Subgroups — #26121456

Order 12032.

998 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688884	C₂	1	4	yes	maximal
4	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
7	C₄×C₄⋊Dic₄₇	C₄	1	4	yes	complemented
8	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
9	#15688969	—	2	2	no	—
10	#15688969	—	2	1	no	—
11	#15688969	C₄	1	4	yes	complemented
12	C₂×C₄₇⋊#95	—	2	2	no	—
13	C₄²×Dic₄₇	—	2	1	no	—
14	#15688884	C₄	1	16	yes	complemented
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₄₇	—	2	2	no	—
18	C₄×M₁₆×C₄₇	C₄	1	4	yes	complemented
19	C₂×#97×C₄₇	—	2	1	no	—
20	C₂×C₄×C₈×C₄₇	—	2	1	no	—
21	C₂³×Dic₄₇	—	2	2	no	—
22	#13093983	—	2	4	no	—
23	#13093983	C₂×C₄	1	8	yes	—
24	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
25	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₂×C₄×Dic₄₇	—	2	2	no	—
28	C₂×C₄×Dic₄₇	—	2	4	no	—
29	C₂×C₄×Dic₄₇	—	2	4	no	—
30	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
32	C₄₇⋊#95	—	2	8	no	—
33	C₄₇⋊#95	—	2	4	no	—
34	C₂³×C₄×C₄₇	D₄	1	2	yes	—
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₄₇	—	2	2	no	—
37	C₂×M₁₆×C₄₇	—	2	4	no	—
38	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
39	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
48	#97×C₄₇	—	2	4	no	—
49	C₄×C₈×C₄₇	—	2	4	no	—
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	4	4	no	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	4	yes	—
64	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	—	2	4	no	—
66	C₂²×Dic₄₇	—	2	8	no	—
67	C₂²×Dic₄₇	—	4	8	no	—
68	M₁₆×C₄₇	—	2	4	no	—
69	M₁₆×C₄₇	—	2	8	no	—
70	M₁₆×C₄₇	C₄²	1	8	yes	complemented
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	8	no	—
73	C₄²×C₄₇	C₄²	1	4	yes	—
74	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
77	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
79	C₂²⋊Dic₄₇	—	2	8	no	—
80	C₄⋊Dic₄₇	—	2	4	no	—
81	C₄⋊Dic₄₇	C₄²	1	8	yes	—
82	C₄×Dic₄₇	—	2	4	no	—
83	C₄×Dic₄₇	—	2	8	no	—
84	C₄×Dic₄₇	—	2	8	no	—
85	C₄×Dic₄₇	—	4	8	no	—
86	C₄×Dic₄₇	C₄²	1	8	yes	complemented
87	C₂³×C₄₇	—	2	4	no	—
88	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₄₇	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	#95	1	4	yes	—
97	C₂×C₄×C₄₇	#95	1	4	yes	—
98	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
99	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#117	1	2	yes	—
106	C₂×C₄×C₄₇	#117	1	2	yes	—
107	C₂×C₄×C₄₇	#117	1	2	yes	—
108	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
109	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
110	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
111	C₂×Dic₄₇	—	2	8	no	—
112	C₂×Dic₄₇	—	2	8	no	—
113	C₂×Dic₄₇	—	4	8	no	—
114	C₂×Dic₄₇	—	4	16	no	—
115	C₂×Dic₄₇	—	4	8	no	—
116	C₈×C₄₇	—	2	8	no	—
117	#10701	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	8	no	—
121	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
122	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
123	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
124	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
125	C₂²×C₄₇	#421	1	4	yes	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	C₄³	1	1	yes	characteristic
129	C₄×C₄₇	#376	1	1	yes	characteristic
130	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
131	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
132	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
133	Dic₄₇	—	4	8	no	—
134	Dic₄₇	—	4	8	no	—
135	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
136	C₄×#117	—	47	1	no	—
137	C₂×#376	—	47	1	no	—
138	C₂×#339	—	47	4	no	—
139	C₂×C₄₇	—	2	2	no	—
140	C₂×C₄₇	—	2	2	no	—
141	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
142	C₂×C₄₇	#1733	1	2	yes	—
143	C₂×C₄₇	#1759	1	4	yes	—
144	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
145	C₂×#117	—	47	2	no	—
146	C₂²×C₄²	—	47	2	no	—
147	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
148	C₄×M₁₆	—	2	2	no	—
149	C₄×M₁₆	Dic₄₇	1	4	yes	complemented
150	C₂×#97	—	2	1	no	—
151	C₂×C₄×C₈	—	2	1	no	—
152	C₄×C₄⋊C₄	—	47	4	no	—
153	C₄×C₂²⋊C₄	—	94	2	no	—
154	#376	—	47	4	no	—
155	#376	—	94	2	no	—
156	#376	—	94	1	no	—
157	C₂×#95	—	94	2	no	—
158	C₄³	—	94	1	no	—
159	#339	—	47	16	no	—
160	C₄₇	#10701	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	—	94	2	no	—
162	C₂³×C₄	D₁₈₈	1	2	yes	—
163	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	C₄×D₄₇	1	8	yes	—
167	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
168	C₂²×C₈	—	2	2	no	—
169	#117	—	47	8	no	—
170	#117	—	94	4	no	—
171	C₂×C₄⋊C₄	—	47	4	no	—
172	C₂×C₂²⋊C₄	—	94	2	no	—
173	C₂×C₄²	—	47	8	no	—
174	C₂×C₄²	—	47	4	no	—
175	C₂×C₄²	—	94	4	no	—
176	C₂×C₄²	—	94	2	no	—
177	C₂×C₄²	—	94	2	no	—
178	C₂×C₄²	—	94	4	no	—
179	C₂×C₄²	Dic₉₄	1	2	yes	—
180	C₂×C₄²	C₄×D₄₇	1	2	yes	—
181	C₂×C₄²	C₄×D₄₇	1	2	yes	—
182	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
183	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
184	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
186	#97	—	2	4	no	—
187	C₄×C₈	—	2	4	no	—
188	#95	—	94	8	no	—
189	#95	—	94	4	no	—
190	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	94	4	no	—
193	C₂²×C₄	—	94	4	no	—
194	C₂²×C₄	—	94	8	no	—
195	C₂²×C₄	—	188	8	no	—
196	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
197	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
198	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
199	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
200	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
204	C₂²×C₄	#10625350	1	4	yes	—
205	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	8	no	—
210	M₁₆	C₄×Dic₄₇	1	8	yes	complemented
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	8	no	—
213	C₄⋊C₄	—	47	8	no	—
214	C₄⋊C₄	—	94	4	no	—
215	C₂²⋊C₄	—	94	8	no	—
216	C₄²	—	47	8	no	—
217	C₄²	—	94	4	no	—
218	C₄²	—	94	8	no	—
219	C₄²	—	94	8	no	—
220	C₄²	—	188	8	no	—
221	C₄²	C₄×Dic₄₇	1	4	yes	—
222	C₄²	Dic₄₇⋊C₄	1	4	yes	—
223	C₄²	Dic₄₇⋊C₄	1	2	yes	—
224	C₄²	Dic₄₇⋊C₄	1	2	yes	—
225	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
226	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
229	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
230	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
231	C₂³	C₄×D₁₈₈	1	4	yes	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	94	8	no	—
237	C₂×C₄	—	94	8	no	—
238	C₂×C₄	—	188	16	no	—
239	C₂×C₄	—	188	8	no	—
240	C₂×C₄	—	188	8	no	—
241	C₂×C₄	C₄₇⋊#95	1	4	yes	—
242	C₂×C₄	C₄₇⋊#95	1	4	yes	—
243	C₂×C₄	C₄²×D₄₇	1	2	yes	—
244	C₂×C₄	C₄²×D₄₇	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
248	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
250	C₂×C₄	#13093981	1	2	yes	—
251	C₂×C₄	#13093981	1	2	yes	—
252	C₂×C₄	#13093983	1	2	yes	—
253	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
254	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
255	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
256	C₈	—	2	8	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
261	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
262	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
263	C₂²	C₄×C₄⋊Dic₄₇	1	2	yes	—
264	C₂²	#15689139	1	4	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	—	188	8	no	—
268	C₄	—	188	8	no	—
269	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
270	C₄	#15688969	1	1	yes	characteristic
271	C₄	C₄×C₂²⋊Dic₄₇	1	2	yes	—
272	C₄	C₄×Dic₄₇⋊C₄	1	2	yes	—
273	C₄	C₄×Dic₄₇⋊C₄	1	2	yes	—
274	C₂	—	2	2	no	—
275	C₂	—	2	2	no	—
276	C₂	C₄×C₄₇⋊#95	1	1	yes	characteristic
277	C₂	#20243099	1	2	yes	—
278	C₂	#20243201	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₄³
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#10701
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
7. C₄×C₄⋊Dic₄₇	C₄	semidirect
11. #15688969	C₄	semidirect
14. #15688884	C₄	semidirect
18. C₄×M₁₆×C₄₇	C₄	semidirect
70. M₁₆×C₄₇	C₄²	semidirect
86. C₄×Dic₄₇	C₄²	semidirect
149. C₄×M₁₆	Dic₄₇	semidirect
160. C₄₇	#10701	semidirect
210. M₁₆	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13093983	2
C₂×#15688969	2
C₂×#15688884	2
C₂×C₄×M₁₆×C₄₇	2
#10701	47