Subgroups — C₄₇×#14335 #26054033

Order 12032.

6494 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#381	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₇×#379	C₂	1	4	yes	maximal, complemented
4	C₄₇×#1438	C₂	1	8	yes	maximal, complemented
5	C₂⁶×C₄₇	C₂²	1	2	yes	—
6	C₂³×D₄×C₄₇	—	2	16	no	—
7	C₂³×D₄×C₄₇	—	2	8	no	—
8	C₂³×D₄×C₄₇	C₄	1	8	yes	complemented
9	C₂³×D₄×C₄₇	C₂²	1	4	yes	—
10	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
12	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	16	yes	complemented
13	C₄₇×#381	—	2	8	no	—
14	C₄₇×#379	—	2	16	no	—
15	C₂×#95×C₄₇	C₂²	1	8	yes	complemented
16	C₂⁵×C₄₇	—	2	8	no	—
17	C₂⁵×C₄₇	—	2	8	no	—
18	C₂⁵×C₄₇	—	4	16	no	—
19	C₂⁵×C₄₇	C₂×C₄	1	8	yes	—
20	C₂⁵×C₄₇	D₄	1	16	yes	—
21	C₂⁵×C₄₇	C₂³	1	4	yes	—
22	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×D₄×C₄₇	—	2	128	no	—
24	C₂²×D₄×C₄₇	—	2	64	no	—
25	C₂²×D₄×C₄₇	—	2	32	no	—
26	C₂²×D₄×C₄₇	—	2	32	no	—
27	C₂²×D₄×C₄₇	—	2	16	no	—
28	C₂²×D₄×C₄₇	C₂×C₄	1	16	yes	—
29	C₂³×C₄×C₄₇	—	2	8	no	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₄₇	D₄	1	8	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	8	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
34	C₂×C₂²⋊C₄×C₄₇	—	2	16	no	—
35	C₂×C₂²⋊C₄×C₄₇	—	2	8	no	—
36	C₂×C₂²⋊C₄×C₄₇	—	2	32	no	—
37	C₂×C₂²⋊C₄×C₄₇	—	4	32	no	—
38	#95×C₄₇	—	2	16	no	—
39	C₂⁴×C₄₇	—	2	8	no	—
40	C₂⁴×C₄₇	—	2	16	no	—
41	C₂⁴×C₄₇	—	2	16	no	—
42	C₂⁴×C₄₇	—	2	64	no	—
43	C₂⁴×C₄₇	—	2	8	no	—
44	C₂⁴×C₄₇	—	2	8	no	—
45	C₂⁴×C₄₇	—	2	32	no	—
46	C₂⁴×C₄₇	—	4	32	no	—
47	C₂⁴×C₄₇	—	4	64	no	—
48	C₂⁴×C₄₇	—	4	64	no	—
49	C₂⁴×C₄₇	—	4	64	no	—
50	C₂⁴×C₄₇	C₂²⋊C₄	1	32	yes	complemented
51	C₂⁴×C₄₇	C₂²×C₄	1	8	yes	—
52	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂⁴×C₄₇	C₂×D₄	1	16	yes	—
54	C₂⁴×C₄₇	C₂×D₄	1	8	yes	—
55	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
56	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₂×D₄×C₄₇	—	2	128	no	—
58	C₂×D₄×C₄₇	—	2	256	no	—
59	C₂×D₄×C₄₇	—	2	64	no	—
60	C₂×D₄×C₄₇	—	2	32	no	—
61	C₂²×C₄×C₄₇	—	2	16	no	—
62	C₂²×C₄×C₄₇	—	2	32	no	—
63	C₂²×C₄×C₄₇	—	2	8	no	—
64	C₂²×C₄×C₄₇	—	2	16	no	—
65	C₂²×C₄×C₄₇	—	2	8	no	—
66	C₂²×C₄×C₄₇	—	2	32	no	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	16	yes	complemented
68	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
70	C₂²⋊C₄×C₄₇	—	4	64	no	—
71	C₂²⋊C₄×C₄₇	—	4	16	no	—
72	C₂³×C₄₇	—	2	8	no	—
73	C₂³×C₄₇	—	2	32	no	—
74	C₂³×C₄₇	—	2	16	no	—
75	C₂³×C₄₇	—	2	8	no	—
76	C₂³×C₄₇	—	2	8	no	—
77	C₂³×C₄₇	—	2	32	no	—
78	C₂³×C₄₇	—	2	32	no	—
79	C₂³×C₄₇	—	2	32	no	—
80	C₂³×C₄₇	—	2	32	no	—
81	C₂³×C₄₇	—	2	64	no	—
82	C₂³×C₄₇	—	4	64	no	—
83	C₂³×C₄₇	—	4	64	no	—
84	C₂³×C₄₇	—	4	64	no	—
85	C₂³×C₄₇	—	4	256	no	—
86	C₂³×C₄₇	—	4	32	no	—
87	C₂³×C₄₇	—	4	32	no	—
88	C₂³×C₄₇	C₂×C₂²⋊C₄	1	16	yes	—
89	C₂³×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
90	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
91	C₂³×C₄₇	#120	1	16	yes	—
92	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
94	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
95	C₂³×C₄₇	H(5,𝔽₂)	1	8	yes	—
96	D₄×C₄₇	—	2	128	no	—
97	C₂×C₄×C₄₇	—	2	8	no	—
98	C₂×C₄×C₄₇	—	2	16	no	—
99	C₂×C₄×C₄₇	—	2	32	no	—
100	C₂×C₄×C₄₇	—	4	8	no	—
101	C₂×C₄×C₄₇	—	4	32	no	—
102	C₂×C₄×C₄₇	—	4	16	no	—
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
104	#14335	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₄₇	—	2	16	no	—
106	C₂²×C₄₇	—	2	8	no	—
107	C₂²×C₄₇	—	2	32	no	—
108	C₂²×C₄₇	—	2	32	no	—
109	C₂²×C₄₇	—	2	16	no	—
110	C₂²×C₄₇	—	4	128	no	—
111	C₂²×C₄₇	—	4	16	no	—
112	C₂²×C₄₇	—	4	16	no	—
113	C₂²×C₄₇	—	4	64	no	—
114	C₂²×C₄₇	#379	1	8	yes	—
115	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	2	yes	—
116	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₄₇	#518	1	4	yes	—
118	C₂²×C₄₇	C₂×#120	1	4	yes	—
119	C₂²×C₄₇	#534	1	16	yes	—
120	C₂²×C₄₇	#535	1	8	yes	—
121	C₄×C₄₇	—	2	8	no	—
122	C₄×C₄₇	—	8	8	no	—
123	C₂⁴×D₄	C₂×C₄₇	1	1	yes	characteristic
124	C₂×#381	C₂×C₄₇	1	2	yes	complemented
125	C₂×#379	C₂×C₄₇	1	4	yes	complemented
126	#1438	C₂×C₄₇	1	8	yes	complemented
127	C₂×C₄₇	—	2	8	no	—
128	C₂×C₄₇	—	4	16	no	—
129	C₂×C₄₇	C₂×#379	1	1	yes	characteristic
130	C₂×C₄₇	#2308	1	4	yes	—
131	C₂×C₄₇	#2309	1	2	yes	—
132	C₂×C₄₇	#2613	1	8	yes	—
133	C₂⁶	C₂²×C₄₇	1	2	yes	—
134	C₂³×D₄	—	2	16	no	—
135	C₂³×D₄	—	2	8	no	—
136	C₂³×D₄	C₄×C₄₇	1	8	yes	complemented
137	C₂³×D₄	C₂²×C₄₇	1	4	yes	—
138	C₂⁴×C₄	C₂²×C₄₇	1	1	yes	characteristic
139	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	4	yes	—
140	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	16	yes	complemented
141	#381	—	2	8	no	—
142	#379	—	2	16	no	—
143	C₂×#95	C₂²×C₄₇	1	8	yes	complemented
144	C₄₇	#14335	1	1	yes	characteristic, direct product factor, Sylow
145	C₂⁵	—	2	8	no	—
146	C₂⁵	—	2	8	no	—
147	C₂⁵	—	4	16	no	—
148	C₂⁵	C₂×C₄×C₄₇	1	8	yes	—
149	C₂⁵	D₄×C₄₇	1	16	yes	—
150	C₂⁵	C₂³×C₄₇	1	4	yes	—
151	C₂⁵	C₂³×C₄₇	1	1	yes	characteristic
152	C₂²×D₄	—	2	128	no	—
153	C₂²×D₄	—	2	64	no	—
154	C₂²×D₄	—	2	32	no	—
155	C₂²×D₄	—	2	32	no	—
156	C₂²×D₄	—	2	16	no	—
157	C₂²×D₄	C₂×C₄×C₄₇	1	16	yes	—
158	C₂³×C₄	—	2	8	no	—
159	C₂³×C₄	C₂×C₄×C₄₇	1	4	yes	—
160	C₂³×C₄	D₄×C₄₇	1	8	yes	—
161	C₂³×C₄	C₂³×C₄₇	1	8	yes	—
162	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
163	C₂×C₂²⋊C₄	—	2	16	no	—
164	C₂×C₂²⋊C₄	—	2	8	no	—
165	C₂×C₂²⋊C₄	—	2	32	no	—
166	C₂×C₂²⋊C₄	—	4	32	no	—
167	#95	—	2	16	no	—
168	C₂⁴	—	2	8	no	—
169	C₂⁴	—	2	16	no	—
170	C₂⁴	—	2	16	no	—
171	C₂⁴	—	2	64	no	—
172	C₂⁴	—	2	8	no	—
173	C₂⁴	—	2	8	no	—
174	C₂⁴	—	2	32	no	—
175	C₂⁴	—	4	32	no	—
176	C₂⁴	—	4	64	no	—
177	C₂⁴	—	4	64	no	—
178	C₂⁴	—	4	64	no	—
179	C₂⁴	C₂²⋊C₄×C₄₇	1	32	yes	complemented
180	C₂⁴	C₂²×C₄×C₄₇	1	8	yes	—
181	C₂⁴	C₂²×C₄×C₄₇	1	2	yes	—
182	C₂⁴	C₂×D₄×C₄₇	1	16	yes	—
183	C₂⁴	C₂×D₄×C₄₇	1	8	yes	—
184	C₂⁴	C₂×D₄×C₄₇	1	4	yes	—
185	C₂⁴	C₂⁴×C₄₇	1	1	yes	characteristic
186	C₂×D₄	—	2	128	no	—
187	C₂×D₄	—	2	256	no	—
188	C₂×D₄	—	2	64	no	—
189	C₂×D₄	—	2	32	no	—
190	C₂²×C₄	—	2	16	no	—
191	C₂²×C₄	—	2	32	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	16	no	—
195	C₂²×C₄	—	2	32	no	—
196	C₂²×C₄	C₂²⋊C₄×C₄₇	1	16	yes	complemented
197	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
198	C₂²×C₄	C₂×D₄×C₄₇	1	8	yes	—
199	C₂²⋊C₄	—	4	64	no	—
200	C₂²⋊C₄	—	4	16	no	—
201	C₂³	—	2	8	no	—
202	C₂³	—	2	32	no	—
203	C₂³	—	2	16	no	—
204	C₂³	—	2	8	no	—
205	C₂³	—	2	8	no	—
206	C₂³	—	2	32	no	—
207	C₂³	—	2	32	no	—
208	C₂³	—	2	32	no	—
209	C₂³	—	2	32	no	—
210	C₂³	—	2	64	no	—
211	C₂³	—	4	64	no	—
212	C₂³	—	4	64	no	—
213	C₂³	—	4	64	no	—
214	C₂³	—	4	256	no	—
215	C₂³	—	4	32	no	—
216	C₂³	—	4	32	no	—
217	C₂³	C₂×C₂²⋊C₄×C₄₇	1	16	yes	—
218	C₂³	C₂×C₂²⋊C₄×C₄₇	1	8	yes	—
219	C₂³	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
220	C₂³	#120×C₄₇	1	16	yes	—
221	C₂³	C₂³×C₄×C₄₇	1	1	yes	characteristic
222	C₂³	C₂²×D₄×C₄₇	1	4	yes	—
223	C₂³	C₂²×D₄×C₄₇	1	2	yes	—
224	C₂³	H(5,𝔽₂)×C₄₇	1	8	yes	—
225	D₄	—	2	128	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	16	no	—
228	C₂×C₄	—	2	32	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	—	4	32	no	—
231	C₂×C₄	—	4	16	no	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	8	yes	—
233	C₂²	—	2	16	no	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	32	no	—
236	C₂²	—	2	16	no	—
237	C₂²	—	2	32	no	—
238	C₂²	—	4	128	no	—
239	C₂²	—	4	16	no	—
240	C₂²	—	4	16	no	—
241	C₂²	—	4	64	no	—
242	C₂²	C₄₇×#379	1	8	yes	—
243	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	2	yes	—
244	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
245	C₂²	C₄₇×#518	1	4	yes	—
246	C₂²	C₂×#120×C₄₇	1	4	yes	—
247	C₂²	C₄₇×#534	1	16	yes	—
248	C₂²	C₄₇×#535	1	8	yes	—
249	C₄	—	2	8	no	—
250	C₄	—	8	8	no	—
251	C₂	—	2	8	no	—
252	C₂	—	4	16	no	—
253	C₂	C₂×C₄₇×#379	1	1	yes	characteristic
254	C₂	C₄₇×#2308	1	4	yes	—
255	C₂	C₄₇×#2309	1	2	yes	—
256	C₂	C₄₇×#2613	1	8	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₇×#381	C₂	semidirect
3. C₂×C₄₇×#379	C₂	semidirect
4. C₄₇×#1438	C₂	semidirect
8. C₂³×D₄×C₄₇	C₄	semidirect
12. C₂²×C₂²⋊C₄×C₄₇	C₂²	semidirect
15. C₂×#95×C₄₇	C₂²	semidirect
50. C₂⁴×C₄₇	C₂²⋊C₄	semidirect
67. C₂²×C₄×C₄₇	C₂²⋊C₄	semidirect
104. #14335	C₄₇	direct (N × H)
124. C₂×#381	C₂×C₄₇	semidirect
125. C₂×#379	C₂×C₄₇	semidirect
126. #1438	C₂×C₄₇	semidirect
136. C₂³×D₄	C₄×C₄₇	semidirect
140. C₂²×C₂²⋊C₄	C₂²×C₄₇	semidirect
143. C₂×#95	C₂²×C₄₇	semidirect
144. C₄₇	#14335	direct (N × H)
179. C₂⁴	C₂²⋊C₄×C₄₇	semidirect
196. C₂²×C₄	C₂²⋊C₄×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₄₇	2
C₂×C₄₇×#381	2
C₂×C₄₇×#379	2
C₄₇×#1438	2
#14335	47

Subgroups — C47×#14335 #26054033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#14335 #26054033