Subgroups — #26246250

Order 12032.

664 conjugacy classes of subgroups.

The subgroup lattice has 306 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	#20251951	C₂	1	1	yes	maximal, characteristic
3	#20244964	C₂	1	8	yes	maximal, complemented
4	C₂×#15688893	C₂	1	2	yes	maximal
5	#20243097	C₂	1	2	yes	maximal
6	C₂×C₄₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	1	yes	characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂×#13093983	C₂²	1	1	yes	characteristic
10	#15689172	—	2	2	no	—
11	#15689172	C₂²	1	4	yes	—
12	C₂×#13093960	—	2	2	no	—
13	#15689151	—	2	2	no	—
14	#15689149	—	2	2	no	—
15	#15689146	C₂²	1	4	yes	—
16	#15688893	C₂²	1	8	yes	—
17	#15688882	C₂²	1	8	yes	—
18	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₄₇	C₂²	1	2	yes	—
20	C₄₇×H(3,ℤ/4)	C₂²	1	4	yes	—
21	C₂×#10625329	—	2	1	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	C₂³	1	4	yes	—
30	C₂×#10625329	C₂³	1	2	yes	—
31	C₂²×C₄₇⋊C₈	—	2	1	no	—
32	C₂²×C₄₇⋊C₈	—	2	2	no	—
33	#13093983	—	2	1	no	—
34	#13093983	—	2	1	no	—
35	#13093983	Q₈	1	2	yes	—
36	#13093983	Q₈	1	2	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	Dic₄₇⋊C₈	—	2	8	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	D₄	1	2	yes	—
46	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₄₇	D₄	1	2	yes	—
48	#117×C₄₇	—	2	2	no	—
49	#117×C₄₇	—	2	2	no	—
50	#117×C₄₇	—	2	2	no	—
51	#117×C₄₇	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄²×C₄₇	—	2	2	no	—
55	C₂⁴×C₄₇	#41	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	2	2	no	—
68	M₁₆×C₄₇	—	2	4	no	—
69	M₁₆×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	2	no	—
72	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₄⋊C₄×C₄₇	—	4	2	no	—
75	C₂²⋊C₄×C₄₇	—	2	4	no	—
76	C₂²⋊C₄×C₄₇	—	4	2	no	—
77	C₄²×C₄₇	—	2	4	no	—
78	C₄²×C₄₇	—	4	2	no	—
79	C₂²⋊Dic₄₇	—	2	2	no	—
80	C₂²⋊Dic₄₇	—	2	2	no	—
81	C₄⋊Dic₄₇	—	2	2	no	—
82	C₄⋊Dic₄₇	—	2	2	no	—
83	C₄×Dic₄₇	—	2	2	no	—
84	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
85	#10625329	—	2	4	no	—
86	#10625329	—	2	4	no	—
87	#10625329	—	2	4	no	—
88	#10625329	—	4	1	no	—
89	#10625329	—	4	1	no	—
90	#10625329	—	4	2	no	—
91	#10625329	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂³×C₄₇	—	8	1	no	—
98	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₄₇	#126	1	2	yes	—
101	C₂³×C₄₇	#126	1	2	yes	—
102	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	—	8	4	no	—
105	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#123	1	2	yes	—
108	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#124	1	2	yes	—
112	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#125	1	2	yes	—
114	C₂×C₄×C₄₇	#125	1	2	yes	—
115	C₂×C₄×C₄₇	#125	1	2	yes	—
116	C₂×C₄×C₄₇	#125	1	2	yes	—
117	C₂×C₄×C₄₇	#126	1	2	yes	—
118	C₂×C₄×C₄₇	#126	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
123	C₂×Dic₄₇	—	2	4	no	—
124	C₂×Dic₄₇	—	8	2	no	—
125	C₈×C₄₇	—	2	4	no	—
126	C₄₇⋊C₈	—	4	2	no	—
127	C₄₇⋊C₈	—	4	4	no	—
128	#23751	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	4	1	no	—
130	C₂²×C₄₇	—	4	2	no	—
131	C₂²×C₄₇	—	4	1	no	—
132	C₂²×C₄₇	—	4	1	no	—
133	C₂²×C₄₇	—	4	2	no	—
134	C₂²×C₄₇	#400	1	2	yes	—
135	C₂²×C₄₇	#400	1	1	yes	characteristic
136	C₂²×C₄₇	#400	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#126	1	2	yes	—
139	C₄×C₄₇	—	8	4	no	—
140	C₄×C₄₇	#400	1	2	yes	—
141	C₄×C₄₇	#400	1	1	yes	characteristic
142	C₄×C₄₇	#400	1	1	yes	characteristic
143	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
144	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
145	C₄×C₄₇	C₂×#125	1	2	yes	—
146	Dic₄₇	—	4	4	no	—
147	C₂×#430	—	47	1	no	—
148	#2905	—	47	1	no	—
149	#2102	—	47	8	no	—
150	C₂×#344	—	47	2	no	—
151	#1733	—	47	2	no	—
152	C₂×H(3,ℤ/4)	D₄₇	1	1	yes	characteristic
153	C₂×C₄₇	—	2	1	no	—
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	—	2	1	no	—
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	—	2	1	no	—
158	C₂×C₄₇	C₂×#400	1	1	yes	characteristic
159	C₂²×M₁₆	—	47	1	no	—
160	C₂²×M₁₆	—	47	2	no	—
161	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
162	C₂×#117	—	47	1	no	—
163	C₂×#117	C₂×D₄₇	1	2	yes	—
164	#430	—	47	4	no	—
165	#430	—	94	2	no	—
166	C₂×#108	—	94	2	no	—
167	#424	—	94	2	no	—
168	#424	—	94	2	no	—
169	#423	—	47	4	no	—
170	#344	—	47	8	no	—
171	#339	—	47	8	no	—
172	H(3,ℤ/4)	C₂×D₄₇	1	4	yes	—
173	C₄₇	#23751	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	47	4	no	—
178	C₂×M₁₆	—	47	2	no	—
179	C₂×M₁₆	—	94	1	no	—
180	C₂×M₁₆	—	94	2	no	—
181	C₂×M₁₆	—	94	2	no	—
182	C₂×M₁₆	—	94	2	no	—
183	C₂×M₁₆	—	94	2	no	—
184	C₂×M₁₆	—	94	2	no	—
185	C₂×M₁₆	—	94	2	no	—
186	C₂×M₁₆	—	94	2	no	—
187	C₂×M₁₆	D₁₈₈	1	2	yes	—
188	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
189	C₂²×C₈	—	94	1	no	—
190	C₂²×C₈	—	94	2	no	—
191	C₂²×C₈	D₁₈₈	1	2	yes	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	—	47	2	no	—
196	#117	—	47	2	no	—
197	#117	—	94	1	no	—
198	#117	—	94	1	no	—
199	#117	C₂²×D₄₇	1	4	yes	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	94	1	no	—
202	C₂×C₂²⋊C₄	—	2	2	no	—
203	C₂×C₂²⋊C₄	—	94	1	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	47	2	no	—
206	#108	—	94	8	no	—
207	C₄⋊C₈	—	94	8	no	—
208	C₂⁴	#10625352	1	1	yes	characteristic
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	94	2	no	—
211	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
213	C₂²×C₄	#10625350	1	2	yes	—
214	C₂²×C₄	#10625350	1	2	yes	—
215	C₂²×C₄	#10625352	1	1	yes	characteristic
216	C₂²×C₄	#10625352	1	2	yes	—
217	C₂²×C₄	#10625352	1	2	yes	—
218	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	94	4	no	—
224	M₁₆	—	94	4	no	—
225	M₁₆	—	94	4	no	—
226	M₁₆	—	188	1	no	—
227	M₁₆	—	188	1	no	—
228	M₁₆	—	188	2	no	—
229	M₁₆	—	188	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	94	2	no	—
233	C₂×C₈	—	94	4	no	—
234	C₂×C₈	—	188	2	no	—
235	C₂×C₈	—	188	2	no	—
236	C₂×C₈	—	188	2	no	—
237	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	94	2	no	—
241	C₄⋊C₄	—	94	2	no	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	94	2	no	—
245	C₂²⋊C₄	—	94	2	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	47	4	no	—
249	C₄²	—	94	2	no	—
250	C₂³	—	8	1	no	—
251	C₂³	#13094004	1	1	yes	characteristic
252	C₂³	#13094004	1	1	yes	characteristic
253	C₂³	#13094017	1	2	yes	—
254	C₂³	#13094017	1	2	yes	—
255	C₂³	C₂×#10625352	1	1	yes	characteristic
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	94	4	no	—
259	C₂×C₄	—	376	2	no	—
260	C₂×C₄	#13094004	1	1	yes	characteristic
261	C₂×C₄	#13094004	1	1	yes	characteristic
262	C₂×C₄	#13094008	1	1	yes	characteristic
263	C₂×C₄	#13094008	1	1	yes	characteristic
264	C₂×C₄	#13094008	1	2	yes	—
265	C₂×C₄	#13094012	1	1	yes	characteristic
266	C₂×C₄	#13094012	1	2	yes	—
267	C₂×C₄	#13094012	1	1	yes	characteristic
268	C₂×C₄	#13094014	1	2	yes	—
269	C₂×C₄	#13094014	1	2	yes	—
270	C₂×C₄	#13094014	1	2	yes	—
271	C₂×C₄	#13094014	1	2	yes	—
272	C₂×C₄	#13094017	1	2	yes	—
273	C₂×C₄	#13094017	1	2	yes	—
274	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
275	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂×#10625352	1	2	yes	—
277	C₂×C₄	C₂×#10625350	1	2	yes	—
278	C₈	—	2	4	no	—
279	C₈	—	188	2	no	—
280	C₈	—	188	4	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	2	no	—
286	C₂²	#15689063	1	1	yes	characteristic
287	C₂²	#15689063	1	1	yes	characteristic
288	C₂²	#15689063	1	2	yes	—
289	C₂²	C₂×#13094004	1	1	yes	characteristic
290	C₂²	C₂×#13094017	1	2	yes	—
291	C₄	—	8	4	no	—
292	C₄	—	188	4	no	—
293	C₄	#15689063	1	1	yes	characteristic
294	C₄	#15689063	1	1	yes	characteristic
295	C₄	#15689063	1	2	yes	—
296	C₄	C₂×#13094008	1	1	yes	characteristic
297	C₄	C₂×#13094012	1	1	yes	characteristic
298	C₄	C₂×#13094014	1	2	yes	—
299	C₂	—	2	2	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	—	2	1	no	—
304	C₂	C₂×#15689063	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13094012
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×H(3,ℤ/4)	C₂
socle	C₂×C₄₇	C₂×#400
prime radical	C₂⁴×C₄₇	#41
hypercenter	C₂×H(3,ℤ/4)	D₄₇
hypocenter	C₄₇	#23751
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
3. #20244964	C₂	semidirect
173. C₄₇	#23751	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
#20251951	2
#20244964	2
C₂×#15688893	2
#20243097	2
C₂×C₄₇×H(3,ℤ/4)	2
#23751	47