Subgroups — #26219715

Order 12032.

914 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×#15689172	C₂	1	1	yes	maximal, characteristic
2	#20251952	C₂	1	2	yes	maximal
3	C₂×M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20244142	C₂	1	4	yes	maximal, complemented
5	#20244137	C₂	1	4	yes	maximal, complemented
6	C₂×#15688882	C₂	1	2	yes	maximal
7	C₄₇×#2911	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	#15689172	—	2	4	no	—
12	C₂×#13093960	C₂²	1	4	yes	—
13	#15689150	—	2	4	no	—
14	#15689150	C₂²	1	8	yes	—
15	#15689147	—	2	4	no	—
16	M₁₆×Dic₄₇	—	2	4	no	—
17	C₂×#13093934	C₂²	1	2	yes	—
18	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
19	#15688882	C₂²	1	4	yes	—
20	#15688882	C₂²	1	4	yes	—
21	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₄₇	C₂²	1	2	yes	—
23	C₄₇×#428	C₄	1	8	yes	complemented
24	C₄₇×#427	C₂²	1	4	yes	—
25	C₂³×Dic₄₇	—	2	1	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	4	no	—
30	C₂×#10625329	—	2	4	no	—
31	C₂×#10625329	D₄	1	4	yes	—
32	C₂×#10625329	C₂³	1	4	yes	—
33	C₂²×C₄₇⋊C₈	—	2	2	no	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	1	no	—
37	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
40	#13093960	—	2	8	no	—
41	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
42	#13093934	—	2	4	no	—
43	C₈×Dic₄₇	—	2	4	no	—
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
47	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	#117×C₄₇	—	2	2	no	—
50	#117×C₄₇	—	2	2	no	—
51	#117×C₄₇	C₂×C₄	1	4	yes	—
52	#117×C₄₇	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₄²×C₄₇	—	2	2	no	—
56	#107×C₄₇	C₂×C₄	1	8	yes	complemented
57	#106×C₄₇	C₂×C₄	1	8	yes	complemented
58	C₂⁴×C₄₇	#41	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	2	4	no	—
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	2	yes	—
63	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	1	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	2	1	no	—
71	C₂²×Dic₄₇	—	4	4	no	—
72	M₁₆×C₄₇	C₄⋊C₄	1	16	yes	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
75	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
78	C₂²⋊C₄×C₄₇	—	2	4	no	—
79	C₂²⋊C₄×C₄₇	—	2	4	no	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	2	4	no	—
82	C₄×Dic₄₇	—	2	4	no	—
83	C₄×Dic₄₇	—	2	1	no	—
84	C₄×Dic₄₇	—	2	1	no	—
85	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
86	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
87	#10625329	—	2	8	no	—
88	#10625329	—	2	4	no	—
89	#10625329	—	2	4	no	—
90	#10625329	—	4	4	no	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	4	4	no	—
95	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
96	C₂³×C₄₇	—	4	2	no	—
97	C₂³×C₄₇	C₄×D₄	1	2	yes	—
98	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
99	C₂³×C₄₇	#125	1	2	yes	—
100	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	—	8	4	no	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
106	C₂×C₄×C₄₇	#121	1	4	yes	—
107	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₄₇	#125	1	2	yes	—
109	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
110	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
116	C₂×Dic₄₇	—	2	4	no	—
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	2	2	no	—
119	C₂×Dic₄₇	—	4	8	no	—
120	C₂×Dic₄₇	—	4	2	no	—
121	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
122	C₄₇⋊C₈	—	2	4	no	—
123	C₄₇⋊C₈	—	4	4	no	—
124	#21165	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	4	2	no	—
128	C₂²×C₄₇	—	4	2	no	—
129	C₂²×C₄₇	#389	1	4	yes	—
130	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×#125	1	1	yes	characteristic
133	C₄×C₄₇	—	4	8	no	—
134	C₄×C₄₇	#389	1	4	yes	—
135	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
137	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
138	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	Dic₄₇	—	2	4	no	—
140	Dic₄₇	—	4	2	no	—
141	C₂×#430	—	47	1	no	—
142	#2911	D₄₇	1	1	yes	characteristic
143	#2905	—	47	2	no	—
144	C₂×C₄×M₁₆	—	47	1	no	—
145	#1946	—	47	4	no	—
146	#1945	—	47	4	no	—
147	C₂×#339	—	47	2	no	—
148	C₂×C₄₇	—	2	1	no	—
149	C₂×C₄₇	—	2	1	no	—
150	C₂×C₄₇	—	2	4	no	—
151	C₂×C₄₇	C₂×#389	1	1	yes	characteristic
152	C₂×C₄₇	#2956	1	2	yes	—
153	C₂²×M₁₆	—	47	1	no	—
154	C₂²×M₁₆	—	47	2	no	—
155	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
156	C₂×#117	C₂×D₄₇	1	2	yes	—
157	C₂²×C₄²	—	47	1	no	—
158	#430	—	94	4	no	—
159	C₂×#108	—	47	4	no	—
160	#428	Dic₄₇	1	8	yes	complemented
161	#427	C₂×D₄₇	1	4	yes	—
162	#424	—	47	8	no	—
163	#424	—	94	4	no	—
164	#423	—	94	4	no	—
165	C₄×M₁₆	—	94	4	no	—
166	C₂×#97	—	47	2	no	—
167	C₂×C₄×C₈	—	47	2	no	—
168	#339	—	47	4	no	—
169	#339	—	47	4	no	—
170	C₄₇	#21165	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	—	94	1	no	—
172	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
173	C₂×M₁₆	—	47	4	no	—
174	C₂×M₁₆	—	47	4	no	—
175	C₂×M₁₆	—	94	2	no	—
176	C₂×M₁₆	—	94	2	no	—
177	C₂×M₁₆	—	94	2	no	—
178	C₂×M₁₆	—	94	4	no	—
179	C₂×M₁₆	—	94	4	no	—
180	C₂×M₁₆	Dic₉₄	1	4	yes	—
181	C₂×M₁₆	D₁₈₈	1	4	yes	—
182	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
183	C₂²×C₈	—	47	2	no	—
184	C₂²×C₈	—	94	2	no	—
185	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	C₂×Dic₄₇	1	4	yes	—
189	#117	C₂²×D₄₇	1	4	yes	—
190	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	47	4	no	—
194	C₂×C₄²	—	47	1	no	—
195	C₂×C₄²	—	47	1	no	—
196	C₂×C₄²	—	94	2	no	—
197	C₂×C₄²	—	94	1	no	—
198	#108	—	94	8	no	—
199	#107	C₂×Dic₄₇	1	8	yes	complemented
200	#106	C₂×Dic₄₇	1	8	yes	complemented
201	C₄⋊C₈	—	94	16	no	—
202	#97	—	94	4	no	—
203	C₄×C₈	—	94	4	no	—
204	C₂⁴	#10625352	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	94	1	no	—
207	C₂²×C₄	—	94	4	no	—
208	C₂²×C₄	—	94	1	no	—
209	C₂²×C₄	—	188	4	no	—
210	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
211	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
212	C₂²×C₄	D₄×D₄₇	1	2	yes	—
213	C₂²×C₄	#10625352	1	1	yes	characteristic
214	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
215	C₂²×C₄	#10625354	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
217	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
218	M₁₆	—	94	8	no	—
219	M₁₆	—	94	4	no	—
220	M₁₆	—	94	4	no	—
221	M₁₆	—	188	4	no	—
222	M₁₆	C₄⋊Dic₄₇	1	16	yes	—
223	C₂×C₈	—	47	4	no	—
224	C₂×C₈	—	94	2	no	—
225	C₂×C₈	—	94	2	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	188	4	no	—
228	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
229	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
230	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	47	2	no	—
238	C₄²	—	47	2	no	—
239	C₄²	—	94	4	no	—
240	C₄²	—	94	1	no	—
241	C₄²	—	94	1	no	—
242	C₂³	—	4	2	no	—
243	C₂³	D₄×Dic₄₇	1	2	yes	—
244	C₂³	#13094004	1	2	yes	—
245	C₂³	#13094015	1	2	yes	—
246	C₂³	C₂×#10625352	1	1	yes	characteristic
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	—	94	4	no	—
250	C₂×C₄	—	94	2	no	—
251	C₂×C₄	—	94	2	no	—
252	C₂×C₄	—	188	8	no	—
253	C₂×C₄	—	188	2	no	—
254	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
255	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
256	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
257	C₂×C₄	#13093999	1	4	yes	—
258	C₂×C₄	#13094004	1	2	yes	—
259	C₂×C₄	#13094015	1	2	yes	—
260	C₂×C₄	#13094023	1	4	yes	—
261	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
264	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
265	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
266	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
267	C₈	—	94	4	no	—
268	C₈	—	188	4	no	—
269	C₈	C₂×C₄⋊Dic₄₇	1	8	yes	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	#15689022	1	4	yes	—
275	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
276	C₂²	C₂×#13094004	1	1	yes	characteristic
277	C₂²	C₂×#13094015	1	1	yes	characteristic
278	C₄	—	4	8	no	—
279	C₄	—	94	4	no	—
280	C₄	—	188	2	no	—
281	C₄	#15689022	1	4	yes	—
282	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
283	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
284	C₄	C₂×#13093999	1	1	yes	characteristic
285	C₄	C₂×#13094023	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	4	no	—
288	C₂	—	2	1	no	—
289	C₂	C₂×#15689022	1	1	yes	characteristic
290	C₂	#20252298	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2911	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2911	D₄₇
hypocenter	C₄₇	#21165
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2911

Complements

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

N	H ≅ G/N	type
4. #20244142	C₂	semidirect
5. #20244137	C₂	semidirect
23. C₄₇×#428	C₄	semidirect
56. #107×C₄₇	C₂×C₄	semidirect
57. #106×C₄₇	C₂×C₄	semidirect
160. #428	Dic₄₇	semidirect
170. C₄₇	#21165	semidirect
199. #107	C₂×Dic₄₇	semidirect
200. #106	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689172	2
#20251952	2
C₂×M₁₆×Dic₄₇	2
#20244142	2
#20244137	2
C₂×#15688882	2
C₄₇×#2911	2
#21165	47