Subgroups — C₂×C₄₇×#1322 #26051137

Order 12032.

466 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	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×C₂²⋊C₁₆	C₂	1	2	yes	maximal
3	C₄₇×#1322	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₄₇	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₄₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₄₇	—	2	2	no	—
8	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
9	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
10	C₂×C₄⋊C₈×C₄₇	C₄	1	2	yes	—
11	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
12	C₄₇×C₂²⋊C₁₆	—	2	2	no	—
13	C₄₇×C₂²⋊C₁₆	C₂²	1	4	yes	—
14	C₂³×C₄×C₄₇	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₄₇	C₈	1	1	yes	characteristic
16	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₄₇	—	2	1	no	—
18	C₂²×C₈×C₄₇	—	2	1	no	—
19	C₂²×C₈×C₄₇	—	4	2	no	—
20	C₂²×C₈×C₄₇	—	4	2	no	—
21	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	D₄	1	2	yes	—
24	C₂²×C₈×C₄₇	D₄	1	2	yes	—
25	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	—	2	1	no	—
27	C₂×C₄²×C₄₇	—	2	1	no	—
28	C₂×C₄²×C₄₇	—	2	1	no	—
29	C₂×C₄²×C₄₇	—	2	1	no	—
30	C₂×C₄²×C₄₇	C₈	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₄₇	—	2	4	no	—
33	C₂×C₁₆×C₄₇	—	4	4	no	—
34	C₄⋊C₈×C₄₇	—	2	4	no	—
35	C₄⋊C₈×C₄₇	—	2	4	no	—
36	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	—	2	1	no	—
38	C₂²×C₄×C₄₇	—	2	1	no	—
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	2	1	no	—
43	C₂²×C₄×C₄₇	—	2	1	no	—
44	C₂²×C₄×C₄₇	—	4	2	no	—
45	C₂²×C₄×C₄₇	C₁₆	1	4	yes	complemented
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	—	2	4	no	—
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	—	4	1	no	—
57	C₂×C₈×C₄₇	—	4	2	no	—
58	C₂×C₈×C₄₇	—	4	1	no	—
59	C₂×C₈×C₄₇	—	4	2	no	—
60	C₂×C₈×C₄₇	—	4	2	no	—
61	C₂×C₈×C₄₇	—	4	2	no	—
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₄²×C₄₇	—	2	2	no	—
66	C₄²×C₄₇	—	2	2	no	—
67	C₄²×C₄₇	—	2	2	no	—
68	C₁₆×C₄₇	—	4	4	no	—
69	C₂³×C₄₇	—	2	2	no	—
70	C₂³×C₄₇	—	2	2	no	—
71	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
72	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
73	C₂³×C₄₇	C₂×C₁₆	1	2	yes	—
74	C₂³×C₄₇	M₃₂	1	2	yes	—
75	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	4	2	no	—
89	C₂×C₄×C₄₇	—	4	2	no	—
90	C₂×C₄×C₄₇	—	4	2	no	—
91	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#99	1	2	yes	—
94	C₂×C₄×C₄₇	#101	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×C₁₆	1	4	yes	complemented
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
98	C₈×C₄₇	—	2	2	no	—
99	C₈×C₄₇	—	2	2	no	—
100	C₈×C₄₇	—	4	2	no	—
101	C₂×#1322	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₄₇	—	2	4	no	—
103	C₂²×C₄₇	—	2	4	no	—
104	C₂²×C₄₇	—	2	1	no	—
105	C₂²×C₄₇	—	2	1	no	—
106	C₂²×C₄₇	—	2	1	no	—
107	C₂²×C₄₇	—	2	1	no	—
108	C₂²×C₄₇	#324	1	2	yes	—
109	C₂²×C₄₇	C₂²⋊C₁₆	1	2	yes	—
110	C₂²×C₄₇	C₂²⋊C₁₆	1	2	yes	—
111	C₂²×C₄₇	#349	1	2	yes	—
112	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
113	C₂²×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
115	C₄×C₄₇	—	2	4	no	—
116	C₄×C₄₇	—	2	1	no	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	—	4	2	no	—
119	C₄×C₄₇	#324	1	2	yes	—
120	C₄×C₄₇	C₂×#99	1	1	yes	characteristic
121	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
122	C₂²×C₄⋊C₈	C₂×C₄₇	1	1	yes	characteristic
123	C₂×C₂²⋊C₁₆	C₂×C₄₇	1	2	yes	—
124	#1322	C₂×C₄₇	1	4	yes	direct product factor
125	C₂×C₄₇	—	2	2	no	—
126	C₂×C₄₇	—	2	1	no	—
127	C₂×C₄₇	—	2	1	no	—
128	C₂×C₄₇	#1322	1	4	yes	direct product factor
129	C₂×C₄₇	C₂×#324	1	1	yes	characteristic
130	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
131	C₂×C₄₇	C₂×#349	1	1	yes	characteristic
132	C₂³×C₈	C₄×C₄₇	1	1	yes	characteristic
133	C₂³×C₈	C₂²×C₄₇	1	1	yes	characteristic
134	C₂²×C₄²	C₄×C₄₇	1	1	yes	characteristic
135	C₂²×C₁₆	—	2	2	no	—
136	C₂×C₄⋊C₈	—	2	2	no	—
137	C₂×C₄⋊C₈	—	2	2	no	—
138	C₂×C₄⋊C₈	C₄×C₄₇	1	2	yes	—
139	C₂×C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
140	C₂²⋊C₁₆	—	2	2	no	—
141	C₂²⋊C₁₆	C₂²×C₄₇	1	4	yes	—
142	C₄₇	C₂×#1322	1	1	yes	characteristic, direct product factor, Sylow
143	C₂³×C₄	C₈×C₄₇	1	1	yes	characteristic
144	C₂³×C₄	C₈×C₄₇	1	1	yes	characteristic
145	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	4	2	no	—
149	C₂²×C₈	—	4	2	no	—
150	C₂²×C₈	C₂×C₄×C₄₇	1	2	yes	—
151	C₂²×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
152	C₂²×C₈	D₄×C₄₇	1	2	yes	—
153	C₂²×C₈	D₄×C₄₇	1	2	yes	—
154	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	—	2	1	no	—
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	C₈×C₄₇	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
161	C₂×C₁₆	—	2	4	no	—
162	C₂×C₁₆	—	4	4	no	—
163	C₄⋊C₈	—	2	4	no	—
164	C₄⋊C₈	—	2	4	no	—
165	C₂⁴	C₂×C₈×C₄₇	1	1	yes	characteristic
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₁₆×C₄₇	1	4	yes	complemented
175	C₂²×C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₈×C₄₇	1	2	yes	—
178	C₂²×C₄	C₂×C₈×C₄₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₈×C₄₇	1	2	yes	—
180	C₂²×C₄	M₁₆×C₄₇	1	1	yes	characteristic
181	C₂²×C₄	M₁₆×C₄₇	1	1	yes	characteristic
182	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	C₂²⋊C₄×C₄₇	1	2	yes	—
192	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₁₆	—	4	4	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	C₂²⋊C₈×C₄₇	1	1	yes	characteristic
201	C₂³	C₂²⋊C₈×C₄₇	1	1	yes	characteristic
202	C₂³	C₂×C₁₆×C₄₇	1	2	yes	—
203	C₂³	M₃₂×C₄₇	1	2	yes	—
204	C₂³	C₂²×C₈×C₄₇	1	1	yes	characteristic
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₂²⋊C₈×C₄₇	1	1	yes	characteristic
221	C₂×C₄	C₂²⋊C₈×C₄₇	1	1	yes	characteristic
222	C₂×C₄	#99×C₄₇	1	2	yes	—
223	C₂×C₄	#101×C₄₇	1	2	yes	—
224	C₂×C₄	C₂×C₁₆×C₄₇	1	4	yes	complemented
225	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
226	C₂×C₄	C₂×M₁₆×C₄₇	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	4	2	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	C₄₇×#324	1	2	yes	—
237	C₂²	C₄₇×C₂²⋊C₁₆	1	2	yes	—
238	C₂²	C₄₇×C₂²⋊C₁₆	1	2	yes	—
239	C₂²	C₄₇×#349	1	2	yes	—
240	C₂²	C₂×C₂²⋊C₈×C₄₇	1	1	yes	characteristic
241	C₂²	C₂²×C₁₆×C₄₇	1	1	yes	characteristic
242	C₂²	C₂×M₃₂×C₄₇	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	2	no	—
247	C₄	C₄₇×#324	1	2	yes	—
248	C₄	C₂×#99×C₄₇	1	1	yes	characteristic
249	C₄	C₂×#101×C₄₇	1	1	yes	characteristic
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	C₄₇×#1322	1	4	yes	direct product factor
254	C₂	C₂×C₄₇×#324	1	1	yes	characteristic
255	C₂	C₂×C₄₇×C₂²⋊C₁₆	1	1	yes	characteristic
256	C₂	C₂×C₄₇×#349	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₄₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₁₆×C₄₇
Frattini, cosocle	C₂²×C₈	C₂³×C₄₇
socle	C₂³×C₄₇	C₂²⋊C₈
prime radical	C₂⁴×C₄₇	C₂×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₇ ◁ C₂³×C₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₇×#1322	C₂	direct (N × H)
45. C₂²×C₄×C₄₇	C₁₆	semidirect
95. C₂×C₄×C₄₇	C₂×C₁₆	semidirect
101. C₂×#1322	C₄₇	direct (N × H)
124. #1322	C₂×C₄₇	direct (N × H)
128. C₂×C₄₇	#1322	direct (N × H)
142. C₄₇	C₂×#1322	direct (N × H)
174. C₂²×C₄	C₁₆×C₄₇	semidirect
224. C₂×C₄	C₂×C₁₆×C₄₇	semidirect
253. C₂	C₄₇×#1322	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₈×C₄₇	2
C₂×C₄₇×C₂²⋊C₁₆	2
C₄₇×#1322	2
C₂×#1322	47

Subgroups — C2×C47×#1322 #26051137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#1322 #26051137