Subgroups — #26368874

Order 12032.

894 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689531	C₂	1	4	yes	maximal
2	#20251785	C₂	1	1	yes	maximal, characteristic
3	#20251749	C₂	1	1	yes	maximal, characteristic
4	#20243555	C₂	1	4	yes	maximal
5	#20243547	C₂	1	1	yes	maximal, characteristic
6	#20243545	C₂	1	1	yes	maximal, characteristic
7	#20243544	C₂	1	1	yes	maximal, characteristic
8	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1845	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094023	—	2	2	no	—
11	C₂×#13094023	C₂²	1	2	yes	—
12	C₂×#13094015	C₂²	1	2	yes	—
13	#15689531	—	2	16	no	—
14	C₂×#13093957	C₂²	1	4	yes	—
15	C₂×#13093956	C₂²	1	4	yes	—
16	C₂×#13093955	C₂²	1	4	yes	—
17	C₂×#13093934	C₂²	1	2	yes	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	#15689022	—	2	2	no	—
21	#15689019	—	2	2	no	—
22	#15688993	C₂²	1	1	yes	characteristic
23	#15688993	C₂²	1	1	yes	characteristic
24	#15688992	—	2	2	no	—
25	#15688992	C₂²	1	2	yes	—
26	#15688988	C₂²	1	2	yes	—
27	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
28	C₂×#107×C₄₇	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#384	C₂²	1	1	yes	characteristic
31	C₄₇×#384	C₂²	1	1	yes	characteristic
32	C₂²×Dic₉₄	—	2	2	no	—
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	#13094023	—	2	8	no	—
35	#13094023	—	4	8	no	—
36	#13094015	—	2	8	no	—
37	C₂×C₄⋊Dic₄₇	—	2	2	no	—
38	C₂×C₄⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄⋊Dic₄₇	—	2	2	no	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×Dic₄₇⋊C₄	—	2	2	no	—
43	C₂×Dic₄₇⋊C₄	—	2	4	no	—
44	C₂×Dic₄₇⋊C₄	—	2	2	no	—
45	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
46	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
47	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
49	#13093957	—	2	8	no	—
50	#13093956	—	2	8	no	—
51	#13093955	—	2	8	no	—
52	#13093934	D₄	1	8	yes	—
53	#13093934	Q₈	1	8	yes	complemented
54	C₄×C₄₇⋊C₈	Q₈	1	4	yes	complemented
55	C₄₇⋊#95	—	2	4	no	—
56	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	#107×C₄₇	—	2	8	no	—
63	C₄×C₈×C₄₇	Q₈	1	4	yes	complemented
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	2	no	—
70	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
73	C₂×Dic₉₄	—	4	8	no	—
74	C₂×Dic₉₄	—	4	4	no	—
75	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
78	C₄⋊C₄×C₄₇	—	2	4	no	—
79	C₄⋊C₄×C₄₇	—	2	4	no	—
80	C₄⋊C₄×C₄₇	—	4	4	no	—
81	C₄⋊C₄×C₄₇	—	4	4	no	—
82	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
83	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
84	C₄⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	4	4	no	—
87	C₄⋊Dic₄₇	—	4	4	no	—
88	C₄⋊Dic₄₇	—	4	4	no	—
89	Dic₄₇⋊C₄	—	4	8	no	—
90	Dic₄₇⋊C₄	—	4	4	no	—
91	Dic₄₇⋊C₄	—	4	4	no	—
92	C₄×Dic₄₇	—	2	4	no	—
93	C₄×Dic₄₇	—	2	4	no	—
94	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
95	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
96	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
97	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
98	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
99	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	4	2	no	—
102	C₂×C₄×C₄₇	—	4	4	no	—
103	C₂×C₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
105	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
106	C₂×C₄×C₄₇	#137	1	4	yes	—
107	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
110	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×Dic₄₇	—	2	2	no	—
112	C₂×Dic₄₇	—	2	2	no	—
113	C₂×Dic₄₇	—	2	2	no	—
114	C₂×Dic₄₇	—	2	2	no	—
115	C₂×Dic₄₇	—	4	2	no	—
116	C₂×Dic₄₇	—	4	2	no	—
117	C₂×Dic₄₇	—	4	4	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	#127	1	2	yes	—
120	C₂×Dic₄₇	#127	1	2	yes	—
121	C₂×Dic₄₇	C₄⋊Q₈	1	4	yes	—
122	Dic₉₄	—	8	8	no	—
123	C₈×C₄₇	C₄⋊Q₈	1	8	yes	complemented
124	C₄₇⋊C₈	C₄⋊Q₈	1	8	yes	complemented
125	#33316	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
130	C₂²×C₄₇	C₂×#137	1	2	yes	—
131	C₄×C₄₇	—	8	2	no	—
132	C₄×C₄₇	—	8	2	no	—
133	C₄×C₄₇	#501	1	8	yes	complemented
134	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
135	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
136	Dic₄₇	—	2	2	no	—
137	Dic₄₇	—	2	2	no	—
138	Dic₄₇	—	2	4	no	—
139	Dic₄₇	—	8	2	no	—
140	Dic₄₇	—	8	2	no	—
141	C₂×#501	—	47	4	no	—
142	#2868	—	47	1	no	—
143	#2859	—	47	1	no	—
144	#1847	—	47	4	no	—
145	#1845	—	47	1	no	—
146	#1845	D₄₇	1	1	yes	characteristic
147	#1844	—	47	1	no	—
148	#1844	—	47	1	no	—
149	C₄×#97	—	47	1	no	—
150	C₂×C₄₇	#2868	1	1	yes	characteristic
151	C₂×C₄₇	#3151	1	1	yes	characteristic
152	C₂×C₄₇	#3152	1	1	yes	characteristic
153	C₂×C₄₇	C₂×#501	1	2	yes	—
154	C₂×C₄₇	#3165	1	2	yes	—
155	C₂×C₄⋊Q₈	—	47	2	no	—
156	C₂×C₄⋊Q₈	—	94	2	no	—
157	C₂×#125	—	47	2	no	—
158	#501	—	94	16	no	—
159	C₂×#107	—	47	4	no	—
160	C₂×#107	C₂×D₄₇	1	4	yes	—
161	C₂×#106	—	47	4	no	—
162	C₂×#106	—	47	4	no	—
163	C₂×#97	—	47	2	no	—
164	C₂×#97	—	47	2	no	—
165	C₂×C₄×C₈	—	47	1	no	—
166	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
167	#389	—	94	2	no	—
168	#389	—	94	2	no	—
169	#384	—	47	2	no	—
170	#384	—	47	1	no	—
171	#384	—	47	1	no	—
172	#384	—	94	2	no	—
173	#384	C₂×D₄₇	1	1	yes	characteristic
174	#384	C₂×D₄₇	1	1	yes	characteristic
175	#382	—	47	2	no	—
176	C₄³	—	47	1	no	—
177	C₄₇	#33316	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₈	—	94	2	no	—
179	C₂²×C₈	—	47	2	no	—
180	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
181	C₄⋊Q₈	—	94	8	no	—
182	C₄⋊Q₈	—	188	8	no	—
183	#125	—	94	8	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	47	2	no	—
187	C₂×C₄⋊C₄	—	47	2	no	—
188	C₂×C₄⋊C₄	—	94	2	no	—
189	C₂×C₄⋊C₄	—	94	2	no	—
190	C₂×C₄⋊C₄	—	94	2	no	—
191	C₂×C₄⋊C₄	—	94	4	no	—
192	C₂×C₄⋊C₄	—	94	2	no	—
193	C₂×C₄⋊C₄	—	94	2	no	—
194	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
195	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
196	C₂×C₄²	—	47	1	no	—
197	C₂×C₄²	—	47	1	no	—
198	C₂×C₄²	—	47	2	no	—
199	C₂×C₄²	—	47	2	no	—
200	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
201	#107	—	2	8	no	—
202	#107	—	94	8	no	—
203	#106	—	94	8	no	—
204	#106	—	94	8	no	—
205	#97	—	47	8	no	—
206	#97	—	47	8	no	—
207	C₄×C₈	—	47	4	no	—
208	C₄×C₈	Dic₉₄	1	4	yes	complemented
209	#95	—	94	4	no	—
210	C₂×Q₈	—	188	4	no	—
211	C₂×Q₈	—	188	8	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	47	2	no	—
215	C₂²×C₄	—	47	1	no	—
216	C₂²×C₄	—	47	1	no	—
217	C₂²×C₄	—	94	2	no	—
218	C₂²×C₄	—	94	2	no	—
219	C₂²×C₄	D₄×D₄₇	1	2	yes	—
220	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
221	C₂×C₈	—	47	4	no	—
222	C₂×C₈	—	47	4	no	—
223	C₂×C₈	—	47	4	no	—
224	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
225	C₂×C₈	D₄×D₄₇	1	4	yes	—
226	C₂×C₈	Q₈×D₄₇	1	4	yes	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	94	4	no	—
232	C₄⋊C₄	—	94	4	no	—
233	C₄⋊C₄	—	188	4	no	—
234	C₄⋊C₄	—	188	4	no	—
235	C₄⋊C₄	—	188	4	no	—
236	C₄⋊C₄	—	188	8	no	—
237	C₄⋊C₄	—	188	4	no	—
238	C₄⋊C₄	—	188	4	no	—
239	C₄²	—	47	4	no	—
240	C₄²	—	47	4	no	—
241	C₄²	—	94	4	no	—
242	C₄²	—	94	4	no	—
243	C₄²	C₂×Dic₉₄	1	2	yes	—
244	C₄²	Q₈×D₄₇	1	2	yes	—
245	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
246	Q₈	—	376	8	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	47	2	no	—
252	C₂×C₄	—	47	2	no	—
253	C₂×C₄	—	47	4	no	—
254	C₂×C₄	—	94	2	no	—
255	C₂×C₄	—	94	2	no	—
256	C₂×C₄	—	94	2	no	—
257	C₂×C₄	—	94	2	no	—
258	C₂×C₄	—	188	4	no	—
259	C₂×C₄	—	188	2	no	—
260	C₂×C₄	—	188	2	no	—
261	C₂×C₄	—	188	4	no	—
262	C₂×C₄	#13094023	1	4	yes	—
263	C₂×C₄	#13094048	1	4	yes	—
264	C₂×C₄	#13094056	1	4	yes	—
265	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
266	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
268	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
269	C₈	—	47	8	no	—
270	C₈	#13094023	1	8	yes	complemented
271	C₂²	#127×D₄₇	1	1	yes	characteristic
272	C₂²	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
273	C₂²	C₂×#13094023	1	1	yes	characteristic
274	C₂²	C₂×#13094048	1	2	yes	—
275	C₂²	C₂×#13094056	1	2	yes	—
276	C₄	—	8	2	no	—
277	C₄	—	8	2	no	—
278	C₄	—	94	2	no	—
279	C₄	—	94	2	no	—
280	C₄	—	94	4	no	—
281	C₄	—	376	2	no	—
282	C₄	—	376	2	no	—
283	C₄	#15689531	1	8	yes	complemented
284	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
285	C₄	C₂×#13094023	1	2	yes	—
286	C₂	#20251785	1	1	yes	characteristic
287	C₂	#20253839	1	1	yes	characteristic
288	C₂	#20253846	1	1	yes	characteristic
289	C₂	C₂×#15689531	1	2	yes	—
290	C₂	#20253931	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1845	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#1845	D₄₇
hypocenter	C₄₇	#33316
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
53. #13093934	Q₈	semidirect
54. C₄×C₄₇⋊C₈	Q₈	semidirect
63. C₄×C₈×C₄₇	Q₈	semidirect
123. C₈×C₄₇	C₄⋊Q₈	semidirect
124. C₄₇⋊C₈	C₄⋊Q₈	semidirect
133. C₄×C₄₇	#501	semidirect
177. C₄₇	#33316	semidirect
208. C₄×C₈	Dic₉₄	semidirect
270. C₈	#13094023	semidirect
283. C₄	#15689531	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689531	2
#20251785	2
#20251749	2
#20243555	2
#20243547	2
#20243545	2
#20243544	2
C₄×#13093934	2
C₄₇×#1845	2
#33316	47