Subgroups — #26110024

Order 12032.

418 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243539	C₂	1	1	yes	maximal, characteristic
2	#20243504	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688875	C₂	1	1	yes	maximal, characteristic
6	C₂×#15688873	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₇×#334	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×Dic₄₇⋊C₈	—	2	2	no	—
11	C₂×#13093934	C₄	1	1	yes	characteristic
12	C₂×#13093934	C₄	1	1	yes	characteristic
13	C₂×#13093934	C₄	1	1	yes	characteristic
14	C₂×#13093934	C₄	1	1	yes	characteristic
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
17	#15688877	—	2	2	no	—
18	#15688876	—	2	2	no	—
19	#15688875	—	2	2	no	—
20	#15688873	—	2	2	no	—
21	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#334	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	—	2	1	no	—
26	C₂²×C₄₇⋊C₈	—	2	1	no	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
32	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
33	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
37	Dic₄₇⋊C₈	—	4	4	no	—
38	#13093934	—	2	2	no	—
39	#13093934	—	2	2	no	—
40	#13093934	—	2	2	no	—
41	#13093934	C₈	1	4	yes	complemented
42	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
43	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
44	C₂²×C₈×C₄₇	—	2	1	no	—
45	C₂²×C₈×C₄₇	—	2	1	no	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₇	—	2	1	no	—
50	C₄⋊C₈×C₄₇	—	2	1	no	—
51	C₄⋊C₈×C₄₇	—	2	1	no	—
52	C₄⋊C₈×C₄₇	—	2	1	no	—
53	C₄×C₈×C₄₇	D₄	1	2	yes	—
54	C₄×C₈×C₄₇	Q₈	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
59	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₄₇	—	2	1	no	—
61	C₂×C₈×C₄₇	—	2	1	no	—
62	C₂×C₈×C₄₇	—	4	1	no	—
63	C₂×C₈×C₄₇	—	4	1	no	—
64	C₂×C₈×C₄₇	—	4	2	no	—
65	C₂×C₈×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
70	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
73	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
74	C₄×Dic₄₇	—	2	1	no	—
75	C₄×Dic₄₇	—	2	1	no	—
76	C₄×Dic₄₇	—	2	2	no	—
77	C₄×Dic₄₇	—	2	2	no	—
78	C₄×Dic₄₇	—	4	2	no	—
79	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
80	C₄×Dic₄₇	M₁₆	1	2	yes	—
81	C₂×C₄₇⋊C₈	—	2	1	no	—
82	C₂×C₄₇⋊C₈	—	2	1	no	—
83	C₂×C₄₇⋊C₈	—	4	1	no	—
84	C₂×C₄₇⋊C₈	—	4	1	no	—
85	C₂×C₄₇⋊C₈	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
88	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
89	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
90	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
91	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄×C₄₇	#137	1	2	yes	—
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	2	2	no	—
108	C₂×Dic₄₇	—	2	2	no	—
109	C₂×Dic₄₇	—	2	2	no	—
110	C₂×Dic₄₇	C₄⋊C₈	1	4	yes	—
111	C₈×C₄₇	—	2	2	no	—
112	C₈×C₄₇	—	8	1	no	—
113	C₈×C₄₇	—	8	1	no	—
114	C₈×C₄₇	C₄⋊C₈	1	4	yes	complemented
115	C₄₇⋊C₈	—	2	2	no	—
116	C₄₇⋊C₈	—	8	1	no	—
117	C₄₇⋊C₈	—	8	1	no	—
118	C₄₇⋊C₈	C₄⋊C₈	1	4	yes	complemented
119	#8808	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	#384	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₄₇	#423	1	1	yes	characteristic
123	C₂²×C₄₇	#428	1	2	yes	—
124	C₂²×C₄₇	#430	1	2	yes	—
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	—	2	1	no	—
127	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₄×C₄₇	#423	1	1	yes	characteristic
130	C₄×C₄₇	#423	1	1	yes	characteristic
131	C₄×C₄₇	#496	1	1	yes	characteristic
132	C₄×C₄₇	#497	1	1	yes	characteristic
133	C₄×C₄₇	#501	1	2	yes	—
134	Dic₄₇	—	2	4	no	—
135	Dic₄₇	—	4	2	no	—
136	#1842	—	47	1	no	—
137	#1831	—	47	1	no	—
138	C₄×#97	—	47	1	no	—
139	C₂×#335	—	47	1	no	—
140	C₂×#335	—	47	1	no	—
141	C₂×#334	—	47	1	no	—
142	C₂×#334	D₄₇	1	1	yes	characteristic
143	C₂×C₄₇	#1565	1	2	yes	—
144	C₂×C₄₇	#1569	1	2	yes	—
145	C₂×C₄₇	#1842	1	1	yes	characteristic
146	C₂×C₄₇	#1847	1	1	yes	characteristic
147	C₂×C₄₇	#1849	1	1	yes	characteristic
148	C₂×C₄⋊C₈	—	47	1	no	—
149	C₂×C₄⋊C₈	—	47	1	no	—
150	C₂×C₄⋊C₈	—	94	2	no	—
151	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
152	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
153	C₂×#97	—	47	1	no	—
154	C₂×#97	—	47	1	no	—
155	C₂×#97	—	47	1	no	—
156	C₂×#97	—	47	1	no	—
157	C₂×C₄×C₈	—	47	1	no	—
158	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
159	C₄³	—	47	1	no	—
160	#336	—	94	2	no	—
161	#335	—	94	2	no	—
162	#335	—	94	2	no	—
163	#334	—	2	2	no	—
164	#334	—	94	2	no	—
165	C₄₇	#8808	1	1	yes	characteristic, complemented, Sylow
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	47	1	no	—
169	C₂²×C₈	—	47	1	no	—
170	C₂²×C₈	—	94	1	no	—
171	C₂²×C₈	—	94	1	no	—
172	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
173	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
174	C₂×C₄²	—	47	1	no	—
175	C₂×C₄²	—	47	1	no	—
176	C₂×C₄²	—	47	2	no	—
177	C₂×C₄²	—	47	2	no	—
178	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
179	C₄⋊C₈	—	2	1	no	—
180	C₄⋊C₈	—	2	1	no	—
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	94	1	no	—
184	C₄⋊C₈	—	94	1	no	—
185	C₄⋊C₈	—	94	1	no	—
186	C₄⋊C₈	—	94	1	no	—
187	C₄⋊C₈	—	188	4	no	—
188	#97	—	47	4	no	—
189	#97	—	94	2	no	—
190	#97	—	94	2	no	—
191	#97	—	94	2	no	—
192	C₄×C₈	—	47	2	no	—
193	C₄×C₈	—	47	2	no	—
194	C₄×C₈	Dic₉₄	1	2	yes	—
195	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
196	C₂²×C₄	—	47	2	no	—
197	C₂²×C₄	—	47	2	no	—
198	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
199	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	47	2	no	—
208	C₂×C₈	—	47	2	no	—
209	C₂×C₈	—	47	2	no	—
210	C₂×C₈	—	47	2	no	—
211	C₂×C₈	—	94	1	no	—
212	C₂×C₈	—	94	1	no	—
213	C₂×C₈	—	188	1	no	—
214	C₂×C₈	—	188	1	no	—
215	C₂×C₈	—	188	2	no	—
216	C₂×C₈	—	188	2	no	—
217	C₂×C₈	C₈×D₄₇	1	2	yes	—
218	C₂×C₈	#10625324	1	2	yes	—
219	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
220	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
221	C₄²	—	47	2	no	—
222	C₄²	—	47	2	no	—
223	C₄²	—	94	1	no	—
224	C₄²	—	94	1	no	—
225	C₄²	—	94	2	no	—
226	C₄²	—	94	2	no	—
227	C₄²	—	188	2	no	—
228	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
229	C₄²	D₄×D₄₇	1	1	yes	characteristic
230	C₄²	Q₈×D₄₇	1	1	yes	characteristic
231	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
232	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	47	4	no	—
235	C₂×C₄	—	94	2	no	—
236	C₂×C₄	—	94	2	no	—
237	C₂×C₄	—	94	2	no	—
238	C₂×C₄	—	94	2	no	—
239	C₂×C₄	—	94	2	no	—
240	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
241	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
242	C₂×C₄	#13094020	1	1	yes	characteristic
243	C₂×C₄	#13094022	1	1	yes	characteristic
244	C₂×C₄	#13094023	1	1	yes	characteristic
245	C₂×C₄	#13094023	1	1	yes	characteristic
246	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
247	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
248	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
250	C₂×C₄	#13094051	1	2	yes	—
251	C₂×C₄	#13094053	1	2	yes	—
252	C₈	—	2	2	no	—
253	C₈	—	8	1	no	—
254	C₈	—	8	1	no	—
255	C₈	—	47	4	no	—
256	C₈	—	94	2	no	—
257	C₈	—	376	1	no	—
258	C₈	—	376	1	no	—
259	C₈	Dic₄₇⋊C₈	1	4	yes	complemented
260	C₂²	#15688992	1	1	yes	characteristic
261	C₂²	C₄⋊C₈×D₄₇	1	1	yes	characteristic
262	C₂²	#15689145	1	1	yes	characteristic
263	C₂²	#15689162	1	2	yes	—
264	C₂²	#15689170	1	2	yes	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	94	4	no	—
268	C₄	—	188	2	no	—
269	C₄	C₄⋊C₈×D₄₇	1	1	yes	characteristic
270	C₄	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
271	C₄	#15689145	1	1	yes	characteristic
272	C₄	#15689146	1	1	yes	characteristic
273	C₄	#15689509	1	1	yes	characteristic
274	C₄	#15689514	1	1	yes	characteristic
275	C₄	#15689530	1	2	yes	—
276	C₂	#20242137	1	2	yes	—
277	C₂	#20242152	1	2	yes	—
278	C₂	#20243539	1	1	yes	characteristic
279	C₂	#20243553	1	1	yes	characteristic
280	C₂	#20243561	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#334	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#334	D₄₇
hypocenter	C₄₇	#8808
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
41. #13093934	C₈	semidirect
114. C₈×C₄₇	C₄⋊C₈	semidirect
118. C₄₇⋊C₈	C₄⋊C₈	semidirect
165. C₄₇	#8808	semidirect
259. C₈	Dic₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20243539	2
#20243504	2
C₄×#13093934	2
C₂×#15688876	2
C₂×#15688875	2
C₂×#15688873	2
C₂×C₄₇×#334	2
#8808	47