Subgroups — #26219714

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	#20251978	C₂	1	1	yes	maximal, characteristic
3	#20251951	C₂	1	2	yes	maximal
4	#20244141	C₂	1	4	yes	maximal, complemented
5	#20244136	C₂	1	4	yes	maximal, complemented
6	C₂×#15688883	C₂	1	2	yes	maximal
7	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂×#13093983	C₂²	1	2	yes	—
10	#15689172	—	2	4	no	—
11	C₂×#13093960	C₂²	1	4	yes	—
12	#15689165	C₄	1	8	yes	complemented
13	#15689161	C₂²	1	4	yes	—
14	#15689151	—	2	4	no	—
15	#15689149	C₂²	1	8	yes	—
16	#15689146	—	2	4	no	—
17	#15688883	C₂²	1	4	yes	—
18	#15688883	C₂²	1	4	yes	—
19	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₄₇	—	2	4	no	—
23	C₂×#97×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	4	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	C₂³	1	4	yes	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	#13093983	—	2	2	no	—
31	#13093983	—	2	2	no	—
32	#13093983	C₂×C₄	1	4	yes	—
33	#13093983	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	#13093960	—	2	8	no	—
38	#13093958	C₂×C₄	1	8	yes	complemented
39	#13093956	C₂×C₄	1	8	yes	complemented
40	Dic₄₇⋊C₈	—	2	16	no	—
41	C₂³×C₄×C₄₇	—	2	1	no	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
47	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
48	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
49	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₄₇	—	2	2	no	—
52	C₂×C₄²×C₄₇	—	2	1	no	—
53	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	#97×C₄₇	—	2	4	no	—
57	C₄×C₈×C₄₇	—	2	4	no	—
58	C₂⁴×C₄₇	#41	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	—	4	4	no	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	4	no	—
72	M₁₆×C₄₇	—	2	8	no	—
73	M₁₆×C₄₇	C₄⋊C₄	1	16	yes	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	2	1	no	—
82	C₄²×C₄₇	—	2	1	no	—
83	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
84	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
85	C₂²⋊Dic₄₇	—	2	4	no	—
86	C₂²⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	2	4	no	—
88	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
89	C₄×Dic₄₇	—	2	4	no	—
90	C₄×Dic₄₇	—	2	4	no	—
91	#10625329	—	2	4	no	—
92	#10625329	—	2	4	no	—
93	#10625329	—	4	4	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	4	4	no	—
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	4	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	4	8	no	—
105	C₂×C₄×C₄₇	—	4	2	no	—
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
109	C₂×C₄×C₄₇	#121	1	4	yes	—
110	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	#125	1	2	yes	—
112	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₄₇	—	2	8	no	—
120	C₂×Dic₄₇	—	8	4	no	—
121	C₈×C₄₇	—	2	4	no	—
122	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
123	C₄₇⋊C₈	—	4	4	no	—
124	#21165	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	4	no	—
126	C₂²×C₄₇	—	2	2	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	—	2	4	no	—
134	C₄×C₄₇	—	4	2	no	—
135	C₄×C₄₇	#389	1	4	yes	—
136	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
139	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	Dic₄₇	—	4	8	no	—
141	C₂×#430	—	47	1	no	—
142	#2911	—	47	1	no	—
143	#2905	—	47	2	no	—
144	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
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	2	no	—
154	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
155	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
156	C₂×#117	—	47	2	no	—
157	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
158	#430	—	94	4	no	—
159	C₂×#108	—	47	4	no	—
160	#428	—	47	8	no	—
161	#427	—	47	4	no	—
162	#424	—	47	8	no	—
163	#424	—	94	4	no	—
164	#423	—	94	4	no	—
165	C₄×M₁₆	—	2	4	no	—
166	C₂×#97	C₂×D₄₇	1	2	yes	—
167	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
168	#339	—	47	4	no	—
169	#339	—	47	4	no	—
170	C₄₇	#21165	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	—	2	1	no	—
172	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	47	4	no	—
176	C₂×M₁₆	—	94	2	no	—
177	C₂×M₁₆	—	94	4	no	—
178	C₂×M₁₆	—	94	4	no	—
179	C₂×M₁₆	Dic₉₄	1	4	yes	—
180	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
181	C₂×M₁₆	D₁₈₈	1	4	yes	—
182	C₂×M₁₆	C₂²⋊D₄₇	1	4	yes	—
183	C₂²×C₈	—	94	2	no	—
184	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
185	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
186	#117	—	47	4	no	—
187	#117	—	47	4	no	—
188	#117	—	94	2	no	—
189	#117	—	94	2	no	—
190	C₂×C₄⋊C₄	—	47	4	no	—
191	C₂×C₂²⋊C₄	—	94	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	94	2	no	—
195	C₂×C₄²	Dic₉₄	1	4	yes	—
196	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
197	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
198	#108	—	94	8	no	—
199	#107	—	47	8	no	—
200	#106	—	47	8	no	—
201	C₄⋊C₈	—	94	16	no	—
202	#97	—	2	4	no	—
203	C₄×C₈	—	2	4	no	—
204	C₂⁴	#10625350	1	1	yes	characteristic
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	94	4	no	—
210	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
211	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
213	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
214	C₂²×C₄	#10625350	1	1	yes	characteristic
215	C₂²×C₄	#10625350	1	2	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
217	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
218	M₁₆	—	2	8	no	—
219	M₁₆	—	94	4	no	—
220	M₁₆	—	94	4	no	—
221	M₁₆	—	188	4	no	—
222	M₁₆	Dic₄₇⋊C₄	1	16	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	—	188	4	no	—
227	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
228	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
229	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
230	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
231	C₄⋊C₄	—	47	8	no	—
232	C₄⋊C₄	—	94	4	no	—
233	C₂²⋊C₄	—	94	4	no	—
234	C₂²⋊C₄	—	94	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	94	4	no	—
239	C₄²	—	94	4	no	—
240	C₄²	C₂×Dic₉₄	1	2	yes	—
241	C₄²	C₂×Dic₉₄	1	2	yes	—
242	C₂³	—	4	2	no	—
243	C₂³	C₄×D₁₈₈	1	2	yes	—
244	C₂³	#13094003	1	2	yes	—
245	C₂³	#13094016	1	2	yes	—
246	C₂³	C₂×#10625350	1	1	yes	characteristic
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	94	8	no	—
253	C₂×C₄	—	376	4	no	—
254	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
255	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
256	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
257	C₂×C₄	#13093998	1	4	yes	—
258	C₂×C₄	#13094003	1	2	yes	—
259	C₂×C₄	#13094016	1	2	yes	—
260	C₂×C₄	#13094024	1	4	yes	—
261	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
264	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
265	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
266	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	—	188	4	no	—
269	C₈	C₂×Dic₄₇⋊C₄	1	8	yes	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	#15689021	1	4	yes	—
275	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
276	C₂²	C₂×#13094003	1	1	yes	characteristic
277	C₂²	C₂×#13094016	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	4	2	no	—
280	C₄	—	188	8	no	—
281	C₄	#15689021	1	4	yes	—
282	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
283	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
284	C₄	C₂×#13093998	1	1	yes	characteristic
285	C₄	C₂×#13094024	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	4	no	—
288	C₂	—	2	1	no	—
289	C₂	C₂×#15689021	1	1	yes	characteristic
290	C₂	#20252299	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#21165
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #20244141	C₂	semidirect
5. #20244136	C₂	semidirect
12. #15689165	C₄	semidirect
38. #13093958	C₂×C₄	semidirect
39. #13093956	C₂×C₄	semidirect
170. C₄₇	#21165	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
#20251978	2
#20251951	2
#20244141	2
#20244136	2
C₂×#15688883	2
C₂×C₄×M₁₆×C₄₇	2
#21165	47