Subgroups — C₄₇×#1802 #20239412

Order 6016.

372 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#102×C₄₇	C₂	1	2	yes	maximal
4	C₄₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#340	C₂	1	2	yes	maximal, complemented
6	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₄₇	—	2	1	no	—
8	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
9	C₄×D₄×C₄₇	—	2	2	no	—
10	C₄×D₄×C₄₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
12	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
13	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₇	—	2	2	no	—
18	#102×C₄₇	—	2	4	no	—
19	C₂⁴×C₄₇	—	2	1	no	—
20	C₂×D₄×C₄₇	—	2	1	no	—
21	C₂×D₄×C₄₇	—	2	1	no	—
22	C₂×D₄×C₄₇	—	2	1	no	—
23	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₄₇	D₄	1	2	yes	—
25	C₂×D₄×C₄₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₄₇	—	2	1	no	—
27	C₂²×C₄×C₄₇	—	2	4	no	—
28	C₂²×C₄×C₄₇	—	2	2	no	—
29	C₂²×C₄×C₄₇	—	4	2	no	—
30	C₂²×C₄×C₄₇	—	4	2	no	—
31	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₄⋊C₄×C₄₇	—	4	2	no	—
41	C₄⋊C₄×C₄₇	—	4	2	no	—
42	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂²⋊C₄×C₄₇	—	2	4	no	—
44	C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₄²×C₄₇	D₄	1	2	yes	—
46	C₂³×C₄₇	—	2	2	no	—
47	C₂³×C₄₇	—	2	2	no	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂³×C₄₇	—	4	1	no	—
50	C₂³×C₄₇	—	4	1	no	—
51	C₂³×C₄₇	—	4	1	no	—
52	C₂³×C₄₇	—	4	1	no	—
53	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
54	D₄×C₄₇	—	2	2	no	—
55	D₄×C₄₇	—	2	2	no	—
56	D₄×C₄₇	—	2	2	no	—
57	D₄×C₄₇	C₄⋊C₄	1	4	yes	complemented
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	4	1	no	—
62	C₂×C₄×C₄₇	—	4	1	no	—
63	C₂×C₄×C₄₇	—	4	8	no	—
64	C₂×C₄×C₄₇	—	4	2	no	—
65	C₂×C₄×C₄₇	—	4	2	no	—
66	C₂×C₄×C₄₇	—	4	2	no	—
67	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
68	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
69	C₂×C₄×C₄₇	D₈	1	2	yes	—
70	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
71	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
77	C₈×C₄₇	—	4	2	no	—
78	C₂²×C₄₇	—	2	4	no	—
79	C₂²×C₄₇	—	4	2	no	—
80	C₂²×C₄₇	—	4	2	no	—
81	C₂²×C₄₇	—	4	2	no	—
82	C₂²×C₄₇	—	4	2	no	—
83	C₂²×C₄₇	—	4	2	no	—
84	C₂²×C₄₇	—	4	2	no	—
85	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄₇	#121	1	1	yes	characteristic
87	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
89	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂²×C₄₇	#137	1	1	yes	characteristic
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	4	2	no	—
94	C₄×C₄₇	—	8	2	no	—
95	C₄×C₄₇	#102	1	4	yes	complemented
96	C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₄₇	#117	1	1	yes	characteristic
98	C₄×C₄₇	#121	1	1	yes	characteristic
99	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
100	#1802	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂×C₄₇	—	4	4	no	—
102	C₂×C₄₇	#380	1	1	yes	characteristic
103	C₂×C₄₇	C₂×#102	1	1	yes	characteristic
104	C₂×C₄₇	#417	1	1	yes	characteristic
105	C₂×C₄₇	#459	1	1	yes	characteristic
106	C₂×C₄₇	#461	1	1	yes	characteristic
107	C₂×C₄₇	#474	1	1	yes	characteristic
108	C₂×C₄₇	#476	1	1	yes	characteristic
109	C₂×C₄×D₄	C₂×C₄₇	1	1	yes	characteristic
110	C₂×C₄⋊C₈	C₂×C₄₇	1	1	yes	characteristic, complemented
111	C₂×#102	C₂×C₄₇	1	2	yes	—
112	#384	C₂×C₄₇	1	1	yes	characteristic, complemented
113	#340	C₂×C₄₇	1	2	yes	complemented
114	C₄₇	#1802	1	1	yes	characteristic, direct product factor, Sylow
115	C₂²×D₄	C₂²×C₄₇	1	1	yes	characteristic
116	C₂³×C₄	—	2	1	no	—
117	C₂²×C₈	C₂²×C₄₇	1	2	yes	—
118	C₄×D₄	—	2	2	no	—
119	C₄×D₄	C₄×C₄₇	1	4	yes	complemented
120	C₂×C₄⋊C₄	—	2	1	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
124	C₂×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
126	C₄⋊C₈	—	2	2	no	—
127	#102	—	2	4	no	—
128	C₂⁴	—	2	1	no	—
129	C₂×D₄	—	2	1	no	—
130	C₂×D₄	—	2	1	no	—
131	C₂×D₄	—	2	1	no	—
132	C₂×D₄	C₂×C₄×C₄₇	1	2	yes	—
133	C₂×D₄	D₄×C₄₇	1	2	yes	—
134	C₂×D₄	Q₈×C₄₇	1	2	yes	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	D₄×C₄₇	1	1	yes	characteristic
141	C₂²×C₄	D₄×C₄₇	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
152	C₂²⋊C₄	—	2	4	no	—
153	C₄²	C₂×C₄×C₄₇	1	2	yes	—
154	C₄²	D₄×C₄₇	1	2	yes	—
155	C₂³	—	2	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	—	4	1	no	—
159	C₂³	—	4	1	no	—
160	C₂³	—	4	1	no	—
161	C₂³	—	4	1	no	—
162	C₂³	C₂×D₄×C₄₇	1	1	yes	characteristic
163	D₄	—	2	2	no	—
164	D₄	—	2	2	no	—
165	D₄	—	2	2	no	—
166	D₄	C₄⋊C₄×C₄₇	1	4	yes	complemented
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
177	C₂×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
178	C₂×C₄	D₈×C₄₇	1	2	yes	—
179	C₂×C₄	SD₁₆×C₄₇	1	2	yes	—
180	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
181	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
183	C₂×C₄	C₂×Q₈×C₄₇	1	1	yes	characteristic
184	C₂×C₄	#41×C₄₇	1	1	yes	characteristic
185	C₂×C₄	#41×C₄₇	1	1	yes	characteristic
186	C₈	—	4	2	no	—
187	C₂²	—	2	4	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
195	C₂²	#121×C₄₇	1	1	yes	characteristic
196	C₂²	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
197	C₂²	C₂×D₈×C₄₇	1	1	yes	characteristic
198	C₂²	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
199	C₂²	AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
200	C₂²	#137×C₄₇	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	8	2	no	—
204	C₄	#102×C₄₇	1	4	yes	complemented
205	C₄	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
206	C₄	#117×C₄₇	1	1	yes	characteristic
207	C₄	#121×C₄₇	1	1	yes	characteristic
208	C₄	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
209	C₂	—	4	4	no	—
210	C₂	C₄₇×#380	1	1	yes	characteristic
211	C₂	C₂×#102×C₄₇	1	1	yes	characteristic
212	C₂	C₄₇×#417	1	1	yes	characteristic
213	C₂	C₄₇×#459	1	1	yes	characteristic
214	C₂	C₄₇×#461	1	1	yes	characteristic
215	C₂	C₄₇×#474	1	1	yes	characteristic
216	C₂	C₄₇×#476	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊C₈×C₄₇	C₂	semidirect
4. C₄₇×#384	C₂	semidirect
5. C₄₇×#340	C₂	semidirect
10. C₄×D₄×C₄₇	C₄	semidirect
57. D₄×C₄₇	C₄⋊C₄	semidirect
95. C₄×C₄₇	#102	semidirect
100. #1802	C₄₇	direct (N × H)
110. C₂×C₄⋊C₈	C₂×C₄₇	semidirect
112. #384	C₂×C₄₇	semidirect
113. #340	C₂×C₄₇	semidirect
114. C₄₇	#1802	direct (N × H)
119. C₄×D₄	C₄×C₄₇	semidirect
166. D₄	C₄⋊C₄×C₄₇	semidirect
204. C₄	#102×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₄₇	2
C₂×C₄⋊C₈×C₄₇	2
C₂×#102×C₄₇	2
C₄₇×#384	2
C₄₇×#340	2
#1802	47

Subgroups — C47×#1802 #20239412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#1802 #20239412