Subgroups — #26106971

Order 12032.

412 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252033	C₂	1	1	yes	maximal, characteristic
2	#20241771	C₂	1	1	yes	maximal, characteristic
3	#20241768	C₂	1	1	yes	maximal, characteristic
4	#20241681	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10625350	C₂²	1	1	yes	characteristic
9	#15689185	—	2	2	no	—
10	#15689176	C₂²	1	1	yes	characteristic
11	#15689174	—	2	1	no	—
12	#15689174	—	2	1	no	—
13	#15689151	C₄	1	2	yes	—
14	#15689091	C₄	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	#15688855	C₄	1	2	yes	—
18	#15688853	C₄	1	2	yes	—
19	#15688852	C₄	1	2	yes	—
20	#15688850	C₄	1	2	yes	—
21	#15688845	—	2	2	no	—
22	#15688845	—	2	1	no	—
23	#15688844	—	2	1	no	—
24	C₄₇×#431	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄₇⋊C₈	—	2	1	no	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
32	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
33	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₄₇	—	2	1	no	—
35	C₄×C₂²⋊D₄₇	—	2	1	no	—
36	#13093983	C₂×C₄	1	1	yes	characteristic
37	#13093981	—	2	1	no	—
38	C₄²×D₄₇	—	2	1	no	—
39	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
40	Dic₄₇⋊C₈	—	2	2	no	—
41	Dic₄₇⋊C₈	—	2	2	no	—
42	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
43	(C₂×D₄₇)⋊C₈	—	2	2	no	—
44	(C₂×D₄₇)⋊C₈	—	2	2	no	—
45	C₄×C₄₇⋊C₈	—	2	1	no	—
46	C₄×C₄₇⋊C₈	—	2	1	no	—
47	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
48	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
49	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
50	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
51	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₄₇	—	2	1	no	—
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₄₇	—	2	1	no	—
56	#97×C₄₇	—	2	1	no	—
57	C₄×C₈×C₄₇	—	2	1	no	—
58	C₄×C₈×C₄₇	—	2	1	no	—
59	C₄×C₈×C₄₇	—	2	1	no	—
60	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₇	—	2	1	no	—
65	#10625350	—	2	2	no	—
66	#10625350	—	2	2	no	—
67	C₂×D₁₈₈	C₄²	1	1	yes	characteristic
68	C₂×C₄×D₄₇	—	2	1	no	—
69	C₂×C₄×D₄₇	—	2	1	no	—
70	C₂×C₄×D₄₇	—	2	1	no	—
71	C₂×Dic₉₄	C₄²	1	1	yes	characteristic
72	M₁₆×C₄₇	—	2	2	no	—
73	M₁₆×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²⋊Dic₄₇	—	2	1	no	—
83	(C₂×D₄₇)⋊C₄	—	2	1	no	—
84	(C₂×D₄₇)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₄₇	C₄²	1	1	yes	characteristic
86	C₄⋊Dic₄₇	C₄²	1	1	yes	characteristic
87	Dic₄₇⋊C₄	—	2	1	no	—
88	Dic₄₇⋊C₄	—	2	1	no	—
89	C₄×Dic₄₇	—	2	1	no	—
90	C₄×Dic₄₇	—	2	1	no	—
91	C₄×Dic₄₇	—	2	1	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
98	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
99	C₂³×C₄₇	#117	1	1	yes	characteristic
100	C₂²×D₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#131	1	2	yes	—
110	C₂×C₄×C₄₇	#131	1	2	yes	—
111	C₂²⋊D₄₇	—	4	2	no	—
112	C₂×Dic₄₇	—	2	1	no	—
113	C₂×Dic₄₇	—	2	1	no	—
114	C₂×Dic₄₇	—	2	1	no	—
115	D₁₈₈	—	2	1	no	—
116	D₁₈₈	—	2	1	no	—
117	C₄×D₄₇	—	4	2	no	—
118	C₄×D₄₇	—	4	2	no	—
119	C₄×D₄₇	—	4	2	no	—
120	Dic₉₄	—	2	1	no	—
121	Dic₉₄	—	2	1	no	—
122	C₈×C₄₇	—	4	2	no	—
123	C₈×C₄₇	—	4	2	no	—
124	C₈×C₄₇	—	4	2	no	—
125	C₄₇⋊C₈	—	2	2	no	—
126	C₄₇⋊C₈	—	2	2	no	—
127	C₄₇⋊C₈	—	4	2	no	—
128	C₄₇⋊C₈	—	4	2	no	—
129	#8188	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₄₇	#433	1	2	yes	—
134	C₂×D₄₇	—	4	1	no	—
135	C₂×D₄₇	—	4	1	no	—
136	C₂×D₄₇	—	4	1	no	—
137	C₄×C₄₇	—	2	1	no	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	#405	1	1	yes	characteristic
140	C₄×C₄₇	#405	1	1	yes	characteristic
141	C₄×C₄₇	#408	1	2	yes	—
142	C₄×C₄₇	#444	1	2	yes	—
143	C₄×C₄₇	#444	1	2	yes	—
144	Dic₄₇	—	4	1	no	—
145	Dic₄₇	—	4	1	no	—
146	Dic₄₇	—	4	1	no	—
147	#2922	—	47	1	no	—
148	#1476	—	47	1	no	—
149	#1476	—	47	1	no	—
150	#1452	—	47	1	no	—
151	C₂×#322	—	47	1	no	—
152	#1278	—	47	1	no	—
153	#1278	D₄₇	1	1	yes	characteristic, complemented
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	#1750	1	1	yes	characteristic
156	C₂×C₄₇	#1765	1	1	yes	characteristic
157	C₂×C₄₇	#1765	1	1	yes	characteristic
158	D₄₇	—	8	1	no	—
159	C₄×#41	—	47	1	no	—
160	#433	—	94	2	no	—
161	#431	—	47	1	no	—
162	#431	—	94	1	no	—
163	#431	—	94	1	no	—
164	#431	C₂×D₄₇	1	1	yes	characteristic
165	#424	—	47	2	no	—
166	#408	—	47	2	no	—
167	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
168	C₂×C₄×C₈	—	47	1	no	—
169	C₂×C₄×C₈	—	47	1	no	—
170	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
171	Q₈⋊C₈	—	47	2	no	—
172	Q₈⋊C₈	—	47	2	no	—
173	D₄⋊C₈	—	47	2	no	—
174	D₄⋊C₈	—	47	2	no	—
175	#322	—	94	1	no	—
176	#322	—	94	2	no	—
177	#322	—	94	1	no	—
178	C₄₇	#8188	1	1	yes	characteristic, complemented, Sylow
179	C₂×#41	—	47	1	no	—
180	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	47	2	no	—
183	C₂²×C₈	—	94	1	no	—
184	C₄×Q₈	—	47	1	no	—
185	C₄×Q₈	—	47	1	no	—
186	C₄×D₄	—	47	1	no	—
187	C₄×D₄	—	47	1	no	—
188	C₄×D₄	—	94	1	no	—
189	C₄×D₄	—	94	1	no	—
190	#117	—	47	1	no	—
191	#117	—	94	1	no	—
192	C₂×C₄²	—	94	1	no	—
193	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
194	C₄⋊C₈	—	47	2	no	—
195	C₄⋊C₈	—	94	2	no	—
196	C₄⋊C₈	—	94	2	no	—
197	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
198	C₂²⋊C₈	—	2	1	no	—
199	C₂²⋊C₈	—	94	2	no	—
200	C₂²⋊C₈	—	94	2	no	—
201	C₂²⋊C₈	—	94	1	no	—
202	#97	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	47	2	no	—
207	C₄×C₈	—	47	2	no	—
208	C₄×C₈	—	47	1	no	—
209	C₄×C₈	—	47	1	no	—
210	C₄×C₈	—	94	1	no	—
211	C₄×C₈	—	94	1	no	—
212	C₄×C₈	C₄×D₄₇	1	2	yes	—
213	#41	—	94	2	no	—
214	#41	—	94	2	no	—
215	C₂×Q₈	—	47	1	no	—
216	C₂×D₄	—	47	1	no	—
217	C₂×D₄	—	94	1	no	—
218	C₂²×C₄	—	94	1	no	—
219	C₂²×C₄	—	94	1	no	—
220	C₂²×C₄	—	94	1	no	—
221	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
222	C₂²×C₄	#10625352	1	1	yes	characteristic
223	C₂²×C₄	#10625354	1	1	yes	characteristic
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	47	2	no	—
231	C₂×C₈	—	47	2	no	—
232	C₂×C₈	—	94	2	no	—
233	C₂×C₈	—	94	2	no	—
234	C₂×C₈	—	94	2	no	—
235	C₂×C₈	—	94	2	no	—
236	C₂×C₈	—	188	2	no	—
237	C₄⋊C₄	—	47	1	no	—
238	C₄⋊C₄	—	47	1	no	—
239	C₄⋊C₄	—	94	1	no	—
240	C₄⋊C₄	—	94	1	no	—
241	C₂²⋊C₄	—	94	1	no	—
242	C₂²⋊C₄	—	94	1	no	—
243	C₂²⋊C₄	—	94	1	no	—
244	C₄²	—	94	1	no	—
245	C₄²	—	94	1	no	—
246	C₄²	—	94	1	no	—
247	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
249	C₄²	D₄×D₄₇	1	1	yes	characteristic
250	C₄²	D₄×D₄₇	1	1	yes	characteristic
251	C₂³	—	94	1	no	—
252	C₂³	#13093982	1	1	yes	characteristic
253	Q₈	—	94	1	no	—
254	Q₈	—	94	1	no	—
255	D₄	—	94	1	no	—
256	D₄	—	94	1	no	—
257	D₄	—	188	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	94	1	no	—
261	C₂×C₄	—	94	1	no	—
262	C₂×C₄	—	94	1	no	—
263	C₂×C₄	—	188	2	no	—
264	C₂×C₄	—	188	2	no	—
265	C₂×C₄	—	188	2	no	—
266	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
268	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
269	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
270	C₂×C₄	#13093986	1	1	yes	characteristic
271	C₂×C₄	#13093986	1	1	yes	characteristic
272	C₂×C₄	#13094030	1	2	yes	—
273	C₂×C₄	#13094032	1	2	yes	—
274	C₈	—	4	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	2	no	—
277	C₈	—	94	2	no	—
278	C₈	—	94	2	no	—
279	C₈	—	188	2	no	—
280	C₈	—	188	2	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	188	1	no	—
284	C₂²	—	188	1	no	—
285	C₂²	—	188	1	no	—
286	C₂²	#15688970	1	1	yes	characteristic
287	C₂²	#15689183	1	2	yes	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	188	1	no	—
291	C₄	—	188	1	no	—
292	C₄	—	188	1	no	—
293	C₄	#15689077	1	1	yes	characteristic
294	C₄	#15689080	1	1	yes	characteristic
295	C₄	#15689088	1	2	yes	—
296	C₄	#15689242	1	2	yes	—
297	C₄	#15689244	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	376	1	no	—
300	C₂	#20243153	1	1	yes	characteristic
301	C₂	#20243229	1	1	yes	characteristic
302	C₂	#20243231	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1278	C₂
socle	C₂²×C₄₇	C₄×C₂²⋊C₄
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1278	D₄₇
hypocenter	C₄₇	#8188
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
4. #20241681	C₂	semidirect
5. C₂×#15688845	C₂	semidirect
6. #20241226	C₂	semidirect
7. C₄₇×#1278	C₂	semidirect
153. #1278	D₄₇	semidirect
178. C₄₇	#8188	semidirect

Maximal subgroups

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

subgroup	index
#20252033	2
#20241771	2
#20241768	2
#20241681	2
C₂×#15688845	2
#20241226	2
C₄₇×#1278	2
#8188	47