Subgroups — C₂×#20241530 #26136485

Order 12032.

294 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688957	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241530	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×#504	C₂	1	1	yes	maximal, characteristic
5	#15688957	—	2	1	no	—
6	#15688957	C₂²	1	2	yes	complemented
7	C₂×C₄₇⋊C₃₂	—	2	1	no	—
8	C₂×C₄₇⋊C₃₂	—	2	1	no	—
9	C₂×C₄₇⋊C₃₂	—	2	1	no	—
10	C₂×C₄₇⋊C₃₂	C₂²	1	2	yes	complemented
11	C₂×#131×C₄₇	C₄	1	1	yes	characteristic
12	C₄₇×#504	—	2	1	no	—
13	C₄₇×#504	—	2	1	no	—
14	C₄₇×#504	C₄	1	2	yes	—
15	C₄₇×#504	C₂²	1	2	yes	—
16	C₂×M₃₂×C₄₇	—	2	1	no	—
17	C₂×M₃₂×C₄₇	C₄	1	1	yes	characteristic
18	C₂²×C₁₆×C₄₇	—	2	1	no	—
19	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇⋊C₃₂	—	2	2	no	—
21	C₄₇⋊C₃₂	—	2	2	no	—
22	C₂×#41×C₄₇	C₈	1	1	yes	characteristic
23	#131×C₄₇	—	2	1	no	—
24	#131×C₄₇	—	2	1	no	—
25	#131×C₄₇	C₈	1	2	yes	—
26	#131×C₄₇	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₄₇	—	2	1	no	—
28	C₂×M₁₆×C₄₇	C₈	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	—	2	1	no	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	M₃₂×C₄₇	—	2	1	no	—
32	M₃₂×C₄₇	—	2	2	no	—
33	M₃₂×C₄₇	—	2	2	no	—
34	M₃₂×C₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₄₇	—	2	1	no	—
36	C₂×C₁₆×C₄₇	—	2	2	no	—
37	C₂×C₁₆×C₄₇	—	2	2	no	—
38	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
44	#41×C₄₇	—	2	1	no	—
45	#41×C₄₇	—	2	1	no	—
46	#41×C₄₇	C₁₆	1	2	yes	—
47	#41×C₄₇	C₂×C₈	1	2	yes	—
48	C₂×Q₈×C₄₇	C₁₆	1	1	yes	characteristic
49	C₂×D₄×C₄₇	—	2	1	no	—
50	C₂×D₄×C₄₇	C₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	1	no	—
52	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
53	M₁₆×C₄₇	—	2	1	no	—
54	M₁₆×C₄₇	—	2	2	no	—
55	M₁₆×C₄₇	—	2	2	no	—
56	M₁₆×C₄₇	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₄₇	—	2	1	no	—
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	—	2	2	no	—
60	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
63	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
64	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
65	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₁₆×C₄₇	—	2	2	no	—
67	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₄₇	—	2	1	no	—
72	C₂³×C₄₇	M₃₂	1	1	yes	characteristic
73	Q₈×C₄₇	—	2	1	no	—
74	Q₈×C₄₇	C₂×C₁₆	1	2	yes	—
75	D₄×C₄₇	—	2	2	no	—
76	D₄×C₄₇	—	2	2	no	—
77	D₄×C₄₇	—	2	1	no	—
78	D₄×C₄₇	C₂×C₁₆	1	2	yes	—
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
86	C₂×C₄×C₄₇	M₃₂	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
88	C₈×C₄₇	—	2	2	no	—
89	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
90	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
91	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
93	C₂×#1401	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	—	2	2	no	—
95	C₂²×C₄₇	—	2	2	no	—
96	C₂²×C₄₇	—	4	2	no	—
97	C₂²×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
98	C₂²×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
99	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
100	C₄×C₄₇	—	2	2	no	—
101	C₄×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
102	C₄×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
103	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
105	C₂×#504	D₄₇	1	1	yes	characteristic
106	C₂×M₆₄	—	47	1	no	—
107	C₂²×C₃₂	—	47	1	no	—
108	#1401	—	47	4	no	—
109	C₂×C₄₇	—	2	1	no	—
110	C₂×C₄₇	—	2	1	no	—
111	C₂×C₄₇	—	4	2	no	—
112	C₂×C₄₇	#1401	1	2	yes	complemented
113	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
114	C₂×#131	Dic₄₇	1	1	yes	characteristic
115	#504	—	2	1	no	—
116	#504	—	2	1	no	—
117	#504	Dic₄₇	1	2	yes	—
118	#504	C₂×D₄₇	1	2	yes	—
119	C₂×M₃₂	—	2	1	no	—
120	C₂×M₃₂	Dic₄₇	1	1	yes	characteristic
121	C₂²×C₁₆	—	2	1	no	—
122	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
123	M₆₄	—	47	2	no	—
124	M₆₄	—	94	1	no	—
125	C₂×C₃₂	—	47	2	no	—
126	C₂×C₃₂	—	94	1	no	—
127	C₂×C₃₂	—	94	1	no	—
128	C₂×C₃₂	—	94	1	no	—
129	C₄₇	C₂×#1401	1	1	yes	characteristic, complemented, Sylow
130	C₂×#41	C₄₇⋊C₈	1	1	yes	characteristic
131	#131	—	2	1	no	—
132	#131	—	2	1	no	—
133	#131	C₄₇⋊C₈	1	2	yes	—
134	#131	C₂×Dic₄₇	1	2	yes	—
135	C₂×M₁₆	—	2	1	no	—
136	C₂×M₁₆	C₄₇⋊C₈	1	1	yes	characteristic
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
139	M₃₂	—	2	1	no	—
140	M₃₂	—	2	2	no	—
141	M₃₂	—	2	2	no	—
142	M₃₂	C₂×Dic₄₇	1	2	yes	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
147	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
148	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
150	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
152	C₃₂	—	94	2	no	—
153	C₃₂	—	94	2	no	—
154	#41	—	2	1	no	—
155	#41	—	2	1	no	—
156	#41	C₄₇⋊C₁₆	1	2	yes	—
157	#41	C₂×C₄₇⋊C₈	1	2	yes	—
158	C₂×Q₈	C₄₇⋊C₁₆	1	1	yes	characteristic
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	C₄₇⋊C₁₆	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
163	M₁₆	—	2	1	no	—
164	M₁₆	—	2	2	no	—
165	M₁₆	—	2	2	no	—
166	M₁₆	C₂×C₄₇⋊C₈	1	2	yes	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
171	C₂×C₈	#10625329	1	1	yes	characteristic
172	C₂×C₈	#10625329	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
175	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂²⋊Dic₄₇	1	1	yes	characteristic
178	C₁₆	C₂²⋊Dic₄₇	1	1	yes	characteristic
179	C₁₆	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
180	C₁₆	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂³	—	2	1	no	—
182	C₂³	#13093964	1	1	yes	characteristic
183	Q₈	—	2	1	no	—
184	Q₈	C₂×C₄₇⋊C₁₆	1	2	yes	—
185	D₄	—	2	2	no	—
186	D₄	—	2	2	no	—
187	D₄	—	2	1	no	—
188	D₄	C₂×C₄₇⋊C₁₆	1	2	yes	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
193	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
196	C₂×C₄	#13093964	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
200	C₈	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
201	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
202	C₈	C₂×#10625329	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
207	C₂²	C₂²⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
208	C₂²	C₂×#13093964	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	C₂²⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
211	C₄	C₂²⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
212	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
213	C₄	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	4	2	no	—
217	C₂	#20241530	1	2	yes	direct product factor
218	C₂	C₂×C₂²⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₂×C₄₇×#504	C₂
socle	C₂²×C₄₇	C₂²⋊C₁₆
prime radical	C₂×#41×C₄₇	C₈
cosocle	C₂×C₁₆×C₄₇	C₂³
hypercenter	C₂×#504	D₄₇
hypocenter	C₄₇	C₂×#1401
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#15688957	C₂	semidirect
2. C₂²×C₄₇⋊C₃₂	C₂	semidirect
3. #20241530	C₂	direct (N × H)
6. #15688957	C₂²	semidirect
10. C₂×C₄₇⋊C₃₂	C₂²	semidirect
112. C₂×C₄₇	#1401	semidirect
129. C₄₇	C₂×#1401	semidirect
217. C₂	#20241530	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×#15688957	2
C₂²×C₄₇⋊C₃₂	2
#20241530	2
C₂×C₄₇×#504	2
C₂×#1401	47

Subgroups — C2×#20241530 #26136485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241530 #26136485