Subgroups — #26216689

Order 12032.

1126 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689159	C₂	1	4	yes	maximal
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20245795	C₂	1	1	yes	maximal, characteristic
4	#20243550	C₂	1	4	yes	maximal, complemented
5	#20243547	C₂	1	2	yes	maximal, complemented
6	#20243544	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×#13093983	C₂²	1	2	yes	—
10	#15689159	—	2	16	no	—
11	C₂×#13093957	C₂²	1	8	yes	—
12	C₂×#13093955	C₂²	1	8	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
17	#15688993	C₂²	1	2	yes	—
18	#15688989	C₂²	1	2	yes	—
19	#15688983	—	2	2	no	—
20	C₄×C₄⋊Dic₄₇	—	2	2	no	—
21	C₂×#117×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#384	C₂²	1	2	yes	—
24	C₄₇×#382	C₂²	1	2	yes	—
25	C₄₇×#380	—	2	2	no	—
26	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
28	C₂²×C₄₇⋊C₈	D₄	1	8	yes	—
29	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	#13093983	—	2	8	no	—
33	C₂×C₄⋊Dic₄₇	—	2	4	no	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	#13093957	—	2	16	no	—
38	#13093955	—	2	16	no	—
39	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
40	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
41	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
42	C₄₇⋊#95	—	2	4	no	—
43	C₂³×C₄×C₄₇	D₄	1	2	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	#117×C₄₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₇	D₄	1	2	yes	—
52	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	#95×C₄₇	—	2	4	no	—
56	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	8	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₄⋊C₄×C₄₇	—	2	8	no	—
70	C₄⋊C₄×C₄₇	—	4	8	no	—
71	C₂²⋊C₄×C₄₇	—	4	4	no	—
72	C₄²×C₄₇	—	4	4	no	—
73	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
74	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
77	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
78	C₂²⋊Dic₄₇	—	4	4	no	—
79	C₄⋊Dic₄₇	—	2	8	no	—
80	C₄⋊Dic₄₇	—	4	8	no	—
81	C₄×Dic₄₇	—	4	4	no	—
82	C₂×C₄₇⋊C₈	C₄⋊C₄	1	32	yes	complemented
83	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
84	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
85	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
86	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
87	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
88	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₄₇	#127	1	2	yes	—
92	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
93	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	16	no	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	—	4	8	no	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₇	#127	1	2	yes	—
103	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
104	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
105	C₂×C₄×C₄₇	#135	1	8	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×Dic₄₇	—	4	4	no	—
111	C₂×Dic₄₇	—	4	8	no	—
112	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
113	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
114	#20797	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	4	no	—
116	C₂²×C₄₇	—	2	4	no	—
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	#384	1	4	yes	—
119	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×#135	1	4	yes	—
123	C₄×C₄₇	—	2	4	no	—
124	C₄×C₄₇	—	8	4	no	—
125	C₄×C₄₇	#384	1	4	yes	—
126	C₄×C₄₇	#427	1	8	yes	complemented
127	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
128	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
129	Dic₄₇	—	8	4	no	—
130	C₂×#427	—	47	4	no	—
131	C₂²×C₄×C₈	—	47	1	no	—
132	#2285	—	47	1	no	—
133	#2285	D₄₇	1	1	yes	characteristic
134	#1846	—	47	4	no	—
135	#1845	—	47	2	no	—
136	#1844	—	47	2	no	—
137	C₂×C₄₇	—	2	4	no	—
138	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
139	C₂×C₄₇	C₂×#427	1	2	yes	—
140	C₂×C₄₇	#3147	1	2	yes	—
141	C₂×C₄₇	#3160	1	2	yes	—
142	C₂³×C₈	—	47	2	no	—
143	C₂×#117	—	47	2	no	—
144	C₂×#117	C₂×D₄₇	1	2	yes	—
145	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
146	#427	—	94	16	no	—
147	C₂×#107	—	47	8	no	—
148	C₂×#106	—	47	8	no	—
149	C₂×C₄×C₈	—	47	8	no	—
150	C₂×C₄×C₈	—	47	1	no	—
151	C₂×C₄×C₈	—	47	1	no	—
152	C₂×C₄×C₈	—	47	2	no	—
153	#384	—	47	2	no	—
154	#384	C₂×D₄₇	1	2	yes	—
155	#382	—	47	2	no	—
156	#382	C₂×D₄₇	1	2	yes	—
157	#380	—	2	2	no	—
158	#380	—	94	2	no	—
159	C₄×C₄⋊C₄	—	2	2	no	—
160	C₄×C₄⋊C₄	—	94	2	no	—
161	C₄₇	#20797	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
163	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
164	C₂²×C₈	—	47	8	no	—
165	C₂²×C₈	—	47	8	no	—
166	C₂²×C₈	—	47	8	no	—
167	C₂²×C₈	—	47	2	no	—
168	C₂²×C₈	—	47	2	no	—
169	#117	—	2	8	no	—
170	#117	—	94	8	no	—
171	C₂×C₄⋊C₄	—	2	4	no	—
172	C₂×C₄⋊C₄	—	47	4	no	—
173	C₂×C₄⋊C₄	—	94	4	no	—
174	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
175	C₂×C₂²⋊C₄	—	2	2	no	—
176	C₂×C₂²⋊C₄	—	94	2	no	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	—	94	2	no	—
179	C₂×C₄²	Dic₉₄	1	4	yes	—
180	C₂×C₄²	C₄×D₄₇	1	4	yes	—
181	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
182	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
184	#107	—	94	16	no	—
185	#106	—	94	16	no	—
186	C₄×C₈	—	47	4	no	—
187	C₄×C₈	—	47	4	no	—
188	C₄×C₈	—	47	8	no	—
189	#95	—	2	4	no	—
190	#95	—	94	4	no	—
191	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	94	4	no	—
195	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
196	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
198	C₂²×C₄	D₄×D₄₇	1	4	yes	—
199	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
202	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
204	C₂×C₈	—	47	32	no	—
205	C₂×C₈	—	47	4	no	—
206	C₂×C₈	—	47	4	no	—
207	C₂×C₈	—	47	4	no	—
208	C₂×C₈	—	47	4	no	—
209	C₂×C₈	—	47	4	no	—
210	C₂×C₈	—	47	4	no	—
211	C₄⋊C₄	—	2	8	no	—
212	C₄⋊C₄	—	4	8	no	—
213	C₄⋊C₄	—	94	8	no	—
214	C₄⋊C₄	—	188	8	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	—	188	4	no	—
217	C₄²	—	4	4	no	—
218	C₄²	—	188	4	no	—
219	C₄²	Dic₄₇⋊C₄	1	8	yes	—
220	C₄²	C₂×Dic₉₄	1	2	yes	—
221	C₄²	C₂×Dic₉₄	1	2	yes	—
222	C₄²	C₂×C₄×D₄₇	1	2	yes	—
223	C₄²	C₂×C₄×D₄₇	1	2	yes	—
224	C₂³	—	2	4	no	—
225	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
226	C₂³	#13094020	1	2	yes	—
227	C₂³	#13094022	1	2	yes	—
228	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	4	8	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	188	8	no	—
234	C₂×C₄	—	188	4	no	—
235	C₂×C₄	C₄⋊C₄×D₄₇	1	8	yes	—
236	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
237	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
238	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
239	C₂×C₄	#13094020	1	2	yes	—
240	C₂×C₄	#13094022	1	2	yes	—
241	C₂×C₄	#13094023	1	8	yes	—
242	C₂×C₄	#13094044	1	8	yes	—
243	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
245	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
246	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
247	C₈	—	47	8	no	—
248	C₈	—	47	8	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	#15688992	1	4	yes	—
253	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
254	C₂²	C₂×#13094020	1	1	yes	characteristic
255	C₂²	C₂×#13094022	1	1	yes	characteristic
256	C₂²	C₂×#13094044	1	4	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	8	4	no	—
259	C₄	—	376	4	no	—
260	C₄	#15688992	1	4	yes	—
261	C₄	#15689159	1	8	yes	complemented
262	C₄	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
263	C₄	C₂×#13094023	1	2	yes	—
264	C₂	—	2	4	no	—
265	C₂	C₂×#15688992	1	1	yes	characteristic
266	C₂	C₂×#15689159	1	2	yes	—
267	C₂	#20253814	1	2	yes	—
268	C₂	#20253898	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#20797
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
4. #20243550	C₂	semidirect
5. #20243547	C₂	semidirect
6. #20243544	C₂	semidirect
13. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
82. C₂×C₄₇⋊C₈	C₄⋊C₄	semidirect
126. C₄×C₄₇	#427	semidirect
161. C₄₇	#20797	semidirect
261. C₄	#15689159	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689159	2
C₂²×C₄×C₄₇⋊C₈	2
#20245795	2
#20243550	2
#20243547	2
#20243544	2
C₄₇×#2285	2
#20797	47