇	—	2	4	no	—
138	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
139	C₂×C₄₇	C₂×#428	1	2	yes	—
140	C₂×C₄₇	#3138	1	1	yes	characteristic
141	C₂×C₄₇	#3139	1	1	yes	characteristic
142	C₂×C₄₇	#3141	1	2	yes	—
143	C₂²×M₁₆	—	47	2	no	—
144	C₂×#117	C₂×D₄₇	1	2	yes	—
145	C₂²×C₄⋊C₄	—	47	2	no	—
146	C₂²×C₄⋊C₄	—	94	2	no	—
147	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
148	C₄×M₁₆	—	47	8	no	—
149	C₂×#97	—	47	2	no	—
150	C₂×C₄×C₈	—	47	2	no	—
151	#384	—	47	2	no	—
152	#384	—	47	2	no	—
153	#384	—	94	2	no	—
154	#384	C₂×D₄₇	1	2	yes	—
155	#382	C₂×D₄₇	1	2	yes	—
156	#380	—	2	2	no	—
157	C₄×C₄⋊C₄	—	2	2	no	—
158	#340	—	47	16	no	—
159	C₄₇	#20764	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₄	—	94	2	no	—
161	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
162	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
163	C₂×M₁₆	—	47	8	no	—
164	C₂×M₁₆	—	94	4	no	—
165	C₂×M₁₆	—	94	4	no	—
166	C₂²×C₈	—	47	4	no	—
167	#117	—	2	8	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	47	4	no	—
171	C₂×C₄⋊C₄	—	94	8	no	—
172	C₂×C₄⋊C₄	—	94	2	no	—
173	C₂×C₄⋊C₄	—	94	4	no	—
174	C₂×C₄⋊C₄	—	94	4	no	—
175	C₂×C₄⋊C₄	—	188	8	no	—
176	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
177	C₂×C₂²⋊C₄	—	2	2	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	Dic₉₄	1	4	yes	—
180	C₂×C₄²	C₄×D₄₇	1	4	yes	—
181	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
182	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
184	#97	—	47	8	no	—
185	C₄×C₈	—	47	8	no	—
186	#95	—	2	4	no	—
187	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	94	4	no	—
192	C₂²×C₄	—	188	8	no	—
193	C₂²×C₄	—	188	4	no	—
194	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
195	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
196	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
197	C₂²×C₄	#10625352	1	4	yes	—
198	C₂²×C₄	#10625354	1	4	yes	—
199	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
202	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
203	M₁₆	—	94	16	no	—
204	C₂×C₈	—	47	8	no	—
205	C₂×C₈	—	94	4	no	—
206	C₂×C₈	—	94	4	no	—
207	C₄⋊C₄	—	2	8	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	—	94	8	no	—
211	C₄⋊C₄	—	188	8	no	—
212	C₄⋊C₄	—	188	4	no	—
213	C₄⋊C₄	—	188	8	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	—	4	4	no	—
216	C₄²	Dic₄₇⋊C₄	1	8	yes	—
217	C₄²	C₂×Dic₉₄	1	2	yes	—
218	C₄²	C₂×Dic₉₄	1	2	yes	—
219	C₄²	C₂×C₄×D₄₇	1	2	yes	—
220	C₄²	C₂×C₄×D₄₇	1	2	yes	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
224	C₂³	#13094011	1	2	yes	—
225	C₂³	#13094013	1	2	yes	—
226	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	16	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	376	8	no	—
234	C₂×C₄	—	188	4	no	—
235	C₂×C₄	—	188	8	no	—
236	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
237	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
238	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
239	C₂×C₄	#13093982	1	8	yes	—
240	C₂×C₄	#13094011	1	2	yes	—
241	C₂×C₄	#13094013	1	2	yes	—
242	C₂×C₄	#13094015	1	8	yes	—
243	C₂×C₄	#13094049	1	4	yes	—
244	C₂×C₄	#13094054	1	4	yes	—
245	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
247	C₂×C₄	C₂×#10625352	1	2	yes	—
248	C₂×C₄	C₂×#10625354	1	2	yes	—
249	C₈	—	94	8	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	#15688988	1	4	yes	—
255	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
256	C₂²	C₂×#13094011	1	1	yes	characteristic
257	C₂²	C₂×#13094013	1	1	yes	characteristic
258	C₂²	C₂×#13094049	1	2	yes	—
259	C₂²	C₂×#13094054	1	2	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	8	4	no	—
262	C₄	—	376	4	no	—
263	C₄	#15688988	1	4	yes	—
264	C₄	#15689166	1	8	yes	—
265	C₄	C₂×#13093982	1	2	yes	—
266	C₄	C₂×#13094015	1	2	yes	—
267	C₂	—	2	4	no	—
268	C₂	C₂×#15688988	1	1	yes	characteristic
269	C₂	C₂×#15689166	1	2	yes	—
270	C₂	#20253764	1	1	yes	characteristic
271	C₂	#20253770	1	1	yes	characteristic
272	C₂	#20253781	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#20764
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #20243521	C₂	semidirect
4. #20243515	C₂	semidirect
10. C₄×#10625329	C₄	semidirect
34. #13093935	C₂×C₄	semidirect
35. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
159. C₄₇	#20764	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625329	2
C₂×#15688993	2
#20243521	2
#20243515	2
#20243109	2
C₄₇×#2285	2
#20764	47