Subgroups — #26133231

Order 12032.

530 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689364	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688926	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#20241479	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#427	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
7	C₂×#13094003	—	2	1	no	—
8	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	#15689364	—	2	2	no	—
10	#15689364	C₄	1	4	yes	complemented
11	C₂×#13093958	C₂²	1	1	yes	characteristic
12	C₂×#13093949	—	2	1	no	—
13	#15688926	—	2	4	no	—
14	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
15	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₄₇	—	2	1	no	—
17	C₄₇×#427	—	2	4	no	—
18	C₂×#107×C₄₇	C₂²	1	2	yes	—
19	C₂×#106×C₄₇	—	2	1	no	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂×Dic₁₈₈	—	2	2	no	—
22	C₂×Dic₁₈₈	—	2	2	no	—
23	C₂×Dic₁₈₈	—	2	1	no	—
24	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
25	#13094003	—	2	4	no	—
26	#13094003	—	4	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×Dic₄₇⋊C₄	—	4	1	no	—
29	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
30	C₂×C₄₇⋊C₁₆	—	2	2	no	—
31	C₂×C₄₇⋊C₁₆	—	2	2	no	—
32	C₂×C₄₇⋊C₁₆	—	2	2	no	—
33	#13093958	—	2	1	no	—
34	#13093958	C₂×C₄	1	2	yes	—
35	#13093949	—	2	4	no	—
36	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₇	—	2	2	no	—
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	D₄	1	2	yes	—
42	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
43	#117×C₄₇	—	2	4	no	—
44	#117×C₄₇	—	4	2	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
47	C₂×C₄²×C₄₇	—	4	1	no	—
48	#107×C₄₇	—	2	2	no	—
49	#107×C₄₇	D₄	1	4	yes	—
50	#106×C₄₇	—	2	4	no	—
51	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	1	no	—
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	—	2	1	no	—
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	4	1	no	—
62	C₂²×Dic₄₇	—	4	1	no	—
63	C₂×Dic₉₄	—	2	2	no	—
64	C₂×Dic₉₄	—	4	1	no	—
65	C₂×Dic₉₄	—	4	1	no	—
66	C₂×Dic₉₄	—	4	1	no	—
67	C₂×Dic₉₄	—	4	1	no	—
68	C₂×Dic₉₄	—	4	1	no	—
69	C₂×C₈×C₄₇	—	2	1	no	—
70	C₂×C₈×C₄₇	—	2	1	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	#41	1	2	yes	—
78	C₄⋊C₄×C₄₇	—	2	4	no	—
79	C₄⋊C₄×C₄₇	—	4	2	no	—
80	C₂²⋊C₄×C₄₇	—	4	4	no	—
81	C₄²×C₄₇	—	4	4	no	—
82	C₂²⋊Dic₄₇	—	4	4	no	—
83	C₄⋊Dic₄₇	—	2	2	no	—
84	C₄⋊Dic₄₇	—	4	1	no	—
85	Dic₄₇⋊C₄	—	4	4	no	—
86	Dic₁₈₈	—	2	4	no	—
87	Dic₁₈₈	—	4	2	no	—
88	C₄₇⋊C₁₆	—	4	2	no	—
89	C₂³×C₄₇	—	2	1	no	—
90	C₂³×C₄₇	—	2	1	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	#102	1	2	yes	—
94	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	8	2	no	—
103	C₂×C₄×C₄₇	—	8	2	no	—
104	C₂×C₄×C₄₇	#102	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
111	C₂×Dic₄₇	—	4	2	no	—
112	C₂×Dic₄₇	—	4	2	no	—
113	C₂×Dic₄₇	—	8	1	no	—
114	C₂×Dic₄₇	—	8	1	no	—
115	C₂×Dic₄₇	—	8	2	no	—
116	Dic₉₄	—	4	2	no	—
117	Dic₉₄	—	4	2	no	—
118	Dic₉₄	—	8	1	no	—
119	Dic₉₄	—	8	1	no	—
120	C₈×C₄₇	—	2	2	no	—
121	C₈×C₄₇	C₄×D₄	1	2	yes	—
122	C₈×C₄₇	#121	1	2	yes	—
123	#12994	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	4	2	no	—
128	C₂²×C₄₇	—	4	2	no	—
129	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
130	C₂²×C₄₇	#447	1	1	yes	characteristic
131	C₂²×C₄₇	#480	1	1	yes	characteristic
132	C₂²×C₄₇	#508	1	2	yes	—
133	C₂²×C₄₇	#510	1	2	yes	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	8	4	no	—
136	C₄×C₄₇	#386	1	1	yes	characteristic
137	C₄×C₄₇	#417	1	1	yes	characteristic
138	C₄×C₄₇	#448	1	1	yes	characteristic
139	C₄×C₄₇	#481	1	1	yes	characteristic
140	Dic₄₇	—	8	2	no	—
141	Dic₄₇	—	8	2	no	—
142	C₂×#467	—	47	1	no	—
143	C₂×#427	D₄₇	1	1	yes	characteristic
144	C₂×#358	—	47	2	no	—
145	C₂×C₂²⋊C₁₆	—	47	1	no	—
146	#1380	—	47	2	no	—
147	C₂×C₄₇	—	4	2	no	—
148	C₂×C₄₇	#1874	1	1	yes	characteristic
149	C₂×C₄₇	#2138	1	1	yes	characteristic
150	C₂×C₄₇	#2141	1	1	yes	characteristic
151	C₂×C₄₇	#2189	1	2	yes	—
152	C₂×C₄₇	#2237	1	2	yes	—
153	C₂²×Q₁₆	—	47	1	no	—
154	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
155	C₂×C₂²⋊Q₈	—	94	1	no	—
156	C₂×#117	—	2	1	no	—
157	C₂²×C₁₆	—	47	2	no	—
158	#467	—	47	4	no	—
159	#467	—	94	2	no	—
160	#427	—	2	4	no	—
161	C₂×#107	—	47	1	no	—
162	C₂×#107	C₂×D₄₇	1	2	yes	—
163	C₂×#106	—	2	1	no	—
164	C₂×#103	—	94	1	no	—
165	#358	—	94	4	no	—
166	C₂²⋊C₁₆	—	94	2	no	—
167	C₄₇	#12994	1	1	yes	characteristic, complemented, Sylow
168	C₂²×Q₈	—	94	1	no	—
169	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂×Q₁₆	—	47	2	no	—
171	C₂×Q₁₆	—	94	1	no	—
172	C₂×Q₁₆	—	94	2	no	—
173	C₂×Q₁₆	—	94	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	C₄×D₄₇	1	2	yes	—
177	C₂²×C₈	D₁₈₈	1	2	yes	—
178	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
180	C₂²⋊Q₈	—	94	4	no	—
181	C₂²⋊Q₈	—	188	2	no	—
182	#117	—	2	4	no	—
183	#117	—	4	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	94	1	no	—
186	C₂×C₄⋊C₄	—	188	1	no	—
187	C₂×C₂²⋊C₄	—	4	1	no	—
188	C₂×C₂²⋊C₄	—	188	1	no	—
189	C₂×C₄²	—	4	1	no	—
190	C₂×C₁₆	—	94	2	no	—
191	C₂×C₁₆	—	94	2	no	—
192	C₂×C₁₆	—	94	2	no	—
193	#107	—	2	2	no	—
194	#107	—	47	2	no	—
195	#107	—	94	1	no	—
196	#107	C₂²⋊D₄₇	1	4	yes	—
197	#106	—	2	4	no	—
198	#103	—	94	4	no	—
199	C₂⁴	#10625334	1	1	yes	characteristic
200	C₂×Q₈	—	94	2	no	—
201	C₂×Q₈	—	188	1	no	—
202	C₂×Q₈	—	188	1	no	—
203	C₂×Q₈	—	188	1	no	—
204	C₂×Q₈	—	188	1	no	—
205	C₂×Q₈	—	188	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	—	188	1	no	—
212	C₂²×C₄	—	188	1	no	—
213	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
214	C₂²×C₄	#10625334	1	1	yes	characteristic
215	C₂²×C₄	#10625336	1	2	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	Q₁₆	—	94	4	no	—
218	Q₁₆	—	188	2	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
224	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
225	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
226	C₂×C₈	#10625350	1	2	yes	—
227	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	94	2	no	—
231	C₄⋊C₄	—	188	4	no	—
232	C₄⋊C₄	—	188	1	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₂²⋊C₄	—	188	4	no	—
235	C₄²	—	4	4	no	—
236	C₁₆	—	188	2	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	1	no	—
241	C₂³	#13093944	1	2	yes	—
242	C₂³	C₂×#10625334	1	1	yes	characteristic
243	Q₈	—	376	1	no	—
244	Q₈	—	376	1	no	—
245	Q₈	—	188	2	no	—
246	Q₈	—	188	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	376	1	no	—
257	C₂×C₄	—	376	2	no	—
258	C₂×C₄	—	376	1	no	—
259	C₂×C₄	—	188	2	no	—
260	C₂×C₄	—	188	2	no	—
261	C₂×C₄	#13093944	1	2	yes	—
262	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
263	C₂×C₄	#13093993	1	1	yes	characteristic
264	C₂×C₄	#13094007	1	1	yes	characteristic
265	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
266	C₂×C₄	#13094050	1	1	yes	characteristic
267	C₂×C₄	#13094055	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	C₄×C₂²⋊D₄₇	1	2	yes	—
270	C₈	#13093998	1	2	yes	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₂×#13093944	1	1	yes	characteristic
277	C₂²	#15689257	1	1	yes	characteristic
278	C₂²	#15689436	1	1	yes	characteristic
279	C₂²	#15689553	1	2	yes	—
280	C₂²	#15689563	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	8	4	no	—
283	C₄	—	376	2	no	—
284	C₄	—	376	2	no	—
285	C₄	#15689002	1	1	yes	characteristic
286	C₄	#15689122	1	1	yes	characteristic
287	C₄	#15689262	1	1	yes	characteristic
288	C₄	#15689441	1	1	yes	characteristic
289	C₂	—	4	2	no	—
290	C₂	#20243705	1	1	yes	characteristic
291	C₂	#20245093	1	1	yes	characteristic
292	C₂	#20245107	1	1	yes	characteristic
293	C₂	#20245328	1	2	yes	—
294	C₂	#20245596	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#427	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#427	D₄₇
hypocenter	C₄₇	#12994
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. C₂×#15688926	C₂	semidirect
4. #20241479	C₂	semidirect
10. #15689364	C₄	semidirect
167. C₄₇	#12994	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689364	2
C₂×#15688926	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20241479	2
C₂×C₄₇×#427	2
#12994	47