Subgroups — #26125497

Order 12032.

310 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245191	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245187	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241330	C₂	1	2	yes	maximal, complemented
4	#20241314	C₂	1	2	yes	maximal
5	C₄₇×#2922	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
9	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
10	#15688899	—	2	1	no	—
11	C₄×C₄₇⋊C₁₆	—	2	1	no	—
12	C₄×#41×C₄₇	C₄	1	1	yes	characteristic
13	C₄₇×#433	—	2	1	no	—
14	C₄₇×#433	—	2	1	no	—
15	C₄₇×#431	—	2	1	no	—
16	C₄₇×#431	—	2	1	no	—
17	C₄₇×#424	C₄	1	1	yes	characteristic
18	C₄₇×#424	C₂²	1	1	yes	characteristic
19	C₄₇×#408	C₄	1	1	yes	characteristic
20	C₄₇×#408	C₂²	1	1	yes	characteristic
21	C₄×M₁₆×C₄₇	C₄	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	C₂×C₄₇⋊C₁₆	—	2	2	no	—
25	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₄×Q₈×C₄₇	C₈	1	1	yes	characteristic
31	C₄×Q₈×C₄₇	C₈	1	1	yes	characteristic
32	C₄×D₄×C₄₇	—	2	1	no	—
33	C₄×D₄×C₄₇	—	2	1	no	—
34	C₄×D₄×C₄₇	C₈	1	1	yes	characteristic
35	C₄×D₄×C₄₇	C₈	1	1	yes	characteristic
36	#117×C₄₇	—	2	1	no	—
37	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	—	2	1	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₄₇	—	2	1	no	—
41	C₄⋊C₈×C₄₇	—	2	1	no	—
42	C₄⋊C₈×C₄₇	—	2	1	no	—
43	C₄⋊C₈×C₄₇	—	2	1	no	—
44	C₂²⋊C₈×C₄₇	—	2	1	no	—
45	C₂²⋊C₈×C₄₇	—	2	1	no	—
46	C₂²⋊C₈×C₄₇	—	2	1	no	—
47	C₂²⋊C₈×C₄₇	—	2	1	no	—
48	#97×C₄₇	—	2	1	no	—
49	C₄×C₈×C₄₇	—	2	1	no	—
50	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
54	#41×C₄₇	—	4	1	no	—
55	#41×C₄₇	—	4	1	no	—
56	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
57	C₂×D₄×C₄₇	—	2	1	no	—
58	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	M₁₆×C₄₇	—	4	1	no	—
66	M₁₆×C₄₇	—	4	1	no	—
67	C₂×C₈×C₄₇	—	2	1	no	—
68	C₂×C₈×C₄₇	—	2	1	no	—
69	C₂×C₈×C₄₇	—	4	1	no	—
70	C₂×C₈×C₄₇	—	4	1	no	—
71	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₄₇	—	2	1	no	—
76	C₄⋊C₄×C₄₇	—	2	1	no	—
77	C₄⋊C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
78	C₄⋊C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
79	C₂²⋊C₄×C₄₇	—	2	1	no	—
80	C₂²⋊C₄×C₄₇	—	2	1	no	—
81	C₂²⋊C₄×C₄₇	—	2	1	no	—
82	C₄²×C₄₇	—	2	1	no	—
83	C₄²×C₄₇	—	2	1	no	—
84	C₄²×C₄₇	—	2	1	no	—
85	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
89	C₄₇⋊C₁₆	—	4	2	no	—
90	C₄₇⋊C₁₆	—	4	2	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
93	Q₈×C₄₇	—	4	1	no	—
94	D₄×C₄₇	—	4	1	no	—
95	D₄×C₄₇	—	4	2	no	—
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	1	no	—
102	C₂×C₄×C₄₇	—	4	1	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	—	4	1	no	—
106	C₂×C₄×C₄₇	—	4	1	no	—
107	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
113	C₈×C₄₇	—	2	1	no	—
114	C₈×C₄₇	—	2	1	no	—
115	C₈×C₄₇	—	2	1	no	—
116	C₈×C₄₇	—	2	1	no	—
117	C₈×C₄₇	—	4	1	no	—
118	C₈×C₄₇	—	4	1	no	—
119	#11464	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	—	2	1	no	—
121	C₂²×C₄₇	—	2	1	no	—
122	C₂²×C₄₇	—	2	1	no	—
123	C₂²×C₄₇	—	4	1	no	—
124	C₂²×C₄₇	—	4	1	no	—
125	C₂²×C₄₇	—	4	1	no	—
126	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₄×C₄₇	—	2	1	no	—
128	C₄×C₄₇	—	2	1	no	—
129	C₄×C₄₇	—	2	1	no	—
130	C₄×C₄₇	—	2	1	no	—
131	C₄×C₄₇	—	4	1	no	—
132	C₄×C₄₇	—	4	1	no	—
133	C₄×C₄₇	—	4	1	no	—
134	C₄×C₄₇	#350	1	2	yes	—
135	C₄×C₄₇	#410	1	1	yes	characteristic
136	C₄×C₄₇	#413	1	1	yes	characteristic
137	#2922	D₄₇	1	1	yes	characteristic
138	#2163	—	47	1	no	—
139	#2162	—	47	1	no	—
140	#1323	—	47	2	no	—
141	#1315	—	47	2	no	—
142	C₂×C₄₇	—	4	1	no	—
143	C₂×C₄₇	—	8	1	no	—
144	C₂×C₄₇	#1458	1	1	yes	characteristic
145	C₂×C₄₇	C₂×#350	1	1	yes	characteristic
146	C₂×C₄₇	#2117	1	1	yes	characteristic
147	C₄×#41	Dic₄₇	1	1	yes	characteristic
148	#433	—	2	1	no	—
149	#433	—	2	1	no	—
150	#431	—	2	1	no	—
151	#431	—	2	1	no	—
152	#424	Dic₄₇	1	1	yes	characteristic
153	#424	C₂×D₄₇	1	1	yes	characteristic
154	#408	Dic₄₇	1	1	yes	characteristic
155	#408	C₂×D₄₇	1	1	yes	characteristic
156	C₄×M₁₆	Dic₄₇	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
158	C₄⋊C₁₆	—	94	1	no	—
159	C₄⋊C₁₆	—	94	1	no	—
160	C₂²⋊C₁₆	—	47	2	no	—
161	C₂²⋊C₁₆	—	47	2	no	—
162	#346	—	94	1	no	—
163	C₄×C₁₆	—	94	1	no	—
164	C₄₇	#11464	1	1	yes	characteristic, complemented, Sylow
165	C₂×#41	C₂×Dic₄₇	1	1	yes	characteristic
166	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
167	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
168	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
170	C₄×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
171	C₄×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	—	2	1	no	—
174	C₄×D₄	C₄₇⋊C₈	1	1	yes	characteristic
175	C₄×D₄	C₄₇⋊C₈	1	1	yes	characteristic
176	#117	—	2	1	no	—
177	#117	C₂×Dic₄₇	1	1	yes	characteristic
178	C₂×C₄²	—	2	1	no	—
179	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
180	C₂×C₁₆	—	94	2	no	—
181	C₂×C₁₆	—	94	2	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	—	2	1	no	—
187	C₂²⋊C₈	—	2	1	no	—
188	C₂²⋊C₈	—	2	1	no	—
189	C₂²⋊C₈	—	2	1	no	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
193	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	C₂×Q₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	#10625329	1	1	yes	characteristic
205	C₂²×C₄	#10625329	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
215	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
220	C₄⋊C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
228	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
229	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
230	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
231	C₁₆	—	188	2	no	—
232	C₁₆	—	188	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	C₂×#10625329	1	1	yes	characteristic
235	Q₈	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
251	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	#15688907	1	2	yes	—
276	C₄	#15689097	1	1	yes	characteristic
277	C₄	#15689107	1	1	yes	characteristic
278	C₂	—	4	1	no	—
279	C₂	—	8	1	no	—
280	C₂	#20241702	1	1	yes	characteristic
281	C₂	C₂×#15688907	1	1	yes	characteristic
282	C₂	#20245016	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2922	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₄₇	C₂×C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2922	D₄₇
hypocenter	C₄₇	#11464
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2922

Complements

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

N	H ≅ G/N	type
1. #20245191	C₂	semidirect
2. #20245187	C₂	semidirect
3. #20241330	C₂	semidirect
164. C₄₇	#11464	semidirect

Maximal subgroups

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

subgroup	index
#20245191	2
#20245187	2
#20241330	2
#20241314	2
C₄₇×#2922	2
#11464	47