Subgroups — C₄₇×#9848 #26049546

Order 12032.

422 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3170	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#1994	C₂	1	2	yes	maximal
3	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1567	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1493	C₂	1	2	yes	maximal
6	C₄₇×#533	C₂²	1	2	yes	—
7	C₄₇×#500	C₄	1	1	yes	characteristic, complemented
8	C₄₇×#494	C₄	1	1	yes	characteristic, complemented
9	C₄₇×#487	C₄	1	2	yes	complemented
10	C₄₇×#467	—	2	2	no	—
11	C₄×Q₁₆×C₄₇	—	2	2	no	—
12	C₄₇×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#339	—	2	4	no	—
17	C₄₇×#336	—	2	2	no	—
18	C₄₇×#335	—	2	2	no	—
19	C₄₇×Q₈⋊C₈	—	2	4	no	—
20	C₂×Q₁₆×C₄₇	—	2	2	no	—
21	C₂×M₁₆×C₄₇	—	2	2	no	—
22	C₂²×C₈×C₄₇	—	2	1	no	—
23	C₂²×C₈×C₄₇	D₄	1	2	yes	—
24	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
26	#125×C₄₇	—	2	2	no	—
27	C₂²⋊Q₈×C₄₇	—	2	2	no	—
28	C₂²⋊Q₈×C₄₇	—	2	2	no	—
29	C₄×Q₈×C₄₇	—	2	2	no	—
30	C₄×Q₈×C₄₇	—	2	2	no	—
31	#117×C₄₇	—	2	2	no	—
32	C₂×C₄²×C₄₇	—	2	1	no	—
33	C₂×C₄²×C₄₇	—	2	1	no	—
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	#107×C₄₇	—	2	2	no	—
37	C₄⋊C₈×C₄₇	—	2	2	no	—
38	C₄⋊C₈×C₄₇	—	2	2	no	—
39	#103×C₄₇	—	2	4	no	—
40	C₄×C₈×C₄₇	—	2	1	no	—
41	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₄₇	D₄	1	2	yes	—
44	C₄×C₈×C₄₇	D₄	1	2	yes	—
45	C₂×Q₈×C₄₇	—	2	2	no	—
46	C₂×Q₈×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
51	Q₁₆×C₄₇	—	4	4	no	—
52	M₁₆×C₄₇	—	4	4	no	—
53	C₂×C₈×C₄₇	—	2	2	no	—
54	C₂×C₈×C₄₇	—	4	2	no	—
55	C₂×C₈×C₄₇	—	4	2	no	—
56	C₂×C₈×C₄₇	—	4	2	no	—
57	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄×C₄₇	—	2	2	no	—
60	C₄⋊C₄×C₄₇	—	2	2	no	—
61	C₄⋊C₄×C₄₇	—	2	2	no	—
62	C₄⋊C₄×C₄₇	—	2	2	no	—
63	C₄⋊C₄×C₄₇	—	4	2	no	—
64	C₄⋊C₄×C₄₇	—	4	2	no	—
65	C₂²⋊C₄×C₄₇	—	4	2	no	—
66	C₄²×C₄₇	—	2	2	no	—
67	C₄²×C₄₇	—	4	2	no	—
68	C₄²×C₄₇	—	4	2	no	—
69	C₄²×C₄₇	—	4	2	no	—
70	C₄²×C₄₇	Q₁₆	1	4	yes	complemented
71	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₇	#41	1	1	yes	characteristic
75	C₂³×C₄₇	#121	1	1	yes	characteristic
76	Q₈×C₄₇	—	4	4	no	—
77	Q₈×C₄₇	—	4	4	no	—
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	—	4	2	no	—
82	C₂×C₄×C₄₇	—	4	2	no	—
83	C₂×C₄×C₄₇	—	4	2	no	—
84	C₂×C₄×C₄₇	—	4	2	no	—
85	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
92	C₂×C₄×C₄₇	#135	1	2	yes	—
93	C₈×C₄₇	—	2	2	no	—
94	C₈×C₄₇	—	2	2	no	—
95	C₈×C₄₇	—	4	2	no	—
96	C₈×C₄₇	—	8	2	no	—
97	#9848	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₄₇	—	2	1	no	—
99	C₂²×C₄₇	—	2	1	no	—
100	C₂²×C₄₇	#390	1	1	yes	characteristic
101	C₂²×C₄₇	#467	1	2	yes	—
102	C₄×C₄₇	—	2	1	no	—
103	C₄×C₄₇	—	2	1	no	—
104	C₄×C₄₇	—	4	4	no	—
105	C₄×C₄₇	—	8	2	no	—
106	C₄×C₄₇	—	8	2	no	—
107	C₄×C₄₇	#421	1	2	yes	—
108	C₄×C₄₇	C₄×Q₁₆	1	2	yes	—
109	C₄×C₄₇	#444	1	2	yes	—
110	C₄×C₄₇	#494	1	1	yes	characteristic
111	C₄×C₄₇	#495	1	1	yes	characteristic
112	#3170	C₂×C₄₇	1	1	yes	characteristic
113	#1994	C₂×C₄₇	1	2	yes	—
114	#1744	C₂×C₄₇	1	1	yes	characteristic
115	#1567	C₂×C₄₇	1	1	yes	characteristic
116	#1493	C₂×C₄₇	1	2	yes	—
117	C₂×C₄₇	—	2	2	no	—
118	C₂×C₄₇	#1963	1	1	yes	characteristic
119	C₂×C₄₇	#1969	1	1	yes	characteristic
120	C₂×C₄₇	#1975	1	1	yes	characteristic
121	#533	C₂²×C₄₇	1	2	yes	—
122	#500	C₄×C₄₇	1	1	yes	characteristic, complemented
123	#494	C₄×C₄₇	1	1	yes	characteristic, complemented
124	#487	C₄×C₄₇	1	2	yes	complemented
125	#467	—	2	2	no	—
126	C₄×Q₁₆	—	2	2	no	—
127	#423	C₂²×C₄₇	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
130	C₄³	C₂²×C₄₇	1	1	yes	characteristic
131	#339	—	2	4	no	—
132	#336	—	2	2	no	—
133	#335	—	2	2	no	—
134	Q₈⋊C₈	—	2	4	no	—
135	C₄₇	#9848	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂²×C₈	—	2	1	no	—
139	C₂²×C₈	D₄×C₄₇	1	2	yes	—
140	C₄⋊Q₈	C₂×C₄×C₄₇	1	2	yes	—
141	C₄⋊Q₈	C₂×C₄×C₄₇	1	2	yes	—
142	#125	—	2	2	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₂²⋊Q₈	—	2	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	#117	—	2	2	no	—
148	C₂×C₄²	—	2	1	no	—
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	D₄×C₄₇	1	2	yes	—
151	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
152	#107	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	#103	—	2	4	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
158	C₄×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
159	C₄×C₈	D₄×C₄₇	1	2	yes	—
160	C₄×C₈	D₄×C₄₇	1	2	yes	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
166	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
167	Q₁₆	—	4	4	no	—
168	M₁₆	—	4	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂²⋊C₄×C₄₇	1	2	yes	—
174	C₂×C₈	C₂²⋊C₄×C₄₇	1	2	yes	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	Q₁₆×C₄₇	1	4	yes	complemented
187	C₄²	C₂²×C₄×C₄₇	1	1	yes	characteristic
188	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
189	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
190	C₄²	#41×C₄₇	1	1	yes	characteristic
191	C₂³	#121×C₄₇	1	1	yes	characteristic
192	Q₈	—	4	4	no	—
193	Q₈	—	4	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
202	C₂×C₄	C₄×D₄×C₄₇	1	1	yes	characteristic
203	C₂×C₄	C₄×D₄×C₄₇	1	1	yes	characteristic
204	C₂×C₄	#121×C₄₇	1	1	yes	characteristic
205	C₂×C₄	#124×C₄₇	1	1	yes	characteristic
206	C₂×C₄	#127×C₄₇	1	1	yes	characteristic
207	C₂×C₄	C₂×Q₁₆×C₄₇	1	2	yes	—
208	C₂×C₄	#135×C₄₇	1	2	yes	—
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	2	no	—
212	C₈	—	8	2	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₄₇×#390	1	1	yes	characteristic
216	C₂²	C₄₇×#467	1	2	yes	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	4	no	—
220	C₄	—	8	2	no	—
221	C₄	—	8	2	no	—
222	C₄	C₄₇×#421	1	2	yes	—
223	C₄	C₄×Q₁₆×C₄₇	1	2	yes	—
224	C₄	C₄₇×#444	1	2	yes	—
225	C₄	C₄₇×#494	1	1	yes	characteristic
226	C₄	C₄₇×#495	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₄₇×#1963	1	1	yes	characteristic
229	C₂	C₄₇×#1969	1	1	yes	characteristic
230	C₂	C₄₇×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₇	#127
commutator	C₄²	C₂²×C₄×C₄₇
Frattini, cosocle	C₂×C₄²	C₂³×C₄₇
socle	C₂²×C₄₇	#390
prime radical	C₂³×C₄₇	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₄₇×#500	C₄	semidirect
8. C₄₇×#494	C₄	semidirect
9. C₄₇×#487	C₄	semidirect
70. C₄²×C₄₇	Q₁₆	semidirect
97. #9848	C₄₇	direct (N × H)
122. #500	C₄×C₄₇	semidirect
123. #494	C₄×C₄₇	semidirect
124. #487	C₄×C₄₇	semidirect
135. C₄₇	#9848	direct (N × H)
186. C₄²	Q₁₆×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#3170	2
C₄₇×#1994	2
C₄₇×#1744	2
C₄₇×#1567	2
C₄₇×#1493	2
#9848	47

Subgroups — C47×#9848 #26049546

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#9848 #26049546