Subgroups — #26219707

Order 12032.

890 conjugacy classes of subgroups.

The subgroup lattice has 286 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	C₂×#15689149	C₂	1	2	yes	maximal
4	#20244141	C₂	1	4	yes	maximal, complemented
5	#20244136	C₂	1	4	yes	maximal, complemented
6	#20243098	C₂	1	2	yes	maximal
7	C₂×C₄₇×#405	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	#15689149	—	2	4	no	—
15	#15689149	C₂²	1	4	yes	—
16	#15689149	C₂²	1	4	yes	—
17	C₂×Dic₄₇⋊C₈	—	2	4	no	—
18	#15688883	C₂²	1	8	yes	—
19	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#405	—	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	—
39	#13093956	C₂×C₄	1	8	yes	—
40	Dic₄₇⋊C₈	—	2	8	no	—
41	Dic₄₇⋊C₈	—	2	8	no	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
45	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
46	C₂²×C₈×C₄₇	—	2	2	no	—
47	C₂²×C₈×C₄₇	—	2	2	no	—
48	C₂²×C₈×C₄₇	D₄	1	4	yes	—
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
52	#117×C₄₇	—	2	2	no	—
53	#117×C₄₇	Q₈	1	4	yes	—
54	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
55	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
56	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
57	#97×C₄₇	—	2	4	no	—
58	C₄×C₈×C₄₇	—	2	4	no	—
59	C₂⁴×C₄₇	#41	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	2	4	no	—
70	M₁₆×C₄₇	C₄⋊C₄	1	16	yes	—
71	C₂×C₈×C₄₇	—	2	8	no	—
72	C₂×C₈×C₄₇	—	2	1	no	—
73	C₂×C₈×C₄₇	—	2	1	no	—
74	C₂×C₈×C₄₇	—	2	1	no	—
75	C₂×C₈×C₄₇	—	2	1	no	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
81	C₄⋊C₄×C₄₇	—	2	4	no	—
82	C₂²⋊C₄×C₄₇	—	2	4	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	C₂×Q₈	1	4	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	—	4	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	8	no	—
117	C₂×Dic₄₇	—	4	8	no	—
118	C₈×C₄₇	—	2	2	no	—
119	C₈×C₄₇	—	2	2	no	—
120	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
121	C₄₇⋊C₈	—	4	4	no	—
122	#21164	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	4	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	4	2	no	—
126	C₂²×C₄₇	—	4	2	no	—
127	C₂²×C₄₇	#389	1	4	yes	—
128	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#125	1	1	yes	characteristic
131	C₄×C₄₇	—	4	4	no	—
132	C₄×C₄₇	#389	1	4	yes	—
133	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
135	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
136	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
137	Dic₄₇	—	4	8	no	—
138	C₂×#430	—	47	1	no	—
139	#2911	—	47	1	no	—
140	C₂×#424	—	47	2	no	—
141	C₂×#405	D₄₇	1	1	yes	characteristic
142	#1946	—	47	4	no	—
143	#1945	—	47	4	no	—
144	#1733	—	47	2	no	—
145	C₂×C₄₇	—	2	1	no	—
146	C₂×C₄₇	—	2	1	no	—
147	C₂×C₄₇	—	2	4	no	—
148	C₂×C₄₇	C₂×#389	1	1	yes	characteristic
149	C₂×C₄₇	#2955	1	2	yes	—
150	C₂²×M₁₆	—	47	2	no	—
151	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
152	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
153	C₂×#117	—	47	2	no	—
154	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
155	#430	—	94	4	no	—
156	C₂×#108	—	47	4	no	—
157	#428	—	47	8	no	—
158	#427	—	47	4	no	—
159	#424	—	47	4	no	—
160	#424	—	47	4	no	—
161	#424	—	94	4	no	—
162	C₂×C₄⋊C₈	—	94	4	no	—
163	#405	—	2	4	no	—
164	C₂×#97	C₂×D₄₇	1	2	yes	—
165	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
166	#339	—	47	8	no	—
167	C₄₇	#21164	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
169	C₂×M₁₆	—	47	4	no	—
170	C₂×M₁₆	—	94	2	no	—
171	C₂×M₁₆	—	94	4	no	—
172	C₂×M₁₆	—	94	4	no	—
173	C₂×M₁₆	Dic₉₄	1	4	yes	—
174	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
175	C₂×M₁₆	C₂²⋊D₄₇	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	94	2	no	—
179	C₂²×C₈	D₁₈₈	1	4	yes	—
180	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
183	#117	—	2	2	no	—
184	#117	—	47	4	no	—
185	#117	—	47	4	no	—
186	#117	—	94	2	no	—
187	#117	—	94	2	no	—
188	#117	Dic₉₄	1	4	yes	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	47	4	no	—
191	C₂×C₂²⋊C₄	—	2	1	no	—
192	C₂×C₂²⋊C₄	—	94	2	no	—
193	C₂×C₄²	—	94	2	no	—
194	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
195	#108	—	94	8	no	—
196	#107	—	47	8	no	—
197	#106	—	47	8	no	—
198	C₄⋊C₈	—	94	8	no	—
199	C₄⋊C₈	—	94	8	no	—
200	#97	—	2	4	no	—
201	C₄×C₈	—	2	4	no	—
202	C₂⁴	#10625350	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	94	4	no	—
205	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
206	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
207	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
208	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
209	C₂²×C₄	#10625350	1	1	yes	characteristic
210	C₂²×C₄	#10625350	1	2	yes	—
211	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
212	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
213	M₁₆	—	94	4	no	—
214	M₁₆	—	94	4	no	—
215	M₁₆	—	188	4	no	—
216	M₁₆	Dic₄₇⋊C₄	1	16	yes	—
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	94	4	no	—
223	C₂×C₈	—	188	4	no	—
224	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
225	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
226	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
227	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
228	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	47	8	no	—
231	C₄⋊C₄	—	94	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	94	4	no	—
234	C₂²⋊C₄	—	94	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	94	4	no	—
237	C₄²	—	94	4	no	—
238	C₄²	C₂×Dic₉₄	1	4	yes	—
239	C₂³	—	4	2	no	—
240	C₂³	C₄×D₁₈₈	1	2	yes	—
241	C₂³	#13094003	1	2	yes	—
242	C₂³	#13094016	1	2	yes	—
243	C₂³	C₂×#10625350	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	94	8	no	—
247	C₂×C₄	—	188	8	no	—
248	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
249	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
250	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
251	C₂×C₄	#13093998	1	4	yes	—
252	C₂×C₄	#13094003	1	2	yes	—
253	C₂×C₄	#13094016	1	2	yes	—
254	C₂×C₄	#13094024	1	4	yes	—
255	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
258	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
259	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
260	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	188	4	no	—
264	C₈	C₂×Dic₄₇⋊C₄	1	8	yes	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	#15689021	1	4	yes	—
270	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
271	C₂²	C₂×#13094003	1	1	yes	characteristic
272	C₂²	C₂×#13094016	1	1	yes	characteristic
273	C₄	—	4	4	no	—
274	C₄	—	188	8	no	—
275	C₄	#15689021	1	4	yes	—
276	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
277	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
278	C₄	C₂×#13093998	1	1	yes	characteristic
279	C₄	C₂×#13094024	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	4	no	—
282	C₂	—	2	1	no	—
283	C₂	C₂×#15689021	1	1	yes	characteristic
284	C₂	#20252292	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#21164
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂×#405

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
167. C₄₇	#21164	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
C₂×#15689149	2
#20244141	2
#20244136	2
#20243098	2
C₂×C₄₇×#405	2
#21164	47