Subgroups — #26219708

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	C₂×#15689150	C₂	1	2	yes	maximal
3	C₂×#15689081	C₂	1	1	yes	maximal, characteristic
4	#20244142	C₂	1	4	yes	maximal, complemented
5	#20244137	C₂	1	4	yes	maximal, complemented
6	#20243097	C₂	1	2	yes	maximal
7	C₄₇×#2911	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×#13093983	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	4	yes	—
15	#15689150	C₂²	1	4	yes	—
16	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
17	#15689081	—	2	4	no	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
20	#15688882	C₂²	1	8	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	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	C₂²×C₄₇⋊C₈	—	2	2	no	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
35	#13093983	—	2	2	no	—
36	#13093983	Q₈	1	4	yes	—
37	C₂×C₄⋊Dic₄₇	—	2	1	no	—
38	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
39	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
40	#13093960	—	2	8	no	—
41	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
42	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
43	#13093934	—	2	4	no	—
44	C₈×Dic₄₇	—	2	4	no	—
45	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₄₇	C₂×C₄	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	#117×C₄₇	—	2	2	no	—
51	#117×C₄₇	—	2	2	no	—
52	#117×C₄₇	C₂×C₄	1	4	yes	—
53	#117×C₄₇	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
55	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
56	C₂×C₄²×C₄₇	—	2	2	no	—
57	#107×C₄₇	C₂×C₄	1	8	yes	—
58	#106×C₄₇	C₂×C₄	1	8	yes	—
59	C₂⁴×C₄₇	#41	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	—	2	4	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	2	no	—
70	M₁₆×C₄₇	C₄⋊C₄	1	16	yes	—
71	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
76	C₂²⋊C₄×C₄₇	—	2	4	no	—
77	C₂²⋊C₄×C₄₇	—	2	4	no	—
78	C₄²×C₄₇	—	2	4	no	—
79	C₄²×C₄₇	—	2	4	no	—
80	C₂²⋊Dic₄₇	—	2	4	no	—
81	C₄⋊Dic₄₇	—	2	4	no	—
82	C₄×Dic₄₇	—	2	2	no	—
83	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
84	#10625329	—	2	4	no	—
85	#10625329	—	2	4	no	—
86	#10625329	—	4	4	no	—
87	C₂×C₄₇⋊C₈	—	2	8	no	—
88	C₂×C₄₇⋊C₈	—	2	1	no	—
89	C₂×C₄₇⋊C₈	—	2	1	no	—
90	C₂×C₄₇⋊C₈	—	2	1	no	—
91	C₂×C₄₇⋊C₈	—	2	1	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	4	4	no	—
94	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
95	C₂×C₄₇⋊C₈	C₂×D₄	1	2	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	—	4	8	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	—	4	4	no	—
118	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
119	C₄₇⋊C₈	—	2	2	no	—
120	C₄₇⋊C₈	—	2	2	no	—
121	C₄₇⋊C₈	—	4	4	no	—
122	#21164	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	4	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	8	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	4	no	—
138	C₂×#430	—	47	1	no	—
139	#2911	D₄₇	1	1	yes	characteristic
140	C₂×#424	—	47	2	no	—
141	C₂×#405	—	47	1	no	—
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	—	47	1	no	—
153	C₂×#117	—	47	1	no	—
154	C₂×#117	C₂×D₄₇	1	2	yes	—
155	#430	—	94	4	no	—
156	C₂×#108	—	47	4	no	—
157	#428	Dic₄₇	1	8	yes	complemented
158	#427	C₂×D₄₇	1	4	yes	—
159	#424	—	47	4	no	—
160	#424	—	47	4	no	—
161	#424	—	94	4	no	—
162	C₂×C₄⋊C₈	—	94	4	no	—
163	#405	—	94	4	no	—
164	C₂×#97	—	47	2	no	—
165	C₂×C₄×C₈	—	47	2	no	—
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	D₁₈₈	1	4	yes	—
175	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
176	C₂²×C₈	—	47	4	no	—
177	C₂²×C₈	—	47	1	no	—
178	C₂²×C₈	—	47	1	no	—
179	C₂²×C₈	—	94	2	no	—
180	C₂²×C₈	—	94	2	no	—
181	C₂²×C₈	—	94	2	no	—
182	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	47	4	no	—
186	#117	—	94	2	no	—
187	#117	C₂×Dic₄₇	1	4	yes	—
188	#117	C₂²×D₄₇	1	4	yes	—
189	C₂×C₄⋊C₄	—	94	1	no	—
190	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	94	1	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	47	2	no	—
195	#108	—	94	8	no	—
196	#107	C₂×Dic₄₇	1	8	yes	—
197	#106	C₂×Dic₄₇	1	8	yes	—
198	C₄⋊C₈	—	94	8	no	—
199	C₄⋊C₈	—	94	8	no	—
200	#97	—	94	4	no	—
201	C₄×C₈	—	94	4	no	—
202	C₂⁴	#10625352	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	94	2	no	—
205	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
206	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
207	C₂²×C₄	D₄×D₄₇	1	2	yes	—
208	C₂²×C₄	#10625352	1	1	yes	characteristic
209	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
210	C₂²×C₄	#10625354	1	2	yes	—
211	C₂²×C₄	C₂²×Dic₄₇	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	C₄⋊Dic₄₇	1	16	yes	—
217	C₂×C₈	—	47	2	no	—
218	C₂×C₈	—	47	2	no	—
219	C₂×C₈	—	94	8	no	—
220	C₂×C₈	—	94	1	no	—
221	C₂×C₈	—	94	1	no	—
222	C₂×C₈	—	94	1	no	—
223	C₂×C₈	—	94	1	no	—
224	C₂×C₈	—	94	4	no	—
225	C₂×C₈	—	188	4	no	—
226	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
227	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
228	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	94	4	no	—
231	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	94	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	47	4	no	—
238	C₄²	—	94	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	D₄×Dic₄₇	1	2	yes	—
241	C₂³	#13094004	1	2	yes	—
242	C₂³	#13094015	1	2	yes	—
243	C₂³	C₂×#10625352	1	1	yes	characteristic
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	—	94	4	no	—
247	C₂×C₄	—	188	4	no	—
248	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
249	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
250	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
251	C₂×C₄	#13093999	1	4	yes	—
252	C₂×C₄	#13094004	1	2	yes	—
253	C₂×C₄	#13094015	1	2	yes	—
254	C₂×C₄	#13094023	1	4	yes	—
255	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
258	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
259	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
260	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
261	C₈	—	94	2	no	—
262	C₈	—	94	2	no	—
263	C₈	—	188	4	no	—
264	C₈	C₂×C₄⋊Dic₄₇	1	8	yes	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	#15689022	1	4	yes	—
270	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
271	C₂²	C₂×#13094004	1	1	yes	characteristic
272	C₂²	C₂×#13094015	1	1	yes	characteristic
273	C₄	—	4	8	no	—
274	C₄	—	188	4	no	—
275	C₄	#15689022	1	4	yes	—
276	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
277	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
278	C₄	C₂×#13093999	1	1	yes	characteristic
279	C₄	C₂×#13094023	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	4	no	—
282	C₂	—	2	1	no	—
283	C₂	C₂×#15689022	1	1	yes	characteristic
284	C₂	#20252291	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	#21164
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
157. #428	Dic₄₇	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
C₂×#15689150	2
C₂×#15689081	2
#20244142	2
#20244137	2
#20243097	2
C₄₇×#2911	2
#21164	47