Subgroups — #26369371

Order 12032.

598 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245195	C₂	1	4	yes	maximal, complemented
2	#20245187	C₂	1	4	yes	maximal, complemented
3	#20245147	C₂	1	2	yes	maximal, complemented
4	#20245002	C₂	1	2	yes	maximal
5	C₄×#13093964	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093964	C₂²	1	2	yes	—
9	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	8	yes	—
11	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	8	yes	—
12	#15688899	C₂²	1	2	yes	—
13	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
14	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
15	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
16	C₂×#131×C₄₇	C₄	1	1	yes	characteristic
17	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
18	C₄×#41×C₄₇	C₄	1	1	yes	characteristic
19	C₄₇×#405	C₄	1	2	yes	—
20	C₄₇×#405	C₄	1	1	yes	characteristic
21	C₄₇×#405	C₂²	1	1	yes	characteristic
22	C₄₇×#405	C₂²	1	2	yes	—
23	C₄×M₁₆×C₄₇	C₄	1	2	yes	—
24	C₄×M₁₆×C₄₇	C₄	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	#13093964	—	2	4	no	—
28	C₂×C₄₇⋊C₁₆	—	2	4	no	—
29	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
30	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
31	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
32	#131×C₄₇	—	2	4	no	—
33	#131×C₄₇	—	2	4	no	—
34	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
42	C₄×Q₈×C₄₇	C₈	1	2	yes	—
43	C₄×D₄×C₄₇	C₈	1	4	yes	—
44	C₄×D₄×C₄₇	C₈	1	2	yes	—
45	#117×C₄₇	C₂×C₄	1	2	yes	—
46	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	#97×C₄₇	C₂×C₄	1	2	yes	—
50	#97×C₄₇	C₂×C₄	1	2	yes	—
51	#97×C₄₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₄₇	C₂³	1	2	yes	—
58	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	#41×C₄₇	—	2	2	no	—
60	#41×C₄₇	—	2	2	no	—
61	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
62	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
63	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₄₇	—	2	2	no	—
69	M₁₆×C₄₇	—	2	2	no	—
70	M₁₆×C₄₇	—	2	4	no	—
71	M₁₆×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	#41	1	4	yes	—
81	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
83	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
84	C₂²⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
85	C₂²⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
86	C₄²×C₄₇	C₂×C₈	1	4	yes	—
87	C₄²×C₄₇	C₂×C₈	1	2	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
91	C₄₇⋊C₁₆	—	2	4	no	—
92	C₄₇⋊C₁₆	—	2	4	no	—
93	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
94	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
95	Q₈×C₄₇	—	2	2	no	—
96	D₄×C₄₇	—	2	4	no	—
97	D₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	#117	1	2	yes	—
103	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
105	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₄₇	—	2	2	no	—
111	C₈×C₄₇	—	2	2	no	—
112	C₈×C₄₇	#117	1	2	yes	—
113	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
114	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
115	C₈×C₄₇	C₂×#41	1	2	yes	—
116	#33420	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	#431	1	2	yes	—
121	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	—	2	4	no	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	#431	1	2	yes	—
126	C₄×C₄₇	#504	1	4	yes	—
127	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
128	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
129	C₄×#131	D₄₇	1	1	yes	characteristic
130	#2164	—	47	4	no	—
131	#2162	—	47	4	no	—
132	#2151	—	47	2	no	—
133	#2113	—	47	2	no	—
134	C₄×M₃₂	—	47	1	no	—
135	C₂×C₄×C₁₆	—	47	1	no	—
136	C₂×C₄₇	—	2	2	no	—
137	C₂×C₄₇	—	4	2	no	—
138	C₂×C₄₇	C₂×#431	1	1	yes	characteristic
139	C₂×C₄₇	C₂×#504	1	2	yes	—
140	C₂×#131	Dic₄₇	1	1	yes	characteristic
141	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
142	C₄×#41	Dic₄₇	1	1	yes	characteristic
143	C₂×M₃₂	—	47	2	no	—
144	C₂²×C₁₆	—	47	2	no	—
145	#405	Dic₄₇	1	2	yes	—
146	#405	Dic₄₇	1	1	yes	characteristic
147	#405	C₂×D₄₇	1	1	yes	characteristic
148	#405	C₂×D₄₇	1	2	yes	—
149	C₄×M₁₆	Dic₄₇	1	1	yes	characteristic
150	C₄×M₁₆	Dic₄₇	1	2	yes	—
151	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
153	C₄⋊C₁₆	—	47	8	no	—
154	C₂²⋊C₁₆	—	47	8	no	—
155	#346	—	47	2	no	—
156	C₄×C₁₆	—	47	2	no	—
157	C₄×C₁₆	—	47	2	no	—
158	C₄×C₁₆	—	47	2	no	—
159	C₄₇	#33420	1	1	yes	characteristic, complemented, Sylow
160	C₂×#41	C₂×Dic₄₇	1	1	yes	characteristic
161	#131	—	2	4	no	—
162	#131	—	2	4	no	—
163	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
164	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
165	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
166	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
167	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
168	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
170	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
171	C₄×Q₈	C₄₇⋊C₈	1	2	yes	—
172	C₄×D₄	C₄₇⋊C₈	1	4	yes	—
173	C₄×D₄	C₄₇⋊C₈	1	2	yes	—
174	#117	C₂×Dic₄₇	1	2	yes	—
175	#117	C₂×Dic₄₇	1	1	yes	characteristic
176	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
177	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
178	M₃₂	—	94	4	no	—
179	C₂×C₁₆	—	47	4	no	—
180	C₂×C₁₆	—	47	4	no	—
181	C₂×C₁₆	—	94	4	no	—
182	#97	C₂×Dic₄₇	1	2	yes	—
183	#97	C₂×Dic₄₇	1	2	yes	—
184	#97	C₂×Dic₄₇	1	2	yes	—
185	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
186	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
187	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
188	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₄₇	1	2	yes	—
191	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	C₂×Q₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
195	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
196	C₂×D₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
198	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	#10625329	1	4	yes	—
208	C₂×C₈	#10625350	1	4	yes	—
209	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
210	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
211	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
212	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
213	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
214	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
215	C₄⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	—
216	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
217	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	—
218	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
219	C₄²	C₂×C₄₇⋊C₈	1	4	yes	—
220	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
221	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
222	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
223	C₄²	C₂²×Dic₄₇	1	2	yes	—
224	C₁₆	—	94	4	no	—
225	C₁₆	—	94	4	no	—
226	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
227	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	#13093983	1	2	yes	—
236	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
237	C₂×C₄	C₂²×C₄₇⋊C₈	1	4	yes	—
238	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
239	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
240	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
241	C₂×C₄	C₂×#10625329	1	2	yes	—
242	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	#13093983	1	2	yes	—
246	C₈	C₂×#10625329	1	2	yes	—
247	C₈	C₂×#10625329	1	2	yes	—
248	C₈	C₂×#10625350	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	#15689176	1	2	yes	—
253	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	#15689176	1	2	yes	—
258	C₄	#15689540	1	4	yes	—
259	C₄	C₂×#13093983	1	1	yes	characteristic
260	C₄	C₂²×#10625329	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	C₂×#15689176	1	1	yes	characteristic
264	C₂	C₂×#15689540	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄×#131×C₄₇	C₂
prime radical	C₂×#41×C₄₇	C₂×C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₄×#131	D₄₇
hypocenter	C₄₇	#33420
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #20245195	C₂	semidirect
2. #20245187	C₂	semidirect
3. #20245147	C₂	semidirect
5. C₄×#13093964	C₂	semidirect
6. C₂×C₄×C₄₇⋊C₁₆	C₂	semidirect
159. C₄₇	#33420	semidirect

Maximal subgroups

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

subgroup	index
#20245195	2
#20245187	2
#20245147	2
#20245002	2
C₄×#13093964	2
C₂×C₄×C₄₇⋊C₁₆	2
C₄×#131×C₄₇	2
#33420	47