Subgroups — C₄₇×#11373 #26051071

Order 12032.

886 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#340	C₂	1	4	yes	maximal
3	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
4	C₄₇×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
7	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×#384	—	2	2	no	—
11	C₄₇×#384	C₄	1	4	yes	complemented
12	C₂×#95×C₄₇	—	2	2	no	—
13	C₄₇×#340	—	2	8	no	—
14	C₄₇×#336	—	2	2	no	—
15	C₂³×C₄×C₄₇	—	2	2	no	—
16	C₂³×C₄×C₄₇	—	2	2	no	—
17	C₂³×C₄×C₄₇	D₄	1	2	yes	—
18	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₄₇	—	2	2	no	—
20	C₂²×C₈×C₄₇	—	2	2	no	—
21	C₂²×C₈×C₄₇	—	2	4	no	—
22	C₂²×C₈×C₄₇	—	2	4	no	—
23	C₂²×C₈×C₄₇	D₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
26	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
27	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
32	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
33	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₄₇	Q₈	1	4	yes	—
35	C₂×C₄²×C₄₇	—	2	4	no	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₄₇	D₄	1	2	yes	—
38	#95×C₄₇	—	4	4	no	—
39	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	—	2	4	no	—
41	C₂²×C₄×C₄₇	—	2	4	no	—
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	4	2	no	—
45	C₂²×C₄×C₄₇	—	4	2	no	—
46	C₂²×C₄×C₄₇	—	4	2	no	—
47	C₂²×C₄×C₄₇	—	4	2	no	—
48	C₂²×C₄×C₄₇	—	4	4	no	—
49	C₂²×C₄×C₄₇	—	4	4	no	—
50	C₂²×C₄×C₄₇	—	4	4	no	—
51	C₂²×C₄×C₄₇	—	4	4	no	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
53	C₂²×C₄×C₄₇	D₈	1	4	yes	—
54	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₇	#41	1	4	yes	—
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂×C₈×C₄₇	—	2	4	no	—
63	C₂×C₈×C₄₇	—	2	8	no	—
64	C₂×C₈×C₄₇	—	4	2	no	—
65	C₂×C₈×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	—	4	4	no	—
67	C₄⋊C₄×C₄₇	—	2	8	no	—
68	C₄⋊C₄×C₄₇	—	2	8	no	—
69	C₄⋊C₄×C₄₇	—	4	16	no	—
70	C₄⋊C₄×C₄₇	—	4	4	no	—
71	C₄⋊C₄×C₄₇	—	4	4	no	—
72	C₄²×C₄₇	—	2	4	no	—
73	C₄²×C₄₇	—	2	4	no	—
74	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
76	C₂³×C₄₇	#123	1	2	yes	—
77	C₂³×C₄₇	#124	1	1	yes	characteristic
78	C₂³×C₄₇	#127	1	1	yes	characteristic
79	C₂³×C₄₇	C₂×D₈	1	2	yes	—
80	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
81	C₂³×C₄₇	#137	1	4	yes	—
82	C₂×C₄×C₄₇	—	2	4	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	8	no	—
85	C₂×C₄×C₄₇	—	2	8	no	—
86	C₂×C₄×C₄₇	—	4	4	no	—
87	C₂×C₄×C₄₇	—	4	4	no	—
88	C₂×C₄×C₄₇	—	4	8	no	—
89	C₂×C₄×C₄₇	—	4	8	no	—
90	C₂×C₄×C₄₇	—	8	4	no	—
91	C₂×C₄×C₄₇	—	8	4	no	—
92	C₂×C₄×C₄₇	—	8	2	no	—
93	C₂×C₄×C₄₇	—	8	2	no	—
94	C₂×C₄×C₄₇	—	8	2	no	—
95	C₂×C₄×C₄₇	—	8	2	no	—
96	C₂×C₄×C₄₇	#102	1	8	yes	—
97	C₂×C₄×C₄₇	#117	1	2	yes	—
98	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₄₇	#121	1	2	yes	—
101	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
102	C₂×C₄×C₄₇	#123	1	4	yes	—
103	C₂×C₄×C₄₇	#123	1	2	yes	—
104	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#125	1	4	yes	—
106	C₂×C₄×C₄₇	#126	1	4	yes	—
107	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
108	C₈×C₄₇	—	4	4	no	—
109	#11373	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₄₇	#381	1	1	yes	characteristic
111	C₂²×C₄₇	#390	1	1	yes	characteristic
112	C₂²×C₄₇	#391	1	1	yes	characteristic
113	C₂²×C₄₇	#399	1	2	yes	—
114	C₂²×C₄₇	#400	1	2	yes	—
115	C₂²×C₄₇	C₂×#102	1	2	yes	—
116	C₂²×C₄₇	#419	1	2	yes	—
117	C₂²×C₄₇	#474	1	4	yes	—
118	C₂²×C₄₇	#475	1	4	yes	—
119	C₂²×C₄₇	#480	1	4	yes	—
120	C₂²×C₄₇	#483	1	4	yes	—
121	C₂²×C₄₇	#486	1	2	yes	—
122	C₂²×C₄₇	#491	1	2	yes	—
123	C₂²×C₄₇	#492	1	1	yes	characteristic
124	C₂²×C₄₇	#493	1	1	yes	characteristic
125	C₂²×C₄₇	#497	1	2	yes	—
126	C₄×C₄₇	—	2	8	no	—
127	C₄×C₄₇	—	8	4	no	—
128	C₄×C₄₇	—	8	4	no	—
129	C₄×C₄₇	#387	1	4	yes	—
130	C₄×C₄₇	#397	1	2	yes	—
131	C₄×C₄₇	#398	1	2	yes	—
132	C₂×#384	C₂×C₄₇	1	1	yes	characteristic
133	C₂×#340	C₂×C₄₇	1	4	yes	—
134	C₂×#336	C₂×C₄₇	1	1	yes	characteristic
135	C₄⋊#95	C₂×C₄₇	1	1	yes	characteristic
136	C₂×C₄₇	#1445	1	1	yes	characteristic
137	C₂×C₄₇	#1823	1	2	yes	—
138	C₂×C₄₇	#1957	1	1	yes	characteristic
139	C₂×C₄₇	#1960	1	1	yes	characteristic
140	C₂×C₄₇	#1988	1	2	yes	—
141	C₂×C₄₇	#2093	1	2	yes	—
142	C₂×C₄₇	#2096	1	2	yes	—
143	C₂×C₄₇	#2099	1	4	yes	—
144	C₂³×C₈	C₂²×C₄₇	1	2	yes	—
145	C₂²×C₄⋊C₄	—	2	2	no	—
146	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
147	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
148	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
149	#384	—	2	2	no	—
150	#384	C₄×C₄₇	1	4	yes	complemented
151	C₂×#95	—	2	2	no	—
152	#340	—	2	8	no	—
153	#336	—	2	2	no	—
154	C₄₇	#11373	1	1	yes	characteristic, direct product factor, Sylow
155	C₂³×C₄	—	2	2	no	—
156	C₂³×C₄	—	2	2	no	—
157	C₂³×C₄	D₄×C₄₇	1	2	yes	—
158	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	—	2	4	no	—
163	C₂²×C₈	D₄×C₄₇	1	4	yes	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	—	2	8	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	4	4	no	—
172	C₂×C₄⋊C₄	—	4	4	no	—
173	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
174	C₂×C₄⋊C₄	Q₈×C₄₇	1	4	yes	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
177	C₂×C₄²	D₄×C₄₇	1	2	yes	—
178	#95	—	4	4	no	—
179	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	4	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	C₂²⋊C₄×C₄₇	1	4	yes	—
193	C₂²×C₄	D₈×C₄₇	1	4	yes	—
194	C₂²×C₄	SD₁₆×C₄₇	1	4	yes	—
195	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
199	C₂²×C₄	#41×C₄₇	1	4	yes	—
200	C₂²×C₄	#41×C₄₇	1	2	yes	—
201	C₂²×C₄	#41×C₄₇	1	2	yes	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	8	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	4	no	—
207	C₄⋊C₄	—	2	8	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	16	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	4	no	—
214	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
215	C₂³	C₂²⋊Q₈×C₄₇	1	2	yes	—
216	C₂³	#123×C₄₇	1	2	yes	—
217	C₂³	#124×C₄₇	1	1	yes	characteristic
218	C₂³	#127×C₄₇	1	1	yes	characteristic
219	C₂³	C₂×D₈×C₄₇	1	2	yes	—
220	C₂³	C₂×SD₁₆×C₄₇	1	2	yes	—
221	C₂³	#137×C₄₇	1	4	yes	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	—	8	4	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	#102×C₄₇	1	8	yes	—
237	C₂×C₄	#117×C₄₇	1	2	yes	—
238	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
239	C₂×C₄	C₄×Q₈×C₄₇	1	2	yes	—
240	C₂×C₄	#121×C₄₇	1	2	yes	—
241	C₂×C₄	C₂²⋊Q₈×C₄₇	1	4	yes	—
242	C₂×C₄	#123×C₄₇	1	4	yes	—
243	C₂×C₄	#123×C₄₇	1	2	yes	—
244	C₂×C₄	#124×C₄₇	1	1	yes	characteristic
245	C₂×C₄	#125×C₄₇	1	4	yes	—
246	C₂×C₄	#126×C₄₇	1	4	yes	—
247	C₂×C₄	C₄⋊Q₈×C₄₇	1	1	yes	characteristic
248	C₈	—	4	4	no	—
249	C₂²	C₄₇×#381	1	1	yes	characteristic
250	C₂²	C₄₇×#390	1	1	yes	characteristic
251	C₂²	C₄₇×#391	1	1	yes	characteristic
252	C₂²	C₄₇×#399	1	2	yes	—
253	C₂²	C₄₇×#400	1	2	yes	—
254	C₂²	C₂×#102×C₄₇	1	2	yes	—
255	C₂²	C₄₇×#419	1	2	yes	—
256	C₂²	C₄₇×#474	1	4	yes	—
257	C₂²	C₄₇×#475	1	4	yes	—
258	C₂²	C₄₇×#480	1	4	yes	—
259	C₂²	C₄₇×#483	1	4	yes	—
260	C₂²	C₄₇×#486	1	2	yes	—
261	C₂²	C₄₇×#491	1	2	yes	—
262	C₂²	C₄₇×#492	1	1	yes	characteristic
263	C₂²	C₄₇×#493	1	1	yes	characteristic
264	C₂²	C₄₇×#497	1	2	yes	—
265	C₄	—	2	8	no	—
266	C₄	—	8	4	no	—
267	C₄	—	8	4	no	—
268	C₄	C₄₇×#387	1	4	yes	—
269	C₄	C₄₇×#397	1	2	yes	—
270	C₄	C₄₇×#398	1	2	yes	—
271	C₂	C₄₇×#1445	1	1	yes	characteristic
272	C₂	C₄₇×#1823	1	2	yes	—
273	C₂	C₄₇×#1957	1	1	yes	characteristic
274	C₂	C₄₇×#1960	1	1	yes	characteristic
275	C₂	C₄₇×#1988	1	2	yes	—
276	C₂	C₄₇×#2093	1	2	yes	—
277	C₂	C₄₇×#2096	1	2	yes	—
278	C₂	C₄₇×#2099	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₄₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂³×C₄	C₂³×C₄₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₇ ◁ C₂²×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₄₇×#384	C₄	semidirect
109. #11373	C₄₇	direct (N × H)
150. #384	C₄×C₄₇	semidirect
154. C₄₇	#11373	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#384	2
C₂×C₄₇×#340	2
C₂×C₄₇×#336	2
C₄₇×C₄⋊#95	2
#11373	47

Subgroups — C47×#11373 #26051071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#11373 #26051071