Subgroups — #26369417

Order 12032.

524 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245204	C₂	1	4	yes	maximal, complemented
2	#20245200	C₂	1	4	yes	maximal, complemented
3	#20245154	C₂	1	2	yes	maximal, complemented
4	C₂×#15688905	C₂	1	2	yes	maximal
5	#20244993	C₂	1	2	yes	maximal, complemented
6	C₄₇×#2873	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093964	C₂²	1	4	yes	—
8	#15688944	C₂²	1	8	yes	complemented
9	#15688905	C₂²	1	4	yes	—
10	#15688905	C₂²	1	4	yes	—
11	#15688901	C₂²	1	8	yes	complemented
12	C₂²×M₁₆×C₄₇	C₄	1	1	yes	characteristic
13	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₄₇	C₄	1	1	yes	characteristic
15	C₄₇×#405	C₄	1	2	yes	—
16	C₄₇×#405	C₄	1	1	yes	characteristic
17	C₄₇×#405	C₄	1	1	yes	characteristic
18	C₄₇×#405	C₂²	1	2	yes	—
19	C₄₇×#405	C₂²	1	1	yes	characteristic
20	C₄₇×#405	C₂²	1	1	yes	characteristic
21	C₄×M₁₆×C₄₇	C₄	1	2	yes	—
22	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
23	#13093964	—	2	4	no	—
24	#13093964	C₂³	1	8	yes	—
25	C₂×C₄₇⋊C₁₆	—	2	4	no	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₄₇	—	2	4	no	—
28	C₂×M₁₆×C₄₇	—	2	4	no	—
29	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
39	#117×C₄₇	—	2	1	no	—
40	#117×C₄₇	—	2	1	no	—
41	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
42	#117×C₄₇	C₂×C₄	1	2	yes	—
43	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	#97×C₄₇	C₂×C₄	1	2	yes	—
48	#97×C₄₇	C₂×C₄	1	2	yes	—
49	#97×C₄₇	C₂×C₄	1	2	yes	—
50	#97×C₄₇	C₂³	1	2	yes	—
51	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₄₇	C₂³	1	2	yes	—
55	C₂⁴×C₄₇	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	M₁₆×C₄₇	—	2	2	no	—
66	M₁₆×C₄₇	—	2	2	no	—
67	M₁₆×C₄₇	—	2	2	no	—
68	M₁₆×C₄₇	—	2	4	no	—
69	M₁₆×C₄₇	—	2	2	no	—
70	M₁₆×C₄₇	—	2	2	no	—
71	M₁₆×C₄₇	M₁₆	1	4	yes	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	#41	1	4	yes	—
80	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₄₇	—	2	2	no	—
82	C₄⋊C₄×C₄₇	—	2	1	no	—
83	C₄⋊C₄×C₄₇	—	2	1	no	—
84	C₂²⋊C₄×C₄₇	—	2	2	no	—
85	C₂²⋊C₄×C₄₇	—	2	1	no	—
86	C₂²⋊C₄×C₄₇	—	2	1	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄₇⋊C₁₆	—	2	8	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	8	1	no	—
93	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
94	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
95	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	8	2	no	—
106	C₂×C₄×C₄₇	#117	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₈×C₄₇	—	2	1	no	—
113	C₈×C₄₇	—	2	1	no	—
114	C₈×C₄₇	—	2	2	no	—
115	C₈×C₄₇	#117	1	2	yes	—
116	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
117	C₈×C₄₇	C₂×#41	1	2	yes	—
118	#33427	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	1	no	—
120	C₂²×C₄₇	—	2	1	no	—
121	C₂²×C₄₇	—	4	1	no	—
122	C₂²×C₄₇	—	4	1	no	—
123	C₂²×C₄₇	—	4	1	no	—
124	C₂²×C₄₇	—	8	2	no	—
125	C₂²×C₄₇	#432	1	2	yes	—
126	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
127	C₄×C₄₇	—	2	1	no	—
128	C₄×C₄₇	—	2	1	no	—
129	C₄×C₄₇	—	4	2	no	—
130	C₄×C₄₇	—	4	2	no	—
131	C₄×C₄₇	#432	1	2	yes	—
132	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
133	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
134	#2873	D₄₇	1	1	yes	characteristic
135	#2166	—	47	4	no	—
136	#2165	—	47	4	no	—
137	#2153	—	47	2	no	—
138	C₂×#349	—	47	2	no	—
139	#2110	—	47	2	no	—
140	C₂×C₄₇	—	2	1	no	—
141	C₂×C₄₇	—	2	2	no	—
142	C₂×C₄₇	—	4	1	no	—
143	C₂×C₄₇	—	4	1	no	—
144	C₂×C₄₇	C₂×#432	1	1	yes	characteristic
145	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
146	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
147	C₂×#117	Dic₄₇	1	1	yes	characteristic
148	C₂×M₃₂	—	47	4	no	—
149	#405	Dic₄₇	1	1	yes	characteristic
150	#405	Dic₄₇	1	2	yes	—
151	#405	Dic₄₇	1	1	yes	characteristic
152	#405	C₂×D₄₇	1	2	yes	—
153	#405	C₂×D₄₇	1	1	yes	characteristic
154	#405	C₂×D₄₇	1	1	yes	characteristic
155	C₄×M₁₆	Dic₄₇	1	2	yes	—
156	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
157	#364	—	47	8	no	—
158	#349	—	47	4	no	—
159	#349	—	47	4	no	—
160	#347	—	47	8	no	—
161	C₄₇	#33427	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
166	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
167	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
168	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
169	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
170	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
171	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
172	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
173	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
175	#117	—	2	1	no	—
176	#117	—	2	1	no	—
177	#117	C₂×Dic₄₇	1	2	yes	—
178	#117	C₂×Dic₄₇	1	1	yes	characteristic
179	#117	C₂×Dic₄₇	1	1	yes	characteristic
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₂²⋊C₄	—	2	1	no	—
182	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
183	M₃₂	—	47	8	no	—
184	M₃₂	—	94	4	no	—
185	C₂×C₁₆	—	94	4	no	—
186	#97	C₂×Dic₄₇	1	2	yes	—
187	#97	C₂×Dic₄₇	1	2	yes	—
188	#97	C₂×Dic₄₇	1	2	yes	—
189	#97	C₂²×D₄₇	1	2	yes	—
190	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
191	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
192	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
193	C₄×C₈	C₂²×D₄₇	1	2	yes	—
194	C₂⁴	#10625329	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	#10625329	1	1	yes	characteristic
199	C₂²×C₄	#10625329	1	1	yes	characteristic
200	C₂²×C₄	#10625329	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	#10625329	1	4	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	#10625350	1	4	yes	—
214	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
215	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
216	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
217	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
218	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
219	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	—	2	1	no	—
226	C₄²	—	2	2	no	—
227	C₄²	C₂²×Dic₄₇	1	2	yes	—
228	C₄²	C₂²×Dic₄₇	1	2	yes	—
229	C₁₆	—	94	8	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	8	1	no	—
232	C₂³	C₂×#10625329	1	1	yes	characteristic
233	C₂³	C₂×#10625329	1	1	yes	characteristic
234	C₂³	C₂×#10625329	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	#13093983	1	2	yes	—
246	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
247	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
248	C₂×C₄	C₂×#10625329	1	2	yes	—
249	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
250	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	2	no	—
254	C₈	#13093983	1	2	yes	—
255	C₈	C₂×#10625329	1	4	yes	—
256	C₈	C₂×#10625350	1	2	yes	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	#15689180	1	2	yes	—
264	C₂²	C₂²×#10625329	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	#15689180	1	2	yes	—
270	C₄	C₂×#13093983	1	1	yes	characteristic
271	C₄	C₂²×#10625329	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	2	no	—
274	C₂	—	4	1	no	—
275	C₂	—	4	1	no	—
276	C₂	C₂×#15689180	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13093983
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2873	C₂
socle	C₂×C₄₇	C₂×#432
prime radical	C₂⁴×C₄₇	M₁₆
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2873	D₄₇
hypocenter	C₄₇	#33427
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ #405 ◁ #2873

Complements

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

N	H ≅ G/N	type
1. #20245204	C₂	semidirect
2. #20245200	C₂	semidirect
3. #20245154	C₂	semidirect
5. #20244993	C₂	semidirect
8. #15688944	C₂²	semidirect
11. #15688901	C₂²	semidirect
161. C₄₇	#33427	semidirect

Maximal subgroups

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

subgroup	index
#20245204	2
#20245200	2
#20245154	2
C₂×#15688905	2
#20244993	2
C₄₇×#2873	2
#33427	47